noise and fluctuations: twentieth international conference on noise and fluctuations

The Ring of Brownian Motion: the good, the bad and the simply silly

Peter Hanggi 1,2

' Institute of Physics, University of Augsburg, 86135 Augsburg, Germany ^ Department of Physics and Centre for Computational Science and Engineering, National

University of Singapore, Republic of Singapore 117542

Abstract. In this plenary talk I give an account on the blossoming role that Brownian motion Theory and Experiment played - and still keeps doing so - in germinating and advancing several, partially diverse physical disciplines. Although the use of Brownian motion concepts generally most favorably impacted those scientific areas there are also some abuses where the application of such concepts may not describe satisfactorily physical reality.

Keywords: Brownian motion, eqiulibrium and non-eqiulibrium statistical mechanics. Stochastic Resonance, Brownian motors PACS: 05.60.-k, 47.61.-k, 81.07.-b, 85.25.-j, 85.35.-p, 87.16.-b

ROLE AND IMPACT OF BROWNIAN MOTION FOR PHYSICS AND RELATED FIELDS

Since the turn of the 20-th century Brownian hiss has continuously disclosed a rich variety of phenomena in and around physics. The understanding of this jittering motion of suspended microscopic particles has undoubtedly helped to reinforce and substantiate those pillars on which the basic modem physical theories are resting. - Its formal description provided the key to great achievements in statistical mechanics, the foundations of quantum mechanics and also astrophysical phenomena, to name only a few [1]. Brownian motion also determines the rate limiting step in most transport phenomena via escape events that help to overcome obstructing bottlenecks [2], or triggers those intriguing oscillatory dynamics occurring in excitable media [3].

My purpose here is as follows: Rather than presenting yet another sketchy overview of Brownian motion phenomena from an abundance of most useful and not so useful apphcations 1 instead prefer to point out a few timely such topical areas which are in the limehght of present and ongoing research activities. Different aspects and perspectives of these have been repeatedly reviewed in the recent literature with comprehensive reviews and features being available, see the cited literature given below. So, rather than presenting yet an additional such account 1 prefer to guide the interested reader to a selection of overviews. Next, 1 shall briefly highhght three such recent activities.

CPn29, Noise and Fluctuations, 20"" International Conference (ICNF 2009) edited by M. Macucci and G. Basso

© 2009 American Institute of Pliysics 978-0-7354-0665-0/09/$25.00

http://85.25.-j

http://87.16.-b

Stochastic Resonance

Although noise is usually thought of as the enemy of order it in fact can - given the necessary conditions - also be of constructive influence. The phenomena of Stochastic Resonance [4] presents one such paradigm. In a nutshell, Stochastic Resonance refers to the phenomenon for which the addition of the appropriate dose of noise can boost a (information-carrying) signal and hence facilitate its detection in a noisy environment. Due to its intrinsic simplicity and robustness the phenomenon has widespread applications and impacts a rich abundance of interdisciplinary fields.

Given its slow start after its invention in the 80's of the last century Stochastic Resonance spurred interest ever since and widened in scope and breadth of apphcations exhibiting still growing momentum. Presently its most salient applications apply to noisy physical biology [4, 5, 6] and to quantum information processing schemes [4, 7, 8].

Brownian motors

Noise can also induce directed transport: In systems possessing spatial or dynamical symmetry breaking, Brownian motion combined with unbiased external input signals, deterministic or random, alike, can assist directed motion of particles and continuous phases at the submicron scales. The by now common terminology for this paradigm of noise-assisted directed transport is "Brownian motors" [9, 10, 11, 12].

The concept of Brownian motors has given rise to novel design and implementation of various transport and separation devices in physics, chemistry, and in physical biology that operate on the nano-and/or microscale [12]. Most importantly, the combination of ever present thermal hiss with additional, unbiased non-equilibrium disturbances enables the rectification of haphazard Brownian thermal noise so that quantum and classical objects can be directed around on a priori designed routes.

Relativistic Brownian motion and relativistic thermodynamics

A commonly less known topic within the community of Brownian motion practitioners is the relativistic generahzation of Brownian motion. The theoretical description of relativistic Brownian motion, relativistic (then necessarily non-Markovian) diffusion processes and relativistic thermodynamics per se has experienced considerable progress over the past decade [13]. The theory of relativity implies that the progression of time experienced by a physical object is tightly linked to its state of motion. This then has salient implications for fast moving Brownian particles. In view of the experimental progress in high energy physics, astrophysics, and cosmology, relativistic Brownian motion concepts will play an increasingly important role in these fields as well. It is, therefore, important to understand how the underlying ideas can be consistently embedded into the theories of special and general relativity. This progress then naturally carries over to improved formulations of relativistic thermodynamics and relativistic statistical mechanics [13].

ACKNOWLEDGMENTS

Financial support by the German Excellence Initiative via the Nanosystems Initiative Munich (NIM), the Volkswagen Foundation (project 1/80424), the DFG-collaborative research centers SFB-486, SFB-631 and SFB-484 is gratefully acknowledged.

REFERENCES

1. W. Ebeling and I. M. Sokolov, Statistical thermodynamics and stochastic theory of nonequilibrium systems, Adv. Stat. Mech., Vol. 8 (World Scientific, Singapore 2005).

2. P. Hanggi, P. Talkner, and M. Borkovec, Reaction-rate theory: fifty years after Kramers, Rev. Mod. Phys. 62, 251-342 (1990).

3. B. Lindner, J. Garcia-Ojalvo, A. Neiman and L. Schimansky-Geier, Effects of noise in excitable systems, Phys. Rep. 392, 321^24 (2004).

4. L. Gammaitoni, P. Hanggi, P. Jung and F. Marchesoni, Stochastic Resonance, Rev. Mod.Phys. 70, 223-288 (1998).

5. P. Hanggi, Stochastic Resonance in Biology: how noise can enhance detection of weak signals and help improve biological information processing, ChemPhysChem 3, 285-290 (2002).

6. K. Wiesenfeld and F. Moss, Stochastic Resonance and the benefits of noise. Nature 373, 33-36 (1995). 7. M. Grifoni and P Hanggi, Driven Tunneling, Phys. Rep. 304, 229-354 (1998). 8. T. Wellens and A. Buchleitner, Stochastic Resonance, Rep. Progr. Phys. 67, 45-105 (2004). 9. R. D. Astumian and P. Hanggi, Brown/anmoto/-*, Physics Today 55 (11), 33-39 (2002). 10. P. Reimann, Brownian motors: noisy transport far from equilibrium Phys. Rep. 361, 57-265 (2002). 11. P. Hanggi, F. Marchesoni, and F. Nori, Brownian motors, Ann. Phys. (Berlin) 14, 51-70 (2005). 12. P. Hanggi and F Marchesoni, Artificial Brownian motors: Controlling transport on the nanoscale.

Rev Mod. Phys. 86,1-56 (2009); arXiv:0807.1283. 13. J. Dunkel and P. Hanggi, Relativistic Brownian motion, Phys. Rep. 471, 1-73 (2009).

Correlation-Fluctuation Effects In Non-Equilibrium Quantum Gas

R. Katilius*, S. V. Gantsevichf, V. D. Kagan^, M. I. Muradov''', M. Ramonas* and M. Rudan**

'Semiconductor Physics Institute, Vilnius 01108, Lithuania, [email protected] ^lojfe Physico-Technical Institute, 194021 St.-Petersburg, Russia **ARCES & DEIS University of Bologna, 1-40136 Bologna, Italy

Abstract. In the quasi-classical approximation, the theory of fluctuations in a degenerate two-or three-dimensional non-equilibrium electron gas is developed from first principles. It is shown that, thanks to the small-angle character of the inter-electron scattering, the theory takes a rather simple and transparent form, in contrast to other types of Fermi gases for which, due to exchange effects, the situation proves to be much more complicated.

Keywords: Noise in degenerate non-equilibrium systems, correlation of fluctuations, quasi-classical approximation PACS: 71.10.Ca, 72.70.+m, 05.30.-d, 05.40.-a

INTRODUCTION

The kinetic theory of fluctuations for a classical non-equilibrium gas with binary collisions was developed 40 years ago and is now expounded in review articles [1, 2], monographs [3, 4], and even in the advanced text-books [5]. In the framework of the correlation-function method, the fluctuations were investigated also in Fermi gas with inter-electron collisions [6-9] . The aim of this paper is to check earlier results by deriving, from first principles, the equations for the two-particle, time-displaced and equal-time, correlation functions for non-equilibrium degenerate quasi-classical gases with pair collisions. A self-contained, rather simple and transparent system of such equations will be obtained for a gas of electrons (or holes). It will be demonstrated that for Coulomb (or nearly Coulomb) interaction potential, thanks to the prevalence of small-angle inter-electron scattering, the theory takes a rather simple form. Namely, the kinetic equations for the two-electron correlation functions governing fluctuations in the degenerate electron gas are self-contained and comparatively simple. This is in contrast to other possible types of interaction in Fermi gases where, due to exchange effects, the situation proves to be much more complicated even in a quasi-classical picture.

PROBLEM STATEMENT

It is known that in the non-equilibrium state of a gas a correlation among particles exists due to particle-particle colhsions (see, e.g. [1]). Indeed, during such a collision, two particles change simultaneously their momenta and this process creates a constant flow

CPn29, Noise andFluctuaUons, 20"" International Conference (ICNF 2009) edited by M. Macucci and G. Basso


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of correlated particles into a pair of states pp'. As a result, the equal-time two particle correlation function of the occupancy fluctuations of the one-particle states acquires a non-trivial part: {5Fp{t)5Fpi{t)) = FpSppi + (Ppp'{t) (here Fp is the stationary non-equilibrium distribution function). The two-particle correlation function (Ppp'{t) satisfies the special kinetic equation [1], where a collision operator of particle pairs appears at the right-hand-side. Such an operator describes the permanently-acting source of correlation in the quasi-momentum space. In turn, the left-hand-side is responsible for the relaxation of correlation in a way similar to that of one-particle kinetics:

{dt + Jp+Jp>)(Ppp> = -npp,{F,F}. (1)

Here the quantities Jp and Jpi are the linearized operators of the one-particle kinetics describing the evolution of small deviations from the non-equilibrium state created by the combined action of an external driving force and collisions [1]. (For a steady state the function cpppi is time-independent, and the time derivative in (1) should be omitted.)

For classical particles n^y is given by

np^{F,F} = Y^iwgFpF^ - wf^FkFy). (2)

Here W , is the probability for two particles with momenta k and k" to get the momenta j9 and j9' after the mutual collision. (To simplify the formulae, we omit here and henceforth all unnecessary details, in particular, we do not write the vector indices).

The correlation source Ilppi has a clear physical meaning, representing a balance between the incoming and outgoing pairs of particles, quite analogous to the similar balance for individual particles in a one-particle collision operator (when summed over p', the operator n^y becomes the usual particle-particle collision operator Ilp{F,F} of the Boltzmann equation.) In equilibrium Ilppt = 0, and correlation among particles vanishes. The quantity TIppi is the main element of the theory of fluctuations in non-equilibrium states of classical particles.

As mentioned above, an attempt to derive its analogue for a degenerate non-equilibrium quasi-classical electron gas with pair collisions was initially made by Kagan [6], and later by Muradov [7] (two-band semiconductor). It was shown in [8, 9] that observation of fluctuations in a non-equilibrium electron gas would allow one to investigate directly and rather simply the macroscopic Pauli-correlation.

While deriving, in the quasi-classical approximation but from first principles, kinetic equations for correlation functions in the degenerate non-equilibrium electron gas, we took advantage of the modernized version [10] of the diagram technique that we had developed earlier (see, e.g., [11 - 13]).

In the general case of a Fermi gas, the terms representing the large-angle scattering appear in the expression for the source, as well as in the left hand side of equation (1). The general quasi-classical theory of fluctuations in a Fermi gas, taking into account all exchange effects, would be much more complicated than the version presented in [6 -9], and is not available yet.

10

RESULTS AND DISCUSSION

We were able to demonstrate that, for long-range inter-particle potentials like the Coulomb potential, the source in the equation for the two-particle equal-time correlation function in non-equilibrium is expressible in terms of the particle-particle collision probabilities and the non-equilibrium one-particle distribution function:

n./ = S«'+n(;\ (3) kV

where O**, is defined by the following expression:

^%,=FpFp,{\ -Fk-Fk>)-FkFk>{\ -Fp-Fp,), (4)

and the additional contribution IlyJ is given by the expression

kk'

The comparative simplicity of our main results (3) and (5) is ensured by a "soft" character of the Coulomb (or nearly Coulomb) interaction between electrons. For that type of interaction, the Fourier transform of the inter-particle potential U{r) (cf [6, 7]) is large enough just for small characteristic values of the momentum transfer q = p — k. Since in (3), (5) the momentum k takes, among others, values close to p, the Fourier transform Uq turns out to be large. On the other hand, the disregarded terms contain the transforms that depend on the large momentum transfer (comparable to p — p'). Such a transform is much smaller than Uq in the region of small enough values of q:

\Up_p,\«\Uq\. (6)

This rather strong inequality enables us to neglect the terms containing the Fourier transforms of the potential U{r) depending on the large momentum transfer. They are less important for the soft potentials in which we are interested here. However, such items should necessarily be taken into account in a theory of fluctuations in a non-equilibrium Fermi gas provided that the inter-particle scattering at large angles is essential. In parallel with sources, the corresponding terms should be included into the response operators, as mentioned above. These problems will be considered elsewhere.

The method we used for investigating fluctuations in a degenerate non-equilibrium electron gas, thanks to the prevalence of small-angle electron-electron scattering, is self-contained. The results are rather transparent. We remind that we define fluctuations of the distribution function, extracting the product of the uncorrected (averaged independently) distribution functions FpFpi from the initial two-particle distribution function. It looks like FpFpi — FpFpi. But in FpFpi some terms needed to form the full independent Fp andi^/ are absent because of the Pauli constraints. This absence reveals itself as an additional fluctuation, and the corresponding expressions emerge with opposite signs in the fluctuation source. Such is the origin of the additional non-trivial correlation in

11

non-equilibrium electron gases. The existence of this correlation phenomenon remained so far unnoticed in the fluctuation theory. Similar phenomena can exist also in other physical situations.

Let us note that the modification in the pair correlation function, induced by the Pauli principle, could reveal itself through macroscopic observables, e.g., current fluctuations in a conductor or ionized plasma (cf. [8, 9]). In fact, the effect is none other than the indirect manifestation of the electronic Hanbury-Brown and Twiss (HBT) effect in the solid-state or gaseous plasma. Indeed, a correlation of occupancies of different j9-states of electrons apparently can be interpreted as a correlation of intensities of "partial" electron beams in a real space. The correlation of particle-state occupancies, conditioned by the Pauli constraints, are not destructible by inter-electron collisions since the latter conserve energy and momentum. On the contrary, for Pauli correlations to be detectable in noise experiments, the effectiveness of inter-electron scattering should be higher than the electron-lattice interaction. The correlation survives, contributing to the noise spectrum, as long as relaxation processes due to interaction with lattice allow.

The Pauli correlation can reveal itself in a macroscopic system via electric noise only provided that the electron system is substantially displaced from the equilibrium with the lattice. This is a quantum effect measured by quasi-classical (kinetic) means. The vanishing at equilibrium is a fundamental property of all the additional terms. Though existing on microscopic level, the Pauli correlation disappears after averaging with the equilibrium distribution (density matrix). Indeed, at equilibrium the average partial fluxes in the momentum space balance out, so the two-particle correlation sources disappear. In contrast to averaging over the equilibrium distribution, the averaging over a non-equilibrium one does not necessarily lead to the vanishing of correlation sources.

Computation of Pauli correlation effect on spectra of current fluctuations is in progress.

REFERENCES

1. S. V. Gantsevich, V. L. Gurevich and R. Katilius, Riv Nuovo Cimento 2 (5), 1-87 (1979). 2. R. Katilius, "An Overview of the Development of the Kinetic Theory of Fluctuations" in Noise

and Fluctuations Control in Electronic Devices, edited by A. A. Balandin, Stevenson Ranch, CA: American Scientific, 2002, pp. 1-10.

3. Sh. M. Kogan, Electronic Noise and Fluctuations in Solids, Cambridge: Cambridge University Press, 1996, Chapter 3.

4. H. L. Hartnagel, R. Katilius and A. Matulionis Microwave Noise in Semiconductor Devices, New York: Wiley, 2001, Chapters 3-5.

5. E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics, Oxford: Pergamon, 1981. 6. V. D. Kagan, Fiz. Tverd Tela (Leningrad) 17, 1969-1977 (1975) [Sov Phys. Solid State 17, 1289

(1975)]. 7. M. I. Muradov,P/;j;s. Rev B 58, 12883-12898 (1998). 8. R. Katilius, Phys. Rev B 69, 245315-1-8 (2004). 9. R. Katilius and M. Rudan, Phys. Rev B 74, 233101-1-4 (2006). 10. R. Katihus, S. V. Gantsevich, V. D. Kagan and M. I. Muradov (to be published). 11. S. V. Gantsevich, V. L. Gurevich, V. D. Kagan and R. Yia<d\ms,phys. stat. sol. (b) 75,407-422 (1976). 12. R. Barkauskas, S. V. Gantsevich and R. Katihus, Zh. Eksp. Teor Fiz. 84, 2082-2091 (1983).

[Sov.Phys.JETP SI, 1212-1216 (1983)]. 13. S. V. Gantsevich, V. L. Gurevich, M. I. Muradov and D. A. Parshin, Phys. Rev B 52, 14006-14017

(1995).

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http://Sov.Phys.JETP

Modeling scaled processes and clustering of events by the nonlinear stochastic differential

equations B. Kaulakys, M. Alaburda and V. Gontis

Institute of Theoretical Physics and Astronomy of Vilnius University, A. Gostauto 12, LT-01108 Vilnius, Lithuania

Abstract. We present and analyze the nonlinear stochastic differential equations generating scaled signals with the power-law statistics, including 1 / / ^ noise and (/-Gaussian distribution. Numerical analysis reveals that the process exhibits some peaks, bursts or extreme events, characterized by power-law distributions of the burst statistics and, therefore, the model may simulate self-organized critical and other systems exhibiting avalanches, bursts or clustering of events.

Keywords: 1/f noise, stochastic differential equations, q-Gaussian distribution PACS: 05.40. .a, 72.70. +m, 89.75.Da

INTRODUCTION

Power-law distributions, including 1 / / noise, are ubiquitous in physics and in many other fields [1, 2, 3]. Despite the numerous models and theories, the intrinsic origin of 1/f noise and other scaled distributions still remain open questions. Starting from the multiplicative point process [4] we obtained the stochastic nonlinear differential equations, which generated signals with the power-law statistics, including 1 / f^ fluctuations [3, 5]. Here the other nonlinear stochastic differential equation generating q-Gaussian distribution of the bursting signal and \/f^ noise is presented and analyzed.

THE THEORY

We consider a nonlinear stochastic differential equation

Ax=(n-^x\{xl,+x^)'^'\At+{xl,+x^Y'^&W, 77 > 1 , A > 1 (1)

generating ^-Gaussian distributed signal

2

with q = I + 2/A. Here W is a standard Wiener process and Xm is the parameter of the ^-Gaussian distribution. Eq. (1) for small x <CXm represents the linear additive stochastic



13

v^r(VV-^'^-+'^'^ v^ r f^V- '^ ^""^

process generating the Brownian motion with the linear relaxation, whereas for x > Xm Eq. (1) reduces to the nonlinear multiplicative equation.

In accordance with Refs. [3, 4] the power spectrum of the process generated by Eq. (2) may be approximated as

with A characterizing the intensity of 1 //'^ noise, /o '- fmin being the frequency for transition of spectrum at low frequencies to the flat spectrum, and

/3 = 1 A - 3

2(77-1]

The autocorrelation function of the process is

C(5) ^ ( / ) c o s ( 2 ; r / 5 ) d / = - ^ f ^ 0 r (p /2 ) V/o

Kk{2nfos),

(4)

(5)

with Ki^{z) being the modified Bessel function and /i = (/3 — l) /2. The second order structural function F2{s) and height-height correlation function F{s) are expressed as

F{s)=Fi{s) = {\x{t + s) -x{tf) = 2[C(0) -C(5)] = A j ^ S{f)^m\nsfW. (6)

Particular cases of Eqs. (5) and (6) are presented in Ref. [3].

NUMERICAL ANALYSIS

We present here the investigation results of the dependence of characteristics of Eq. (1) solutions on the nonlinearity parameter r\ for the fixed parameter A = 3, i.e., for the pure 1//noise.

FIGURE 1. Examples of the numerically computed signals according to Eq. (1) with the parameters A = 3, Xm = 10^^, whereas 7] = 1.5 (left figure) and 7] = 2.5 (right figure).

As examples, in figure 1 we show the illustrations of the signals generated according to Eq. (1). We see bursts of the signal. In figures 2 and 3 the numerical calculations of the distribution density, P{x), power spectral density, S{f), autocorrelation function.

14

FIGURE 2. Distribution density, P{x), and power spectral density, S{f), for solutions of Eq. (1) with A = 3, Xm = 10^^ and different values of 7] = 1.5 (circles), J] = 2 (squares) and 7] = 2.5 (triangles) in comparison with the analytical results (solid lines) according to Eqs. (2) and (3), respectively.

. r -î^pijaiigafEât^wîr^fjr-—-^ ^ f f l

j ^ ^ i ^ M I ^ M

: 1

^^m

10 10 10 10 10 10 10 10

FIGURE 3. Autocorrelation function, C{s), and the second order structural function, F2 (s), for solutions of Eq. (1) with the same parameters as in figure 2 in comparison with the analytical results (solid lines) according to Eqs. (7) and (8), respectively.

C{s), and the second order structural function, F2{s), for solutions of Eq. (1) with A = 3, Xm = 0.01 and different values of the parameter 77 are presented. We see rather good agreement between the numerical calculations and the analytical results for /3 = 1,

C{s) =-A[r+\n{n/os)] (7)

- • -d

v^'

* '

10"^ 10"^ 10"^ 10"' 10" 10' 10^ 10^ 10^

FIGURE 4. Dependence of the burst size 5' as a function of the burst duration T and distributions of the burst size, P{S), for the peaks above the the threshold value x^ = 0.1. Calculations are as in figures 2 and 3 with the same parameters.

15

FIGURE 5. Burst duration, P{T), and interburst time, P(6), for the peaks above the the threshold value Xii, = 0.1. Calculations are as in figure 4 with the same parameters.

F2{s) = ^2A[\n{nf,^^s)-r], (8)

where 7 = 0.577216 is Euler's constant and fmax is the cutoff of the 1 / / spectrum at high frequency. Figures 4 and 5 demonstrate that the size of the generated bursts S is approximately proportional to the squared burst duration T, i.e., S <x T^, and asymptotically power-law distributions of the burst size, P{S) '-^ S^^-^, burst duration, P{T) r^ r-i"* and interburst time, P(0) '-^ 9^^-^, for the peaks above the threshold value Xth of the variable x{t). These dependencies slightly depend on the degree of nonlinearity exponent 77 of the stochastic equation and are similar to those discovered [3] for the q-exponential distributions.

CONCLUSION

The nonlinear stochastic differential equations may generate ^-Gaussian distributed signals with 1 //'^ power spectrum, exhibiting bursts, similar to the crackling processes [6] and observable long-term memory time series [7, 8].

ACKNOWLEDGMENTS

We acknowledge the support by the Agency for International Science and Technology Development Programs in Lithuania and EU COST Action MP 0801.

REFERENCES

M. E .J. Newman, Contemp. Phys. 46, 323 (2005). Scholarpedia, h t t p : / /www. s c h o l a r p e d i a . o r g / a r t i c l e / l / f _ n o i s e (2009). B. Kaulakys and M. Alaburda, J. Stat. Mech. P02051 (2009). B. Kaulaliys, V. Gontis, and M. Alaburda, Phys. Rev. E 71, 051105 (2005). B. Kaulaliys, J. Ruseckas, V. Gontis, and M. Alaburda, PhysicaA 365, 217 (2006). J. P. Sethna, K. A. Dahmen, and C. R. Myers, Nature 410, 242 (2001). A. Bunde, J. F. Eichner, J. W. Kantelhardt, and S. Havlin, Phys. Rev. Lett. 94, 048701 (2005). R. Blender, K. Fraedrich, and F. Sienz, Nonlin. Processes Geophys. 15, 557 (2008).

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Enlargement of a low-dimensional stochastic web S.M. Soskin*+',* LA. Khovanov***, R. Mannella^ and P.V.E. McClintock*

* Institute of Semiconductor Physics, 03028 Kiev, Ukraine ^Abdus Salam ICTP, 34100 Trieste, Italy

** School of Engineering, University of Warwick, Coventry CV4 7AL, UK ^Physics Department, Lancaster University, Lancaster LAI 4YB, UK

^Dipartimento di Fisica, Universita di Pisa, 56127Pisa, Italy

Abstract. We consider an archetypal example of a low-dimensional stochastic web, arising in a 1D oscillator driven by a plane wave of a frequency equal or close to a multiple of the oscillator's natural frequency. We show that the web can be greatly enlarged by the introduction of a slow, very weak, modulation of the wave angle. Generalizations are discussed. An appUcation to electron transport in a nanometre-scale semiconductor superlattice in electric and magnetic fields is suggested.

Keywords: stochastic webs, separatrix chaotic layer, chaotic transport, semiconductor superlattice PACS: PACS numbers: 05.45.-a,72.20.Ht

INTRODUCTION

In weakly perturbed integrable Hamiltonian systems, small areas of the phase space near resonances are chaotic [1]. The stochastic web concept dates back to the early 1960s when Arnold showed [2] that, in non-degenerate Hamiltonian systems of dimension exceeding 2, resonance lines necessarily intersect, forming a web of infinite-size in the Poincare section. It provides in turn for a slow chaotic (sometimes called "stochastic") diffusion over infinite distances in the relevant dynamical variables.

It was discovered at the end of 1980s [3-5] that, in degenerate or nearly degenerate systems, stochastic webs can arise even for dimension 3/2. An archetypal example of such a low-dimensional stochastic web arises when a ID harmonic oscillator is perturbed by a weak traveling wave whose frequency coincides with a multiple of the natural frequency of the oscillator [1, 4, 6]. Perturbation plays a dual role: on the one hand, it gives rise to a slow dynamics characterized by an auxiliary Hamiltonian that possesses an infinite web-like separatrix; on the other hand, the perturbation destroys this self-generated separatrix, replacing it with a thin chaotic layer. Such a low-dimensional stochastic web may be relevant to a variety of physical systems and plays an important role in the corresponding transport phenomena: see [1, 4, 6] for reviews on relevant classical systems. There are also quantum systems the dynamics of whose transport reduces to that of the classical model described above. One example is a nanometre-scale semiconductor superlattice with an applied voltage and magnetic field [7, 8]. Moreover, there is evidence that, if a classical system possesses a stochastic web, then transport in the quantum analogue of that system is much stronger than where the classical system does not possesses a web [9, 10], a finding that may be relevant e.g. to the transport of ultra-cold atoms in optical lattices [9].

One might assume that, like the Arnold web, the low-dimensional stochastic web



17

described above should be infinite, so that it would provide for transport between the centre of the web and states situated arbitrarily distant in coordinate and momentum. However this is not the case. Provided that the perturbation is not exponentially strong, the real web is confined to the region within just a few inner loops of the infinite web-like resonant separatrix (cf. [1, 4, 6]). The reason is as follows. The resonant Hamiltonian possesses a single, infinite, web-like separatrix only in the first-order approximation of the averaging method [11]; with account taken of the next-order approximations, the separatrix splits into many separate complex loops successively embedded within each other. Non-resonant terms of the perturbation dress the separatrices by exponentially narrow chaotic layers. If the perturbation is not small, the chaotic layers manage to connect neighbouring separatrix loops situated close to the centre. However, the width of the chaotic layer decreases exponentially fast with increasing distance from the centre [1, 4, 6]. As a result, coalescence of the chaotic layers associated with adjacent loops only occurs among the few loops closest to the centre.

If the resonance between the perturbation and the oscillator is inexact, or if the oscillator is nonlinear, the separation of neighbouring loops is typically much larger: it is already apparent in the first-order approximation of the averaging method [1, 5, 6]. So, the number of the loops connected to the centre by chaotic transport is even smaller [1, 5, 6] than in the case of an exact resonance.

How can one modify the perturbation in order that the transport should become unlimited or, at least, significantly extended? One possible answer was obtained at the very beginning of the studies of low-dimensional webs [3]: if the perturbation consists of short kicks that are periodic in space and time, and if the frequency of the kicks is equal to a multiple of the natural frequency, then a so called uniform web is formed covering the whole phase space. Such a perturbation seldom applies in practice, however, and even where it does the chaotic transport is still exponentially slow [1,6].

Is it possible to obtain a web of form similar to the original one [4] but substantially extended in the phase space? Our present work leads to a positive answer based on the following simple idea. The chaotic layer in the web is exponentially narrow because the frequency of the non-resonant perturbation of the resonant Hamiltonian is necessarily much higher than the frequency of small eigenoscillations in the cell of the web-like separatrix [1,3-6]. So, we need to modify the perturbation in such a way that the resonant Hamiltonian does not change while its perturbation contains, in addition to the conventional terms, a low-frequency one. One may do this by modulating the wave angle at a low frequency or by adding one more wave of frequency shifted slightly from the original one. The latter option will be considered elsewhere together with a generalization for the uniform web (leading to a huge enhancement of the chaotic transport through it). Our present work concentrates on the first option since it may have immediate apphcations to nanometre-scale semiconductor superlattices in electric and magnetic fields [7, 8].

SLOW MODULATION OF THE WAVE ANGLE

Fig. 1 demonstrates the efficiency of our method. We integrate the equation

^ + ^ = 0.1sin[15^-4?-/jsin(0.02?)], (1)

first for /j = 0 (i.e. for the conventional case with parameters as in [1,4,6]), and secondly for /j = 0.1. Although the modulation in the latter case is weak (its amplitude is small compared to the 2K period of the wave angle), the resultant increase in size of the web in coordinate and momentum is large: a factor of about 6x.

To account for these results we develop the analytic theory, generahzed for the off-resonant case [1, 5, 6], using a general method developed recently in [12]. It is anticipated that the method can also be generahzed for uniform webs [1, 3, 6], leading to an exponentially strong enhancement of chaotic transport through them.

6

4

O- 0 •a

-2

-4

-6

z^M^

M -^;^:^=?F^-=:^

^ ^

S^ -

^^^^ir r i r t : :^ '^! "^^-^~S=^==^

^ ^ ^ ^ \ 8' w*" ^^-

-0.5 0 q/2n

0.5

FIGURE 1. The Poincare section for a trajectory of the system (1) with initial state q = 0.1, q = 0 (at instants i„ = nT where T = In/Q.Ql is the period of the modidation and n = 1,2,3, ...600000) for h = Q (left panel) and h = Q.\ (right panel). A symplectic integration scheme of the fourth order is used, with an integration step tint = jlfjo ~ 1.57 x lO^'*, so that the inaccuracy at each step is of the order of Hnt ~ X10^' '- The left panel corresponds to the conventional case considered in [1,4,6]. The right panel demonstrates that the modidation, although weak, greatly enlarges the web sizes (note the different axes scales), thereby greatly enhancing the chaotic transport.

APPLICATION TO SEMICONDUCTOR SUPERLATTICES

We now consider application to quantum electron transport in nanometre-scale ID semiconductor superlattices (SLs) subject to a constant electric field along the SL axis and to a constant magnetic field [7, 8]. The spatial periodicity gives rise to minibands for the electrons. In the tight-binding approximation, the electron's energy £ as a function of its momentum p in the lowest miniband is given by

E{p) = A[l-cos(/7;c^//j)] PI- -PI (2)

2 2OT* '

where x is the direction along the SL axis, A is the miniband width, d is the SL period, m* is the electron effective mass for the motion in the transversal (i.e. y — z) direction.

Thus, the quasi-classical motion of an electron of charge e in an electric field F and a magnetic field B is described by:

f = -4^+[Vp-£( /^ )x5]} . (3)

It was shown in [7] that, for constant electric field along the SL axis F = (—Fo,0,0) and constant magnetic field with a given angle 9 to the axis B = (5cos(0),O,5sin(0)), the dynamics of the z-component of momentum p^ reduces to the equation of motion of an auxiliary harmonic oscillator in a plane wave. At certain values of the parameters, the ratio of the wave and oscillator frequencies takes integer values (as in Eq. (1) with h = 0) giving rise to the onset of the stochastic web. This leads in turn to a delocahzation of the electron in the x-direction and, consequently, to an increase of the dc-conductivity along the SL axis. The experiment [8] appears to provide evidence in favor of this exciting hypothesis.

At the same time, the finite size of the web and, yet more so, the exponentially fast decrease of the transport rate as the distance from the centre of the web increases, seem to place strong limitations on the use of the effect. We now suggest a simple and efficient way to overcome these limitations: if we add to the original (constant) electric field FQ a small time-periodic (ac) component FacCOs(QacO' then the wave angle in the equation of motion of p^ is modulated by the following term (cf. Eq. (1)):

hcos{Q.t) = -——-cos[-—t], Qo = ^ — • (4)

This allows us to increase drastically the size of the web and the rate of chaotic transport through it. For example, for the case shown in Fig. 1, where we have an increase of the web size by the factor of 6x, it is sufficient to add an ac component of electric field of frequency 0.02 x QQ and amplitude Fac = 0.1 x 0.02 x FQ, i.e. an amplitude that is 500x smaller than the original constant field FQ!

REFERENCES

1. G.M. Zaslavsky, Physics of Chaos in Hamiltonian systems, Imperial CoUedge Press, London, 2007. 2. V.I. Arnold, Dokl Acad. Nauk SSSR 156, 9 (1964). 3. G.M. Zaslavsky et al., Sov. Phys. JETP 64, 294 (1986); A.A. Chernikov et al., Nature 326, 559

(1987). 4. A.A. Chernikov et al., Phys. Lett A 122, 39 (1987). 5. A.A. Chernikov et al., Phys. Lett A 129, 377 (1988). 6. G.M. Zaslavsky, R.D. Sagdeev, D.A. Usikov and A.A. Chernikov, Weak Chaos and Quasi-Regular

Patterns, Cambridge University Press, Cambridge, 1991. 7. T.M. Fromhold et al., Phys. Rev. Lett 87, 046803 (2001). 8. T.M. Fromhold et al., Nature 428, 726 (2004). 9. W.K. Hensinger et al., Nature 412, 52 (2001); D.A. Steck, W.H. Oskay, M.G. Raizen, Science 293,

274 (2001); R.G. Scott et al., Phys. Rev. A 66, 023407 (2002). 10. A.R.R. Carvalho, A. Buchleitner, Phys. Rev Lett 93, 204101 (2004). 11. N.N. Bogolyubov, Yu.A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillators,

Gordon and Breach, New York, 1961. 12. S.M. Soskin, R. Mannella, and O.M. Yevtushenko, Phys. Rev E11, 036221 (2008).

20

Acceleration of the chaotic and noise-induced transport in adiabatically driven spatially

periodic systems

S.M. Soskin*'1', R. Mannella**, O.M. Yevtushenko* andM. Filiasi**

'Institute of Semiconductor Physics, 03028 Kiev, Ukraine Ubdus Salam ICTP, 34100 Trieste, Italy

"Dipartimento diFisica, Universita diPisa, 56127Pisa, Italy "^Physics Department, Ludwig-Maximilians-UniversitdtMUnchen, D-80333 Miinchen, Germany

Abstract. We show that, in an ac-driven spatially periodic Hamiltonian system, the width of the chaotic layer associated with a separatrix diverges in the adiabatic limit. A similar behaviour is observed for the layer of transient chaos in the presence of a weak dissipation. If noise is added, this mechanism may have as a conseguence that the noise-induced transport is greatly accelerated by the adiabatic AC-drive.

Keywords: separatrix chaos, adiabatic, noise-induced, diffusion, threshold devices PACS: 05.45.Ac,05.40.-a,05.45.Pq,66.30.-h

INTRODUCTION

A time-periodic perturbation of a Hamiltonian system possessing a separatrix destroys the separatrix itself, leading to the onset of chaos in its vicinity [1-3]. Until recently (cf. [1-3]) it was assumed that, for a small amplitude of the perturbation, the separatrix chaotic layer should necessarily be narrow (see in particular [4-6], where adiabatic chaos was studied). However we have discovered [7] that any spatially periodic Hamiltonian system (for instance, a pendulum) driven by an AC-force, i.e. by a time-periodic coordinate-independent force, possesses a remarkable property: the upper energy boundary of the chaotic layer diverges as the driving frequency goes to zero (Fig. 1). The origin of this counterintuitive feature is as follows. Let the system stay initially at the separatrix. If the weak AC-force changes slowly, then a small acceleration of the system caused by this force keeps its sign for a long time, namely for a half-period of the force. If the period is sufficiently long, then the system acquires a large velocity by the end of this time. During the next half-period, the acceleration changes to deceleration so that, by the end of the period, the system again approaches the vicinity of the separatrix, where some chaotization occurs: the new acceleration-deceleration round starts then with a small random shift of the AC-force angle from a multiple ofln. Thus, the trajectory is chaotic on sufficiently large time-scales (Fig. 2). The smaller the frequency, the larger the maximum absolute value of the velocity; therefore the faster the chaotic transport becomes (cf the inset of Fig. 2). We have developed an explicit adiabatic theory [7-9] which nicely describes results of simulations (Fig. 1(c)). We have also developed a qualitative description of the chaotic diffusion and explained its acceleration in the adiabatic



21

limit [7-9], but its rigorous description as well as a thorough numerical study have not been done. Some, mostly heuristic, theoretical study of the adiabatic chaos in the same system was recently undertaken in [10].

Coordinate q (mod 271:) Frequency, lo8((0j.>

FIGURE 1. AC-drivenpendulum [7]:i/ = ;72/2-ffl^cos(^) + /;ffl ^sin(ffii;-<) with fflo = l and/; = 0.01. (a). Trajectory in the stroboscopic (fori = «2;i:/(»/withw = 0,1,2,...) Poincare section for <»/ = 0.01 and q{Q) = n, p{Q) = 0. (b). Spectral dependence of the maximum excess of energy E = p'^/2 — OQ COS((/) over the barrier level OQ = 1. (c) As in (b) but compared with our adiabatic theory (solid line) and its asymptotes for relatively low and high adiabatic frequencies (dashed and dotted lines respectively).

FIGURE 2. The frajectory of the same system as in Fig. 1(a), on a large time scale [8]. The inset compares frajectories for different cOf.

EFFECT ON NOISE-INDUCED PHENOMENA

In the presence of a weak dissipation, Hamiltonian chaos is replaced by transient chaos [1]. But the adiabatic divergence of the width of the chaotic area in the phase plane and the related enhancement of the deterministic transport are still valid [8].

Moreover, the adiabatic acceleration of the deterministic transport described above suggests that noise-induced transport may also be greatly accelerated by the adiabatic AC-drive. Our simulations and qualitative estimates support this hypothesis. Below, we consider two characteristic examples: spatial diffusion and threshold devices.

Spatial diffusion

Consider an underdamped pendulum driven by a weak AC-force (cf. that in Fig. 1) and by a weak white noise:

22

q = p^ p = -sm{q)-rp-hsm{a)ft)+fn{t) (/nW) = o, (/„(0/„(0)) = 2rr5(0, r < i , / i < i , r < A f / = 2.

(1)

For the system (1) initially at the bottom of one of the potential wells, we have computed the statistical distribution over wells at some given instant by computer simulations of Eq. (1), without (Fig. 3) and with (Fig. 4) AC-drive. Simulations demonstrate that the AC-drive may greatly accelerate the spatial diffusion, especially if the driving is adiabatic. Moreover, there is an optimal frequency (Fig. 4(c)): if cof decreases below it, the diffusion slows down (see the detailed discussion, including a criterion for the optimal frequency, in [8, 9]). Acceleration of the spatial diffusion by the adiabatic AC-drive applied to strongly underdamped spatially periodic systems was suggested earlier in [11] but neither the chaotic mechanism nor the correct criterion for the optimal frequency were given.

FIGURE 3. Noise-driven underdamped pendulum (Eq. (1) with T = 0.005, T = 0.5 and h = 0): the distribution over wells at some given instant (t = 10^), for the initial state being in the well 0 (units at the vertical axis are arbitrary).

FIGURE 4. The same as in Fig. 3 but in the presence of the AC-drive with h = 0.2 and cOf equal to: (a) 2.0, (b) 0.1, (c) 0.005, (d) 0.001. As cOf decreases, the diffusion first accelerates ((a)-(c)) but then slows down ((c)-(d)), thus revealing the existence of an optimal small frequency.

Threshold devices

Threshold devices are reset to an initial state if a given dynamical variable reaches a preset threshold. Such devices, linked to various stochastic systems, were considered in the context of stochastic resonance (see e.g. [12]). We suggest to use them in a different context: the adiabatic AC-drive may greatly increase the mean rate of resets. Consider the model (1) and a threshold in kinetic energy, Kth', Fig. 5 shows an example of the relevant scheme. Let Kth be higher than the barrier height AU and let the temperature T

23

Josephson j unction

-^—r drive

M^v^

threshold device

FIGURE 5. Example of the relevant scheme with a treshold device.

be smaller than both AU and Kth — ^U. In the absence of the AC-drive, the mean rate of resets has an activation-like dependence on T:

Rocexv{-Kth/T), h = 0, T^Kth. (2)

If the adiabatic AC-drive is applied, then the system starting from the top of the barrier may follow the deterministic trajectory reaching a high kinetic energy'. This means that the activation barrier reduces to AU, i.e. the reset rate should drastically increase.

We have tested this in computer simulations. The parameters are the same of Fig. 3. The threshold kinetic energy is Kth = 4.5. In the undriven case, R ^ 69 while, for the adiabatically driven case ( « / = 0.1, /i = 0.2), R « 240100, about 3500 times larger. This huge increase is obtained despite a rather non-small temperature: for smaller temperatures it would be exponentially higher. The origin of the effect is the adabaticity of the AC-drive: for the non-adiabatic drive (cof = 2),R^ 7060, about 35 times smaller.

Such a strong sensitivity of the reset rate may be used for control or as a new kind of sensors.

REFERENCES

9. 10. 11. 12.

A.J. Lichtenberg and M.A. Liebermann, Regular and Stochastic Motion, Springer, New York, 1992. G.M. Zaslavsky, Physics of Chaos in Hamiltonian systems. Imperial CoUedge Press, London, 2007. G. N. Piftankin, and D.V. Treschev, Russian Math. Surveys 62, 219-322 (2007). A.I. Neishtadt, Sov. J. Plasma Phys. 12, 568-573 (1986). J.R. Gary, D.F. Escande, J.L. Tennyson, P/;j;s. Rev. A 34, 4256-4275 (1986). Y. Elskens and D.F. Escande, Nonlinearity 4, 615-667 (1991). S.M. Soskin, O.M. Yevtushenko, R. Mannella, Phys. Rev Lett. 95, 224101 (2005). S.M. Soskin, R. Mannella, and O.M. Yevtushenko, in Noise and Fluctuations in Circuits, Devices and Materials, edited by M. Macucci, L.K.J. Vandamme, C. Ciofi, M.B. Weissman, Proceedings of SPIE 6600, 660008 (2007). S.M. Soskin, R. Mannella, O.M. Yevtushenko, Commun. Nonlinear Set Numer Simulat., in press. D. Hennig, L. Shimansky-Geier, and P Hanggi, Eur Phys. J. B 62,493-503 (2008). J. Kallunki, M. Dube, T. Ala-Nissila, Surface Science 460, pp. 39-48, 2000. N.G. Stocks, Phys. Rev Lett. 84, 2310-2313 (2000).

The deterministic trajectory coincides with the regular-like part of the chaotic trajectory though chaotic-ity is irrelevant

24

New Approach To The Treatment Of Separatrix Chaos

S.M. Soskin*'^ and R. Mannella**

* Institute of Semiconductor Physics, 03028 Kiev, Ukraine Uhdus Salam ICTP, 34100 Trieste, Italy

**Dipartimento diFisica, Universita di Pisa, 56127Pisa, Italy

Abstract. For a time-periodically perturbed ID Hamiltonian system, we match the separatrix map and the resonance Hamiltonian dynamics for the frequency ranges where the separatrix chaotic layer (SCL) possesses the largest width. This allows us to describe the boundaries of the SCL in the phase plane, in particular high peaks in the frequency dependence of the SCL width in energy.

Keywords: Hamiltonian chaos, separatrix, resonances PACS: 05.45.Ac,05.45.Pq

INTRODUCTION

Separatrix chaos is the seed of chaos in most Hamiltonian systems [1-3]. Consider a system of dimensionality 3/2, for which a 1D Hamiltonian Ho{p,q) perturbed by a weak time-periodic perturbation could be an archetypal example. If//o possesses a separatrix, then the perturbation destroys the separatrix, which is replaced by a chaotic layer. If HQ does not possess a separatrix, the perturbation generates resonances, i.e. areas in the phase plane where the deviation of the angle of HQ from the angle of the perturbation performs regular oscillations [1-3]. These areas are separated from the areas where the absolute value of the deviation regularly grows with time by thin chaotic layers which may be considered as the separatrices generated by the resonant part of the perturbation which are destroyed by the non-resonant part of the perturbation [1-3]. Thus, typically, chaos in 3/2 D systems is immediately related to separatrix chaos. In many higher-dimensional cases, the origin of chaos is similar.

Outer boundaries of the separatrix chaotic layer (SCL) in a Poincare section can be easily found numerically by integration of the Hamiltonian equations with a set of initial conditions in the vicinity of the separatrix: it is easy then to distinguish regular and chaotic trajectories, e.g. using that the former ones lie on lines i.e. ID objects while the latter trajectories occupy layers i.e. 2D objects. But it is important to be able also to describe and to predict various properties of the SCL theoretically. This especially concerns the SCL width, either in energy on in related quantities. There is a long and rich history of the corresponding studies. The results may be classified as follows.

I. Heuristic analytic results. Consider a ID Hamiltonian system perturbed by a weak time-periodic perturbation:

H = Ho{p,q)+hV{p,q,t), V{p,q,t+ 2n/(0f) = V{p,q,t), / i < 1, (1)



25

where Ho{p,q) possesses a separatrix and, for the sake of notation compactness, all relevant parameters of//Q and V, except possibly « / , are assumed to be '- 1.

There were a few heuristic criteria set by physicists (see e.g. [1-3]) which gave qualitatively similar results for the SCL width in terms of energy E = Ho{p,q):

A£'(tO/)-/KKi/|e|, | e | - l for t O / - l , (2) |e| oc exp(-a(Kiy) <c 1, U'-^l for ft)y>l.

The quantity h\e\ is called the separatrix split [2] (see also Eq. (4)): it determines the maximum distance between the perturbed incoming and outcoming separatrices [1-5] .

It follows from (2) that the maximum of Ai? lies in the frequency range (Of'~ 1 while the maximum itself is '~ h:

AE^^ = max{AE{(Of)} ~ h, ^ " " ^ ~ • (3)

II. Mathematical results and accurate analytic physical results. There were many works which studied the SCL by methods typical of mathematicians.

Thus, for the range « / > 1, there were many works studying the separatrix splitting (see the review [4] and references therein) and the SCL width in terms of the so called normal coordinates (see the review [5] and references therein). Though quantities studied in these works typically differ from those studied by physicists [1-3], they implicitly confirm the qualitative conclusion from the heuristic formula (2) in the high frequency range: if « / > 1 the SCL width is exponentially small.

There were also several works studying the SCL in the opposite i.e. adiabatic limit (i)f -^ 0: see e.g. [6-9] and references therein. In the context of the SCL width, it is most important that /sE{(i)f -^0) ^ h for most of the systems [6-8]. For a particular class of systems, namely for ac-driven spatially periodic systems (e.g. the ac-driven pendulum), the width of the SCL part above the separatrix diverges in the adiabatic limit [9]: the divergence develops for Of <C 1/ ln(l//i).

Finally, there was a qualitative estimate of the SCL width for the range Of ^Iby one of the mathematical methods [5]. Its result appears to be the following: the width in this range is of the order of the separatratrix split while the latter is of the order ofh.

Thus, it could seem to follow from the above results that, for most of the systems (i.e. for all systems except the ac-driven spatially periodic systems), the maximum of the SCL width is ^ h and occurs in the range (Of ^ \, quite in agreement with the heuristic result (3). In any case (even for the ac-driven spatially periodic systems), this conclusion seemed to apply to the width of the SCL part below the separatrix, for the

whole frequency range, and of the SCL part above the separatrix, for Of ^ l / ln(l/ / i) . III. Numerical evidences of high peaks in AE{cOf) and their rough estimates.

The aforementioned conclusion does not agree with several numerical studies carried out during the last decade (see e.g. [9-15]) which revealed the existence of sharp high peaks in AE{cOf) in the frequency range l/ln(l// i) ^^ (i>f ^ 1- Intuitively, they were related by authors of [10-15] to the absorption of nonlinear resonances by the SCL. For some partial case, rough analytic estimates for the position and magnitude of the peaks were suggested in [10, 15].

26

IV. Accurate analytic description of the peaks. Accurate analytic estimates for the peaks were lacking until recently. They have been

done in recent works [16-18], where a new approach to the theoretical treatment of the separatrix chaos for the relevant frequency range has been developed and applied to the double-separatrix cases [16, 17] and to single-separatrix ones [17, 18].

The basic ideas of the approach are described below.

BASIC IDEAS OF THE APPROACH

The motion near the separatrix may be approximated by the separatrix map (SM) [1-3,5,16,17]. Its actual form may vary, depending on the system under study, but its relevant features in the present context are similar for all systems. For the sake of clarity, let us consider a concrete case when the separatrix of Ho{p, q) possesses a single saddle and two symmetric loops while V = qcos{cOft). Then the SM reads as [16] (cf. [1-3,5]):

Ei+i = Ei + aihssm{(pi), (4) (Of 7t{3 - sign{Ei+i - Es))

2Q){Ei+i) Oi+i = aiSign{Es-Ei+i),

e = e{Q)f) = sign{dHo/dp\f^_^) / dt dHo/dp\^^sm{Q)ft),

Ei = Ho{p,q)l,, (pi = (Ofti, ai = sign{dHo/dpl.),

where Es is the separatrix energy while co{E) is the frequency of oscillation with the energy E in the unperturbed case (i.e. for h = 0).

Consider the two most general ideas of our approach. 1. If the SM trajectory includes a state with E = Es, for any cp ^ nn (where n is an

integer) and any a, then the trajectory is chaotic. Indeed, the angle cp of such a state is not correlated with the angle of the state of the previous step of the map, due to the divergence of co^^ [E -^ Es). Therefore, the angle at the previous step may be arbitrary and, hence, the deviation of energy of the state at the previous step from Es may take an arbitrary value within the interval [—/i|e|,/i|e|]. Obviously, for the state E = Es the variable a is not correlated with that at the previous step either.

2. As weU known [1-3,10,15-18], the frequency of eigenosciUations as a function of the energy near the separatrix is proportional to the reciprocal of the logarithmic factor:

bncoo 3-sign{E-Es) (0{E) = . ^^ ^ , b= ^ '-, (5)

ln( ^ \E-E,

\E-Es\^AH = Es-Est,

where Est is the energy of the stable states. Given that the argument of the logarithm is large in the relevant range of ii, the func

tion co{E) is nearly constant for a substantial variation of the argument. Therefore, as the

27

SM maps the state {EQ = Es,(po,(7o) onto the state with E = E\ =Es + Oohe sin(ô), the value of co{E) for the given sign(c7oesin(ô)) is nearly the same for most of the angles ^0 (except in the close vicinity of multiples of n), namely

&)(£•)« ft)r = (o{Es±h) for sign(c7oesin(ô)) = ±1- (6)

Moreover, if the deviation of the SM trajectory from the separatrix increases fur

ther, (o{E) remains close to oy provided the deviation is not too large, namely if

Indii - Es\/h) <C \n{AH/h). If (Of ^ (Or , then the evolution of the map (4) may be regular-like for a long time until the energy returns to the close vicinity of the separatrix, where the trajectory is chaotized. Such a behavior is especially pronounced if the

perturbation frequency is close to (Or or (Or or to one of their multiples of relatively low order: the resonance between the perturbation and the eigenoscillation gives rise to an accumulation of energy gain for many steps of the SM, which results in a deviation of E from Eg that greatly exceeds the separatrix split /i | e |. As a function of «/ , along the SCL boundary the largest deviation from the separatrix takes its maximum at frequencies close to 0)r or (Or', for the upper or lower boundary of the SCL respectively. This corresponds to the absorption of the Ist-order nonlinear resonance by the SCL.

CONCLUSIONS

We have developed an approach which allows one to describe the separatrix chaotic layer in the frequency range where the layer width takes its maximum due to the absorption of a nonlinear resonance by the layer. The approach has numerous applications.

REFERENCES

1. A.J. Lichtenberg and M.A. Liebermann, Regular and Stochastic Motion, Springer, New York, 1992. 2. G.M. Zaslavsky, Physics of Chaos in Hamiltonian systems, Imperial CoUedge Press, London, 2007. 3. G.M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, 2008. 4. V.G. Gelfreich, and V.F. Lazutkin, Russian Math. Surveys 56, 499-558 (2001). 5. G. N. Piftankin, and D.V. Treschev, Russian Math. Surveys 62, 219-322 (2007). 6. A.I. Neishtadt, Sov J. Plasma Phys. 12, 568-573 (1986). 7. Y. Elskens and D.F. Escande, Nonlinearity 4, 615-667 (1991). 8. A.I. Neishtadt, V.V. Sidorenko, and D.V. Treschev, Chaos 7, 2-11 (1997). 9. S.M. Soskin, O.M. Yevtushenko, R. Mannella, Phys. Rev Lett. 95, 224101 (2005). 10. I.I. Shevchenko, P/;j;s. Scr 57, 185-191 (1998). 11. A.C.J. Luo, K. Gu, and R.PS. Han, Nonlinear Dyn. 19, 3 7 ^ 8 (1999). 12. S.M. Soskin, R. Mannella, M. Arrayas and A.N. Silchenko, Phys. Rev £ 63, 051111 (2001). 13. A.C.J. Lao, Appl Mech. Rev 57, 161-172 (2004). 14. V.V. Vecheslavov, Tech. Phys. 49, 521-525 (2004). 15. I.I. Shevchenko, Phys. Lett. A 372, 808-816 (2008). 16. S.M. Soskin, R. Mannella, and O.M. Yevtushenko, Phys. Rev E11, 036221 (2008). 17. S.M. Soskin, R. Mannella, and O.M. Yevtushenko, "Separatrix chaos: new approach to the theoretical

treatment", in Chaos, Complexity and Transport: Theory and Applications, edited by C. Chandre, X. Leoncini, and G. Zaslavsky, World Scientific, Singapore, 2008, pp. 119-128.

18. S.M. Soskin, and R. Mannella, "Maximal width of the separatrix chaotic layer", submitted to Phys. Rev E.

28

A Basis Set for Characterizing Transient Random Phenomena

Roy M. Howard

Department of Electrical and Computer Engineering, Curtin University of Technology, GPO Box U1987, Perth, 6845, Australia.

Abstract. An orthogonal basis consistent with transient random phenomena is proposed and applied to market data and 1//"noise. For 1//"noise the power-rate spectrum is flat. The basis set leads to a power-rate spectrum and can be used to facilitate the detection of specific signal forms.

Keywords: Transient random phenomena, basis set, power-rate spectrum, power spectral density. PACS: 02.50.Fz, 05.10.Gg, 05.40.-a

INTRODUCTION

The scientific endeavour related to characterizing random phenomena has a long and important history, e.g. [1]. One important, and widely used, method for characterizing random phenomena is via waveform decomposition based on utilizing a basis set for a specified interval of interest, e.g. [2]. Usually, the power, or energy, of the constituent waveforms arising from the decomposition are calculated and utilized to define a power spectral density or a time-scale representation for the random process. Common basis sets include the sinusoidal basis set, which leads to the standard, and widely used, power spectral density function, a basis set consistent with a Karhunen-Loeve decomposition e.g. [3] and [4], a wavelet basis set, e.g. [5], which leads to a time-scale characterization, etc. In general, the basis sets used are such that the waveforms, or sets of waveforms as in wavelet analysis, exhibit the same form over the interval of interest. As such they are not optimally suited to the characterization of transient random phenomena. For transient random phenomena of a specific type, a custom approach, where the signals in the basis set match the form of the underlying random phenomena, is likely to lead to a better characterization and, hence, a simpler model of the random phenomena.

There have been several papers on orthogonal basis sets for the transient case and typically for the interval [0, oo], e.g. [6], [7] and [8]. Such basis sets usually consist of Laguerre, or similar functions, which exhibit exponential decay as infinity is approached. Such basis sets can, naturally, represent transient phenomena of all types. However, in general, the decomposition is likely to be non-optimal when the random phenomenon yields signals which are inconsistent with the basis set signals. In this paper a basis set for characterizing transient phenomena, and consistent with random walk type phenomena is proposed.



29

http://02.50.Fz

http://05.10.Gg

BASIS SET FOR TRANSIENT RANDOM PHENOMENA

N Consider the orthogonal basis set, {b,/.R -^R}, for the interval [0, 2 At], illus-

trated in Figure 1 and a signal x:R^R. The associated data vector X = {XQ, Xp ..., X ^ } can be written, using the basis set, as

N N-l 2^" ' - l N-l

X, = x(kAt) = ^ c,,b,,(kAt)+ ^ ^ c,jb,j(kAt) = ^^+ X ^^(^^^^ (1)

i=0 i=l j=l i=l

N for A: e {0, 1, ..., 2 - 1} , where c,,. is the coefficient associated with b,Xt), and

^fc = S ^^0ô(*^0, r.(kAt)= ^ c^jbij(kAt), ie{l,...,N-l} (2)

i=0 j=l

Accordingly, the data vector can be written as

N-l

X=^+ Y^R., n = L o , H i , . . . , n ^ ^ _ 1, R. = lr.(0),...,r.[(2^-l)At]\(3)

i = I

The first component, \x, represents a 'dynamic' mean component consistent with the basis signals b^ît), b^ît), ..., b^Q{t). The vector R^ is defined by samples of a

n 611(0^12(0 ^^^(^) i ' i4(0 b^^{t) ^16(^)^17(0 ' -O • - 0 • - 0 •—O • - 0 • - 0 • - 0

i(0 = Z ^1/1/0 7 = 1 f 4 - « • • • • • • • • t r i(

/ 2M 4At 6At HAt lOAt UAt UAt l6At 6 I Q ( 0 « - O • - 0 • - 0 • - 0 • - 0 • - 0 • - 0

ôo(0 • ho^'^h^ît) 622(0 623(0 N-2

1^ • • o • O • O ^ "1 t '•iW = Z ^ 2 / 2 / 0

I 2At 4At 6At %At 10At l2At UAt 16At .^, - 1 * • o • o • o ^

h (n ^4o(0

A i'3o(o • o r^(t) = -^ Cyby(t)

7 = 1

• • • • • • • 7H» • < , lAt 4At 6At HAt lOAt UAt UAtdl6At ^* • ' O

FIGURELBasisse t {6oo(0,6,^.(0}> ; e {1, •••,W}>/e {0, . . . , 2 ^ ' - ! } for [0, 2Â0 and

for the case of W = 4 . All waveforms have unity amplitude but some are offset for clarity of

display. The basis signals 6QQ(0, 6 I Q ( 0 , •••, ÂTQCO have a non-zero mean; all other basis signals

have a zero mean. The signals r^ , r^ and fj define the first three signalling random processes.

30

signalling random process r^{t), [9], which is based on the basis functions

bf^{t), ..., b ^_; (t) and has a rate of X. = 1/2 A?. Its first 2 components are

zero consistent with transient phenomena. Given a data vector X it can be readily shown that the coefficients in the basis decomposition are given by

2 ' - ' - l

CQO - ^ 0 ' 1

k=0 2'-'+l

ie{l,2,...,N} (4)

2

• 2 ' - ' - l 2 ' - ' - l

^ 2i+k ^ *• / 7 - 1

2 / + 2 +^ fc = 0 fc = 0

/ e { l , . . . , 7 V - l }

7 e { l , . . . , 2 - 1 }

The average power, P^, in the ith signalling random process r^{t), over the interval

i N [2 At, 2 At], where it is non-zero, is

l^At 2"-'-!

Pi = {2^ -2') At

\ r^(t)dt = 2 ^ - ' - - l

Z ^[ %J (6)

2 At i = l

2 2 2 where E is the expectation operator. For the case of E[c--] = E[c-j^] = a. , for k^j,

2 it follows that P- = a- . For a random walk with a step size of +A, and a step variance

2 2 of a = A , it can be shown, utilizing the independence between step increments, that the average power in the ith signalling random process R- is:

2 ^i 2 P = ^ + 2 ^ . I 1

' 4 12 , / - l

2 2 a A to I , 4 12X,' , / - !

(7)

for ; e {1, ..., A^-1}. The second form arises as X. = 1/2 At and, thus, the

power-rate spectrum exhibits a 1/rate form.


Results taken from Forex data (AUD vs USD) and 1 / / noise are shown in Figure 2 and clearly show, for the Forex data, 'reasonable market' random walk behaviour at a macro level. Note that the power-rate spectrum for 1 / / noise exhibits, consistent with the model proposed in [10], a flat power-rate spectrum. Thus, relative to the proposed

31

basis set 1 / / noise appears as 'white' noise. The following is a summary of power-rate spectral forms: White noise (flat power spectral density) yields a power-rate spectrum varying according to rate, 1 / / noise yields a flat power-rate spectrum and a \lf (a power spectral density consistent with a random walk - see ch. 6 of [9]) yields a 1 /rate power-rate spectral form. The proposed basis set can be used to facilitate a test of whether a random phenomena is consistent with a random walk, or has a component which is consistent with a random walk or 1 / / noise. Finally, the proposed basis set can yield greater sensitivity to certain types of random phenomena than a standard power spectral density.

power-rate spectrum

0.00005

0.00001 5.X10-'

5.X10"'

l . x l O - '

" 1

ffl

• •

.. *' m

1 1 1

• .

«i

normalized rate

FIGURE 2. Left: Theoretical power-rate spectrum for a random walk (continuous curve) and power-rate spectrum of 4096 values of AUD:USD Forex data (dots) taken with a 10 minute step interval. Right: averaged power-rate spectrum from 1 / / noise simulated consistent with [10] and then filtered. 100 averages and 65536 samples have been used. The roU-off is due to the filtering.

CONCLUSION

An orthogonal basis consistent with transient random phenomena has been proposed and applied to market data and 1 / / noise. The basis set leads to a power-rate spectrum and for a random walk the associated power-rate spectrum exhibits a 1/rate form whilst for 1 / / noise the associated power-rate spectrum is constant. The proposed basis set can facilitate the detection of specific signal forms.

REFERENCES

L. Cohen, IEEE Signal Processing Magazine, 20-45, Nov. (2005). L. Debnath and P. Mikusinski, Introduction to Hilbert Spaces with Applications, Academic Press, 1999. M. Loeve, Probability Theory: Part II, Springer Verlag, 4th ed. 1978, p. 144. J. Ardnt, H. Hertzel, S. Bose, M. Falcke and E. SchoU, E. (1997), Chaos, Solitons & Fractals, 8, 1911-1920(1997). T. K. Sarkar, C. Su, R. Adve, M. Salazar-Pahna, L. Garcia-Castillo and R. Boix, IEEE Antennas and Propagation Magazine, 40, 49-70 Oct., 36-49 Dec. (1998). H. L. Armstrong, IRE Transactions on Circuit Theory, 4, 286-287 (1957). H. L. Armstrong, IRE Transactions on Circuit Theory, 7, 351-354 (1959). J. W. Head, Proceedings of the Cambridge Philosophical Society, 52, 640-651 (1956). R. M. Howard, Principles of Random Signal Analysis and Low Noise Design: The Power Spectral Density and its Applications, WUey, 2002, ch. 5.

10. R. M. Howard and L. A. Raffel, 'General models for 1/f noise' in Noise in Devices and Circuits, edited by M. J. Deen et. al, SPIE vol. 5113, SPIE, Bellingham, WA, 2003, pp. 282-293.

32

Stochastic Dynamics of Road-Vehicle Systems and Related Bifurcation Problems

Walter V. Wedig

Universitat Karlsruhe, D-76128 Karlsruhe Institute of Technical Mechanics

Abstract. The paper investigates stochastic dynamics of road-vehicle systems and related bifur

cation problems. Running on rough roads, cars generate vertical vibrations. They are resonant for

critical car speeds and completely tranquilized when the car velocity is increasing infinitely. In

case of weak damping and progressive wheel suspension the car vibrations become unstable when

the car velocity reaches the parameter resonance near twice the critical speed of the linear system.

Keywords. Road spectra, resonant car speeds, Lyapunov exponents, parameter resonances

Pacs. 02.50.Ey 02.70.Hm, 02.70.Kn, 05.10.-a

INTRODUCTION TO ROAD-VEHICLE SYSTEMS

To introduce problems of interest let us consider the simple quarter car model, shown in fig. 1. It is riding with constant velocity v = const, on harmonically or randomly shaped roads with surfaces Z,, determined in dependence of the way coordinate s.

a

Al Xt

Z, = Z COS ujs ^ i ^ Zt

" s = vt

Figure 1: Scheme of road-vehicle-systems

Running on roads, vertical car vibrations are generated described by the equation of motion

Xt + 2Doj,[Xt - Zt) +OJI[1 + 7(Xt - Ztf][Xt - Zt) = 0. (1)

Herein, uji = \Jcjni denotes the natural frequency of the car, D > 0 is its dimensionless damping measure and 7 > 0 determines the cubic spring characteristic of the wheel suspension. In the classical case, the road surface Z^ is harmonically modelled with amplitude Z and middle frequency GJ = 271/1 determined by the wave length £.

CPl 129, Noise and Fluctuations, 20 International Conference (ICNF 2009) edited by M. Macucci and G. Basso

2009 American Institute of Physics 978-0-7354-0665-0/09/$25.00

33

SPECTRAL ANALYSIS OF ROADS AND CARS

In stochastic problems, way-noise models [1] are introduced by the increment equation

dZs = -nZJs + adWs , E{dW^) = ds. (2)

Herein, ds denotes the way increment, dWg is the Wiener increment, a determines its intensity and 0 is the associated corner frequency replacing the middle frequency of the harmonic case, mentioned above. The transformation of (2) into the time domain is performed by means of the increments ds = vdt and dWg = \fv dWf. This gives the time increment equation

dZt = -QvZtdt + a^/v dWt E{dW^) = dt.

In the stationary case, the spectral analysis of (3) leads to the power spectrum [2]

„2„, „2

SAuj) = y202 ^ ( ^ ) - - - 2 0

(3)

(4)

that is integrated over all spectral frequencies |w| < oo in order to obtain the root mean square, respectively the rms-value a^ of the stationary base excitation Zt- Note, that the road spectrum is vanishing for sufficiently slow or fast car speeds, meanwhile the rms-value of the road is constant and independent on the car speed.

T

OxjOz

0

[ / a o n i

D = .01

D = m

D = .10

0 0.4 0.8 1.25 2.5 oo —> V = vfl/uji

Figure 2: Rms-ratios of road and car versus related car speeds

In the linear case 7 = 0, the spectral analysis above can be extended to the quarter car equation (1) that leads to the following spectral density of the road vehicle system.

Sr.(oj) = [{UJI - uj^y + {2Dujiujy]{uj^ + nV) •

(5)

This spectrum can easily be integrated over all spectral frequencies |w| < 00 in order to obtain the following vehicle-road rms-ratio in dependence on the related car velocity ly.

2Z) + (1 + 4Z)2),y

2D[1- 2D)i n

with v = V — (6)

34

Figure 2 shows theoretical and numerical evaluations of both, the excitation (x) and response (o) rms-values, plotted versus the related vehicle speed for the three different damping measures D = .01, D = .05 and D = . 10. Herein, the applied jj — v scaling is linear with ;/ = /i in the under-critical range 0 < /i,;/ < 1 and rationally scaled by;/ = 1/(2 —/i) in the over-critical ranges 1 < /i < 2 and l < v < oo, respectively. Note, that the car vibrations become resonant for the critical speed Vc = l and they are completely tranquilized for infinitely increasing speeds. The measurements of fig. 2 are obtained by Monte-Carlo simulations applying a simple forward Euler scheme with the scan rate Ai = .001 for A = 10^ sample points. The Wiener increments are approximated by normally distributed l\Wn = VArNn.

MC-SIMULATIONS OF NONLINEAR CAR VIBRATIONS

To extend the simulations above to the nonlinear case, the equation of motion (1) is rewritten into a first order form introducing the state process of the displacement Xt and the vertical velocity Xt = uJiYt. Subsequently, Zt is replaced by (3) and all three state processes are nor-mahzed by [Zt, Xt, Yt) = a^{Zt, Xt, Yt). Finally, dimensionless time and noise are introduced by ujidt = dr and ^JuTydWt = dW^, that reduces the five parameters in (1) and (3) to the three ones !/, D and 7. Therewith, the following dimensionless equations are obtained:

dXj- = YT dr,

dYr = -2D{Yr

dZr = —lyZrdr + v 2;/ dW^,

uZr)dT + 2DV2^dWr - [1 +

+j{Xr - Zrf]{Xr " Zr)dT, w i t h 7 = ^fal

(7)

(8)

Figure 3 shows numerical results (o) of the rms-ratios in dependence on the related car speed for the parameter D = 0.1, 0.2, 0.5 and 7 = 0.2. The results are obtained by means of Euler schemes with the scan rate A r = 0.001 applied for A = 10® samples. Straight lines without markers represent the results of the linear S5^em with 7 = 0.

2.5

0"x/0"2

1.5

0.5

0

D=0.1

. D = 0 . 2

D=0.5

0 0.4 0.8 1.25 2.5 00 —y V = vfl/uji

Figure 3: Rms-ratios (o) of the nonlinear car dynamics for 7 = 0.2

35

STABILITY OF THE VERTICAL CAR VIBRATIONS

The stability of the nonlinear car vibrations is investigated by the variational equations of (7) and (8), derived by the perturbation set-up Xt = Xt + AXt and Yt = Yt + AYf.

d{AXr) = AYrdr, AX^, A K « 1, (9)

d{AYr) = -[2DAYr + AXr + 37(X^ - ZrfAXr]dT. (10)

Applying the polar coordinates AX^ = A^ cos $^ and Ay,- = A^ sin $^, the linear perturbation equations (9) and (10) are transformed to the phase and In-amplitude equation

d^r = -dr - [37(X^ - Zrf cos $^ + 2D sin $^] cos $^(ir, (11)

d[\nAr) = -[i^iiXr - Zrf cos<^r + 2D sm<^r]sin<^rdT. (12)

According to the multiplicative ergodic theorem of Oseledec, the amplitude equation (12) gives l-T

1 Xtop = lim 7= / [-37(X^ - Zr) cos $^ - 2D sin $^] sin $^(ir. (13)

Figure 4 contains numerical evaluations of the top Lyapunov exponent Xtop for the values D = 0.07,0.05 and 7 = 0.1,0.12. The scan rate AT = 0.001 was applied for N = 10* sample points. This analysis shows the Kramers effect [3] that for weak positive damping the stationary car vibrations can be destabilized bifurcating into stochastic chaos.

0.05

^top

0.0

-0.025

-0.05 0

fl5= m 7 = 0.1

D = m 7 = 0.12

'D = m 7 = 0.1 chaos

Stat.

0.4 0.8 1.25 2.5 00

Figure 4: Parameter resonance aXvp = 2 near twice the critical speed Vc = l

References

[1] W. Wedig, Dynamics of Cars Driving on Stochastic Roads. In: Computational Stochastic Mechanics ed. by P.D. Spanos & G. Deodatis, Millpress, Rotterdam, 647-654, 2003.

[2] K. Popp & W. Schiehlen, Fahrdynamik, B.G. Teubner, Stuttgart, 1993.

[3] L. Arnold & P. Imkeller, The Kramers Oscillator Revisited. Lecture Notes in Physics, 280-291, 2000.

36

Mathematical Background of 11/Fluctuations

Toshimitsu Musha

Brain Functions Laboratory, Inc. 4259-3 Nagatsuta, Midori-ku, Yokohama, JAPAN226-8510

Abstract. Energy of harmonic oscillators in equilibrium decays exponentially in time when they are coupled in quadratic forms in amplitudes. In reality, however, their Hamiltonian includes higher-order coupling terms. Not all of the higher-order coupling terms contribute to the energy decay of oscillators after averaging over reservoir oscillators, and we find that one of the lowest higher-order terms makes a finite contribution to the energy decay. This effect is equivalently represented by a modified coupling coefficient of quadratic coupling terms. This modification works as a positive feedback to the action-reaction process between oscillators. Eventually the modified coupling terms generate 1//" fluctuations in energy partition among oscillators in equilibrium. It is concluded that l//"type of energy partition is observable with harmonic oscillators if they obey the Bose-Einstein statistics regardless of whether the collective system is classical or quantum mechanical regime.

Keywords: 1/f fluctuations, harmonic oscillator, energy partition, Bose-Einstein statitics. PACS: 40.30Nk, 05.20JJ, 05.30Jp, 05,40Ca, 05.45Xt, 0570Ce

INTRODUCTION

Ubiquity of the 1//" fluctuation is still one of the mysteries in science. The main stream of research in ICNF series is about fluctuations of electric conductivity, but there are many other interesting off-stream 1//"fluctuation phenomena which suggest a possible origin of this type of fluctuations. For instance, fluctuations in traffic currents on an expressway, ' density modulation of biological action potential impulses propagating on nerve axons, ^ frequency fluctuations of extremely temperature-stabilized quartz oscillators, ^ human heart rates,'' neuron discharge intervals of snail, ^ hand tapping intervals, phonon excitation in a temperature-stabilized quartz crystal,^ imaginary part of a dielectric constant,® and so on. The present mathematical approach suggests existence of a universal mechanism for generation of such mysterious fluctuations. The previous theoretical approach ^ is not satisfactory although the final result is supported by experiments and needs no correction. Here an abstract of the full version mathematical manipulations will be described.

FORMULATION

Energy fluctuations of a dynamical system in contact with a heat bath are associated with dissipation processes as is shown by the fluctuation-dissipation theorem and the problem associated with energy dissipation of an oscillator in equilibrium with a

CPn29, Noise andFluctuaUons, 20* International Conference (ICNF 2009) edited by M. Macucci and G. Basso


37

reservoir oscillators was already discussed in detail by Louisell' °, in which he starts with the Hamiltonian for creation and annihilation operators of harmonic oscillators.

H = hcoaâ + ^ha>jb'jbj + Mkjab'j + k'a^bj) + hkjfi'jb,. (1)

We will focus our attention on dynamic behavior of oscillator a and its number operator n (= aâ). All the oscillators in equilibrium will be subject to the same statistical behavior. Other oscillators b/s are regarded as forming a reservoir to a, and their initial values bj{0) are randomized such that <bj{0)> = 0 in making ensemble averaging. Throughout the following manipulations we make <bj{0)> = 0. The Heisenberg picture and the Schroedinger picture will be used as the case may be.

The equation of motion of osciUator a is da / dt = (1 / ifi){aH - Ha), and each of a

and bj is separated into slowly varying part and quickly oscillating part as

a{t) = A(t)e'""' ,bj(t) = Bj{f)e'""''. The equations of motion including n = N= aâ is

N(t) = (l/ih){a*aH^ - Hâ*a)= -ij^ (- kjUb* + k'ja^bj) (2)

b = -icDjbj -ikja (3)

Bj{t) = -ikjA{f)e'""'-"''^' . (4)

Integrating Eq.4 gives

BAt)-Bm = -ik, f rf?M(r)e"*"""^"' (5)

from which Eq.2 is rewritten as

N{f) = -YîkjA{t)(k'j^'dfA\f)e^'"''"'^^'"'A + HA) = -yNit) (6)

under the Wigner-Weisskopff approximation (HA denotes Hermitian adjoint):

Y}k^'^{Mne'"°'''°'^"'"^dt' = {r/2)A(t). (7)

Here 7 is a relaxation frequency. In deriving Eq.6 coupling terms as a^bj play an essential role. Therefore, a simple relaxation process comes from the action-reaction processes between oscillator a and the reservoir oscillators, in which mutual coupling of the reservoir oscillators is ignored. In reality reservoir oscillators are mutually coupled and some part of energy which has been transferred to them from oscillator a diffuses over the reservoir oscillators. This diffusion process works to maintain a part of energy coming from a and feeds it back to a through the action-reaction process. This effect can be taken into account by adding higher-order coupling terms to the Hamiltonian. We investigated effects of terms as aâbj, a^bjbi, aâabj , aâb/bi,

ab^b^b, and aabjb^ and found out that only the term of aâabj{= nabj) has a finite

contribution to dN/dt after ensemble averaging of the reservoir states. Therefore, kj is modified as (1 + n)kj. The additional higher-order Hamiltonian and its effect on dN/dt are described as

H= hriY^kjab] +HA = hY^kââab] + HA (8)

and dN^/dt = {1/ ih^NH-H N)= mY,kjab] +HA

38

= -N{t}^\kjf^'A\t')e"''"''"'^'''"'^dt'A{t} + HA = -rN{t}N{t}. (9)

Finally Eq.6 is reformed as dNldt = -rN-rNN. (10)

In the Schroedinger picture, \t) is a state of the oscillator system at time t, and \t), is

a state of oscillatory at time t. Then

\t) = Y\\t)r (11) In the Schroedinger picture we rewrite as N(t)N(t) = {t\NN\ O) in Eq.9;

(; |AW| 0) = j ' {t\N\ T)P(T){T \N\ 0)dT = j ' {t\N\ r) — {T\N\ 0)dT, (12)

where p{T)dTis the number of states of \t) betweenr and z + dz, which is replaced

by dz{d I dz)). This is inserted inEq.lO;

sN{s) -1 = -yN{s) - yN{s)[sN{s) - 1 ) , (13)

where N{s) = N(t)e'"dt and N(t) is normalized by MO) to simplify the following Jo

treatment. Eq.l3 has two solutions,

ms)=j-\-i±f^. (14) These solutions are approximated in the two extreme cases:

N(s) = l/s or l/y for \s/y\»l,and N(s) = l/-ylys f o r | 5 / 7 | « l . (15)

N{s) refers to an impulse response of oscillators and its Fourier transform is given by

N{in) where /2 is a Fourier frequency. The effective time length of this impulse

response is \l y. With v such oscillators, the power spectral density of the total sum

of normalized power (= v) of these oscillators is given by

i~ |2 fvr /Q^ ( ^ » r )

s^n) = vY\N{in)\ =r ' ) ^[ (16) Suppose each oscillator has a power of £A (0) . Then the PSD of the power

fluctuations must be multiplied by (£^^(0))^ on Eq.l6, namely (£A (0)) .S'(/2). The

fractional PSD is obtained by diving PSD by a squared total power {yeN(^)f.

Therefore, a fractional PSD, Sfraa (•^), is given as

^ v^\ I \\lvQ; {Q«Y). [Mb)

Conclusively the so-called 1//spectrum appears at frequencies below v, above which it is of \lf type. Vibration modes of materials are decomposed into harmonic oscillators and they are always accompanied by nonlinear coupling of mode. Therefore, the present theory predicts that 1//fluctuations should be observed. We cannot get rid of this mysterious fluctuations.

39

DISCUSSION and CONCLUSION

Let the impulse response function be simply given by g{t) (=N{tyN{Oy), then inverse Laplace transformation of Eq.l3 generates the equation of motion as,

^ = -r,(t)-rUt-r)^dr. (18) dt Jo dr

The second term works to slow down the exponential decay in time. Factor g{T)dT is an infinitesimal amount of energy given to the reservoir in dr at timer, and it will decay by git-r) in a remaining time interval t - r up to time t of observation. Therefore, the remaining amount g{t - r) g{T)dT is continuously fed back to the original oscillator, which slows down the otherwise exponential loss rate. This is the meaning of Eq.l8.

It is concluded that as long as a collective system is subject to the Bose-Einstein statistics, energy partition will have 1//fluctuations and its power spectral density is given by Eq.l7 regardless of whether it is in the classical mechanical regime or quantum mechanical regime, and from the observed PSD we can estimate how many elements (molecules) are involved in generation of the observed fluctuations.

REFERENCES

1 . T. Musha and H. Higuchi, Jap. J. Appl Phys. 15 , 1271-1275 (1976); T.Musha andH.Higuchi, Jap. J Appl. Phys. 17, 811-816(1978). 2 . T. Musha, Y. Kosugi, G Matsumoto, M. Suzuki, IEEE Trans. On Biomed. Eng. BME-28, 616-623(1981);

R Graeneis, M. Nakao, M. Yamamoto, T. Musha and H. Nakahama, Biological Cybernetics 60, 161-169(1989).

3 . Y Noguchi, Y Teramachi and T. Musha, Appl Phys Lett. 40,, 872-873(1982). 4 . M. Kobayashi and T. Musha, IEEE Trans. Biomed. Eng BME-29, 456-457(1982). 5. T. Musha, H. Takeuchi and T. Inoue, IEEE Trans. Biomed. Eng BME-30, 1943-197(1983). 6. T. Musha, K. Katsurai and Y Teramachi, IEEE Trans Biomed. Eng BME-32, 578-582(1985). 7 . T. Musha, G Borbely and M. Shoji, Phys Rev. Lett, 64,2394-2397(1990).

T. Musha and G Borbely, .Jpn J Appl. Phys.il, no.3B, L370-L371(1992). 8. T. Musha, A. Nakajima and H. Akabane, Jpn J Appl. Phys 27, L311-L313(1988). 9. T. Musha and M. Tacano, Physica A 346, 339-346 (2005). 1 0 . W.H.Louisell, Quantum Statistical Properties ofRadiation,Nsw York: John Wiley & Sons, 1990, p.420.

40

Non Equilibrium Fluctuations In The Degenerated Polarizable Plasma

V.V. Belyi* and Yu.A. Kukharenko'''

'Department of Theoretical Physics, IZMIRAN, Russian Academy of Sciences, Troitsk, 142190, Russia

^ UIPTP, Russian Academy of Sciences, Moscow, Russia

Abstract. The quantum plasma of Bose and Fermi particles is considered. A scheme of equation linearization for density matrix with the exchange interaction taken in account is proposed and the equation solution is found. An expression for Hartree- Fock dielectric permittivity with the exchange interaction is obtained. This interaction is taken into account in the exchange scattering amplitude. With the use of obtained solutions the non-equilibrium spectral function of electric field fluctuations in presence of exchange interaction and medium polarization is found. It is shown that in the state of thermodynamic equilibrium a Fluctuation-Dissipation Theorem holds. An expression for the system's response to an external electric field in presence of exchange interaction is given.

Keywords: Fluctuation phenomena in plasma, polarization and exchange interaction, Hubbard and related models PACS: 05.40.-a, 05.30.Fk, 52.25.Gj, 72.70.+m, 71.10.Fd

INTRODUCTION

Quantum correlation effects in the non-equilibrium plasma could be traced also in mesoscopics and plasma diagnostics. Recently [1] it was found that the inter-electron collisions in non-equilibrium degenerated plasma create additional quantum correlations and produce inputs to the equal-time non-equilibrium correlation. However, not only the initial correlations, but also dielectric properties of the degenerated plasma determine the fluctuation spectrum. The electron form-factor - spectral function of the electron density fluctuations plays an essential role in determining the spectrum of the scattering of an electromagnetic field by a non-equilibrium plasma [2]. Strongly coupled plasma is characterized by its exchange interaction. In the works [3], on the basis of the BBGKY-hierarchy, we found a solution for the one-time pair correlation function and obtained the kinetic equation describing the one-particle distribution function relaxation with the exchange interaction and polarization taken into account. In this paper these results are generalized to the case of two-time correlation functions. For this purpose we start with the dynamic equation for a one-particle density matrix in the Heisenberg representation. Our approach develops Pines-Schriffer method [4] of accounting for the exchange interaction. Our scheme allows us accounting for not only the self-consistent Hartree field, but the exchange Fock field as well. With the use of the found solution of this linearized equation we obtain an expression for the one-particle density matrix fluctuations spectral function. The expression for the electric field spectral functions contain the exchange scattering ampltude and dielectric permittivity, which exactly takes into account the exchange interaction of particles. It is shown that this expression



41

http://52.25.Gj

in the thermodynamic equilibrium satisfies the Fluctuation-Dissipation Theorem. A general expression for the response function in presence of exchange interaction is given. Approximate solutions of the equation for the exchange scattering amplitude are used for concrete estimates.

THE LINEARIZED EQUATION FOR THE DENSITY MATRIX

We start with the dynamic equation for a one-particle density matrix

N{t,p,p') = a+{t,p')a{t,p), p = (p,cj), (1)

where a{t,p') and a'^{t,p) (p = (p, <T)) are particle creation and annihilation operators in a state with momentum p and spin a . The complete statistical description of the quantum system is determined by the collection of all matrix (1) moments:

{N{t,p,p')), {N{t,pup[)N{t,p2,p'2)). (2)

This system of moments satisfies a chain of equations analogous to the BBGKY-hierarchy. However it is more convenient not to solve the system for coupled equations, but to use the equation of motion of the operator N{t,p,p'). This exact nonlinear equation describes the dynamics of quantum systems. In this work we will be interested in the two-time correlation function {5N{t,pi,p[)5N{t',p2,P2)) where 5N{t,p,p') = N{t,p,p') — {N{t,p,p')) is the density matrix fluctuation. We can linearize equation for operator N{t,p,p'). For this purpose let us represent

N{t,px,p[)N{t,p2,p'2)-{N{t,px,p[)N{t,p2,p'2))

= 5N{t,p2,p[)5{pi -p'2) + 5N{t,pup[) {N{t,p2,p'2)) + 8N{t,p2,p'2) {N{t,pup'i))

+ri5N{t,p2,p[){N{t,puP2)) + ri5N{t,pup'2){N{t,p2,p[)), (3)

where rj is 1 for Bose statistics and -1 is for Fermi statistics. Approximation (3) corresponds to taking exchange into account in the BBGKY-hierarchy as in [3]. The linearized equation for fluctuation has the form

mjm{t,pup'i) = [E{p,)-E{p[)\ 5N{t,pup[)

+ \f{t,p'i)-f{t,Pi)]fdp[U{pi-p\) + nU{p)]8N{t,p,+p,p\+p), (4)

where f/(q) is the interaction potential, and£'(p) = p^/2m + r7/fifp'f/(2^)/(p') is the quasi-particles' energy. The integral sign in (4) denotes integration over momenta and summation over spins. Introduce two new variables p = (pi + p'l)/2, k = (pi — pj )/'h and a new function 5f{t, k, p) = 5N{t, p + ^k/2, p — ^k/2). This i^mction is the Fourrier transformed with respect to the space variables Wigner density matrix.

42

FLUCTUATION SPECTRAL FUNCTIONS

The fluctuation spectral function of a density matrix can be represented in the form

(5/«(p)5/,(p'))a,,k (5)

where Raa'{p,(l,^,z,t) - the resolvent of the Hartree-Fock equation (4), satisfying

[nz + A],Ea{p)]Raa'{T?,(lXz,t)

= eaAkfa{p)J,ecjdp'[(P(k) - 5aM^^)]Rca'{p'AXz,t) + 5aa'5{p-q). (6)

Here we put Uab{k) = ^^^^^(k) = ^ ^ ^ and r] = - 1 . The solution of eq. (6) takes the form

where we introduced the notations: Ak£'a(p) = Ea{p+^) — Ea{p—^), Akya(p) =

fa{p+f)-fa{p-f),

^ , (p) - e ,y # _ _ _ _ _ _ xp , (p ) _ g^,y rfp _ _ _ _ , (8)

and the dielectric function with exchange interaction

e"P{co,k) = l + 0(k)5^e , [dp¥^\p). (9) a •'

The exchange scattering amplitude ra(p,p') for eqs. (7-9) satisfies an integral equation, which contains only the exchange interaction potential

r„(P.p') ^ . , p - p ' , + 4 j r ^ i ^ / < ' P " * ( ! ^ ) r , ( p " . p ' ) (lO)

Ta (p, p') depends on k and z as on parameters and is similar to the vertex-function, well-known in many-particle perturbation theory. Integrating (5) over momenta and summing with ea over sorts of particles we get an expression for the spectral function of the non-equlibrium electrostatic field fluctuation with the exchange interaction taken into account

(5E5E))«,k (11) 2

= An'nY^^{}^yjdq jdpTa{p,({) e^^(co,k) 5(^co-Ak£'a(q))[/6(q + W 2 ) + / 6 ( q - W 2 ) ] .

43

In the thermodynamic equilibrium, when the distribution function is Fermi, a fluctuation-dissipation relation follows from (11)

(5E5E))«,k = ^a" (co ,k ) r t / i | ^ , (12)

where

2

a"{co,k) = -4n'J^0{k)ei dq JdpTa{p,q)

rHF (co,k) Ak/a(q)5(^co-Akiia(q)) (13)

is the imaginary part of the generalized response function. The spectral function (11) and the response function (13) are expressed by the amplitude of the scattering interaction ra(p,p') which satisfies the linear integral equation (10). The solution of this equation in the case of Coulomb interaction of the particles is difficult and requires an appropriate approximation. The simplest approximation is the replacement of the Fock potential un-

derthe integral of (10) by the averaged over the impulse value ^{^^-jf-) = 0(k)G(z,k). One form of G(z,k) was found using a variation procedure in [5]. Then, the dielectric function taking into account exchange interaction particles, takes the form

e^^(z,k) = 1+P(z,k)[l -P(z,k)G(z,k)]- \ (14)

where P(z,k) = 0(k)Xae^/^pAi_/a(p)/[fe —Ai£'a(p)] is the plasma polarization. In this case (11) takes the form

{5E5E))a>,^, = 4n^nJ^(^,^^ f 5{nco-A^Ea{q))[Mq + nk/2)+Mq-nk/2)]dq, e(z,k) •'

(15) where _

e(z,k) = l+P(z ,k) ( l -G(z ,k) ) , (16) ~ 2

e(z,k) in (15) plays the role of the screening of the interaction potential 0(k).

In the equilbrium state a(co,k) = —47t/e^^{(0,k).

and

REFERENCES

1. R. Katilius and M. Rudan, Phys. Rev. B 74, 233101 (2006). 2. V. V. Bs\y\,Phys. Rev. Lett. 88, 255001 (2002). 3. V. V. Belyi and Yu. A. Kukharenko, Contrib. Plasma. Phys. 47, 240-147 (2007); V. V. Belyi and Yu. A.

Kukharenko, J. Stat Mech. (in press) (2009). 4. D. Pines and J. R. Schriffer, Phys. Rev 125, 804 (1962). 5. F. Brosens, L. M. Lemmens and J. T. Devrees, Phys. Stat. Sol. (b) 81, 551 (1975).

44

Modified two-state approximation for classical stochastic resonance

A. A. Dubkov

Radiophysics Department, Nizhniy Novgorod State University, 23 Gagarin Ave., Nizhniy Novgorod, 603950 Russia

Abstract. We propose the modified two-state approximation to investigate the nonUnear regime of stochastic resonance phenomenon. The new model corresponds to some non-trivial cumulant truncation scheme. A comparison between the residts of ordinary cumidant truncation schemes (Gaussian and excess approximations) and obtained in the framework of new method for the power amplification factor is performed.

Keywords: Stochastic resonance, NonUnear regime. Two-state approximation, Cumidant truncation schemes PACS: 05.40.-a,02.50.-r, 05.10.Gg

INTRODUCTION

The stochastic resonance phenomenon (SR) discovered in the beginning of eighties of last century found its theoretical solution after one decade (see well-known review [1] and references therein). Probably, it was basic effect with obvious constructive role of noise, and further a number of publications in this area showed an exponential growth because of wide apphcation in different areas of science. At the same time, the analytical results beyond the usually employed linear response approximation [2] was exclusively obtained in the framework of two-state approximation [3, 4]. Some recent results obtained by precise numerics [5] request to develop new analytical approaches for analysis of the nonlinear regime of stochastic resonance. We mean a possibility to calculate more exactly the correlation function of output signal, the signal-to-noise ratio (SNR), and the SR gain. Here we offer new modified method to investigate the SR nonlinear regime observed in experiments.

MODIFIED TWO-STATE APPROXIMATION AND CUMULANT TRUNCATION SCHEMES

We consider the classical case of stochastic resonance phenomenon in the form of overdamped Brownian motion in symmetric quartic bistable potential under the action of external harmonic field

— = -x^+x+^{t)+AsmQ.t, (1)



45

http://05.10.Gg

h.O=Q .noijemkoiqqs neiseueO

FIGURE 1. The dependence of power amplification factor T) on noise intensity D for different values of signal magnitude A in Gaussian approximation. The frequency of input signal is fJ = 0.1.

where: x(t) is the displacement of Brownian particle, E,(t) is white Gaussian noise with zero mean and intensity 2D, A and Q. are the magnitude and the frequency of external signal respectively. The basic idea of our method consists in the approximation of nonstationary probabihty distribution of output signal x{t) in the form of two delta-functions (as in two-state approximation) but with varying positions of maxima

W{x,t)=p{t)5{x + a{t))+q{t)5{x-b{t)). (2)

Thus, according to Eq. (2) and the normalization condition p{t) +q{t) = 1, we have three parameters to characterize the power amplification factor

ri = (3)

where Pi = (A^+Bf) /2 is the power of the first harmonic of periodic output signal mean value

Ao MO) = ^ + ^A„cos(MQ?)+5„sin(MQ?) (4)

n = l

and Po = A^/2 is the power of input signal. We use the fact that the probabihty density function P{x) of arbitrary random variable

^ generates a sequence of orthonormalized polynomials P„{x) with the weight P{x) [6]

XP„ (x) = S„P„ (x) + V ^ ^ P „ + l (x) + VR~nPn-l W (5)

where —0° < S„ < +0°, /?„ > 0. The generating function of the parameters S„ and Rn can be represented in the form of continuous fraction

h(z) 1 1

^-z/ So-(6)

Si-z- "2

46

r.O=Q .roitBmixoiqqE stBle-ov/T

FIGURE 2. The dependence of power amplification factor T) on noise intensity D for different values of signal magnitude A in the modified two-state approximation. The frequency of input signal is fJ = 0.1.

corresponding to the probability density function P{x). For model probabihty distribution (2) we have in Eq. (6) only three first non-trivial parameters, which are coupled with three unknown parameters in Eq. (2). As a result, from Langevin Eq. (1) we can obtain the following closed set of equations for parameters So(t), Ri (t) and Si (?)

So = So-Sl-Ri{2So + Si)+Am\Q.t,

Ri = 2Ri{l-Ri-Sl-SoSi-Si 2D, (7)

Si 2D,

Si-Si -Ri{So + 2Si) +AsmQ.t {Si-So). Ri

The above-mentioned procedure fits with some specific cumulant truncation scheme. Thereby, a comparison of our results with well-known Gaussian approximation for two first cumulants of random process x{t) obeying Eq. (1)

ki = -3K2Ki-Kl + Ki+AsmQ.t,

K2 = -6l4-6K2Kf + 2K2+2D

and the next (excess) approximation for four first cumulants

kl = -K3-3K2Ki-Kl + Ki+AsmQ.t,

k2 = -2K4-6K3Ki-6l^-6K2Kf+2K2 + 2D,

ks = -9K4Ki-2lK3K2-9K3Kf -I8K2K1 + 3K3,

kl = -48K4K2-l2K4Kf-36Kj-72K3K2Ki-24K2+4K4

(8)

(9)

were performed. To solve the systems (7), (8) and (9) and to find the power amphfication factor (3) we use the Fast Fourier Transform (EFT) program and the program ODE45 from MathLab package.

The dependence of power amphfication factor on the noise intensity for different values of signal magnitude in the framework of Gaussian approximation is depicted

47

FIGURE 3. A comparison of Gaussian approximation, excess approximation and modified two-state approximation for the power amplification factor. The parameters are A = 0.1,f2 = 0.1.

in Fig. 1. The same dependence for the modified two-state approximation is shown in Fig. 2. At last, in Fig. 3 we demonstrate a comparison of dependencies for three approximative schemes (7)-(9). As it is seen from Fig. 3, the value of amphfication is practically the same for Gaussian approximation and two-state approximation schemes whereas the position of the characteristic maximum is different. A comparison with some experimental data and the results giving by the Linear Response Theory (LRT) is also performed.

CONCLUSIONS

We offered new procedure to investigate the nonlinear regime of stochastic resonance, namely, the modified two-state approximation scheme. We compared the results for the power amplification factor given by the ordinary cumulant truncation schemes and the method proposed. The main distinction consists in the maximum position in the dependence of power amplification factor versus noise intensity.

ACKNOWLEDGMENTS

This work was supported by Russian Foundation for Basic Research (project 08-02-01259).

REFERENCES

1, L. Gammaitoni, P. Hanggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998). 2, P Jung and P Hanggi, Phys. Rev. A 44, 8032 (1991), 3, B, McNamara and K, Wiesenfeld, Phys. Rev A 39, 4854 (1989), 4, J, Casado-Pascual et al, Phys. Rev Lett. 91, 210601 (2003), 5, J, Casado-Pascual et al, Phys. Rev E 67, 036109 (2003); 68, 061104 (2003); 69, 067101 (2004), 6, Yu, E, Kuzovlev and G, N, Bochkov, Radiophys. Quant. Electr. 20, 1036 (1977),

48

Resonance with Temporal Stochasticity and Non-locality

Toru Ohira

Sony Computer Science Laboratories, Inc., Tokyo 141-0022, Japan and

Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan

Abstract. We present here a simple model which shows a "resonant" phenomena through stochasticity and non-locality on the time axis. This model incorporates "stochastic time" and "delayed feedback" into its dynamics. We argue that a combination of temporal stochasticity and non-locality can provide rather complex behaviors.

Keywords: Non-locality, Stochasticity, Stochastic Resonance, Time, Delay PACS: 05.40.-a,02.50.-r,01.55.4-b

INTRODUCTION

Contemplations on the nature of "Time" have accumulated various thoughts [1]. For example, the theory of relativity, which has conceptually brought space and time closer to receiving equal treatment, continues to fascinate and attract discussion in diverse fields. In this paper, we attempt a very modest approach to this topic by presenting a view point for considering the concepts of "stochasticity" and "non-locality" on the time axis[2, 3,4]. Normally, these are concepts we are familiar with in space, but not in time. However, we can obtain rather complex dynamicsl behaviors by considering a combination of temporal stochasticity and non-locality. We present a simple dynamical model which includes noise in the time variable but not in the "space" variable, which is opposite to the normal description of stochastic dynamics. We call it "stochastic time." Also, the model incorporates temporal non-locality in the form of delayed dynamics [5, 6, 7, 8, 9]. We shall see that this model can exhibit behaviors which are similar to stochastic resonance[10, 11, 12].

MODEL

The differential equation of the delayed dynamics with stochastic time we discuss here is

-^^ = -ax{t) + jix{t-r). (1)

Here, x is the dynamical variable, and T is the delay. The difference from the normal dynamical equation appears in the existence of delay, and in "time" f, which contains stochastic characteristics. We can define f in a variety of ways as well as the value of T. In order to avoid ambiguity and for simplicity, we focus on the following dynamical



49

X„-t

.0. p i-p

FIGURE 1. A schematic view of the model.

map system incorporating the basic ideas of the above equation.

Xni^i = (1 — ax„j^ )+px„^^x, rik+i = rii + ^i (2)

Here, ^^ is the stochastic variable which can take either +1 or — 1 with certain probabilities. We associate "time" with an integral variable n. The dynamics progress by incrementing integer k, and n occasionally "goes back a unit" with the occurrence of § = — 1. Let the probability of § = — 1 be p for all k, and we set «o = 0. Then naturally, with p = 0, this map reduces to a normal delayed map with «^ = k. We update the variable x„ with the larger k. Hence, x„ in the "past" could be "re-written" as n decreases with the probability p.

Qualitatively, we can make an analogy of this model with a tele-typewriter or a tape-recorder, whose recording device occasionally moves back on a tape. A schematic view is shown in Figure 1. The recording head writes on the tape the values of x at a step, and "time" is associated with positions on the tape. When there is no fluctuation (p = 0), the head moves only in one direction on the tape and it records values of x for a normal delayed dynamics. With probability 0 < p,it moves back a unit of "time" to overwrite the value of x. The question is how the recorded patterns of x on the tape are influenced as we change p.

ANALYSIS

We consider the case of a = 0.03, p = —0.05, and T = 15. We have found, through computer simulations, that an interesting behavior arises. The tuned noise in the time flow gives the system a tendency for oscillatory behavior. In other words, adjusting the value of p controlling § induces an oscillatory dynamical path. Some examples are shown in Figure 2. With increasing probability for the time flow to reverse, i.e., with p increasing, we observe oscillatory behavior both in the sample dynamical path as well as in the corresponding power spectrum. However, when p reaches beyond an optimal value, the oscillatory behavior begins to deteriorate. This phenomenon resembles stochastic resonance. A resonance with delay and noise, called "delayed stochastic resonance"[13], has been proposed for an additive noise in "space". Analytical understanding of the mechanism is yet to be explored for our model. However, this mechanism of stochastic time flow is clearly of a different type and new.

50

(a)

PiX)

(b)

125 ISO 175 200 PiX)

(c)

PiX)

5 10 IS 20 25 30 35 40

PiX)

(e) ,,s

P(;.)

FIGURE 2. Dynamics (left) and power spectrum (right) of delayed dynamical model with stochastic time. This is an example of the dynamics and associated power spectrum through the simulation of the model given in Eq. (3) with the probability p of stochastic time flow varied. The parameters are set as a = 0.03, /3 = —0.05, T = 15 and the stochastic time flow parameter p are set to (a) p = 0, (b) p = 0.15, (c) p = 0.25, (d) p = 0.3, and (e) p = 0.48. We used the boundary condition that x„^ = x n when «s < miniji]^ :k < s), and set x n = 0.5. The simulation is performed up to L = 512 steps and 50 averages are taken for the power spectrum. The unit of frequency K is set as j , and the power P{K) is in arbitrary units.

51

DISCUSSION

We would like to now discuss a couple of points with respect to our model. First, we may extend this model to include non-locality and fluctuations in the space variable x. In this way, we have a picture of dynamical systems with non-locality and fluctuations on both the time and space axes. The analytical framework and tools for such a description need to be developed.

The question of whether this approach can extend to quantum mechanics and/or leads to an alternative understanding of such properties as time-energy uncertainty relations also requires further investigation. In relation to the notion of "causality", this model faces a problem. There are, however, works such as Dirac theory [14, 15, 16] and the Wheeler-Feynman theory [17] related to electrons, which include the effect of state in "future". Also, there is a theory of elementary particles with a fluctuation of space-time, where the noise term is added to the metric[18]. If we can connect our approaches here to such theories remains to be seen.

Finally, this resonance may be used as an experimental indication for probing non-locality or stochasticity in time, if there are some aspects of reality. We have previously proposed "delayed stochastic resonance"[13], a resonance that results from the interplay of spatial noise and delay. It was theoretically extended[19], and the effect was experimentally observed in systems with a feedback loop[20, 21]. It is left for the future to see if an analogous experimental test could be developed with respect to temporal stochasticity and non-locality.

REFERENCES

1. p. Davies, About Time, (Simon and Schuster, New York, 1995). 2. T. Ohira, "Stochastic Time," in Frontiers of Fundamental Physics (FFP8), Eighth International Sym

posium, edited by B. G. Sidharth, A. Alfonso-Faus, and M. J. Fullana, AIP Conference Proceedings 905, American Institute of Physics, New York, 2007, pp. 191-194.

3. T. Ohira, Physica A 379, 483^90 (2007). 4. T. Ohira, J. Stat Mech. P01032 (2009). 5. M. C. Mackey and L. Glass, Science 197, 287-289 (1977). 6. K. L. Cooke and Z. Grossman, J. Math. Anal, and Appl. 86, 592-627 (1982). 7. J. G. Milton, et al., J. Theo. Biol. 138, 129-147 (1989). 8. T. Ohira and T. Yamane, Phys. Rev. E 61, 1247-1257 (2000). 9. T. D. Frank and P J. Beek, Phys. Rev. E 64, 021917 (2001). 10. K. Wiesenfeld, and F Moss, Nature 373, 33-36 (1995). 11. A. R. Bulsara and L. Gammaitoni, Physics Today 49, 3 9 ^ 5 (1996). 12. L. Gammaitoni, P Hanggi, P Jung, and F Marchesoni, Rev. Mod. Phys. 70, 223-287 (1998). 13. T. Ohira and Y Sato, Phys. Rev. Lett 82, 2811-2815(1999). 14. P A. M. Dirac, Proc. Roy. Soc. (London) A 167, p. 148 (1938). 15. F Rohrlich, Classical Charged Particles, (Addison-Wesley, Reading, Mass., 1965). 16. B. G. Sidarth, arXive:physics/0701237(2007). 17. J. A. Wheeler and R. P Feynman, Rev. Mod. Phys. 17, p. 157 (1945); 21, p.425 (1949) 18. Y Takano, Prog. Theor Phys. 26, 304-314(1961); Y Takano, ibid, 38 (1967) 1185-1186. 19. L.S. Tsimring and A. Pikovsky, Phys. Rev. Lett. 87 250602 (2001). 20. C. MasoUer, Phys. Rev. Lett. 88 034102 (2002). 21. M. Misono, T. Todo, K. and Miyakawa, J. Phys. Soc. Jpn., 78, 014802 (2008)

52

Dynamics Of Interaction Of Quantum System With Stochastic Fields

E.A.Sobakinskaya, A.L.Pankratov, V.L.Vaks

Institute for Physics ofMicrostnictures of Russian Academy of Science, GSP-105, Nizhny Novgorod, Russia.

Abstract. Analytical and numerical calculations of dynamics of polarization and population difference for two-level quantum system, interacting with stochastic fields are presented. For the stochastic fields we consider two cases: phase-diffusion and colored-noise field of arbitrary intensity. Stationary values and characteristic evolution times of polarization and population difference are defined.

Keywords: quantum system, dynamics, colored-noise field, phase-diffusion field. PACS: 02.50.Ey, 05.10.Gg, 42.50.Md, 42.65.Sf

INTRODUCTION

Studying of interaction of quantum systems (atoms and molecules) with various stochastic fields is of great interest both for fundamental and applied science. For example, modification of atom and molecular spectrum under an influence of cosmic background is very important in astrophysics. Thermal fields generated by solid surface can affect processes absorption and desorption, that must be taken into account in hi-tech. Development of quantum computers based on employing of single molecules also requires information about noise impact on a "device" operation. Understanding of dynamics of interaction is crucial for these and many other problems. In the literature there are only few papers dedicated to some general points of temporal evolution of density matrix that is not enough for understanding.

Q U A N T U M SYSTEM-FIELD M O D E L

In the presented paper we considered a two-level system driven by coherent probe field and interacting with stochastic field. The entire system is described by the following Hamiltonian:

H(t) = H,+H(t) + V(t), H(t) = d^^E(t),V(t) = di^E(t) (1)

where, Hgis the Hamiltonian for isolated system, and ff{t) and V(t) are interactions between the quantum system and stochastic and regular fields correspondingly, du is the matrix element of the electric dipole between the two states.

For analytical calculations in the case of phase-diffusion field (Wiener-Levy stochastic process) the equation of motion for the density matrix operator p with Hamiltonian (1) is written as:



53

http://05.10.Gg

dN , ^ ^ -r 1 ^ —- = -2x{t) Im A2 + — («o - ^ ) dt i j

2 / I \

where no is the equilibrium population difference, N=pu-p22 is the population

difference, ^(/) = —^(ff(/)e""+g*(/)e ""+£'„cos«/), ito is an amplitude of the probe h

field and sit) = \e(t)\e"''''' is the fluctuating complex amplitude, with \s{t'\ and (p(t) being the real amplitude and phase, respectively. For calculations we consider a stationary Markov process. In this case the first-order correlation function is given by:

l.s*{t,)e{t,)) = elexp(-k - f J / r J , s] = (\s(t)f) (3)

Analytical studies of a quantum system behavior interacting with colored-noise field are based on a paper [1]. In this case the Hamiltonian (1) is rewritten with Pauli matrices, rf.

Ho—^r,, H(f) = J]e,r,E(f), ViOîhY^rMae-""-aê""), 7.=]î (4)

where ro is a unitary matrix and e, are given by:

•> Qj •> Q-,

1 ' 1 ' ii ' 1 The studies in this case are done for weakly non-Markovian noise with correlation

function (3). The numerical calculations (both for phase-diffusion and colored-noise fields with

correlation function (3)) use equations (2) and are made for a system with parameters (incgs): coo= 10.430 GHz, di2=7-10"'', no=3.3-IO', Ti=T2=10"^.

In computer simulations of dynamics the Heun algorithm has been used, which allows to generally solve quantum equations of two-level system (in time-domain) even for high frequencies and intensive stochastic fields.

RESULTS

Phase-diffusion Field

It is shown that the presence of phase-diffusion field does not affect the stationary value of polarization, p'l^, but the relaxation time T2 is decreased:

^. nfiAi , — = —+ i:^ (^\

T'n\ ( « -«„ ) ' + —+ ^ ' T' AT'

Dynamics of the polarization for various intensities £„ is presented on Fig. 1 for the

case Tc=I2.5-10"'° (Fig.Ia) and TC=5T0"' (Fig.Ib) (in cgs). It is shown that in the short time regime, when t«i:c, phase-diffusion field act like a coherent signal, inducing

54

polarization in quantum system. At the moment t-Zc maximum polarization is induced, and then it decays with characteristic time T^ defined by (3). The stationary value of

pl'2 is determined by (3). If T* < T^ , then polarization can be induced several times.

Pu(t)

illlll^

(a) (b) FIGURE 1. Computer simulation of behaviour of Pj" as a function of time:

(a)xc=12.5-10-"',ej=4-10-^;(b)xc=5-10-', ej=4-10-^; r;=1.7-10-l

It is also demonstrated that estimation of the induced polarization can be made by a model of coherent field. Analysis of population difference dynamics shows that presence of the noise field leads to decrease of the population difference and can be considered as heating of quantum system (Fig.2).

FIGURE 2. Computer simulation of behaviour of A as a function of time:

(a) Xc=12.5-10-"', EI =4-10-^; (b)Xc=5-10-', el =4-10-^

Colored-noise Field

It is shown that absorption line shape is not changed only in case of where di2=d2u dn=d22 and y»coo, then the profile retains Lorentzian shape and relaxation time is given by:

1 1 Id^.sl (6) T T - ' 2 -"2 xi+</r)

55

Dynamics of yOj" for this case is presented on Fig.2 (in cgs): y =1-10 , Eo=5-10" . Maximal polarization is induced in t~ T^'. In this case colored noise destroys induced polarization and heats the system.

,-n j ^^^*^^PI^Wi^WWPBP -5 000E+8

(a) (b) FIGURE 3. Computer simulation of behaviour of p"^ as a function of time: (a) red curve e^ =0, blue

curve e ^ =20, T^ = 6.5 • 10"" (b) red curve e^ =0, blue curve e ^ U, r, =1.6-10^

Therefore, the results of studies show that stochastic fields can strongly modify dynamics of a quantum system. It is also demonstrated that crucial role is taken by statistics properties of noise.

ACKNOWLEDGMENTS

This work is financially supported by ISTC 3174, CRDF R U C 2 - 2 8 6 7 - N N -0 7.

REFERENCES

1. K.Faid, R.F.Fox, Phys.Rev., 34, 4286,1986.

56

Noise-assisted quantization in sensor networks

Shin Mizutani*, Kenichi Aral*, Peter Davis*, Naoki Wakamiya''' and Masayuki Murata'''

*NTT Communication Science Laboratories, NTT Corporation, 2-4 Hikaridai Seika-cho, Soraku-gun, Kyoto, 619-023 7 Japan

^Graduate School of Information Science and Technology, Osaka University, 1-5 Yamadaoka, Suita, Osaka, 565-0871 Japan

Abstract. We propose the use of sensing noise as a practical way to introduce heterogeneity in a homo

geneous sensing network, and thus improve sensing performance. In particular, we show that in a sensor network which aggregates the results from multi-level quantizing sensors each having the same input signal, adding independent noise to the sensors can reduce the error between the input signal and output obtained by simply combining the outputs from all the sensors. We can show that this noise-assisted quantization always occurs when the quantizers of the sensors have identical sets of thresholds. This condition is suboptimal as a quantizer system. We also show the case of the optimal quantizer system in which the error monotonically with the noise intensity.

Keywords: stochastic resonance, noise-assisted signal quantization, sensor network, quantization PACS: 05.40.-a, 02.50.-r, 89.20.Bb

INTRODUCTION

A sensor network is a system composed of many separate sensors and a data fusion center that gathers the output from all the sensors. Often the sensors are simple devices with low sensing resolution or quality. However, even if individual sensors are simple, it is possible to obtain higher-sensing performance by aggregation and fusion of the outputs from many sensors. One well known example is quantizing sensors that have analog input and digital output. Even if a quantizer has low resolution, higher resolution can be obtained by combining the outputs from multiple quantizers. Heterogeneity among quantizers can improve the resolution. For example, it is known that setting different values of input bias for each quantizer can give improved resolution, although the optimal control of the bias inputs may be complicated (Papadopoulos et al. [1]). However, for practical reasons it is often preferred that all the sensors are identical and have simple controls.

Here we propose the use of sensing noise as a practical way to introduce heterogeneity in a sensing network, and thus improve sensing performance. In particular, we show that in a sensor network which aggregates the results from many multi-level quantizing sensors each having the same input signal, adding independent noise to the sensors can reduce the error between the input signal and output obtained by simply combining the outputs from all the sensors.



57

MODEL

We consider a network of quantizing sensors as shown in Fig. 1 (a). Each sensor works in parallel to output a discrete multi-level quantized output corresponding to a common analog input and a data fusion center decides a global discrete output based on the local discrete outputs from all quantizers. As shown in Fig. 1(a), the ;-th quantizer receives the common input signal x{t) and individual noise ^^{t) (i = 1,...,A0 and transmits its output y^{t) to the data fusion center. Individual output y^{t) is 7 — 1 if 0j- •_) < x{t) + |,(?) < 0,-,• U = l)---!®- We note that the local quantizer has the threshold parameter set 0. .. N is the number of local quantizers and Q is the number of discrete output levels of each quantizer. The internal state of the data fusion center y{t) (= SiLi7,(0) takes a non-negative integer value (0 - N{Q — 1)). The global output z{t) of the data fusion center is obtained from the internal state y{t) using an output table which assigns the value z^ (A: = 0,... ,7V(g — 1)) when the value of the internal state y{t) is k. We note that the center has the parameter set, z . We also note that the sensor network can correspond to one quantizer which can have up to N{Q— 1) + 1 discrete levels when local quantizers have different thresholds. We focus on the error in the output z and its dependence on the standard deviation a of the noise distribution. We evaluate quantization performance using mean square error E as follows.

N(Q-i) .^

E = % ix-zfq{k\x)pix)dx. (1) k=0 ''-°°

Here j9(x) is the probability that the common input signal has value x and q{k\x) indicates the conditional probability that the data fusion center has an internal state k when an input signal value is x.

RESULTS

First, we want to show that the error between input signal and output can be reduced by adding noise when each quantizer has an identical threshold set (0. . = OX Next, we also show the case of the optimal system in which the error increases monotonically with the noise intensity. In the optimal system the local quantizers have different sets of threshold values. The use of different sets of threshold values means that there is no redundancy in the quantizers.

Here we assume that the input signal is a random signal uniformly distributed over a finite range (p{x) : U[—1/2,1/2]), and the independent noise has Gaussian distribution (N(0, C7 )). As a particular example of a system with homogeneous thresholds, we show results for a network with TV = 4 local quantizers and 2 = 4 four equally spaced threshold levels, with O- = —1/2 + j/A. This set of threshold values is independently optimal for a single quantizer with respect to the given input signal without noise - optimal in the sense that it minimizes E in the case of a single (A = 1) quantizer with center parameters z- = —1/2 — 1/(2 x 4) +7 /4 . For the A = 4 network, we set the center parameters z^ = — l / 2 + l / ( 2 x 13) +A:/13. This output value set corresponds to the set

58

of values which would be optimal (i.e. minimize E) in the case when there was no noise and sensors were allowed to have different optimally assigned threshold values, namely 9,j = -l/2+U + 3x{i-l))/l3.

Figure 1 (b) shows the dependence of the mean square error E of the discrete output signal on the noise intensity a. Note that the error, shown by the line, corresponding

( Noise J

( ig" i ) -C l ter )

X

—I ^ Quantizer I—

Data &sion canter

I Siiml-> Output

tabie Output)

( Noise J (a) (b) FIGURE 1. (a) Our sensor network model. Each sensor works in parallel and a data fusion center decides global discrete output based on the local discrete outputs from each quantizer The data fusion center has a summing part and output table part. The internal state of the data fusion center has the non-negative integer value obtained by summing the individual integer outputs of each quantizer The global output of the data fusion center is obtained from the internal state using an output table, (b) Mean square error E between input signal and output with increases of noise intensity a. Line shows error using an identical (homogeneous) threshold set for local quantizers. Dotted line shows error using the optimal parameter set when local quantizers have different (heterogeneous) threshold values.

to non-optimal homogeneous thresholds can be reduced by adding noise, and there is a finite noise intensity which gives the least error. We call this noise-assisted quantization. We can show that this noise-assisted quantization always occurs when the quantizer of sensors has an identical threshold set and each sensor senses the same signal with independent noise. On the other hand, when the network is optimal, with heterogeneous thresholds, noise only increases the error as shown by dotted line in Fig. 1(b). For the network to be optimal it is necessary for each quantizer to have different threshold values. We can also see that the two lines in Fig. 1 (b) have a cross point and the error by the homogeneous quantizers becomes smaller than the error by the optimal system at a larger a region. The reason is that the threshold set of local quantizer in both cases is not optimized for each noise intensity value and it is optimal only in the absence of noise. This property of the homogeneous quantizers shows robustness against noise intensity.

The sufficiency condition for noise-assisted quantization is still unclear. However, we can show that a necessary condition for noise assisting quantization is that the system is not optimal, where by optimal we mean that the quantization parameters (0.°P ,z°P )

minimize E in the absence of noise such as (0.°P ,z°P ) = argmin^g ^ ii. In particular,

noise-assisted quantization occurs even though each quantizer has a threshold which is independently optimal with respect to the input signal, as the redundancy (homogeneity) in the threshold values in different quantizing sensors means that the system is subop-timal as a whole. On the other hand, when the network is optimal, noise only increases the system error, i.e. degrades the sensing performance. For the network to be optimal it is necessary for each quantizer to have different threshold values.

59

CONCLUSIONS AND DISCUSSIONS

We proposed the use of sensing noise as a practical way to introduce heterogeneity in a sensing network, and thus improve sensing performance. We shown that in a sensor network which aggregates the results from multi-level quantizing sensors each having the same input signal, adding independent noise to the sensors can reduce the error between the input signal and output obtained by simply combining the outputs from all the sensors. We could show that this noise-assisted quantization always occurs when the quantizer of each sensor has an identical threshold set. Even if each quantizer has a threshold which is independently optimal with respect to the input signal, a system of multiple sensors with identical threshold sets is suboptimal. We also showed the case of the optimal system in which the error increases monotonically with the noise intensity. In an optimal system the local quantizers have different sets of threshold values, so there is no redundancy in the quantizers.

Adding noise to improve signal quality is also used in the technique known as "dithering". For example dithering is used to remove spurious edges in digital images and spurious frequency peaks in digital sounds caused by quantization. Dithering makes digital images and sounds more natural for human senses. This dithering effect should be discriminated from our noise-assisted quantization. The relevant evaluation measure in the case of dither is different from that in noise-assisted quantization.

Our model of quantization in sensor networks is a natural extension of a summing network of TV threshold devices studied by Stocks [2]. We can show that the noise-assisted quantization described by Stocks can also be explained by the concept of suboptimality. This explanation simplifies the understanding of noise-assisted quantization in general, and also clarifies the connection of noise-assisted quantization with other cases of noise-assisted signal processing.

ACKNOWLEDGMENTS

This research was supported in part by "Special Coordination Funds for Promoting Science and Technology: Yuragi Project" of the Ministry of Education, Culture, Sports, Science and Technology, Japan.

REFERENCES

1. H. C. Papadopoulos, G. W. Womell and A. V. Oppenheim, "Sequential signal encoding form noisy measures using quantizers with dynamic bias control,"/iiiiii Trans. IT, vol.47, no.3, pp.978-1002, (2001).

2. N. G. Stocks, "Suprathreshold stochastic resonance in multilevel threshold systems,"P/;ys. Rev. Lett., vol.84, no. 11, pp.2310-2313 (2000).

60

stability under influence of noise with regulated periodicity

O. A. Chichigina*, B. Spagnolo^, D. Valenti and A. A. Dubkov**

'Physics Department, Lomonosov Moscow State University, 119992, Moscow, Russia ^INFM and Dipartimento di Fisica e Tecnologie Relative, Group of Interdisciplinary Physics,

Universitd di Palermo, Viale delle Science, pad. 18, 1-90128 Palermo, Italy "Radiophysics Department, Nizhniy Novgorod State University, 23 Gagarin Ave., Nizhniy

Novgorod, 603950 Russia

Abstract. A very simple stochastic differential equation with quasi-periodical multiplicative noise is investigated analytically. For fixed noise intensity the system can be stable at high noise periodicity and unstable at low noise periodicity.

Keywords: StabiUty condition, Noise with regulated periodicity PACS: 05.40.-a,02.50.-r, 05.10.Gg

INTRODUCTION

A great variety of phenomena such as nuclear decay, autocatalytic chemical reactions, beginning of laser generation, spread of epidemics, population dynamics, after a linearization near some critical point can be described by the following first-order differential equation

f = -««/. (1) at

The sign of the parameter UR defines the behavior of system (1) around the critical point OR = 0: the system tends asymptotically to a stable state or has some kind of explosion. But really this parameter is not constant in time and experiences fluctuations: OR^t) = a + ^ (?), with a constant (real value) and ^ (?) noise. Moreover, the probabihty distribution and the time characteristics of the noise ^{t) play a crucial role in the behavior of such a stochastic system.

We consider a random process in the form of 5-pulse train

^{t) = fl^d{t-ti), (2) i

where the random intervals between neighboring 5-pulses l^i = U — ti-i are mutually independent and identically distributed with the probability density w (Q, that is the so-called renewal process [ 1,2, 3]. The effective (average) period of such a noise excitation is r = (Q. Note that E, (t) represents the derivative of the well-known continuous time random walk (CTRW) model process with fixed value of jumps.



61

http://05.10.Gg

GENERAL CONDITION OF MOMENTUM STABILITY

We analyze the stabihty of the system described by a simple stochastic differential equation

^ = -aI + ^{t)I, / ( 0 ) = / o (3)

with multiplicative noise < (?) in the form of pulse train (2), under the conditions r <C 1/a and / <C 1. In particular, the evolution of the species density in population dynamics or the number of infected people in epidemics [4] may be described by equation (3). Using the exact solution of Eq. (3)

I{t)=Ioe-'"exp(l^^z)dz\ (4)

we can calculate the m-order moment of the random process / (?)

{r{t))=I^e-"""(exp\m ['^{z)dz\\=I^e-"""Y.Pn{t)e"'f", (5) n=0

where P„ (?) is the probabihty to have n pulses in the time interval (0,?). According to the theory of renewal processes, the Laplace transform P„ (s) of this probability can be expressed in terms of the Laplace transform w (s) of waiting times distribution

Pn{s) = ^[l-W{s)]. (6)

Performing the Laplace transform of Eq. (5) and using Eq. (6) we arrive to the following exact result for the Laplace transform of m-order moment

^'"^'^^^^{s + ma)[l-e'»fw{s + ma)]' ^^^

which is vahd under the condition

e'"f\w{s + ma)\<l. (8)

Because of \w{s + ma)\ < w{ma) the m-momentum stability condition for arbitrary probability distribution of intervals between pulses takes the form

/ < - l n — ^ . (9) m w{ma)

In particular, for the mean value (m = 1) the sufficient stability condition reads

/ < l n ^ . (10) w{a)

62

FIGURE 1. The pidse train presenting peaks with parameter of periodicity C,olT equal to 0.8 (high periodicity). The period T is equal to 100 s. After each pidse there is the delay o = 80 s (some time interval) during which the next pidse is impossible. After this delay the rate r of pulse train is equal to 0.05. This process is suitable to obtain noise sources with varying degree of randomness.

NOISE WITH MEMORY

Afterwards we analyze the random process (2) with the following probability distribution of intervals between pulses

w(0 = pe

0, C<Co, (11)

where p and C,o are some positive values. As it is evident from Eq. (11), within the time interval o> after each delta-pulse, the occurrence of a new pulse is forbidden (like in neurons). The random process (2) with the probability distribution w(Q of waiting times is known as dead-time-distorted Poisson process and can be considered as an example of renewal process. In Refs. [5] and [6] similar dead-time-distorted Poisson processes are considered as an example of renewal process in quantum optics. In Ref. [7] periodical properties of the process are proved by means of experiments with radiation of 22Na source.

The rate of pulse train p can be expressed in terms of the mean waiting time T = {Q

(0 = Co + 1

(12)

For fixed effective period T, the rate p increases with increasing the memory o (see Fig. 1) and p ^ ^ when o -^ T. So, in such a case from Eq. (11) we arrive to pure deterministic periodical excitation with the probability distribution of time intervals between pulses w(Q = 5 (^ — ^o)- In the opposite situation, i.e. o = 0, the random process ^ (?) becomes the white Poissonian noise. Thus, the parameter of periodicity 1^0/T ranges from 0 to 1.

It is easy to find from Eq. (11) the Laplace transform of probabihty distribution

w{s) p + s

(13)

63

and after substituting it in the stability condition (9) we get

f <at;o + -\n[l+ma{T -t;o)]. (14)

In the case of white Poissonian noise ( o = 0) we have

Mi + mar) ^ ^ m

For periodical pulse excitation, i.e. ll,o = T,p ^ ^ , Eq. (14) transforms into f <aT. From Eq. (14) we obtain the following critical value of the parameter a, corresponding to the transition from stable to unstable state regarding the m''' moment of the system (3)

flc = - + ^ ^ I 1 - — 1 • (16)

When the noise with periodicity is included in Eq. (3), we can see qualitatively different behavior of the system, depending on the values of noise amplitude and periodicity.

CONCLUSIONS

The stabihty problem for systems under influence of quasi-periodical noise is solved analytically. A special noise source model, consisting of a pulse sequence at random times with memory, is presented. By varying the memory we can obtain variable randomness of the stochastic process. The variable delay time between pulses produces different kinds of correlated noise ranging from white noise, without delay, to periodical process, with delay equal to the average period of the pulses.

ACKNOWLEDGMENTS

This work was supported by Russian Foundation for Basic Research (project 08-02-01259), and MIUR.

REFERENCES

1, D. R. Cox, Renewal Theory, Chapman and Hall, New York, 1967. 2, CGodreche, J.M.Luck Journal of Statistical Physics, 104, No, 3/4 (2001) 489-523, 3, O, C, Akin, P, Paradisi, and P Grigolini, J, Stat, Mech, (2009) P01013, 4, O, Chichigina, D, Valenti, B, Spagnolo, FNL, 5, No, 2 (2005) L243-L250, 5, L,M,Riccardi, REsposito, Kybemetic 3, Bd,, Heft 3, (1966) 148-152, 6, E, Jakeman and J, H, Jefferson, OPTICA ACTA 33, No, 5, (1986) 557-576, 7, CFaraci and A,R,Pennisi, Phys,Rev,A 33, No, 1, (1986) 583-588,

64

Complexes Of Defects As The Source Of 1/F Noise In GaAs Based Devices

Evgeny I. Shmelev, Arkady V. Yakimov

Lobachevsky State University, Gagarin Avenue 23, Nizhniy Novgorod 603950, Russia Fax: +7-8312-656416; E-mail:[email protected]

Abstract. Within the framework of the determination of the nature of 1//noise in GaAs based devices, the structure and the spatial bistability mechanism of complexes of defects originated by donor-acceptor pairs are researched. Theoretical analysis has shown that the simplest example of complexes potentially existing in n-GaAs:Si are such pairs as SiAsSioa. VoaSioa, SIASISI • The mechanism of spatial bistability of the entire complex of defects is linked with the influence of Jahn-Teller effect on the complex or one of its elements. The ability of VoaSioa to be one of the sources of 1/f noise in research samples is analyzed.

Keywords: Jahn-Teller effect, bistability, complexes of defects, defect displacement, electron mobility, 1//" noise. PACS:61.72.Bb;72.70.+m

INTRODUCTION

Nowadays the problem of the relationship between 1//noise (for example, voltage noise) in semiconductor devices and fluctuations of either carrier density (f5«-model) or mobility (f5//-model) is discussed.

This work is devoted to research of the structure and spatial bistability mechanism of defect complexes originated by donor-acceptor pairs, which should be one of the potential sources of mobility fluctuation dfi in GaAs based devices.

The absence of reliable information about the sources generating \lf noise in mobility {dfi noise) imparts relevance to the problem under consideration.

THE MODEL OF BISTABLE DEFECTS

For an explanation of \lf noise in semiconductor devices, the model of bistable defects (within the framework of f5//-model) located in a crystal lattice is widely used [1].

According to this model, the defect can exist in one of two different structural configurations " 1 " and "0" at the same charge state (in other words, it forms a two-level system), separated by a potential energy barrier it (Fig. la).



65


a) b)

FIGURE 1. (a) Energy diagram of the bistable defect; (b) movable defect complex of paired type in GaAs lattice

Under the thermal vibration of the lattice, defects can stochastically change spatial configurations, moving between their local states (Fig. lb). Changes of the bistable defect spatial configuration are manifested through modulation of the carrier mobility by the random telegraph process. Ensemble of such bistable defects can lead (under well-known conditions [1]) to l//"noise in mobility // and, consequently, to 1/f noise in the output device voltage.

THE STRUCTURE OF BISTABLE COMPLEXES OF DEFECTS IN N-GaAs:Si

Previous investigation [2] of the noise and electro-physical characteristics of GaAs epitaxial films and GaAs submicron planar FETs with Schottky gate has shown that, according to the model of bistable defects, 1//"noise can be induced by some mobile neutral defect complexes of paired type (Fig. lb). These complexes are defined as neutral because variation of the number of defects has shown no effect on the concentration of free electrons in the conduction band. Bistability of such complexes can be revealed through a spontaneous change in orientation when one of the pair defects jumps into the neighboring energy minimum (Fig. la).

Theoretical analysis has shown that the simplest example of complexes, which potentially exist in n-GaAs:Si [3, 4], are donor-acceptor pairs (with a dipole structure) such as SiAsSioa, VoaSioa (Fig. 2). Also there can be defect complexes consisting of atoms situated at the interstices of a crystal lattice (for example, the interstitial atom of silicon Isi, forming SIASISI pair).

SIASSIQ;

FIGURE 2. Spatial structure of SiAsSiGa,VGaSiGa defects in n-GaAs:Si.

66

More information about the nature of bistable defects can be obtained from studying the energy E necessary for defect spatial reorientation (Fig. la). As it is known, the process spectrum has the form of 1// (where / - frequency) if the local barrier heights E are uniformly distributed in a sufficiently wide range from E\ to E2. In case of symmetrical two-level systems (difference in depth Eh = 0) the lowest and highest comer frequencies are defined by the relation

where/r - frequency of lattice thermal vibrations. Since in the analyzed GaAs FETs [2] the 1//" spectrum was observed at frequencies

from 1 Hz to 20 kHz, it is necessary to have a set of bistable defects with reorientation energies E in the range from £1=0.2 to i?2=0.5 eV in order to explain the form of the measured spectrum.

THE POTENTIAL SPATIAL BISTABILITY MECHANISM OF COMPLEXES OF DEFECTS

Researching the possible defects structure in GaAs, which are capable of forming a system with minimum two metastable states, let us consider the defects influenced by Jahn-Teller effect.

Jahn-Teller effect is that under the influence of electron-nuclear interaction a highly symmetrical configuration of a polyatomic system with orientation electronic degeneracy becomes unstable and spontaneously deforms [5]. Jahn-Teller effect is manifested in intrinsic defects and atoms of doping impurities in semiconductors. Presently, the following Jahn-Teller defects have been identified in GaAs structures: CUGa, A g G a , A u G a , VGa [ 6 ] .

Thus, the vacancy VGa, which is a part of VGaSiGa complex, is displaced from the node of the ideal crystal lattice due to Jahn-Teller distortion. Reference [7] shows that VGa' (with 7 electrons and 1 hole on the vacancy orbitals) should have up to four positions for displacement (shown by arrows in Fig. 2). Each displacement corresponds to its own metastable state. In case of an isolated vacancy, these four locally stable states have equal energy {Eh=0). Whereas presence of SiGa donor near VGa influences the spatial location and depth of the energy minimums, thus destroying the initial energy equivalence of the local states.

Analysis of theoretical and experimental data cited in [7, 8] has shown, that for VGaSiGa Containing movable VGa' , the energy barrier E needed for the reorientation of the entire complex can be estimated as E>0.2 eV.

If we compare the magnitude of i? obtained for VGaSiGa complex with the estimated El and E2 for two-level systems laying in the range from 0.2 to 0.5 eV, we can conclude that VGaSiGa structure is probably one of the bistable (multistable) defects that can contribute to generation of I//noise observed in the analyzed devices.

67

Generally, the defect reorientation energy under the influence of Jahn-Teller effect can be in the range from 0.1 to 0.6 eV, and defect displacements from the symmetric position in the lattice can reach 1 A.

The latter testifies to possible existence of a mechanism, according to which the defects influenced by Jahn-Teller effect can act as sources of 1//"noise.

Thus, further research is needed to determine Jahn-Teller distortion of lattice and the energy necessary for reorientation of various complexes of defects in GaAs based devices.

CONCLUSIONS

It was shown that the simplest example of complexes potentially existing in n-GaAs:Si are such pairs as SiAsSioa , VoaSiGa , SIASISI • The mechanism of spatial bistability of the entire complex of defects is linked with the influence of Jahn-Teller effect on the entire complex or one of its elements. According to this approach, the ability of VoaSiGa to be one of the sources of 1/f noise in research samples is analyzed.

The obtained additional information supports the earlier suggested hypothesis about anisotropically scattering spatially bistable defects being a source of 1//noise in GaAs based devices.

ACKNOWLEDGMENTS

The presented research was carried out in Nizhniy Novgorod State University in frames of the Priority National Project "Education", Educational-Scientific Center "Information-Telecommunication Systems: Physical Bases and Software", Laboratory "Modem Systems of Signal Processing".

REFERENCES

1. A. V. Yakimov, Radiophysics and Quantum Electronics 42(6), 521-524 (1999). 2. A. V. Moryashin, S. V. Obolensky, M. Yu. Perov, and A. V.Yakimov, Radiophysics and Quantum

Electronics 50(21 135-145 (2007). 3. A. E. Kunitsyn, V. V. Chaldyshev, S. P. Vul', et al.. Semiconductors 33(10), 1080-1083 (1999). 4. I. A. Bobrovnikova, M. D. Vilisova, I. V. Ivonin, et al.. Semiconductors 37(9), 1047-1052 (2003). 5. LB. Bersucer, Electron structure and properties of coordination compounds. Introduction in theory,

Leningrad: Himiya, 1971, 352 p. 6. N. S. Averkiev, A. A. Gutkin, M. A. Reshchikov, Semiconductors 33(11), 1196-1201 (1999). 7. N. S. Averkiev, A. A. Gutkin, S. U. Il'inskii, Physics of the Solid State 42(7), 1231-1235 (2000). 8. A. A. Gutkin, T. Pitrowski, J. Pultorak, et al.. Semiconductors^2(1), 33-39 (1998).

1/f Performance Limits of Scanning Tunneling Microscopes

Amanda M. Truong^ and Peter H. Handef

Department of Physics and Astronomy and Center for Nanoscience University Of Missouri, Saint Louis, 1 University Boulevard, Saint Louis, MO 63121, USA

Abstract. The feedback loop of the Scanning Tunneling Microscope (STM) maintains a constant current throughout the circuit and utiUzes a piezoelectric crystal to control the tip-sample separation distance. The piezoelectric crystal is a source of quantum 1/f noise which can reduce detection limits. Both the fluctuations in the timneUng current and the spatial fluctuations of the piezoelectric crystal result in deviations in tip-sample separation distance. Here, for the first time, using the quantum 1/f theory, the fluctuations in the tip-sample separation distance due to the fluctuating relaxation rate, r, of the piezoelectric crystal are calculated.

Keywords: Quantum 1/f noise, piezoelectric. Scanning Tunneling Microscopes, resolution limits. PACS: 87.64.Dz;; 68.37.Ef; 77.65.J

QUANTUM 1/F NOISE IN PIEZOELECTRICS

The phonon emission within the piezoelectric quartz crystal can be caused by scattering events with higher frequency phonons or by scattering events on defects or impurities occurring in the crystal. The phonon emission rate T is equal to 1/T , where T is the relaxation rate of the quartz. Fluctuations in the relaxation rate, T, produce fluctuations in the Young's modulus of the crystal, Eo , which ultimately give rise to fluctuations in tip position.

First, we define the velocity change of a phonon in the crystal due to dissipation as

•S,_CT{Z,J Cli, 2 , 2

2ps^ \+ay ^'^

where C is the specific heat, T is the temperature, y is the Gruneisen constant, So is the speed of sound in the absence of dissipation, Q.o is the resonance frequency of the crystal, p is the crystal density, and x is the relaxation time [1]. To simplify, we let AE = CT <y^> = constant and also use the low frequency Young modulus Eo = p So = constant. Now after letting x = Q.o^x^, Eq. (1) can be written



69

^ ^ ' +1 (2) £•„ 2£'„ 1 + x

Differentiating, and assuming E - E o « l , we obtain an expression for the variation in the Young modulus 5E, which is proportional to the change in the relaxation time 5T

dE=AE y— (3) (l + xf t ^ '

Applying a force to the crystal, or applying a voltage across the crystal, causes spatial displacement, X, in the vertical direction, which we take to be the direction of crystal length. Fluctuations in the Young modulus, Eo, produce tip-sample distance fluctuations 5X/X = -hEolEo, which gives the spectral density of fractional fluctuations as

1

X En ^0

The relation between quartz quality (Q-factor), relaxation time and frequency given by [1], is Ex/AE~Q^ which fluctuations to be written as [1], is Ex/AE~Q^, which allows for the spectral density of fractional spatial

X

f ^2 x' 4

5. « — 5 \^ J

4 ^ ^ 4 ^ (5) (1+xr T Q

Recall that the phonon emission, or interaction, rate T is equal to 1/T , where T is the relaxation rate of the quartz. Therefore, the fractional fluctuations in phonon emission rate are equal to the fractional fluctuations in relaxation rate: 5 sx/x = 5 sr/r- According to the general quantum 1/f formula, F"^ 5 r ( / ) = laAJf, where a = e^/hc is the fine structure constant and A = 2(AJ/ec)^/37i is the quantum 1/f effect in any physical process rate F [2], [3]. The current discontinuity AJ in the Ampere-Maxwell equation equals the discontinuity AdP/dt in the rate of the crystal dipole moment change dP/dt. The spectral density of fluctuations in the rate F of phonon emission /removal from the main resonant oscillation mode of the crystal can be written

5^(/) = rMc^(AP) /3;reV/ (6)

where (AP )^ is the square of the discontinuity in the dipole moment change rate, in the process causing the removal of a phonon from the main oscillator mode. To calculate this change, we write the energy W of the interacting resonator mode <co> in terms of dP/dt

70

W=nh<co> =2(Nm/2)(dx/dt)^ =(Nm/e)^(edx/dt)^ = (m/Ne^)(dP/dt)^,

The factor 2 includes the potential energy contribution. Here m is the reduced mass of the elementary oscillating dipoles, e their charge, and N their number in the crystal. Considering An=l for 1 phonon, we obtain the fractional energy variation

An/n =2IAdP/dtl /(dP/dt), or AdP/dt =(l/2n) dP/dt

Taking AdP/dt fro the expression of W, substituting it into the last expression here, we

obtain an expression for the jump in the rate of change I AP I = (Nh<co>/n)^'^(e/2).

Substituting AP into Eq. (6), the expression for the spectral density is then

Y-^S^{f) = Nah{o}^^)l37rnMc^f ^^

where N is the number of oscillating dipoles in the crystal, n is the number of thermal phonons at the resonant frequency of the quartz (which we set to unity), M is the reduced mass of the elementary oscillating dipoles, and c is the speed of light. <cOeff > replaces the crystal resonator mode frequency, which is usually much smaller. The frequency <C0eff > is the frequency that gives the average of the quantum 1/f contributions. Indeed, the quantum 1/f effect calculated from Eq. (7) includes contributions from all 3 phonons involved in the process in which a phonon from the main resonator mode combines with a thermal phonon of much higher frequency, yielding another thermal phonon of similar high frequency. The largest contribution will not come from the main resonator 1/n factor, but from the 1/n factor corresponding to the high-frequency thermal modes that have n close to 1. Furthermore, n is the number of thermal phonons at the frequency <C0eff > of the quartz, which we set equal to unity [2], [3].

Each coherence volume e of the quartz crystal has independent dissipation fluctuation, with e being the phonon coherence length. Their variances add, and cause a (e^/V) factor to be present in the spectral density of the fractional legth fluctuations for the whole crystal.

Using eq. (7), the fractional spatial fluctuations of equation (5) can be written in terms of the fractional fluctuations of the phonon emission rate T

s 4 £•'

X Q'v' 'iTtnMc^f

4 £•'

Q'v {f)an{ (o. efS

iTinMc'f

' ^ 6 4 ^

fyQ' (8)

Here V is the quartz crystal volume, (N/V) is the number of SiOa molecules in the volume of one unit cell of the quartz crystal, ^ is the squared phonon coherence volume. In this final form, we can make an approximation to the spatial fluctuations in the STM due to the phonon emission rate F of the piezoelectric quartz crystal which controls the feedback loop.

71

DISCUSSION

As a first approximation, assume the piezoelectric element is a cylinder with diameter and length of 1 cm, giving a crystal volume of 7.85 x 10"'' m . Solving kT = h < COeff > , we obtain < cOeff > = 6.3 x 10 ^ /s. A phonon coherence length of 0.1mm is assumed, along with a quartz lattice constant of 4.9 A. If there is only one SiOa molecule in the volume of the quartz unit cell, then (N/V) = 8.5 xlO^'' m" . The reduced mass of the oscillating dipoles was taken to be 10" ^ kg, at room temperature, where kT = 4 x 10" ^ J. Substituting n = kT/h < cOeff > for thermal phonons, and these assumed values into Eq. (8), we obtain fractional spatial fluctuations of Ssx/x = 3.6 x 10" ^ / Hz for Q = 1000 and Ssx/x = 3.6 x 10"" / Hz for Q = 100.

Finally, to calculate the rms spatial fluctuations along the length of the piezoelectric crystal, 5X, we use the following relation, in the case of Q = 1000,

5X r— 3.6x10"'' 6x10"" = A % = \ TTT- = —fTT- (9) X y ^ \ Hz siHz

The rms spatial fluctuations are obtained by multiplying by the crystal length, X, which in this case is 1 cm.

{sx) 6x10"" ^ 6xl0""(lcm) 0.6x10"'° X = 1= = / cm (10)

'Hz ylHz ylHz Therefore, the quantum 1/f fluctuations in this spatial direction are 0.006 Angstroms per square root of Hertz. This means that the piezoelectric crystal in the STM contributes 10 times more error in tip sample separation distance compared with the quantum 1/f fluctuations occurring in a stationary needle without the feedback loop engaged [4].

ACKNOWLEDGMENTS

The support of the Army Research Office and Army Research Laboratory is thankfully acknowledged.

REFERENCES

1. J.J. Gagnepain, J. Uebersfeld, G. Goujon, P.H. Handel, "Relation between 1/f Noise and Q-factor in Quartz Resonators at Room and Low Temperatures," 35"" Annual Symposium on Frequency Control, Philadelphia, pp. 476-483 (1981).

2. P.H. Handel, "Noise, Low frequency," Wiley Encyclopedia of Electrical and Electronics Engineering, vol. 14, pp. 428-449, John Wiley & Sons, Inc., John G. Webster, Editor, (1999).

3. P.H. Handel, A.G. Tournier, "Nanoscale Engineering for Reducing Phase Noise in Electronic Devices," Proceedings of the IEEE Vol. 93, No. 10, October 2005, pp. 1784-1814

4. A. Truong, P. Handel, and P. Fraundorf " Quantum 1/f Biochemical Detection Limits in THz Signatures Revealed by STM Currents", IEEE Sensors Journal, vol. 8 No.6, pp. 1020-1027 (2008).

72

The Impact of CoUisional Broadening on Noise in Silicon at Equilibrium

Christoph Jungemann* and Mihail Nedjalkov'''

*EIT4, Bundeswehr University, 85577 Neuhiherg, Germany ^IHE, TU Wien, A-1040 Wien, Austria

Abstract. The impact of coUisional broadening on noise in silicon is investigated at equilibrium. The Levinson equation is solved exactly for velocity fluctuations and the results are compared to solutions of the Boltzmann equation, which is based on instantaneous scattering. Significant differences are found only at frequencies above 150GHz at room temperature. For weaker scattering the effect will be stronger (e.g. lower temperatures or high mobility materials).

Keywords: Noise, Levinson equation, Boltzmann equation, coUisional broadening, silicon PACS: 72.10.Bg,72.30.+q,72.70+m

INTRODUCTION

In the Boltzmann equation (BE) scattering events are instantaneous and the power spectral density (PSD) of the corresponding Langevin force for scattering is therefore white [1,2]. A generalization of the BE can be derivedfromthe workof I. Levinson [3], where the quantum treatment of scattering leads to a finite duration of the scattering events. This is called coUisional broadening and its impact on transport has been widely studied (e.g. [4]). The question remains, how this affects noise.

THEORY

Here, we use the form of the Levinson equation (LE) given in Ref. [5] for spherical and parabolic bands (e = Tp'Ji^/2m), homogeneous real space conditions, zero electric field, non-degenerate conditions, and constant matrix element phonon scattering (silicon)

df{k,t) _ Q

dt K'h{2Kf ^

with

II* c^cos[a^(X|l')(?-?')]/(A',?')

— c,7Cos[a,7 {k'\k){t-t')]f{k,t')&^l<!&t' (1)

a^(k\k') = he(k)-e(i')-n(a^]. (2)

hcDr^ is the energy transfer of the scattering process r\ and c,, the corresponding scattering constant, where optical and elastic acoustic phonon scattering is considered [6]. Under

CPn29, Noise and Fluctuations, 20"" International Conference flCNF 2009) edited by M. Macucci and G. Basso


73

stationary conditions the convolution integral reduces to an energy conserving delta-function

1 cos.[a^(k\k){t-t')\At' = 5 a,,(X|A') =5 e(k)-e(l<')-%(0^ (3)

and the BE is recovered. Thus, at least for zero fields the impact of coUisional broadening vanishes under stationary conditions.

Noise can be calculated by solving the LE for the autocorrelation function of velocity. Under the above mentioned conditions the PSD of velocity can be calculated exactly in the frequency domain by analytical means with

S^{oi) = 49t {2nf

v{k)'^Fy{k,a))d^k (4)

and the Fourier transformed LE for the conditional probability of velocity

io)Fy{k,o)) = feJe{k))v{k)

d{ar,{k\k') -0)) + d{ar,{k\k') + (o) 1(0

nia^jiklt) - (0^ Fy{k',a))

5{aT^(k'\k)-(o) + 5{aT^(k'\k) + (o) ICO

n[a^{k'\k)-(o2 Fy(k,co)}d^k'. (5)

Based on the out-scattering part of the collision integral a complex valued microscopic relaxation time (MRT) can be defined for the velocity-randomizing scattering processes

1 Q

TLE(e,(s) n{2ny

^ 1)^ d{ar,{k'\k)-a)) + d{ar,(k'\k) + a)) ICO

2 n[a?j{k'\k)-(o2

S{e-tio))+S{e + fio)) . S{e - fico) - S{e + hco

d ¥

, (6) 2 2

where the total scattering rates depend only on energy for the given scattering processes

Q S{e) =

S{e) =

Multiplication of (5) with

(2^)^ IF

—-g^^y c^5 [e(A:') + (e + ^«^

T^E(e,(a)v(A:)

^ I CT^S e(k')-{e + na)T^) d^k",

d'k'.

(7)

(8)

(9)

74

OJ

E ,o,

^ 10° o o > o

Q

CL

10

10

10° 10^ 10^ 10^ lo" 10^ 10®

Absolute frequency [GHz]

FIGURE 1. Power spectral density of the velocity fluctuations for silicon at 77, 300 and 500K (only phonon scattering).

and integration over k yields an exact expression for the PSD

^LE^2

3 \ 1 + lft)T^^ eq^

In the limit of instantaneous scattering (BE) the MRT is given by

TBE(e) TLE(e,O) = 0) ^ '

and real valued [7]. Eq. (10) holds as well.

(10)

(11)

RESULTS

For zero frequency the PSD based on the LE reduces to the result of the BE. Thus, also in the case of noise the BE is recovered for low frequencies. Huge differences between both approaches can be expected only at higher frequencies (Fig. 1), where the PSD of the LE decays with frequency only with the power of —1.5 instead of —2, because at high frequencies the real and imaginary parts of the inverse microscopic relaxation time are both proportional to the square root of the frequency. The relative difference of both results is shown in Fig. 2 and at room temperature the error of the assumption of instantaneous scattering exceeds one percent only at frequencies above 150GHz.

CONCLUSIONS

We have presented for the first time an exact analytical solution of the LE for noise at equilibrium. This result gives more insight into the problem of collisional broadening

75

10° 10 10^ 10^ 10^ 10= 10^

Absolute frequency [GHz]

FIGURE 2. Absolute value of the relative deviation of the two PSDs (S^E _ s^^)/S^^.

than numerical calculations. The BE is recovered in the stationary limit for transport and at zero frequency for noise. In silicon at room temperature a non-negligible impact of colhsional broadening is found only for frequencies above about 150GHz. For weaker scattering (e.g. lower temperatures) and thus higher mobilities the impact increases.

REFERENCES

1. S. Kogan, Electronic Noise and Fluctuations in Solids, Cambridge University Press, Cambridge, New York, Melbourne, 1996.

2. S. V. Gantsevich, V. L. Gurevich, and R. Katilius, Nuovo Cimento, 2, 5 (1979). 3. I. Levinson, Sov. Phys. JETP, 30, 362 (1970). 4. L. Reggiani, P. Lugli, and A. P Jauho, Phys. Rev. B, 36, 6602-6608 (1987). 5. M. Nedjalkov, H. Kosina, R. Kosik, and S. Selberherr, Microelectronic Engineering, 63, 199-203

(2002). 6. R. Brunetti, C. Jacoboni, F. Nava, L. Reggiani, G. Bosman, and R. J. J. Zijlstra, J. Appl. Phys., 52,

6713-6722(1981). 7. W. Brauer, and H. W. Streitwolf, Theoretische Grundlagen der Halhleiterphysik, Vieweg, Braun

schweig, 1977, 2nd edn.

76

Modeling and Measurements of Low Frequency Noise in Single-Walled Carbon Nanotube Films

with Bulk and Percolation Configurations Ashkan Behnam, Ant Ural, and Gijs Bosman

Department of Electrical and Computer Engineering, University of Florida, Gainesville, Florida 32611, USA

Abstract. Monte Carlo simulations and noise modeling are used to study the scaling of 1//noise in single-walled carbon nanotube films as a function of device parameters and film resistivity. This computational approach provides a general framework for the characterization of 1//noise in nanotube films and explains experimental observations.

Keywords: 1/f noise, percolation, carbon nanotube films. PACS: 73.50.-h,73.50.Td, 73.63.Fg.

INTRODUCTION

Single-walled carbon nanotube (CNT) two-dimensional networks and three-dimensional films attract significant research attention due to the fact that they are transparent, conductive, flexible, and have uniform physical and electronic properties since individual variations in nanotube diameter and chirality are ensemble averaged [1]. As a result, the reproducibility and reliability problems found in devices based on individual nanotubes are resolved, and CNT film based devices can be mass produced in a cost effective manner. Several promising device applications of CNT films have been demonstrated, such as thin film transistors, flexible microelectronics, optoelectronic devices, and chemical sensors [2].

It has been shown that for both single nanotubes (regardless of their intrinsic parameters like diameter and chirality) and CNT films, 1// noise levels can be quite high compared to other conventional materials [3]. As a result, determining the device 1//noise level, its origin and its scaling with various CNT film parameters will assist in validating the applicability of CNT films for purposes where device noise cannot be ignored.

In this paper the results of Monte-Carlo low frequency noise simulations as a function of film thickness, width and length will be presented and compared with experimental data. The film feature sizes are chosen in such a way that transitions from bulk to percolation transport can be observed both in terms of resistivity and 1/f noise magnitude.

Whereas the low frequency noise in the bulk domain scales with feature sizes as expected from Hooge's law [4] and the resistivity is a constant, a variety of power laws for the dependence of both the relative 1/f noise magnitude and resistivity on critical feature sizes are observed approaching percolation.



79

MONTE CARLO SIMULATION

Simulation of the noise properties of the single-walled carbon nanotube frlm was performed by randomly generating the carbon nanotubes in the film using a Monte Carlo process, finding the locations of the tube-tube junctions in the generated film, and solving the current continuity equations for these junctions in a matrix format, following a similar approach as explained in detail previously [5, 6].

Briefly, each nanotube in the film is modeled as a "stick" with fixed length ICNT-

The position of one end of the nanotube and its direction on a two-dimensional (2D) plane are generated randomly. Each nanotube is assigned randomly to be either metallic or semiconducting with the ratio of the semiconducting to metallic nanotubes set to 2:1, as typically observed experimentally. This process is repeated until the desired value for the nanotube density per layer n in the 2D layer is achieved. Additional 2D layers are generated randomly using the same approach and stacked vertically to form the 3D nanotube film. In the 3D frlm, it is assumed that nanotubes in a given layer can also form junctions with nanotubes lying in the first and second nearest neighbor layers (i.e. two layers above and two layers below).

To model the physical properties of the film, the resistance of an individual nanotube is calculated by RCNT = Ro l/-^, where / is the length of the nanotube, A is the mean free path (assumed to be 1 }im in our simulations), and Ro = h/4e^ is the theoretical tube resistance at the ballistic limit (-6.5 kQ) [6]. The resistance of the tube-tube junctions depends on whether the junction is metallic/semiconducting (MS), semiconducting/semiconducting (SS), or metallic/metallic (MM). Each different type of tube-tube junction was modeled by a different contact resistance [6]. For computing the 1// noise in the CNT film, we have used a model which takes into account the noise contributions from both the nanotubes themselves and the tube-tube junctions in the film. Assuming independent noise sources (i.e. uncorrelated fluctuations), the relative 1/f noise magnitude of the film, Aeq, can be written as

j2 - yljll^^n^nK' (1)

where i„ is the current, A„ is the relative 1/f current noise magnitude, and r„ is the resistance of the tube or junction associated with the «th individual noise source.

Based on experimental results, we have used A„ = IQ- " Rg/l for the 1// noise amplitude of individual nanotubes, where / is expressed in microns and RQ = 6.5 kQ.[7].

Unlike individual nanotubes, determining A„ for tube-tube junctions based on the available experimental literature is rather difficult. In this work, a relationship \ = ar^ was assumed with a = 10" ° Q.'^.

EXPERIMENTAL SETUP

The CNT films used in our study were deposited by vacuum frltration on a (100) silicon wafer with a 500 nm layer of thermally grown SiOi on top [6]. From AFM

measurements it was estimated that most nanotubes in the film had lengths in the range of 1-10 jim. The CNT film was subsequently patterned and etched to produce four point probe structures with film lengths L between 7 and lOOOjim, width W between 200nm and lOOjim, and film thickness t between 15 and 75 nm. Low frequency noise measurements between IHz and 100 kHz were carried out using a low noise amplifier and spectrum analyzer. Strong 7/f noise spectra were observed scaling with the DC current squared.

SIMULATION AND EXPERIMENTAL RESULTS

Figure 1 shows the log-log plot of the noise amplitude normalized to resistance A/R versus device length L for a single-layer nanotube network, where filled circles denote experimental data points from Snow et al. [8] and open circles denote our simulation results. The simulation results are in excellent agreement with the experimental data, clearly indicating that A/R is a strong function of device length.

10 100 Length (lam)

FIGURE 1. Log-log plot of the noise amplitude normalized by resistance (AIR) versus device length for a single-layer nanotube network. Experimental data from 2D nanotube networks of Snow et al. [8] is shown by the filled circles. Our simulation data points for single-layer devices with W =2 |am, Icm = 2 |am, n= 5 |am , and L ranging from 2 to 20 |am are shown by the open circles.

The dashed line in Fig. 1 is the power-law fit to the experimental data, yielding that A/R - L'^with a critical exponent a= -1.3, in agreement with the simulation data for 8 < L < 20 jim. The decrease in the noise amplitude with device length is consistent with Hooge's classical empirical law [4] where the 1/f noise amplitude A varies inversely with the number of charge carriers N in the device, i.e. A - 1/N. However, since the resistance of the CNT film device is given by R=pLAVt, where p is the resistivity, and A'' scales with the device volume, i.e. A'' - LWt, A/R is expected to scale as A/R - L~ .

Previously, it was suggested that the deviation from this ideal result is due to nonuniformity of the CNT network [8]. Our results, on the other hand, suggest that the observed exponent is probably due to the effect of other device parameters on the 1/f noise amplitude.

81

To illustrate this point further. Fig. 2 shows how the film thickness t affects the scaling of A/R with L.

10^°

10- ^

--10-^^ a D:10-" 5

10-'*

10-'=

lo-"'

ftbn

^

r

r

r

f

• . . .

1 I W g o ^ * ^''P

™ni|^ ^ ^ ^ g.|^ jg layers) , ^ ^ ^ - ° Sim (3 layers)

^ ^ Fit to Sim V 1

V •

V 1 N 1

N

^ 1 •

• V.

1 10 1000 100 Length (lam)

FIGURE 2. Log-log plot of noise amplitude normalized by resistance {AIR) versus device length for multi-layer nanotube films. Filled circles represent our own experimental measurements of CNT film devices with -15 nm thickness, 50 ^,m and 1000 tm device length, and device widths ranging from 2 to 50 ^,m. Open circles and squares are simulation data points for devices with a film thickness of i = 16 nm (8 layers) and t = 6 nm (3 layers), respectively, where the other simulation parameters are W = 2 \un, IcNT = 2 \un, n = 1.25 ^,m , and L ranging from 2 to 14 ^,m.

Two curves are illustrated for films consisting of 3 and 8 layers, respectively. The extracted critical exponents from the power-law fits to the simulation data for L > 6}im are a= -0.8 and a= -1.9, respectively. For comparison with the simulation data, our own experimental measurements of the 1/f noise amplitude in CNT film devices with 15 nm thickness, 50 }im and 1000 \na device length, and device widths ranging from 2 to 50 \na are also shown in Fig. 2 as filled circles.

As can be seen, the simulation results for the f = 16 nm CNT film are in excellent agreement with the experimental data, and both exhibit a critical exponent which is very close to the ideal bulk case of a= -2.

The resistivity scaling with device width close to the percolation threshold has been experimentally observed to be significantly more pronounced than that with device length [6]. This point is also evident in the simulation data shown in the inset of Fig. 3. The results of simulations to investigate the scaling of 1/f noise with device width in the CNT film are shown in the main panel of Fig. 3.

Two regions can be distinguished. For W > 2 }im, A is inversely proportional to W (the power-law exponent extracted from the dashed line fit to the data is W^'^). This variation is expected, since A ~ 1/N, and the number of carriers A increases linearly with device width and the resistivity is constant in this regime, as seen in the inset of Fig. 3. However, for W < 1 }im, there is a strong power-law relationship between A and W with a critical exponent extracted from the fit equal to -5.6. This shows that the variation of resistivity has a strong effect on the noise in this region.

82

10 =

10-

10"'

• Sim ^ Fit \

•

1

1 Width (nm)

0.5 Width (nm)

FIGURE 3. Log-log plot of computed A versus W for a device with L = 5 ^,m, t = 16 nm and other parameters same as in Fig. 2. There are two separate scaUng regimes. The extracted exponent of the dashed Une fit for large widths (where resistivity is constant) is -1.1. The extracted exponent of the dashed line fit for small widths is -5.6. The inset shows a log-log plot of resistivity versus device width for the same device as in the main panel.

As we have seen in Fig. 2 CNT film thickness has a strong effect on the noise scaling with device length as confirmed experimentally by Soliveres et al. [9]. Next, we investigate this dependence by our simulations. The inset of Fig. 4 shows a log-log plot of resistivity versus number of layers (i.e. thickness) where resistivity is almost constant for films with 10 layers or more, while strong inverse power law dependence of resistivity on thickness exists for thin films near the percolation threshold.

10-^

-g-10"^

c , X

< 10""

10'

•

•

" -._

10

' '

0.1

Thickness (Layers)

_ ^ • ^

. >. .ts ^^

= n •ro A | C ^ ^

^

• Sim Fit

•

. 0.02 0.1

Resistivity (n-cm)

FIGURE 4. Log-log plot of computed A x t versus resistivity, resistivity versus device thickness for the same device.

0.5

The inset shows a log-log plot of

The main panel shows the log-log plot of the 1/f noise relative amplitude normalized by thickness, A x f, versus resistivity computed for the same CNT film device as in the inset. The normalized amplitude A x fis used because A varies with thickness linearly in the regime when resistivity is constant. The simulation data can be fit by A X f ~ /»'', where the extracted critical exponent is V = 1.8.

83

CONCLUSIONS

The data show that the relative 1/f noise amphtude of CNT films may depend strongly on device dimensions and on the film resistivity, following a power-law relationship with resistivity near the percolation threshold after properly removing the effect of device dimensions. Furthermore, the noise-resistivity and noise-device dimension critical exponents extracted from the power-law fits are not universal invariants, but rather depend both on the parameter that causes the change in resistivity and noise, and the values of the other device parameters.

ACKNOWLEDGMENTS

This work was funded in part by the University of Florida Research Opportunity Seed Fund.

REFERENCES

1. Z. Wu, Z. Chen, X. Du, J. Logan, J. Sippel, M. Nikolou, K. Kamaras, J. Reynolds, D. Tanner, A. Hebard, and A. Rinzler, Science 305, 1273 (2004).

2. E. Bekyarova, M.E. Itkis, N. Cabrera, B. Zhao, A.P. Yu, J.B. Gao, and R.C. Haddon, J. Am. Chem. Soc. 127, 5990 (2005).

3. P. G. Collins, M. S. Fuhrer, and A. Zettl, Appl. Phys. Lett. 76, 894 (2000). 4. F. N. Hooge, Phys. Lett. A 29, 139 (1969). 5. S. Kumar, J. Y. Murthy, and M. A. Alam, Phys. Rev. Lett. 95, 066802 (2005). 6. A. Behnam and A. Ural, Phys. Rev. B 75, 125432 (2007). 7. Y. M. Lin, J. Appenzeller, J. Knoch, Z. Chen, and P. Avouris, Nano Lett. 6, 930 (2006). 8. E. S. Snow, J. P. Novak, M. D. Lay, and F. K. Perkins, Appl. Phys. Lett. 85, 4172 (2004). 9. S. Soliveres, J. Gyani, C. Delseny, A. Hoffmann, and F. Pascal, Appl. Phys. Lett. 90, 082107 (2007).

84

1/f noise, transport and percolation in carbon nanotube film field-effect transistors: simulation

and experiments

S. Soliveres, F. Martinez, A. Hoffmann, F. Pascal

IIES - UNWERSITE MONTPELUERII- UMR CNRS 5214

Place E. Balaitton, 34095 Montpettier Cedex 5, Erance

Abstract. In this paper we present a model for electrical properties of carbon nanotube film field-effect transistors. The model, based on carbon nanotube physics uses Landauer formalism and tight binding calculation. The total film is described as an electrical network .A modified nodal analysis provides DC and noise characteristics. Theses simulations are in good agreements with experimental results.

Keywords: carbon nanotube film field-effect transistor, 1/f noise, transport. PACS: 73.63.Fg

INTRODUCTION

The properties of an individual nanotube make it interesting to use in microelectronic as transistor (CNFET). If the fabrication of high quality and reproducible CNFETs remains a difficult challenge, sensor apphcations using large quantities of CNTs are already available. The sensors consist of field-effect transistors based on films of CNTs arranged randomly. Although transport properties in individual nanotube have been intensively studied, little is known about structures with large quantities of CNTs. In this paper, we show that percolation dominates the transport and the noise properties of these films. A transport model is presented which allow to reproduce experiment and to predict the behavior of noise and conduction in CNTs films transistors.

THEORY AND MODELING

Experiments have shown that nanotube films seem to behave as a small band gap semiconductor (sc) with p-type conduction. However, it's not possible to describe a NT film as a crystalline semiconductor compound. Due to the absence of covalent bond and to the cylinder nature of NTs, there is a weak couphng between NTs. However, the film has to be described by considering the properties of each nanotube and their interactions. Two microscopic charge transport mechanisms occur in the film: the transport along nanotube themselves and the transport between crossed



85

nanotubes. Considering the large mean fi-ee path in CNTs and the weak coupling between NTs, we assume that the contacts between NTs dominate the transport through the film. From a macroscopic point of view, the NT film is modeled as a percolation network. Percolation networks have electrical properties that vary rapidly at the vicinity of the percolation threshold and follow power laws related to the density of NTs.

In order to get a microscopic understanding of the transport phenomenon, we have developed a simulator for the electrical properties of CNTs films. The objective is to provide a physical interpretation of experiments and to predict DC and noise behaviors of CNTs film transistors. Nanotubes are set randomly on a surface and wide electrodes are defined to form source and drain with a low contact resistance. Each intersection between tubes (metal/metal, metal/sc or sc/sc) is modeled as a dipole with a dynamic resistance re. The hopping from a tube to another is modeled as a perturbation in the transmission probabilities in the framework of Landauer formalism. Transmission probability depends on the nanotube charge and on the energy barriers between nanotubes. Charge inside a tube is calculated self consistently with potential taking into account the gate voltage and the density of states. The energy barrier is obtained by a first neighbor tight-binding calculation. From these values, the transmission probability is obtained by WKB approximation (to reduce computational time).

In the considered experimental conditions each junction exhibits only 1/f noise source. The current noise spectral density is expressed as Sj (f )=KP/f with

K=10''°r^ [1]. Figure 1 shows the simulated current of a SC-SC junction of crossed nanotube controlled by the gate bias. Finally, the total CNT film is modeled as an electrical network. A modified nodal analysis provides DC and noise characteristics.

6e-8

5e-8

4e-8

3e-8

2e-8

1e-8

0.2 0.4 0.8 o.e

VG(V)

FIGURE 1. Simulated current in a NT junction versus gate bias.

RESULTS

Devices and measurements

In order to observe the impact of percolation process, we have fabricated films with different tube densities [2]. Due to the random characteristic of the network below the percolation threshold a large number of devices are necessary to obtain average results. The conductivity and the low-frequency noise coefficient K have been measured for the different densities. Results are reported on figure 2a and 2b. These

two quantities follow power laws with critical exponent 4 for the conductivity and K for the noise, as expected in percolation processes. Figure 2b-inset shows that K is related to the conductivity by an exponentw = KIt^, which lead to a CNT film noise

model function of the design K • RS

L where R is the film resistance, and S the

section area. This expression explains the different empirical relations found in literature [3].

x5

(b)

i.

^ 10-= TO" TO"

^ " u 1

V

•\ = 10^ TO-' TO" TO' 1(^ l t f KC 1

E

1

-, 3*- l

• " • v ^ ^ ^ CTsp-^m) ^

- \

0 10 10 19 14 16 18

NL NL

FIGURE 2. a: Conductivity (x) obtained from measurements and power law (-) versus the number of deposited layers, b: Noise coefficient (x) obtained from measurements and power law (-) versus the

number of deposited layers.

Simulations

Device simulations are performed using the model presented previously. Figure 3a,b shows the simulated potential along a device for VDS=1 V. For high CNTs densities the potential decreases linearly. For low CNTs densities, the potential drop is not homogeneous and there is dispersion in nanotube potentials. 1/f noise is more important in the non homogeneous parts and contributes mainly to the total noise. Figure 4a and 4b present simulated conductivity and the noise coefficient K for the different densities. The curves clearly exhibit the same behavior than experimental results (figure la and lb), with similar exponents. Figure 5 presents a simulation of the drain current versus gate voltage for a CNT film composed of semiconductor nanotubes only. It indicates that transistors with very good lON/IOFF ratio could be competitive if it were possible to sort metallic and semiconductors nanotubes.

1

0.8

£o.6

• 0.4 i **

(b)

0 1 2 3 4 5

L(m) ^ ,p- . L(m| ^.,0-6

FIGURE 3. NT potentials along the film for a density of (a) 125 NT/jim^ and (b) 25 NT/jim^

87

10=

10=

„ 10'

'E 103

a 102 b

10'

10=

10'

- power law with tc=2 1 simulation

200 400 600 800 1000 1200 1400

NT density (NT/Mrrf)

200 400 eOO 800 1000 1200 1400 NT density {UJI\inf)

F I G U R E 4. a: Simulated conductivity (o) and power law (-) versus the N T density, b: Simulated noise (o) coefficient and power law (-) versus the N T density. W / L = l n m / 4 n m .

0.4 0.6 0.8

F I G U R E 5. Simulated current and noise coefficient versus the gate bias for a CNT film field-effect

transistor, for 60 NT/jim^. W / L = l / 4 n m .

CONCLUSION

In summary, we have shown that percolation greatly influence the noise and the conductivity for CNTs film. Non-homogeneous films lead to very high noise. We have developed a simulator based on random generated CNT film. Contact between tube compact model lead to simulation of DC and low-frequency noise behaviour, taking into account percolation transport. A very good agreement with experiments is observed. Prospective simulations on transistors have been performed showing potentially good performances for such devices.

REFERENCES

1. A. Benham, G. Bosman, A. Ural, Phys. Rev. B 78, 085431 (2008) 2. S. Soliveres, J. Gyani, C. Delseny, A. Hoffmann and F. Pascal, Appl. Phys. Lett. 95, 082107 (2007) 3. E. S. Snow, J. P. Novak, M. D. Lay and F. K Perkins, Appl. Phys. Lett 85, 4172 (2004)

Analysis Of Current Noise During The Resistive Transition Of MgB2 Thin Films Produced By

The Application Of An External Magnetic Field

V.Andreoli^ P.Mazzettf, A.Stepanescu^ M.Rajteri'', C.Portesi'', E.Monticone'', E.Taralli'', C.Gandinf and A.Masoero''

"Dipartimento di Fisica, Politecnico di Torino, Corso Duca degUAhruzzi 24, 10129 Torino, Italy Istituto Nazionale di Ricerca Metrologica INRIM, Strada delle Cacce 91, 10135 Torino, Italy 'Dipariimento di Scienze e Tecnologie Avanzate, Universita del Piemonte Orientals "Amedeo

Avogadro", Via Bellini 25/G, 15100 Alessandria, Italy

Abstract. The excess noise observed during the resistive transition of a superconducting material can be used to shed light on the microscopic processes underlying the transition itself. In a previous paper [1] it has been proposed a model to explain the large noise observed during the resistive transition of MgB2 superconducting films, obtained by slowly increasing the specimen temperature across its critical value when a d.c. bias current is applied. The amplitude and frequency behaviour of the noise power spectrum, simulated with this model, are in good agreement with the experimental data. The model is based on the onset of correlated transition of large sets of grains forming resistive layers through the specimen cross section, giving rise to resistance steps. It is assumed that these events produce the large noise, of \lf type in the low frequency range, observed in the experiments. To compare this model with alternative ones, based on dissipative effects produced by fluxoid dynamics, new results, obtained by producing the transition by varying an applied magnetic field, are reported in this paper. The fact that the transition noise remains practically unchanged, even if the fluxoid density is much increased by the magnetic field, suggests that the transition mechanism proposed in the described model is more appropriate than the one based on fluxoid dynamics.

Keywords: resistive transition, magnetic field, dynamic noise PACS: 72.70.+m, 74.40.+k, 74.70.Ad

INTRODUCTION The analysis of current noise, produced in stationary and non stationary conditions

during the resistive transition of polycrystalline samples, can be used to understand the microscopic processes at the base of the transition itself, and to improve the signal to noise ratio in bolometers and transition-edge sensors.

In a previous paper [1] it has been proposed a simple model to explain the noise observed during the resistive transition of MgBa polycrystalline films, which is alternative to other models based on fluxoid dynamics.

According to this model, the transition process is characterized by a large noise, due to avalanche transition processes among the grains, giving rise to abrupt resistance steps. In the model these steps correspond to the creation of resistive layers across the specimen, formed by grains which have undergone the resistive transition. By



assuming a Gaussian distribution for the grains critical currents, the power spectrum of the transition noise of a nanogranular MgBa film could be calculated and found in good agreement with the experimental data. In particular, the large noise at low frequency appears to exclude alternative models based on fluxoids motion within a matrix of pinning centers, which would justify a transition noise not much larger than the stationary one.

To make a further check of the model, measurements of the transition noise power spectrum were taken in the presence of a static magnetic field [2]. In that paper it was proved that, while the application of the field shifts the transition towards smaller temperatures, it does not alter the mechanism at the base of the transition process, since no variations of the noise power spectrum intensity and frequency behavior was observed, as expected if fluxoid depinning and motion would be responsible of the transition process.

In the present paper additional proofs of the validity of the model are reported by producing the transition by varying the external magnetic field and keeping the bias current and temperature constant. The results show that, when the transition occurs with the same rate of change of the film resistance, the noise power spectrum is identical to the one produced in the absence of magnetic field by slowly increasing or decreasing the specimen temperature.

EXPERIMENTAL

The reported experiments have been performed on a MgBa film prepared at the I.N.Ri.M. institute. The film has been prepared by a growth process characterized by a simultaneous evaporation of Mg and B on a SiN substrate kept at 570K. Mg and B precursors were annealed in situ at 773K for 300s in argon atmosphere at a pressure of Imbar. The film thickness was 150nm and the active area of the film (1 x l)mm^. The average grain size was estimated of being between lOnm and 20nm.

The specimen was sealed within a copper container filled with helium, put in a cryocooler and located between the poles of an electromagnet powered by a Kepco power amplifier driven by a low frequency oscillator .

A standard four contact technique was used for the resistance and noise power spectrum measurements during the transition process. An extra low noise step-up transformer was used for the noise signal detection and as a high-pass filter to suppress the dc component of the power spectrum. The stepwise voltage pulses, related to the formation of resistive layers, are changed by the transformer in exponential-like pulses. Fig.l shows the transition curves for two different values of applied field. Fig.2 shows the noise power spectra without applied field and for an applied field varying from 56G to 1560G. The external field was applied perpendicular to the film surface.

These results show that the magnetic field reduces the material critical current density, but it does not have any effect on the transition noise. This fact is in good agreement with the transition mechanism proposed in [1], since it is expected that the field simply reduces the superconducting grain critical current, shifting the transition curve towards lower temperatures, as shown in Fig.l.

90

/=10mA,if=800G

33 34 35 36 37 38 39

temperature (K)

FIGURE 1. Shift of the transition curves along the temperature axis due to a change of the applied field.

DISCUSSION AND CONCLUSIONS

A brief account of the resistive transition model for polycrystalline disordered high Tc superconductors, described in [1], is now reported, with the purpose of explaining the above experimental results. The model is based on the assumption that the specimen crystalline grains have a Gaussian distribution of critical currents and that the transition occurs through the formation of layers of resistive grains crossing the whole specimen, when, by effect of temperature or magnetic field increase, the sum of the critical currents of all the grains forming a layer become smaller than the applied bias current. The formation of each layer correspond to a step in the resistance increase of the whole specimen, and thus to a voltage pulse. The noise is generated by the superposition of these pulses, which correspond to the simultaneous transition of a large quantity of grains, and is thus much larger than the 1// noise produced in stationary conditions. As shown by numerical simulations reported in [1], the 1//^ behavior of the power spectrum of the transition noise in the low frequency range occurs because the stepwise voltage pulse has a rounded trailing edge. This is due to fact that the grains forming the layer gradually shift from an intermediate resistive state to a fully resistive one during the slow decrease of their critical current by effect of the temperature or field increase. Rectangular voltage steps, distributed according to a Poisson distribution, would have given a l/f^ spectrum.

Alternative models to explain the noise produced in high Tc superconductors, both in stationary or non stationary conditions, are based on density fluctuations of fluxoids, moving in a matrix of pinning points [3-5]. According to these models, the noise produced during the transition is expected to be not much different from the l/f noise detected in stationary condition, and its intensity to be dependent on the fact that the resistive transition is produced by slowly varying the temperature in the absence of a magnetic field, or by producing the transition at constant temperature by applying a variable magnetic field. This is obviously not the case for the model described above, where the formation of resistive layers is independent of the way the transition is produced.

91

' N '

>

B

O

lE-l l

lE-12

lE-13

lE-14

lE-15

lE-16

lE-17

lE-18

1 / = 0 mA, H=0, r=35.5 K

2 / = 10 mA, H=0, T: 35.5-36.5 K

3 /,= 10 mA, i?: 56-1560 G, r=35.5 K

100 1000

frequency (Hz)

FIGURE 2. Power spectra of the transition noise taken in the absence of the magnetic field and with a temperature change between 35.5K and 36.5K (curve 2) and under a variation of magnetic field

from 56G to 1560G at a constant temperature of 35.5K, giving the same resistance variation of the specimen in the same time interval (curve 3). Curve 1 represents the background noise which has been

subtracted from the others curves. The presence of the large fluctuation peaks in the high frequency part of the spectrum of curve 3 are due to induced emf related to small vibrations of the specimen immersed

in the magnetic field within the cryocooler.

As a conclusion, it can be stated that this model seems more appropriate for describing the transition mechanism in polycrystalline high Tc superconductors when the transition is produced at low bias current and near the critical temperature of these materials. The other model, based on fluxoid creation and depinning, is certainly more appropriate when the transition occurs by effect of a strong current density at temperatures much lower than the critical one. However, in that case, noise measurements become quite difficult, if not impossible.

ACKNOWLEDGMENTS

This work has been carried out with the contribution of Tecnomeccanica SRL -Novara (Italy), to the financing of a post-doctoral grant, in memory of Guglielmo Agradi.

REFERENCES

1. p. Mazzetti, C.Gandini, A. Masoero, M. Rajteri and C. Portesi, Phys. Rev. B 11, 064516 (2008). 2. M. Rajteri, C. Portesi, M. Accardo, E. Taralli, E. Monticone, C. Gandini, A. Masoero, V. Andreoli

and P. Mazzetti, IEEE Trans. Appl Supercond., in press. 3. D. Daghero, P. Mazzetti, A. Stepanescu, P. Tura and A. Masoero, P/ij's. iJev. S 66, 184514(2002). 4. L. Cattaneo, M. Celasco, A. Masoero, P. Mazzetti, I. Puica and A. Stepanescu, Physica C 261, Ill-

US (1996). 5. C. Heiden and G. I. Rochlin, Phys. Rev. Lett. 21, 691-694 (1968).

92

Flow Noise of Driven Vortex Matter in Amorphous Superconducting Films

S. Okuma", Y. Suzuki", and N. Kokubo' '

'^ResearchCenter for Low Temperature Physics, Tokyo Institute of Technology, 2-12-1, Ohokayama, Meguro-ku, Tokyo 152-8551, Japan

Center for Research and Advancement in Higher Education, Kyushu University, 4-2-1, Ropponmatsu, Chuoh-ku, Fukuoka, Fukuoka 810-0044, Japan

Abstract. Voltage noise >?v(/) generated by current-driven vortices is studied as a function of flow voltage Fand magnetic field for a superconducting amorphous Mo^Gei., film. For high V (or velocity) in the flux-flow state, SY at low frequency (/=100 Hz) exhibits a sharp peak just below a peak field of the critical current. For low V'm the plastic-flow state, an additional broad peak oi SY appears at lower field. Possible origin responsible for large flow noise is discussed. We show that for moderate V, SY clearly detects the structural phase transition of vortex matter.

Keywords: Mixed state; Flux-flow noise; Peak effect. Amorphous films PACS: 74.40.+k, 74.25.Dw, 74.78.Db

INTRODUCTION

In the mixed state of type-11 superconductors, vortex states near the peak-effect (PE) regime, where the critical (depinning) current /c shows a peak with magnetic field B or temperature T, have attracted considerable interest for many years. While the PE was considered to originate from softening of vortex lattice just prior to the upper critical field and random pinning due to quenched disorder, it has become clear that it marks the structural phase transition of vortex matter from the weakly pinned ordered phase (OP) to the strongly pinned (amorphouslike) disordered phase (DP) [1]. This transition is called an order-disorder transition (ODT). We have recently found for amorphous (a-)MOj:Gei.j: films [2] that voltage noise Sy induced by current(/)-driven vortices exhibits a sharp peak just below the peak field 5p of /c, indicating the existence of ODT of vortex matter and metastable vortex states just below 5p [1]. Here we show that SY is a very useful method to detect the phase transition in the vortex-solid phase, where the linear resistance is always zero.

In general, measured noise contains contributions from different physical origins [3-6] and is crucially dependent on the average velocity of driven vortices, which is proportional to the flow voltage V. Therefore, to detect the phase transition clearly, we must choose an appropriate value of V in measuring Sy. We will show that the shape of Sy-B curves is significantly dependent on V and moderately large V is needed to probe ODT clearly. More detailed results concerning present work have been pubhshed elsewhere [2].



93

EXPERIMENTAL

We prepared an a-MoxGci-x film with thickness 330 nm by rf sputtering on a Si substrate held at room temperature [2,5]. The mean-field transition temperature TM and the zero-resistivity temperature T^ are 6.3 and 6.2 K, respectively. The arrangement of the Corbino-disk (CD) electrical contacts is shown schematically in the inset of Fig. 1(c). The current flows between the contact +C of the center and that -C of the perimeter of the disk, which produces radial current density inversely proportional to the radius r [1-3]. The vortices induced by an applied field B rotate around the center of the disk without crossing the sample edges and their dynamics at r is probed using voltage contacts +V and -V placed at r. We measure S^iJ) over a broad/range (1 Hz-40 kHz) by using a fast-Fourier transform spectrum analyzer. We obtained the excess noise spectra S^iJ) by subtracting the background contribution, which was measured with 7=0. All the data shown in this paper were taken at 3.3 K in zero-field-cool mode.


First, we measure I-V characteristics in different fields in the range 5=0.1-5.5 T within the solid phase. The critical current h is defined as a threshold current at which the vortices start to move, where we use a 10" V criterion [2]. In Fig. 1(a), h thus obtained is plotted against B. The peak of/c(5) is clearly visible at 5=4.8 T ( = 5p) before h vanishes at a field of 5.5 T ( = 5c), whereas a sharp rise in I^ (B) at Bp, as reported in NbSe2 crystals for the CD geometry [1], is not visible. The I-V curve exhibits a strong nonlinearity at low currents (/ >/c). We call this region a plastic-flow regime. At high currents above a certain characteristic current, the / -F curve shows a hnear dependence. This region is identified with a flux-flow regime. It is well accepted that, in the presence of random pinning, several dynamic phases of driven vortices with different temporal, positional, and orientational orders are reahzed depending on the amplitude of the driving force and the velocity of vortices. Some of them, such as plastic flow, accompany large broadband noise (BBN) due to pinning induced incoherent flow of vortices, as seen in numerical simulations [6,7].

Next, we measure 5'v(/) generated by current-driven vortices. In general, SyiJ) is strongly dependent on / as well as B. With increasing / in the PE regime below Bp, detectable noise appears just above the onset of V. With further increasing / (>/c), SY/V (SY divided by V) grows rapidly and, after showing a small peak, it eventually becomes / independent or weakly dependent on / in the flux-flow regime. In this work we select three characteristic voltages F=0.1, 0.5, and 2 mV in measuring Syif) as a function of B. We have confirmed that over the broad B region in the solid phase, driven vortices are in the plastic-flow state for F=0.1 mV, while they are in the flux-flow state for V=2 mV [2].

In Fig. 1(b) we show the noise spectra 5'v(/)/Ftaken at F=0.1 mV in different fields, which correspond to OP (0.5, 1.5, and 4.0 T), the onset of the coexisting (OP-DP) phase (4.5 T), ODT [4.75 T(«5p)], and the melting (depinning) transition (5c=5.5 T). A dashed line represents the location of the background level, which corresponds to

94

3.3 K (b) 0.1 mV

FIGURE 1. (a) Field dependence oil, at 3.3 K. (b) Noise spectra5v(/)/Ftaken at F=0.1 mV (i.e., in the plastic-flow regime) in different fields at 3.3 K. A dashed line represents the background level.

5v(/)/I^in 5.5 T is near the background level, (c) Field dependence of 5v(100Hz) /Ftaken at 0.1 mV (circles), 0.5 mV (triangles), and 2 mV (squares). Inset: Schematic illustration of the arrangement of the electrical contacts of CD, in which the vortices rotate around the center of the sample. A vertical

dashed line in (a) and (c) (main panels) marks the location of Bp. All other lines are guides for the eye.

Svif)« 10-'' V^/Hz. BBN of substantial magnitude is clearly seen over the broad fields except for 5.5 T (=5c), which reflects incoherent flow of vortices in the plastic-flow state. As a whole, the spectral shape is of Lorentzian type, although for 4.0 T the shape is shghtly degraded by the appearance of a small hump, whose origin has not been specified. Possibly, the spectrum may be composed of two components of Lorentzian spectra with different comer frequencies.

Shown in Fig. 1(c) is the field dependence of 5'v(/)/Fat^l00 Hz measured for V=2 mV (squares), 0.5 mV (triangles), and 0.1 mV (circles). For either V, an abrupt drop of 5'v(100Hz)/F is clearly seen just above 5p=4.8 T, indicative of the sharp phase transition (ODT). In particular, for the largest V=2 mV, 5'v(100Hz)/Fexhibits a single narrow peak with a sharp rise followed by an almost vertical drop of 5'v(100Hz)/F at 5«5p. Therefore, evidence for ODT is most clearly seen at this voltage. For lower V, we also find the peak of SY/VJUSI below B^: For F=0.5 mV, Sy/V has a very long tail which extends toward the low-5 region, while for the lowest V=0.1 mV, in addition to the sharp peak at ~Bp, an additional very broad peak appears at around 4 T. The similar feature was reported previously in NbSe2 single crystals at lower V. Origin of large noise in the vicinity of Bp was reasonably attributed to vortex instabilities due to DP coming from the sample edges and intermixed with OP in the bulk [1]. This is called edge contamination (EC) effects. We have recently found that the EC effects are not present in our a-MoxGci-x films through comparative studies using CD and striplike configurations [2]. However, the notion of the coexisting phase as an origin

95

for large noise at «5p is not unreasonable, considering a supercooled metastable DP [1,8-11] and/or possible shght inhomogeneities in the sample.

In the meantime, a stationary vortex configuration composed of OP and DP has been revealed in the PE regime of NbSe2 single crystals by shaking the vortex system [12]. The shaking technique is often used to reorder the metastable vortex system. The result found in Ref 12 contradicts the picture of supercooling. To clarify whether large noise observed just prior to 5p in our a-MoxGci-x films indeed results from supercooled DP, we are now conducting shaking experiments. We use a 30 kHz ac drive current with amplitude comparable to that of the dc drive current. Prehminary data shows that no significant change in S^if) is visible after shaking, rejecting the scenario of supercoohng. We thus seek an alternative interpretation. Our film is highly homogeneous, judging from the sharp resistive transition with typical transition width of {TarTc)ITay=Q.Q\5. However, a subtle spatial variation of/c that may be present in the sample will lead to the coexistence of OP and DP in the vicinity of ODT (5p).

Let us summarize the results and interpretations presented in this paper. For the lowest voltage F(=0.1 mV), the sharp peak in the Sy/V-B curve at «5p originates from the coexisting phases in the vicinity of ODT, while the broad peak in Sy/V-B at lower 5 («4 T) is attributed to plastic flow, i.e., pinning-induced incoherent flow of vortices. For higher F(=0.5 and 2 mV), i.e., the higher flow velocity, the pinning effect becomes less effective, resulting in a decrease in plastic-flow noise, whereas Sy/V in the vicinity of ODT survives. The present results clearly show that flow noise Sy is a very useful method to probe the equilibrium phase transition (ODT) of vortex matter, as well as the dynamic transition, which cannot be detected by ordinary static transport measurements, such as, linear resistivity and /c. The results also show that, in order to detect clearly the ODT from the Sy/V-B data, we should measure Sy in the flux-flow state using moderately large / to suppress plastic-flow.

This work was partly supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology of Japan and by the CTC program under JSPS.

REFERENCES

1. Y. Paltiel, E. Zeldov, Y. Myasoedov, M.L. Rappaport, G. Jung, S. Bhattacharya, M.J. Higgins, Z.L. Xiao, E.Y. Andrei, P.L. Gammel, and D.J. Bishop, Phys. Rev. Lett. 85, 3712-3715 (2000).

2. S. Okuma, K. Kashiro, Y. Suzuki, andN. Kokubo, Ptiys. Rev. B 11, 212505 (2008): S. Okuma, Y. Suzuki, andN. Kokubo, Proceedings of 21st Int. Symposium on Superconductivity, in press.

3. S. Okuma and M. Kamada, Phys. Rev. B 70, 014509 (2004). 4. R. Eggenhoffner, E. Celasco, V. Ferrando, and M. Celasco, Appt Phys. Lett 86, 022504 (2005). 5. S. Okuma, J. Inoue, and N. Kokubo, Phys Rev. B 76, 172503 (2007). 6. C.J. Olson, C. Reichhardt, and F. Nori, Phys Rev. Lett 81, 3757-3760 (1998). 7. A.B. Kolton, D. Dominguez, and N. Gronbech-Jensen, Phys Rev. Lett 83, 3061-3064 (1999). 8. B. Rosenstein and V. Zhuravlev, Phys Rev. B 76, 014507 (2007). 9. Z.L. Xiao, O. Dogru, E.Y. Andrei, P. Shuk, and M. Greenblatt, Phys Rev. Lett 92, 227004 (2004). 10. C.J. van der Beek, S. Colson, M.V. Indenbom, and M. Konczykowski, Phys Rev. Lett 84, 4196-

4199(2000). 11. X.S. Ling, S.R. Park, B.A. McClain, S.M. Choi, D.C. Dender, and J.W. Lynn, Phys Rev. Lett 86,

712-715(2001). 12. G. Pasquini, D. Perez Daroca, C. Chiliotte, G.S. Lozano, and V. Bekeris, Phys. Rev. Lett. 100,

247003 (2008).

96

Ultra-low conductivity noise in metallic nanowires

Amrita Singh and Arindam Ghosh

Department of Physics, Indian Institute of Science, Bangalore INDIA 560012.

Abstract. By modifying the electrodeposition technique, we have stabilized the silver nanowires (AgNWs) in high-energy hexagonal closed packed (hep) structure. The conductivity noise measurements show that the noise magnitude in hep silver nanowires is several orders of magnitude smaller than that of face centered cubic (fee) silver nanowires, which is obtained by standard over potential electrodeposition (OPD) technique. The reduction of noise can be attributed to the restricted dislocation dynamics in hep AgNWs due to the presence of less number of slip systems. Temperature dependent noise measurements show that the noise magnitude in hep AgNWs is weakly temperature dependent while in fee AgNWs it is strong function of temperature.

Keywords: erystallinity, conductivity noise, dislocation PACS: 61.05.J-,63.22.Gh, 61.46.Df

INTRODUCTION

Low frequency 1/f noise is a sensitive tool to probe the quality and rehability of electrical devices. As the system size is reduced, 1/f noise increases due to the surface adsorbates as well as the atomic scale structural fluctuations, which restricts the performance of nanoscale devices. Nanowires can be used as interconnects in nanoelectronic devices [1] and also in several proposals of biological and mechanical sensors, so until a way to minimize the noise is found, their applicability is limited. There has been a lot of work on electronic properties of nanowires, but a little is known about their noise characteristics [2]. The main aim of this work is to understand the microscopic origin of noise in nanowires and to find if by tuning the growth conditions properly, the noise level in nanowires can be reduced. Being a highly conducting metal, silver nanowires are the best candidate for this purpose. Normally metals like silver and gold exhibit fee crystal structure but some results show that they acquire high-energy hep crystal structure as the dimension is reduced below about 25 nm, due to competition between surface and internal packing energies [3,4]. By modifying the electrochemical growth parameters, we have stabilized silver nanowires in hep crystal structure, even in nanowires of diameter 100 nm, while the nanowires, grown in standard OPD technique, acquire fee phase [5].

The excess noise in single crystal systems can be due to the defect diffusion along the dislocations, movement of dislocations etc, which are highly crystal structure dependent. Since by tuning the electrochemical growth parameters, we have stabihzed the silver nanowires in both fee and hep crystal structure with unprecedented control, it could be shown, how the electrical noise in these nanowires, with same geometry and size, is affected by the erystallinity and hence the dislocation kinetics.



97

(a) r _ 4 ^ ^ ^ • U I ^ ^ H A M J

FIGURE 1. (a) Schematic of electrodeposition set-up. Selected area electron diffraction (S AED) pattern of (b) LPED silver nanowires showing hep crystal structure and the inset shows the cross-sectional scanning electron microscopy (SEM) image of nanowires (c) OPD silver nanowires showing fee crystal structure and the inset shows the cross-sectional SEM image of nanowires. (d) Resistance-temperature plot of fee (blue triangle) and hep (black circle) silver nanowire.

EXPERIMENTAL RESULTS

Experimental Procedure

Nanowires were electrodeposited in self organized nanopores of polycarbonate (PC) and alumina (AAO) templates, purchased from Whatman International Ltd, with average pore size ^ 60 nm and pore length 6 /im. One side of the template was sputtered with gold film, which serves the purpose of cathode and micro-capillary of Pt or stainless steel was used as anode. Schematic of the electrodeposition set-up is shown in Fig la. AgNWs were fabricated with a modified dc electrodeposition technique, where the nanowires were grown at deposition potentials as low as 10 mV (Nernst potential of silver 780 mV, LPED) with 2M AgNOs solution. Since in this case the electrodeposition takes place at potential, much smaller than the standard Nernst potential, we name this process as low potential electrodeposition(LPED) technique. In standard OPD process, the inter-electrode potential and electrolyte concentration were 850 mV and 2mM respectively. These nanowires were characterized by high resolution transmission electron microscopy (HRTEM, FEI), scanning electron microscopy (SEM) and x-ray diffraction (XRD, Philips XTPERT Pro Diffractometer). The electrical noise was measured with ac five probe technique in an electromagnetically shielded set-up that is sensitive to extremely small voltage fluctuations with power spectral densities as low as 10~^^ V^/Hz. Noise was measured by including the sample in one arm of the Wheatstone bridge, followed by amplification of the error signal, digitization and digital signal processing, which allows simultaneous measurement of background as well as bias dependent sample noise [6]. In Fig lb. and Ic. the selected area electron diffraction (SAED) patterns

10 100 1000 Growth potential (mV)

(b) o J

*

300K 210K 130K

0.1 1 f(Hz)

0" " •

Q-,3.

0 " " •

(c) s -*

4

* 215K o 300K » 125K

'^. %

°'' f(Hz) 1

FIGURE 2. (a) Variation of yn (Hooge parameter)as a function of eletrodeposition potential for a large number of samples in form of band, variation of Normalized power spectral density of (b) fee and (c) hep silver nanowires.

show that the LPED AgNWs acquire hep crystal structure and the OPD AgNWs exhibit fee phase, respectively. In the insets the SEM micrographs of corresponding nanowires are shown. In Fig Id. resistance vs. temperature plot of both hep and fee crystalline silver nanowires show that the hep AgNWs have lower residual resistivity ratio(/?3ooK/^4.2K)-

Ultra low noise silver nanowires

In Fig 2a. the noise magnitude for number of samples, grown at different electrodepo-sition potentials, has been shown in form of bands. We observe that the noise magnitude in hep AgNWs, grown in LPED regime, is two to six orders of magnitude smaller than that of OPD AgNWs [7]. This huge suppression of noise could be attributed to the less number of slip systems in hep nanowires, which results in restricted dislocation dynamics, as less number of slip systems provide less number of ways for dislocations to propagate. Moreover, the lower RRR in hep AgNWs suggests the presence of larger disorder and these excess defects could also lock the dislocation motion, for example, by forming Cottrell atmosphere around the dislocations, which in turn can reduce the noise even further [8].

Fig 2b. and 2c. show the temperature dependent noise behavior in both type of nanowires. We observe that the noise in fee silver nanowires is a strong function of temperature, which could be originated due to the thermally activated dislocation kinetics. A striking observation is that some of the LPED samples show weak temperature

dependence of noise, which can not be explained in terms of Dutta-Hom model [9]. This weak temperature dependence of noise could be related to the high-energy crystal structure, where the disorder might take the system out of thermal equilibrium. In such a case, the dislocation motion between metastable states, separated by high energy barrier, possibly could occur under the influence of internal stress rather than temperature fluctuation resulting in weak temperature dependence of noise.

CONCLUSIONS

We have stabilized silver nanowires in pure hep crystal structure, by using modified electrodeposition technique, even for the nanowires of larger diameter ( lOOnm). We observed that the hep AgNWs exhibit ultra-low electrical noise and this suppression of noise in hep nanowires could be attributed to the restricted dislocation dynamics in basel plane due to less number of slip system as well as pinning of dislocations by point defects. We also observed the weak temperature dependent noise behavior in LPED AgNWs, which could be linked to the high energy hep crystal structure.

REFERENCES

1, G, Snider, and R, S, Williams, Nanotechnology 18, 035204 (2007), 2, A, Bid, A, Bora, and A.K, Raychaudhuri, Phys. Rev. B 72, 113415 (2005), 3, P, Taneja, R, Banerjee, P, Ayyub, and G, K, Dey, Phys. Rev. B 64, 033405 (2001), 4, X, Liu, J, Luo, and J, Jhu, Nano Lett. 6, 408 (2006), 5, A,Singh, and A, Ghosh, /, Phys. Chem. C 112, 3460 (2008), 6, M,B,Weissman, Rev Mod. Phys. 60, 537 (1988), 7, A,Singh, and A, Ghosh, /, Phys. Chem. C93, 102107 (2008), 8, D, Hull, and D, J, Bacon, Introduction to Dislocations (Pergamon, Elsevier , 1984), 9, PDutta and PM,Hom, Rev Mod Phys. 53, 497 (1981),

100

Defect Noise Spectroscopy Results for GaN Nanowires

Erdem Cicek, Jason L. Johnson, Ant Ural and Gijs Bosman

Department of Electrical and Computer Engineering, University of Florida Gainesville, FL 32611 USA

Abstract. Measurements of the spectral density of the vohage fluctuations of GaN nanowires as a function of temperature are reported. The noise data show the presence of at least three electron trap levels. The activation energies of the defects are calculated from the slope of the characteristic times of the generation-recombination noise components versus inverse temperature. The results show energy levels at approximately £,.-03, Ei.-Q.2, and f^-O.H eV respectively. These trapping levels are most likely caused by Nitrogen Vacancy, Carbon and Magnesium sites.

Keywords: GaN nanowires, 1/f Noise, generation-recombination noise. PACS: R70, 73, 73.21.-b, 73.21.Hb

INTRODUCTION

Research interest in GaN nanowires has intensified in recent years because it is realized that for sufficiently thin nanowires, quantum confinement effects can be observed possibly enabling novel applications. They are also strong candidates for power sources [1], gas sensors [2], light emitting diodes [3] and advanced transistors [4]. Although basic logic functions have been performed with these nanowires, the electrical noise produced in these structures limit their analog performance and this prevents the fault-free mass fabrication of electronics based on nanowires.

The focus of this paper is on the electrical noise produced in GaN nanowires and its' low temperature analysis. Once the noise sources have been identified, strategies can be devised to reduce the influence of the electrical noise on device performance.

EXPERIMENTAL M E T H O D S

A. Growth and Sample Preparation

Before growing the GaN nanowires, approximately 15 A of gold was deposited by e-beam evaporation onto a clean wafer of (100) Si with 100 nm of thermally grown oxide. Next, the Ga metal source (99.999%) was poured into a quartz boat and placed into a tube furnace. The growth substrate was inserted and positioned within 3 cm down stream of the Ga metal source. The growth chamber was purged with Ar for 10 minutes at room temperature to remove any residual oxygen. The substrate was heated up to 850°C and annealed for 15 minutes under Ar ambient.



101

FIGURE 1. Left: SEM image of the aligned wires transferred to a substrate by rubbing. Right: Image of a nanowire with contact pads.

The annealing is required for the successful formation of Au catalyst nano-particles on the sample surface. After the anneahng step, high purity NH^ (99.999%) and H2 (99.999%) were introduced into the growth chamber. The GaN nanowires were grown for ~5 hours at 850°C. Finally, the sample was removed from the chamber when its temperature had dropped below 100°C to prevent oxidation. Typical nanowires used in the electrical measurements are shown in Fig. 1. Wires were measured to be between 2 and 10|im in length and between 50 and 150nm in diameter. For ohmic contacts, Ti(20nm)/Al(80nm)/Pt(40nm)/Au(80nm) layers were deposited on top of the nanowires by e-beam evaporation and patterned by photolithography and lift-off to form sets of contact pads with 2, 4, 6 and 10 |j,m spacing. The contacts were annealed at 350°C for 60 s in flowing N2 ambient in a rapid thermal annealing furnace. Further details in materials characterization can be found in the references [5, 6].

B. Current-Voltage Measurements

The current-voltage characteristics were measured on wafer using an HP4145B semiconductor parameter analyzer with tungsten probes. Linear devices with a single nanowire connection between the contact pads spaced at 2 and 4 |im were investigated over voltage range of -3V to 3V. A minimum resistance of l.SkQ. was observed at room temperature.

C. Noise Measurement Set-up

An HP3561A low-frequency spectrum analyzer was used to measure the voltage spectral density in the frequency range of 10 Hz to 100 kHz. Since the noise of the device under test is below the detectable range of the spectrum analyzer, a low noise amplifier was used. The system was powered by a car battery in order to lower external noise at the front of the set-up.

102

EXPERIMENTAL RESULTS and DISCUSSION

Below lOOkHz, 1/f like low-frequency noise is observed which could be decomposed into Lorentzians as shown in Fig. 2. In order to find the noise producing defect levels in the band-gap, the noise data was examined at various temperatures. We observed that the number, the magnitude and the characteristic time of the different g-r components varied as a function of temperature.

V^ / Hz

FIGURE 2. The spectral intensity of a.c open-circuited voltage fluctuations Si^y(f) of a typical GaN wire versus frequency

l E - 1 7

l E - l B

l E - 1 9

1E-20

l E - 2 1

l E - 2 2

l E - 2 3

l E - 2 4

sav/Vg ^

/

i! '. 3 4

V \

5 6 7 E

D e v i c e A

Device B

\ \

lOOO/T '

9 10 1 1

FIGURE 3. The low frequency g-r noise plateau values 5^(0) divided by VQ of the different g-r noise components versus 1000/T. The triangles represent the measured values of device A and circles device B whereas solid and dashed lines are meant to guide the eye.

In Fig. 3 the values of Sj,(0)/7o^ of the various generation-recombination (g-r) noise components are presented as a function of 1000/T for I00<T<300 K for two GaN devices. 5j,(0) is the plateau value of a g-r noise component and VQ is the voltage across the device terminals at that particular temperature. Three maxima at T= 300, 200, 100 K are observed which can be attributed to three distinct trap levels.

103

The positions of the energy levels are found from the slope of the characteristic times of the different g-r noise components versus 1000/T as depicted in Fig. 4. Trap level energies at ~ E^-OJ, E^-O.!, E^-O.H eV are associated with the maxima at T=300, 200 and 100 K respectively.

1 OOE-01 •

1.00E-O2 -

1.00E-0 3 -

1.00E-O4 -

T i m e C o n s t a n t ( s )

if* A

// /

>-.

/ /

/ . /

^

—

^ 1 — -

Device A

Device B

IDOO/T

FIGURE 4. The characteristic times of the different g-r components plotted versus 1000/T. The triangles are data points collected from device A. The circles are data points collected from device B (300-220 Kelvin) whereas the dashed and solid lines are meant to guide the eye.

CONCLUSION

In conclusion we note that the g-r noise measurement technique is a powerful tool for the spectroscopic study of deep level impurities in GaN wires.

A C K N O W L E D G E M E N T S

Funding for this project was provided by the University Scholars Program of the University of Florida.

REFERENCES

1. Z. L. Wang, Compound Semicond. 13, 16 (2007). 2. L. M. Li, C. C. Li, J. Zhang, Z. F. Du, B. S. Zou, H. C. Yu, Y. G. Wang, and T. H. Wang,

Nanotechnology 18, 225504 (2007). 3. H.-Y. Cha, H. Wu, M. Chandrashekhar, Y. C. Choi, S. Chae, G. Koley, and M. G. Spencer,

Nanotechnology 17, 1264 (2006). 4. C. Y. Nam, D. Tham, and J. E. Fischer, Nano Lett. 5, 2029 (2005). 5. J. L. Johnson, Y. Choi, and A. Ural, Journal of Vacuum Science and Technology B 26, 1841

(2008). 6. J. L. Johnson, Y. Choi, A. Ural, W. Lim, J. S. Wright, B. P. Gila, F. Ren, and S. J. Pearton, Journal

of Electronic Materials, 38,490 (2009).

104

Low Frequency Noise Sources in Ge Resistances elaborated on GeOI Wafers

J. Gyani', S. Soliveres', F. Martinez", M. Valenza", C. Le Royer'' and E. Augendre''

'lES - UNIVERSITE MONTPELLIER II - UMR CNRS 5214 Place E. Bataillon, 34095 Montpellier Cedex 5, France

CEA-LETI Minatec- 17, rue des Martyrs, 38054 Grenoble Cedex 9, France

[email protected]

This contribution presents a low frequency noise characterization of germanium (Ge) semiconducting bars, directly relating to the quality of the Ge film in GeOI transistors. We propose an experimental method to dissociate the intrinsic noise of the semi-conducting bar from the access region noise by using 4 point probe (Kelvin) structures. An accurate value of Hooge's parameter for the studied Ge technology is extracted. We obtain a of 2.6 10 which is an indication of the good quality of the studied Ge semiconductor material.

Keywords: low frequency noise, noise, germanium, access resistance. PACS: 73.50.Td, 74.40.4-k

INTRODUCTION

Due to its better transport properties compared to Sihcon (Si), Germanium (Ge) is a promising material in order to achieve sub-32nm node high performance MOSFET channels. In some apphcations, low frequency noise (LFN) can appear as a limiting factor in achieving high performance devices. It is well known that Hooge's parameter (a), extracted from LFN measurements, is used as a figure of merit for a given technology to determine the quality of a semiconductor. As the structures of MOSFETs are complex, it is easier to extract the intrinsic a value of the semiconductor by measuring the noise in a simple homogeneous layer semiconducting bar. However, if the magnitude of the excess noise generated by the access regions is not negligible compared to the LFN of the main bar, the parameter a cannot be accurately determined. Therefore, in order to accurately characterize the noise behaviour of the Ge bar, it is necessary to dissociate the intrinsic noise and access path noise.

In this contribution, we propose an experimental method to dissociate the intrinsic noise of a semi-conducting bar and the excess noise of the access regions using 4 point probe (Kelvin) structures.



105


EXPERIMENT

The studied resistances are elaborated on 200inm GeOl wafers with a 60-80 nm thick Ge active layer obtained using SmartCut^'^ technology, upon which Ge mesa structures were patterned. Samples with different widths were measured. Prior to noise analysis, the electrical resistances of the main semi-conducting bar and the access regions were extracted from dc measurements. Current LFN measurements between Ci and C2, as well as output voltage LFN measurements between contacts Ci and C2, as well as between Ci and K2 were performed. The device structure and the experimental setup are shown in figure l.a and l.b, respectively. The studied structures have a length L=240 |am and a width W between 0.25 |am and 10 |am.

Main semi-conclucting bar

Acces region

Ci

K,

Kelvin plug -:;

r

^

C2

K2

\A<P Voltage amplifier

FIGURE 1. Diagram of the studied samples with Kelvin structures (left), experimental biasing circint for noise measurements (right).


From DC characterization we have extracted the main semi-conducting bar resistances and the access resistances for each structure. For both resistances we obtain a linear evolution versus the inverse of the width:

R. 2,55.10"

Q. and R„ 0.076

+ 18Q. w "" w

Figure 2.a shows the variation of the normalized current power spectral density (PSD) measured at 1 Hz across Ci and C2 (main region in series with the access regions), as a function of current for various device geometries. The noise behaviour exhibited in each device is in accordance with the empirical relation proposed by Hooge [1,2]. However, the variation of the normalized current PSDs@lHz as a function of device width varies as shown in fig 2.b (a -1 slope followed by a plateau), which, at first view, is not predicted by Hooge's relation.

106

• I •

•

W=5^m

W=20^m W=50^m W=0 5^m W=0 25^m

. ^ •

e-12

1e-13

• * \,

\ *

• e^p.rir..n1.l

(Plateau)

* • ^

FIGURE 2. Normalized current PSD@lHz a), versus current for various device geometries and, b) versus width W of the devices.

The total current PSD across Ci and C2 can be expressed as follows,

Rr

v R c + R a c c /

R.

v R c + R a c c /

where S^ is the PSD of the main region and Siacc is the PSD of access regions. These

can be expressed by the relationships S, a - ^ I ' and S, fP ° '"

-• Kin. where P=P(W) and

is the total number of holes in the semi-conducting bar taking part in the current, and

K is some factor. For widths W below 1 10^ im we have—^

R^ and S,^ dominate the expression for S,^^

S,

J_ w

, meaning that

For widths W above 1 10 im we

have-r

W , meaning that R^^^ and S, dominate the expression for S,

The slope (-1) is in accordance with empirical Hooge expression, whereas the plateau indicates the contribution of excess noise from access path noise sources.

In order to separate the noise contribution of the main channel from that of the access path, we determined the analytical noise expressions derived from the voltage noise equivalent circuits in the two measurement configurations (across Ci and C2, and across Ci and K2). From these two expressions as well as voltage noise measurements in the two configurations, it is possible to extract the values of the main bar noise and the access region. All PSDs exhibited pure 1/f noise with y=l.

The extracted normalized main bar and access path PSDs@lHz as a function of device width W are reported figures 5 and 6, respectively. It can be seen that the normahzed main bar noise S varies as W" whereas the access path normahzed

noise S,^ is constant. This result shows that using this simple method, it is possible to

extract low levels of intrinsic noise that would normally be hidden by excess noise. To validate the method of extraction, the extracted voltage PSD@lHz values of

the main bar and the access regions are reinserted into the derived current noise measurement equivalent circuit expression. These values were then compared to the direct measurements of current noise; values showed an excellent agreement.

107

An accurate value of the Hooge's parameter for the studied Ge technology is then extracted. We obtain a of 2.6 10"'*, which is an indication of the good quality of the Ge semiconductor. Note that the range of the observed values in bulk silicon resistors is between 2 10" and 4 10"'* [3]. To the author's knowledge, no a values for GeOl structures have been published in the hterature.

FIGURE 3. Separation of different noise sources : (a) Normalized main bar current PSD@lHz versus current, for various devices (b) normalized access path current spectral density versus current, for devices .

ACKNOWLEDGMENTS

The authors thank French OSEO organization for financial support.

REFERENCES

1. F. N. Hooge, Phys. Lett. 29A, 139 (1969) 2. F. N. Hooge, Physica, B 83,14 (1976) 3. L. K. J. Vandamme and F. N. Hooge, IEEE electron devices, vol. 55, n° l l , 3070 (2008)

108

Noise maximum at trap-filling transition in polyacenes

M. Tizzoni*, A. Carbone*, C. Pennetta^ and L. Reggiani^

*Dipartimento di Fisica, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino, Italy

^Dipartimento di Ingegneria deWInnovazione and CNISM, Universitd del Salento, Lecce, Italy

Abstract. We consider a trapping - detrapping noise model to explain the recently observed maximum in the spectral density of current fluctuations in organic semiconductors (tetracene, pentacene), under space-charge-limited-current conditions. The ratio u = ntjNt of filled to total traps is obtained from the current-voltage characteristics and is used to evaluate the current noise spectral density at the trap-filling transition.

Keywords: Noise model, organic semiconductors

INTRODUCTION

Current noise experiments in organic semiconductors (polyacenes) have evidenced a significant increase of the spectral density at voltages corresponding to the trap filling transition (TFT) between Ohmic and Space Charge Limited Current (SCLC) regimes [1, 2]. These results have been interpreted in terms of a continuous percolation model between the two regimes, considered as different electronic phases. The noise increase was attributed to the transition region responsible of a clustering of insulating regions among which current paths are constrained, thus leading to a substantial increase of the noise in analogy with the increasing of fluctuations near a structural phase transition.

Here, we present an alternative interpretation based on the presence of trapping -detrapping processes of injected carriers through the trapping centers in the TFT regime, which extends from the voltage V = Vtio the voltage V ^2Vt, which correspond to the extremes of the range where the quasi-Fermi level crosses the deep trap level. As a significant validation of the model we consider the simplest case of experiments: the tetracene.

In a seminal paper of 1978 [3], Kleinpenning investigated the sharp cross-over between Ohmic and SCLC transport regimes of 1 / / noise associated with mobihty fluctuations and in the absence of any trap filling transition (TFT). In this case, the excess noise was found to decrease proportionally to the apphed voltage in the SCLC region. Here we discuss the effect of TFT which is found to be responsible of a sharp increase of excess noise with a maximum corresponding to the highest value of the current voltage (I-V) slope.



109


Current voltage characteristics. To evaluate the trapping - detrapping noise from the experimental data, we start by considering the 1-V characteristic measured in [1] for the tetracene (sample b) and reported as symbols in Fig. 1. The sample consists of a thin film sandwiched between Au and Al electrodes with an area A = 0.1 en? and a thickness L = 0.65 jUOT. Charge transport is assumed to be driven by holes already existing or injected from the Au contact. At increasing voltage, three regimes are identified as: Ohmic (linear 1-V), TFT (strong superlinear 1-V), SCLC (quadratic 1-V). In the Ohmic regime the 1-V characteristic is described by Ohm's law

eiiUnA In = -^^y (1)

with e the unit charge, jU the carrier mobihty, and MQ the carrier concentration attributed to a full ionized shallow level. The SCLC 1-V characteristics is given by the Mott -Gumey law

W ^ ^ ^ ^ V ^ , (2)

where e and eo are, respectively, the relative dielectric constant of the material and the vacuum permittivity and 0 = n/rit the ratio between the concentrations of the total free carrier and the total deep traps. A numerical fitting of the experimental 1-V characteristics is obtained by considering the current as the sum of the Ohmic component with that of the SCLC regime, this last weighted by the fraction of filled trap u{V) which is function of the applied voltage and limited within the values 0 < u{V) < 1. Accordingly, it is:

I = In + u{V)IscLC • (3)

where u = u{V) is obtained by best fitting the experimental curve. The results of the fit is reported in Fig. 1. For the relevant parameters of the 1-V characteristics, we find: L/AenojX = 3 • 10"Q and QAeoe iUO/SL^ = 6 • lO^ii A/V^. From the fit of the SCLC region, we obtain Qji = 4.7 • 10^^° crn^/sV, and from the value of the Ohmic resistance we obtain WojU = 1.4 • 10^ {mVsy^. In the absence of an independent determination of mobility (or of 0), we discuss the possible range of values for the relevant parameters of interest. Accordingly, by assuming for jU = 1 crn^/Vs, the value of the single crystal, we find 0 = 10^^° and MQ = lO' cm^^. By using the theoretical expression for 0

0 = — e x p [ - £ , / f e r ] (4) gnt

where ^ = 2 is the trap degeneracy factor, riv the density of states in the valence band, Et the energy of the trap level, kT the thermal energy, for riy = 10^^ cm^^, fit = 10^* cm^^ we find Et = 0.7 — 0.8 eV. All these values compare well with the range of values reported in the hterature and summarized Table 1.

Noise. The experimental noise measured at 20 Hz is obtained by subtracting to the total relative noise the 1/f contribution of 1.8 ps at 20 Hz, which is clearly evidenced to be voltage independent in the Ohmic regime. The data so obtained are reported in Fig.

110

10^

10^

10^

10°

„-1

Ohmic regime Trap-fillinc transition"

^^r

SCLC regime

-

Exp +

Fit

V, 1 2V,

V[V]

FIGURE 1. Current voltage characteristics at room temperature for a tetracene thin film of length L= 0.65 [im and cross-sectional area A= 0.1 cm^. The characteristic evidences an Ohmic region at the lowest voltage, a TFT region between V = Vt ~ 0.8V and V a; 2Vt and a SCLC region at the highest voltages. Symbols refer to experiments, curve to the theory.

TABLE 1. Values of the physical parameters of tetracene, taken from Uterature [1,4] .

Physical parameters of tetracene

Density of traps n, 10 — 10^^ cm^^ Total density of states in the valence band riy 10^' ctrT^ Energy trap level from the valence band Et 0.3 —0.7 eV Carriers'mobiUty (for single crystals [4]) jl 0.1 — 1 ctrP'/sV Relative dielectric constant £r 3.5 — 5

2. In the SCLC regime the observed noise decreases approximately as l/V, according to a noise suppression mechanism analogous to that observed in vacuum tubes. For the theoretical interpretation of experimental data in the TFT regime we use the standard formula for the relative trapping-detrapping noise with a lorentzian spectrum:

% ( / ) N? Nt I+ {27tfzy

3(1- Bu{l (5)

where Nt is total number of traps inside the volume of the device, T the lifetime of the carrier, MQ = 1 — M the fraction of ionized carriers, / the sweeping frequency. Using the u{V) obtained from the I-V characteristics, the best fit between theory and experiment is reported in Fig. 2.

Since the Eq.(5) can only describe a noise maximum which exceeds the equilibrium noise (thermal noise) by a factor 0.25, the constant B has been rescaled to reproduce the total excess noise in the TFT regime. This is obtained for B = Smax/0.25 = 2.8 x 10^^° s.

111

V[V]

FIGURE 2. Log-log plot of the relative trapping - detrapping noise spectral density at 20 Hz vs applied voltage for the tetracene sample in Fig. 1. Data are obtained by subtracting the relative resistance 1/f noise contribution at low voltages to the total excess noise data of Ref [1], Symbols refer to experiments, curve to theory.

where Smax is the maximum experimental value of the noise.

CONCLUSIONS

By introducing a trapping - detrapping noise contribution associated with the TFT regime, we have developed a microscopic interpretation of relative current noise in tetracene. Points of further investigations remain:

1. The high level of noise enhancement in the TFT region, compared to the thermal noise, which is due to the high correlation degree of the trapping processes and cannot be accounted for in a quasi-equilibrium model. This would require the introduction of a noise gain mechanism in the Eq.(5).

2. The explanation of the close 1 / / dependence of the excess current noise in the TFT region.

3. The possibility to interpret the data with multiple trapping centers on similar grounds.

REFERENCES

1. A. Carbone, B. K. Kotowska and D. Kotowski, Phys. Rev Lett. 95, 236601 (2005) 2. A. Carbone, B. K. Kotowska and D. Kotowski, Eur. Phys. J. B 50, 77 - 81 (2006) 3. T. G. M. Kleinpenning, Physica B,C 94, 141,(1978) 4. R . W. I. de Boer et al.J. Appl. Phys. 95, 1196, (2004)

112

Time dependent thermal properties of disordered solids

Moyuru Ochiai

Faculty of Science and Engineering, Waseda University, Okubo3-4-l,Shinjuku-ku,Tokyo 169-8555, Japan

Abstract. In experiments, the entropy of glass is measured by heat flow, and this leads the fact that the entropy determined by cooling and heating shows respectively lower and upper bounds of the entropy defined by statistical thermodynamics. This report presents a new theoretical approach by stochastic theory to the above phenomenon obtained by experiments of glass and makes clear the relation between calorimetric entropy and statistical mechanical one caused by the non-equilibrium process of a glass state with heat exchange. The method shown here can be applied not only so-called glass but also disordered solids. In this theory, a master equation used in non-equilibrium statistical mechanics is basic. Furthermore, a canonical distribution of fluctuations extended to the time-dependent case and detailed balance equation are the key of our theory.

Keywords: glass, entropy, calorimetry, time-dependent canonical form, master equation, detailed balance PACS:61.43.-j

1. INTRODUCTION

The glass transition is a universal phenomenon. Then the work to be presented here has been originated in an effort to understand a lot of studies devoted to making clear the non-equilibrium or irreversible process of a glass state. Glasses are out of equilibrium. A glass state is thus a kind of a relaxation process, that is, an irreversible one which varies very slowly because molecules of glass are apparently kept frozen under glass transition temperature T g . In a glass state, the entropy derived from the theory of statistical mechanics, becomes different from the measured entropies, is one problem. Another is that glass has residual entropy, that is non zero zero-temperature entropy. Nemst's theorem does not work in this case.

Thermodynamics explains the result as follows'-'-'. In order to define the temperature of the system in non-equilibrium, we consider an isolated system composed of a glass and a heat reservoir, of which total entropy increases with time. This model described by thermodynamics gives the relation;

S.(Ta ) ^ S . „ . ( T a ) ^ S , (Ta ). (1) Statistical mechanical approach to this relation will be shown here.



113

2. TIME DEPENDENT CANONICAL DISTRIBUTION

A glass is out of equilibrium and a kind of very slowly relaxation process which tends to an equilibrium state. In order to determine the temperature, we introduce an isolated system which comprises a glass and a heat reservoir. Equilibrium statistical mechanics shows that the probability distribution of the system contact with a heat bath has a canonical form.

In case of a glass state, we assume that a canonical distribution is extended to the following time dependent form. At time t, this would be of the functional form of

\F(t)-E^

kT{t) In spite that this has not been proved, the above mentioned property (2) is natural

as a general asymptotic law of a macroscopic state variable'-^-'.

^ ; ( 0 = exp (2)

3. MASTER EQUATION AND DETAILED BALANCE

A master equation is one of the most suitable starting points of any theory of time-dependent stochastic processes'-^-'. A master equation

dt where stands for a time-dependent transition rate from a state n to m, and is basic.

The detailed balance condition is introduced as follows;

2KnP:(o-w„mP:w]=o- w Thus, the master equation (3) and Eq.(4) give the following expression

2^^n(p:(t)ip:{t))pjt)=2^.n(p:(t)ip:{t))pst). (5)

4. STATISTICAL ENTROPY

Eq.(3) can be written by the expression of the time-dependence of the statistical entropy defined by

S{t) = -k2PJt)lnPJt). (6)

Eq.(3) takes the form

^ = -k2W.nPM^P.(')-^^Pn(t)l (7) m,n

From Eq.(5), Eq.(7) comes to

S = A + B (8) where we put

114

A = -k^PlnP^{t), (9) and B = -k2w„„P:(t)y X X 1 + In—

Using the functional form of Eq.(2), we get 1

A: T

^iiPAt)EJ-^pJ''^ at ^-^ at

m

It may be worthwhile for us to remind the relationship between a generalized drive forth X and its conjugate displacement x appears in thermodynamics. Then Eq.(l I) is reduced as follows:

, 1

(10)

(11)

"^E + ^X/x^ Idt. (12)

Using the energy conservation law of thermodynamics for a closed system, dE=6Q + 6W = 6Q-^X/x^ ,

p

and making a notice that X signifies the drive forth which causes a displacement.

we thus obtain

A = -6Qldt^-Q.

Since the term B of the expression (10) is positive definite; B>Q,

Eq.(8) satisfies the condition

5 > A

(13)

(14)

(15)

5. APPLICATION TO GLASS AND RESULTS

The time evolution of the statistical entropy is presented by (7) and from the expressions (9) and (15), we have the significant results:

'S>'QIT. (16)

As an example, we now consider the entropy of a glass. 5^(7]) is defined by cooling the liquid from the temperature T^ through the glass transition point to the some lower glass state temperature 7]. The measured entropy S^ (T,) is given by

SXTd = S,{TJ+ ^QITdt.

On the other hand, the given expression (16) leads

SST^-S,{TJ>^QITdt

(17)

(18)

Comparing Eq.(I7) and the relation (I8),we obtain 5(r,)>5,(r,). (19)

In case of heating, the story is same as the above mentioned process. So that, defining 5^(7,) as the measured entropy in case of heating, we have

S{T,)<S,{T,). (20)

115

The inequalities (19) and (20) give the relation S^iT,)<SiT,)<S,iT,) (21)

This is the relation we would like to verify in this report.

REFERENCES

1. S.A.Langer, J.P.Sethanaand E.R.Grannan, Phys.Rev.B41,2261(1990) 2. R.Kubo, in Synergetics (Proc. Symp. Synergetics, 1972, Schloss Elmou);

N.G.van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981)

3. M.Ochiai, A.Holz,Y.Yamazaki and R.Ozao, 11 Nuovo Ciment 108B, 709 (1989) and references cited here ; C.W.Gardiner, Handbook of Stochastic Methods (Springer, 2nd ed. 1985)

116

Broadband Noise of Driven Vortices at the Mode-Locking Resonance

S. Okuma", J. Inoue", and N. Kokubo''

^Research Center for Low Temperature Physics, Tokyo Institute of Technology, 2-12-1, Ohokayama, Meguro-ku, Tokyo 152-8551, Japan

^ Center for Research and Advancement in Higher Education, Kyushu University, 4-2-1, Ropponmatsu, Chuoh-ku, Fukuoka, Fukuoka 810-0044, Japan

Abstract. Vortex-flow noise across the mode-locking (ML) resonance has been studied in an amorphous Mo ^Gei. ^ film. We have found that broadband noise (BBN) is remarkably suppressed, when the driven vortex system undergoes ML (dynamic ordering), where current-voltage characteristics show the step-like behavior. The result is consistent with the view of ML freezing that the mode-locked state is a frozen solid pinned in the moving frame of reference. By changing the amplitude of the ac drive, we find the correlation between the sharpness of the ML resonance and suppression of BBN at the ML step.

Keywords: Flux-fiow noise; Dynamic ordering; Phase locking; Amorphous films PACS: 74.40.+k, 74.25.Qt, 74.78.Db

INTRODUCTION

When an object moves in a periodic potential in the presence of combined dc and ac forces, step-like structure analogous to Shapiro steps appears in the force-velocity (F-v) characteristics. This phenomenon called a mode-locking (ML) resonance has been observed in several physical systems with many degrees of freedom [1-12], which include a driven vortex system in type-II superconductors. The steps in the F-v curves appear, when the internal frequency of the system locks to the external frequency xt of the ac drive. The ML resonance for driven vortices has been observed not only in superconductors with periodic pinning [7] but also in those with random pinning, such as amorphous films studied in this paper, where a periodicity can be induced dynamically as a result of the coherent motion of a vortex lattice [8-12].

In our recent work using amorphous {a-)MoxGQ\-x films we have found that broadband noise (BBN) is remarkably suppressed, when a driven vortex system undergoes the ML resonance (i.e., dynamic ordering) [11]. This is consistent with the view of the ML freezing that the mode-locked state is a frozen solid pinned in the moving frame of reference [12]. It is known that the shape of the ML steps in the current-voltage (/-F) curves depends crucially on the ac drive, /rf, as well as the field. In fact, the current width of the ML steps Al, which reflects the sharpness of the ML resonance, exhibits an oscillatory behavior with /rf. These results lead to an interesting question as to whether the sharpness of the ML resonance and that of ML freezing

CPl 129, Noise and Fluctuations, 20^ International Conference (ICNF 2009) edited by M. Macucci and G. Basso

© 2009 American Institute of Physics 978-0-7354-0665-0/09/$25.00

117

(suppression of BBN) correlate to each other. It is reasonable to expect that in a situation where a larger number of vortices take part in ML, both the ML resonance and ML freezing are more pronounced. As far as we know, however, no experimental verification has been reported so far. Here, we perform simultaneous measurements of /-F characteristics and voltage noise spectra Syif) across the (first) ML resonance for the a-MoxGQux film as a function of the ac drive. We find the correlation between the current width AI of the ML step and suppression of BBN (Sy) at ML.

EXPERIMENTAL

We prepared the 330 nm-thick a-Mo^GQi-x film by rf sputtering on a Si substrate held at room temperature [10,11]. The superconducting-transition (zero-resistivity) temperature is 5.78 K and the upper critical field at 4 K defined by the 95% criterion of the normal-state resistivity is 5.2 T [13]. We measured the linear resistivity, I-V characteristics, and the voltage noise spectrum Sy(f) induced by the currents using a four-terminal method. The ac (rf) current /rf was applied through an rf transformer. The frequency y xt of/rf was 10 MHz. In measuring Sy(f) over a broad frequency range J=l Hz-40 kHz, the voltage enhanced with a preamplifier was analyzed with a fast-Fourier transform spectrum analyzer. We obtained the excess noise spectra Sy(f) by subtracting the background contribution, which was measured with /=/rf =0 [14,15]. The sample was directly immersed into liquid " He. The field B was applied perpendicular to the plane of the film.


In Figs. 1(a)-1(e) (top panels), we show the / dependence of the differential resistance dV/dl measured at 4.0 K in 3.8 T in the absence of/rf [1(a)] and in the presence of (10 MHz) /rf of different amplitudes; /rf=0.129 [1(b)], 0.230 [1(c)], 0.367 [1(d)], and 0.458 mA [1(e)]. For /rf=0, the dV/dl vs / curve is smooth and no anomaly is visible, while for nonzero /rf, the dip structure indicative of the ML resonance, resulting from the ML "step" in the I-V characteristics, is clearly seen. From the dip position (/ML) indicated with a vertical dashed line in each figure, we obtain a ML-step voltage F(/ML), which is independent of the amplitude of /rf. Thus, we immediately find that the driven vortex matter is a triangular vortex array moving in the direction parallel to one side of the triangle(s), i.e., the lattice period in the direction of vortex motion is equal to the nearest neighbor distance of the triangular vortex array, (20Ql4?>Bf^, where O^ is the flux quantum [11].

The shape of the ML steps in the I-V characteristics depends crucially on the amplitude of the ac drive /rf, as well as the field B. The sharpness of the ML resonance is reflected by that of the step structure in the / -F curves. We thus use the current width Al of the ML step to quantify the sharpness of the ML resonance, where Al is obtained by integrating the peak of the dl/dV vs V curves (not shown here) with respect to the flux-flow baseline [10]. As seen in Fig. 1(f), Al exhibits an oscillatory behavior with /rf. A detailed analysis shows that Al (/rf) is well reproduced by a squared Bessel function of the first kind with /rf [16], consistent with earlier work [9].

118

T=4K B = 3.ST /e^=10MHz

0.2 0.4 0.6

/rf(mA)

FIGURE 1. / dependences of dV/dl (top) and Sy(f)/V for J^40 kHz (bottom) at 4.0 K in 3.8 T (a) in the absence of/^ and in the presence of superimposed (10 MHz) (b) 1^^0.129, (c) 0.230, (d) 0.367, and (e) 0.458 mA. A vertical dotted line in each figure (b-e) represents the location of the first ML step (/ML) and that in (a) indicates the location of/ML expected when /rf>0. The background level is indicated by shading, (f) /^ dependence of the current width AI of the first ML step. A line is guide for the eye.

(g) Noise spectra Sy(f)/VSit 4.0 K in 3.8 T in the presence of (10 MHz) /i.f-0.230 mA and / (listed in the figure) that covers the first ML region (/ML=0.14 mA). The background level is indicated by shading.

In Fig. 1(g) we display the voltage noise spectra SY(f)/V (Sy divided by V) at 4.0 K and 3.8 T in the presence of ac /rf =0.230 mA and dc / covering the first ML region (7=0.093-0.186 mA). We representatively show the data measured at /rf =0.230 mA that gives the maximum value of AI (/rf) [see Fig. 1(f)]. With increasing / from 0.093 to 0.143 mA, which corresponds to the first ML /ML(=0.14 mA), Sy(f)/V Sit high/(>10 kHz) decreases from ^ ( l - 2 ) x 10" ^ V/Hz down to near the background level («10"^^ V/Hz), as indicated with shading. With further increasing /, Sy(f)IV then shows an increase. In order to see the change in BBN across ML in more detail, we plot Sy(f)/V at y^40 kHz against / in Figs. 1(b)-1(e) (bottom panels). In Fig. 1(a) we show 5'v(40kHz)/F measured with /rf =0 to compare with the data taken for /rfÔ. A vertical dashed line in Fig. 1(a) indicates the location of ML expected when /rf^ 0. Of course, no anomaly is visible at '7ML". By contrast, for nonzero /rf, one can clearly see suppression of 5'v(40kHz)/Fat ML, indicative of ML freezing [12].

Let us focus on the degree of completeness of ML freezing. For this purpose, we compare 5'v(40kHz)/F at the (first) mode-locked state, 5'V,ML/^=5'V(/ML)/^, with that (SyûMiJV) at the mode-unlocked state. Here, the value of 5'v,uML/f is extracted from the smooth interpolation of the data of 5'v(/)/F outside the ML (dip) region to /^• /ML,

such as shown in the inset of Fig. 1(b) in Ref. 11. We find a trend for SyûMiJV to

119

increase monotonically with increasing /rf. However, the minimum value ofSy^uiJVat ML shows a nonmonotonic /rf dependence. We quantify the degree of ML freezing [suppression of 5'v(40kHz)/F at ML] by the ratio of r = 5'V,ML/5'V,UML. The smaller r means more pronounced ML freezing.

It is seen from Figs. 1(b)-1(e) (bottom) that ML freezing (suppression of BBN) occurs most remarkably at the first peak of Al (/rf) for /rf =0.230 mA, where r takes the smallest value of 5V,ML/^V,UML=1.2X 10"^ /(8x 10"^^0.15. At larger /rf =0.367 mA, which gives the local minimum of Al (/rf) between the first and second peaks, r takes the largest value around 0.6 and hence ML freezing is much less pronounced. As /rf is increased further up to 0.458 mA, where the second peak of A/(/rf) occurs, r again becomes small and a tendency toward ML freezing recovers. These results suggest that there is the correlation between Al at the ML steps, which represents the sharpness of the ML resonance, and suppression of BBN, which reflects completeness of ML freezing. To prove this fact more convincingly, we are now conducting experiments including the higher ML-step regions.

ACKNOWLEDGMENTS

This work was partly supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology of Japan and by the CTC program under JSPS.

REFERENCES

1. S. Bhattacharya, J.P. Stokes, M.J. Higgins, and R.A. Klemm, Phys Rev. Lett. 59, 1849 (1987): M.J. Higgins, A.A. Middleton, and S. Bhattacharya, ibid. 70, 3784 (1993).

2. M.S. Sherwin and A. Zettl, Phys. Rev. B 32, 5536 (1985). 3. S.N. Coppersmith and P.B. Littlewood, Phys Rev. Lett. 57, 1927 (1986): A.A. Middleton, O. Biham,

P.B. Littlewood, and P. Sibani, ibid. 68, 1586 (1992). 4. H. Matsukawa and H. Takayama, J. Phys. Soc. Jpn. 56, 1507 (1987): H. Matsukawa, Syn. Met. 29,

343 (1989). 5. E. Barthel, G. Kriza, G. Quirion, P. Wzietek, D. Jerome, J.B. Christensen, M. Jorgensen, and

K. Bechgaard, Phys Rev. Lett. 71, 2825 (1993). 6. A. T. Fiory, Phys. Rev. Lett. 11, 501 (1971). 7. L.Van Look, E. Rosseel, M.J. Van Bael, K. Temst, V.V. Moshchalkov, and Y. Bruynseraede, Phys.

T ev. 5 60, R6998( 1999). 8. C. Reichhardt, R.T. Scalettar, G. Zimanyi, and N. Gronbech-Jensen, Phys. Rev. B 61, Rl 1914 (2000). 9. N. Kokubo, R. Besseling, and P.H. Kes, Phys. Rev. B 69, 064504 (2004):

N. Kokubo, K. Kadowaki, and K. Takita, Phys. Rev. Lett. 95, 177005 (2005). 10. N. Kokubo, T. Asada, K. Kadowaki, K. Takita, T.G. Sorop, and P.H. Kes, Phys. Rev. B 75, 184512

(2007). U . S . Okuma, J. Inoue, and N. Kokubo, Phys. Rev. B 76, 172503 (2007). 12. A.B. Kolton, D. Dominguez, and N.Gronbech-Jensen, Phys. Rev. Lett. 86, 4112 (2001). 13. S. Okuma, Y. Imamoto, and M. Morita, Phys. Rev. Lett. 86, 3136 (2001). 14. S. Okuma and M. Kamada, Phys. Rev. B 70, 014509 (2004). 15. S. Okuma, K. Kashiro, Y. Suzuki, and N. Kokubo, Phys. Rev. B 11, 212505 (2008). 16. A. Schmid and W. Hauger, J. Low Temp. Phys. 11, 667 (1973).

120

Monte Carlo Study of Diffusion Noise Reduction in GaAs Operating under Periodic Conditions

D. Persano Adorno*, N. Pizzolato^ and B. Spagnolo^

*Dipartimento di Fisica e Tecnologie Relative and CNISM, ^ Dipartimento di Fisica e Tecnologie Relative and CNISM, Group of Interdisciplinary Physics,

Viale delle Scienze, Ed. 18,Palermo, Italy

Abstract. The effects of an external correlated source of noise on the intrinsic carrier noise in a low-doped GaAs bulk, operating under periodic conditions, are investigated. Numerical residts confirm that the dynamical response of electrons driven by a high-frequency periodic electric field receives a benefit by the constructive interplay between the fluctuating field and the intrinsic noise of the system. In particidar, in this contribute we show a nonmonotonic behavior of the integrated spectral density, which value critically depends on the correlation time of the external noise source.

Keywords: Monte Carlo, Noise, Semiconductors,Transport properties theory PACS: 72.70.4-m, 72.30.4-q, 05.40.Ca

INTRODUCTION

Important studies about the constructive aspects of noise and fluctuations in different non-linear systems have shown that the addition of external random perturbations to systems with intrinsic noise may affect the dynamics of the system in a counterintuitive way, resulting in a possible reduction of the total noise of the system [1]. A reduction of the diffusion noise in semiconductors driven by a static electric field is theoretically expected when a gaussian noise is added to the external field [2]. A detailed study of the electron transport dynamical response in semiconductor materials, working under cyclostationary conditions, has revealed that the addition of a fluctuating component to the driving high-frequency periodic electric field can reduce the total noise power [3,4]. This result is explained in terms of the noise enhanced stability phenomenon (NES), arising from the fact that the transport dynamics of electrons in the semiconductor receives a benefit by the constructive interplay between the fluctuating electric field and the intrinsic noise of the system [4]. Furthermore, a nonlinear behavior of the integrated spectral density (ISD) with the noise intensity has been found [4].

In the present work we analyze the modification of the intrinsic carrier diffusion noise induced in a n-type low-doped GaAs semiconductor by an external source of random perturbations added to a driving alternate electric field, as a function of the noise correlation time. A Monte Carlo procedure is used to numerically solve the transport equation by keeping into account aU the possible scattering phenomena of the hot electrons in the medium. Numerical results show that, strictly depending on the correlation time, the presence of the external noise modifies the electron average velocity and significantly affects both the correlation function of its fluctuations and the internal noise spectrum of the system.



121

MODEL AND NOISE CALCULATIONS

The electron dynamics in the GaAs bulk embedded in a periodic electric field is simulated by a Monte Carlo algorithm which follows the standard procedure described in Ref.[5]. The conduction bands of GaAs are represented by the T valley, the four equivalent L-valleys and the three equivalent X-valleys. The parameters of the band structure and scattering mechanisms are also those of Ref.[5]. Our computations include the effects of the intravalley and intervalley scattering of the electrons, in multiple energy valleys, and of the nonparabolicity of the band structure. Electron scatterings due to ionized impurities, acoustic and polar optical phonons in each valley as well as all intervalley transitions between the equivalent and non-equivalent valleys are accounted for. We assume field-independent scattering probabihties. Accordingly, the influence of the external fields is only indirect through the field-modified electron velocities. All simulations are calculated with a free electrons concentration n = 10^^ cm^. To neglect the thermal noise contribution and to highhght the partition noise effects we have chosen a lattice temperature of 80 K. We assume that all donors are ionized and that the free electron concentration is equal to the doping concentration.

The semiconductor bulk is driven by an electric field with two components, a periodic and a fluctuating one

E{t) = Eocos{m + <j>) + ri{t) (1)

where / = co/ln and EQ are the frequency and the amplitude of the periodic component, respectively . The random component of the electric field is modeled with an Omstein-Uhlenbeck (OU) stochastic process rj{t), which obeys the following stochastic differential equation:

dr]{t) dt

(2)

where TC and D are, respectively, the correlation time and the variance of the OU process, and E, (?) is the Gaussian white noise with autocorrelation < E, (t)E, it') >= 5(t — t'). The OU correlation function is < r]{t)r]{t') >=Dexp(- | ? -? ' | /Tc) .

The changes of the intrinsic noise properties are investigated by the statistical analysis of the two-time symmetric autocorrelation function of the velocity fluctuations and of its mean spectral density, as described in Refs [6,7, 8]. Intrinsic noise has been investigated also by estimating directly the electron velocity variance. This calculation has been performed separately for each energy valley, following the same method of equivalent time moments described in Refs [4, 6].

MONTE CARLO RESULTS AND CONCLUSIONS

The frequency of the fluctuating periodic electric field has been set to / = 500 GHz. According to a preliminary analysis on the variance of velocity fluctuations and on the spectral density So{E) at zero frequency, as a function of the amplitude of the oscillating field [4], we have chosen the amplitude value of the driving electric field £o = 10 kV/cm.

122

Te=0.01T

Xo=SO T

Tc=5 MT

( a )

«--E.

•

o

:& w c

» • a

1 u o Q .

0.25

0.20

0.15

0.10

0.05

?,

:K

'• *i A *• •- \

t j , O.OIT

Xc=5 MT

500 1000 1500

F requency [GHz ]

FIGURE 1. (a) Correlation functions of single particle velocity fluctuations; (b) Spectral density of electron velocity fluctuations as a function of the frequency. Continuous line corresponds to the results obtained with Tc = 0.01 T; dotted Une with Tc = 50 T; dashed line with Tc = 5 MT, where T is the period of the oscillating electric field.

Since in this contribute we investigate the role of the correlation time on the intrinsic noise modification, we keep the intensity of the external correlated noise at the constant value£)i/2 = 5kV/cm.

Figure 1(a) reports the results of Monte Carlo simulations of the autocorrelation function of the velocity fluctuations for three different values of Tc . Differently from the steady-state operation, under cyclostationary conditions, the autocorrelation curves exhibit damped oscillations around zero with approximately the same frequency of the periodic forcing field. In the presence of the added noise the autocorrelation function shows two similar decreasing trends for noise correlation times equal to 0.01 and 5 • 10^ periods T of the external field. In these both cases the whole relaxation process takes about 10 ps (5 times T). On the contrary, for Tc = 50 T, the correlation function continues to oscillate around zero exactly with frequency / for longer times.

In figure 1(b) we show that the superimposition of an external source of correlated noise affects the spectral density of the electron velocity fluctuations, in a way that critically depends on the OU correlation time. In particular, the two spectra characterized by Tc = 0.01 T and Tc = 5 • 10^ T appear to be very similar each other, while the spectrum obtained with noise at Tc = 50 T shows a narrow peak at 500 GHz," signature" of a strong resonant response.

In order to estimate the total noise power of the system we have computed the Integrated Spectral Density (ISD). The diagram in figure 2 shows a very interesting nonmonotonic behaviour of the ISD as a function of the correlation time. In particular, the total noise power shows an exponential-like decreasing trend with the increasing of the correlation time until Tc becomes about ten times the period T of the forcing field. For higher values, we find a wide minimum of the ISD vs. noise correlation time diagram. Then, for values of Tc greater than lO'* periods T of the external field, the ISD starts to rise again. Finally, for very high values of Tc, the ISD becomes independent from the external source of noise.

123

. — - \ y—--,

\ y \ -' \ ^ \ ^ \ - \ y

•^ , '

\ ' \ / •> ' "

Vv - • • * -

\ — •

*-' •

10-2 l o ' lo" lo^ 10= 10' lo" 10' 10* 10' T^ [ in u n i t o f T ]

FIGURE 2. Integrated spectral density of electron velocity fluctuations as a function of the correlation time of the external source of noise, in unit of T.

As shown in figure 4 of Ref.[4], depending on the intensity of the external source of correlated noise, the fluctuating electric field pushes the electrons to visit regions of the momentum space characterized by a smaller variance of the velocity fluctuations. In this work we have shown that this circumstance is strongly dependent also on the correlation time of the added noise source. In particular, for a given noise amplitude, the intrinsic noise reduction is verified for a wide range of values of the correlation times in which the system undergoes a resonant behavior. In this regime the external fluctuations constructively contribute to force the electrons to oscillate at exactly the same frequency of the periodic field, performing a more ordered dynamics which is confirmed by a lower total noise power.

ACKNOWLEDGMENTS

This work was partially supported by MIUR and CNISM-INFM.

REFERENCES

J, M, G, Vilar, and J, M, Rubi, Phys. Rev. Lett, 86, 950-953 (2001), L. Varani, C. Palermo, C. De Vasconcelos,J. F. MiUithaler, J. C. Vaissiere, J. P. Nougier, E. Starikov, P. Shiktorov, and V. Gruzinskis, in Unsolved Problems of Noise and Fluctuations:UPoN2005, edited by L. Reggiani et al., AIP Conference Proceedings, American Institute of Physics, New York, 2005, pp. 474-479. D. Persano Adomo, N. Pizzolato, and B. Spagnolo, Acta Phys. Pol. A 113 985-988 (2008). D. Persano Adomo, N. Pizzolato, and B. Spagnolo, /. Stat. Mech. P01039-10 (2009). D. Persano Adomo, M. Zarcone, and G. Ferrante, Laser Physics 10 310-315 (2000). T. Gonzalez, S. Perez, E. Starikov, P. Shiktorov, V. Gruzinskis, L. Reggiani, L. Varani, and J. C. Vaissiere, Proc. ofSPIE 5113 252-266 (2003). P. Shiktorov, E. Starikov, V. Gruzinskis, L. Reggiani, L. Varani, and J. C. Vaissiere, Appl. Phys.Letters 80 4759-4761 (2002). P. Shiktorov, E. Starikov, V. Gruzinskis, L. Reggiani, L. Varani, and J. C. Vaissiere, Phys. Rev. B 67 165201-10 (2003).

124

A systematic study of the impact of geometry on the low frequency noise in patterned

LaQjSrQ^MnO^ thin films at 300 K

S. Wu*, B. Guillet^ L. Mechin^ and J.M. Routoure(*)t

*GREYC, CNRS UMR 6072, ENSICAEN, Umverstty of Caen Basse Normandie 6, Bd du marechal Juin 1^050 Caen Cedex. France. *[email protected] Phone

(+33)231452722 / Fax (+33)231452698 "^GREYC, CNRS UMR 6072, ENSICAEN, University of Caen Basse Normandie 6, Bd du

marechal Juin 14050 Caen Cedex. France. * [email protected] / Phone (+33)231452722 / Fax (+33)231452698

Abstract. We report systematic measurements of low frequency noise performed at room temperature in LaojSro^MnOj, (LSMO) thin films (thickness =150 nm) patterned with different lengths {50jxm to 300/JOT) and widths {lOjxm to AOOjxm). Noise measurements were performed using two probe configuration, four probe configuration and even six probe configuration. Different 1/f noise contributions were observed for the film, for the current contacts and also for the voltage contacts. For the smallest devices, the noise spectral density of the film contribution does not follow the classical quadratic dependence with the DC voltage. The current contact contribution is due to current crowding at the metal/LSMO interface as already reported. The voltage contact contribution could be attributed to DC current circulation into the voltage contacts.

Keywords: 1/f noise, manganite, thin films PACS: 73.50 T, 75.47 Lx

INTRODUCTION

Manganite thin films are promising for next generation devices and sensors[l]. In the i>ao.7' ^o.3^wC'3(LSMO) case, it has been demonstrated that high sensitivity bolometers can be realized [2, 3]. High performance room temperature spintronic devices, magnetoresistances as well as strain gauge could also be obtained due to the low value of the low frequency (LF) noise[4]. A systematic study of the impact of length, width and thickness and appropriated geometries on the low frequency (LF) noise level is therefore necessary for future sensor developments [5].

Systematic measurements of LF noise at room temperature in patterned LaojSro^MnOj, (LSMO) thin films of thickness 150 nm, various lengths L (50 m, 100^m, ISOiim, 200^m and 300^m) and two widths W (20^m and lOO^m) have been performed. The spatial homogeneity of the film was checked by resistivity measurements; magnetic characterizations versus temperature have revealed a Curie temperature close to 350 K and the temperature of maximal resistance was found to be about 390 K. These values are very close to the bulk ones proving the good crystallinity of the thin films. The noise measurements have been performed directly on prober using a Karlsuss PM5 with two, four or six probes.



125



EXPERIMENTAL CONDITIONS

Devices geometry

Deposition process as wall as the patterning technique of the devices can be found in [5]. An optical photography of one device is given in the figure (1). It consists of bridges with different lengths and widths with two current pads IP and IM and various voltage pads (VI to V4 on one side of the bridge and VI' to V4' on the other side) which allows various device lengths. The metallic pads are realized using gold. The bridge lengths between VI -V2, V2 - V3 and V3 -V4 are lOOjim, SOlim and 150/im respectively.

FIGURE 1. Optical photography of one 100 jlm width bridge with the two current probes IP and IM and 2 x 4 voltage probes (VI...V4, VI'...V4') on each side of the bridge. The bridge lengths between VI -V2, V2 - V3 and V3 -V4 are lOOjlm, SOjlm and ISOjlm respectively.

Two probe, four probe and six probe configurations

The experimental set-up mainly consists in one low noise high output impedance DC current source previously described in [6] and up to two identical voltage amplifiers. It is assumed here that the DC current source is ideal: its output impedance is infinite and its noise contribution is negligible. We also suppose that the noise contribution of the voltage amplifier is negligible and that the input impedance of the amplifier is very high so that no DC current fiows in its inputs. The device is connected at the output of the DC current source and the DC voltage and the fiuctuations are measured using the voltage amplifiers. A spectrum analyzer (HP3562A or HP89410A) is connected at the output of the amplifiers. It calculates the noise spectral density or the cross spectral density in the case of the six probe configuration.

For all the configurations, the DC current source is connected between IP and IM probes. The mathematical derivation is not described here but it can be shown that :

• in the two probe configuration, one voltage amplifier is connected to the IP and IM probes. The noise contribution of the film and the current contact can he estimated.

126

• In the four probe configuration, one voltage amplifier is connected to tiie voltage probes. The noise contribution of the film and of the voltage contact can be estimated.

• In tiie six probe configuration, two voltage amplifiers are connected on each side of the bridge. With a spectrum analyzer that calculates the cross spectral density, all the uncorrelated noise sources disappear and only the contribution of the film can be estimated.

The LF noise spectral densities or cross spectral densities of the devices were measured for different DC current across the film in the two probe {Sy2p), four probe {Sv4p) and six probe configurations(iS'//6/?)-

RESULTS AND ANALYSIS

Only the results obtained for the device with L= 100 jim and ^ = 100 jlm will be shown and discussed. The same behavior has been measured for the other lengths and widths. In figure (2a) are plotted Syip, SvAp and Syep for / = 1 Hz versus the current in the device. It clearly shows that the three contributions for the film, the current contact and the voltage contact have to be taken into account.

LSMO W=100 ^m, L=100 |im, thickness=150 nm, T=300 K, f=1 Hz

^

Slope 2

V

® s V4p

• s V6p

^

X ' y \

DQI

Voltage V (V)

a. b.

FIGURE 2. a. Spectral density measured in the bridge with L = 100 \im and ^ = 100 \im in the two probe (Syip)^ the four probe (Syip) and the six probe configuration (Sy^p). b. Sketch showing the left part of the device with the IP pad and the two first voltage pads and the path of the current lines. The current lines close to the edge of the bridge penetrate more or less deeper in the LSMO part of the voltage contact depending on the shape of the voltage contact and the width of the bridge.

For the contact contribution the current contact noise level has a very high value as already reported [7, 8]. It is found that the noise spectral density follows a quadratic current dependency. It has been attributed to current crowding at the edge of the gold/LSMO interface.

The film contribution does not follow a quadratic DC current dependency. This may be attributed to lorentzian contributions in the spectra. Whatever the slope is, the bridge contribution is 2 times smaller than the voltage contact one. It shows

127

another time that pubhshed normahzed values of ajn (with a the Hooge constant and n the concentration ) may have been overestimated.

The last point concerns the fact that a LF noise contribution is found for the voltage contacts. In a first analysis, it is assumed that no DC current flows into the voltage contact since the input impedance of the voltage amplifier is very high: no f/f noise should be found in the voltage contact. As shown in the figure (2b ), the current line penetrates more or less deeper in the voltage contact depending on the shape of the LSMO part of the voltage contact. This point has been verified using numerical simulation of the current flow in the device. This current circulation in the voltage contacts creates voltage contact noise contributions. These noise contributions disappear when the devices are characterized in the six probe configuration because the noise contributions that appear on each side of the bridge are not correlated.

CONCLUSION

The impact of the geometry of patterned LSMO thin film with different lengths and widths on the LF noise has been reported. It has been shown that three contributions exist for the film, the current contact and the voltage contact. The last one has been attributed to current flowing more or less deeper in the LSMO part of the voltage contact. This study shows that in order to obtain the correct value of the film noise, six probe configuration has to be used. Moreover, in the frame of sensor development, the shape of the voltage contact has to be designed carefully in order to limit as much as possible the LF noise contribution of the voltage contact.

REFERENCES

1. (2008), URL h t t p : / / w w w . i t r s . n e t . 2. F. Yang, L. Mechin, J.-M. Routoure, B. Guillet, and R. Chalialov, J. Appl. Phys. 99 (2006). 3. M. Bibes, and A. Barthelemy, IEEE Transactions on Electron Devices 54, 1003 (2007). 4. L. Mechin, F. Yang, J.-M. Routoure, B. Guillet, S. Flament, and D. Robbes, Applied Physics

Letters 87 (2005). 5. L. Mechin, J. Routoure, S. Mercone, F. Yang, S. Flament, and R. Chakalov, Journal of Applied

Physics 103, 083709-083709 (2008). 6. J.-M. Routoure, D. Fadil, S. Flament, and L. Mechin, "A low-noise high output impedance DC

current source," in Proceedings of the 19th International conference on Noise and fluctuations, ICNF 2007, edited by M. Tacano, Y. Yamamoto, and M. Nakao, AIP, 2007, vol. 922, pp. 419-424.

7. C. Barone, A. Galdi, S. Pagano, O. Quaranta, L. Mechin, J.-M. Routoure, and P. Perna, Review of Scientific Instruments 78, 093905 (2007).

8. C. Barone, S. Pagano, L. Mechin, J. Routoure, P. Orgiani, and L. Maritato, Review of Scientific Instruments 79, 053908 (2008).

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http://www.itrs.net

The Low-frequency Noise in Al Doped ZnO Films

Bela Szentpali'', Agoston Nemeth", Zoltan Labadi'' and Gyorgy Kovacs

"Hungarian Academy of Sciences, Research Institute for Technical Physics and Materials Science, Konkoly-Thege M. ut 29-33, Budapest, Hungary, postal code: H-1121

Department of Material Physics, Eotvos Lordnd University, Pdzmdny Peter setdny 1/a, Budapest, Hungary, postal code: H-1117

Abstract. The electric noise of Al doped ZnO layers were measured. The optically transparent layers were prepared by reactive sputtering. In the low frequency range pure 1/f noise was observed. Extremely large Hooge parameters were obtained.

Keywords: ZnO, Hooge parameter, noise, reactive sputtering PACS: 72.70+m, 72.80Ey

Transparent conducting films are used widely in different optoelectronic devices as light-emitting diodes, laser diodes, different photo sensors, solar cells or even in flat panel displays. The candidate materials are high band-gap oxide semiconductors as the widely known indium tin oxide. The Al doped ZnO is a promising alternative due to its inexpensiveness. The cheapest and most effective method for deposition of these layers is the reactive sputtering of metallic alloy target. There are considerable efforts for increasing the electric conductivity of the Al doped ZnO films and maintaining its good transmittance at the same time. The published conductivity values improve from year to year. Some recent results'"^ are listed in Table 1.

TABLE 1. Recently published electric parameters of Al doped ZnO layers deposited by reactive sputtering in different laboratories. The transparency for the visible light was

reported at about 80% in each case. Specific resistivity

p milcm

0.24 0.35 0.63 1,3 -5

Density of mobile electrons

n lO'-cm' 6.8 4

n.a. 4.5 n.a.

HaU fi

1 mobility cmWs

38 8.5 n.a. 10

n.a.

Ref.

1 2 3 4 5

The authors has not found any report on the noise properties of this material, however, it would have importance when it is apphed in sensors. There are publications on the noise properties of ZnO nanorods^'^, but that is a quite different structure. In the present work we are going to report the electric parameters and low

CPn29, Noise and Fluctuations, 20 International Conference (ICNF 2009) edited by M. Macucci and G. Basso


129

frequency noise properties obtained on the Al doped ZnO layers prepared by ourselves.

EXPERIMENTS AND RESULTS

Al doped ZnO layers were deposited by pulsed DC reactive magnetron sputtering. The target was a high purity Zn 99,95%, alloyed with 2% (m/m) Al of the size of 110x440 mm . The depositions were performed in Ar/02 atmosphere. There was no closed loop process control apphed during deposition in order to stabihze the process. Details of the process were discussed elsewhere^ The typical structure of the layers is shown in Figure l.in planar and cross-sectional views. The pictures show a columnar structure grown together. The diameters of the columns are in the 20-50 nm range. Some columns are as long as the full layer thickness (186 nm); however the most of them are only about 100 nm.

* ^ ' _ ™ . " - " * " " ' mi*»-MM ?7r "?J^T ?rf^--'--';F';'™^;-T.„. r-!' -.— '\ ^

FIGURE 1. The scanning electron microscope picture of the top (a) and the cross-section (b) of sample No. 9.

The transparency of the samples treated here are over 80% between the wavelengths 500 nm and 800 nm.

The ohmic contacts for the electrical measurements were made by evaporating a 2D array of Cr/Al dots. The roughly 10 nm Cr layer ensured the good adhesion of the Al layer. The diameters of the dots were 3 mm and the lattice constant of the square matrix was 9mm. The contact resistances were measured by the three point method. They proved to be completely hnear.

For Hall-effect measurements squares were cut having edges of 9 mm, the measurement was performed in the van der Pauw configuration. The cutting was done in that way that no shunting formed across the Si wafer. The Hall resistance was measured in the function of the changing magnetic field of the iron cored electromagnet. The field strength was measured by a calibrated Hall sensor (FC-32). A typical measurement is shown in figure 2. The resistance at zero field is not zero, this is due to the slightly asymmetrical shape of the sample. The asymmetry can be estimated by comparing this resistance to the sheet resistance. In the case shown in the figure the asymmetry is 8,23£2/181£2= 4.5 %, i.e. about 0.4 mm in geometry. The

130

measurements were fulfilled on both diagonals of the sample and also at both polarities of the current. The average of the four measurements is given in the Table 2.

8.32

8.30

8.28

8.28

8.24

: 8.22

8.20

8.18

8.18

8.14

D09

^

R=8

> ^

23+0.'.057*B

^

^ /•

-0.20 -0.15 -0.10 -005 0.00 0.05 0.10 0.15 0.20

B[T]

FIGURE 2. The measured Hall resistance in the function of the magnetic induction. The biasing current was 1 mA. The sample No. is D9.

However the I-V characteristics of the contacts were good ohmic, the noise measurements were made in the 4 point configuration. The voltage noises were measured by the SR785 Dynamic Signal Analyzer on strips holding the 4 contacts. Fig. 3. shows the measured noise spectra.

E-15-

::::!: • Sample: DC • w ,

_''^'-".-v,.. ^ ^ ^ ^ w\

• •

Tf

9

"^^S*. •w

• •

ov 0.174^

. 0.343 ^ • 0 601 ^

/

1

'"fifes ^^ f»

• ^p

" ^ L Hz

FIGURE 3. The noise spectra at different biases. The continuous line stands for the 1/f dependence.

The series resistance was always at least twenty times higher than the resistance between the current points of the sample. The bias change during the measurements was less than 1% due to the high-capacitance of the lead batteries. After a few biasing

131

cycle the noise values increase and become weakly bias-dependent. The effect of the atmosphere was investigated by placing the samples in pure nitrogen instead of air. No significant difference has been observed so far. The samples recover in a few days. Then the repeated measurements result in spectra similar to the firsts. It seems that the phenomenon is some bias induced change in the electronic structure, perhaps trapping at the grain boundaries. The noise spectra were evaluated by the Hooge-formula':

^r - ^f, (1)

, where an is the dimensionless Hooge-parameter and N is the number of electrons participating in the conduction. N was calculated from the geometry and the electron density obtained from the Hall measurements. The exponent of the DC bias was 2 within the measuring error. Table 2 summarizes the measured parameters.

TABLE 2. The summary of the data obtained from the Hall- and noise-measurements. SAMPLE NO. D09 D12 Thickness [nm] 186 177 Free electron cone, [cm"'] 8.6*10" 3.7*10" Hall mobility [ c m V s - ' ] 22 41 P 1.02...1.06 1.13...1.23 OH 7300 4500

ACKNOWLEDGMENTS

This work was supported by the Hungarian Research Found (OTKA) under contract No. 73424. The assistances in the sample preparation of Mr. Tamas Szabo and Mr. Attila Nagy are also acknowledged.

REFERENCES

1. J. Hiipkes, B. Rech, S. Calnan, O. Kluth, U. Zastrov, H. Siekmann and M. Wuttig, Thin Solid Films 502, 286-291 (2006)

2. F. Ruske, A. Pflug, V. Sittinger, W. Werner, B. Szyszka and D.J. Christie, Thin Solid Films doi.:10.1016/j.tsf.2007.06.019 (2007)

3. S.J. Jung, B.M. Koo, Y.H. Han, J.J. Lee and J.H. Joo, Surface & Coatings Technology 200, 862-866 (2005)

4. L. Li, L. Fang, X.M. Chen, J. Liu, F.F. Fang, Q.J. Li, G.B. Liu and S.J. Feng, Physica E 41 169-174(2008)

5. C. May, R. Menner, J. Striimpfel,. M. Oertel and B. Sprecher, Surface & Coatings Technology 169-170, 512-516 (2003)

6. H.D. Xiong, W. wang, Q. Li, C. A. Richter, J.S. Suehle, W-K. H, T. Lee, D. M. Fleetwood, Applied Physics Letters 91, 053107-1 (2007)

7. J. Lee, I. Han, B-Y. Yu, G-C. Yi, G. Ghibaudo, J. Korean Physical Society, 53, 339-342 (2008)

8. A. Nemeth, Cs. Major, M. Fried, Z. Labadi, I. Barsony, Thin SolidFilms, Volume 516, Issue 20, 30 August 2008, Pages 7016-7020

9. F.N. Hooge, Physica (Utrecht) 60, 130, (1975)

132

Low Frequency Noise In Electrolyte-Gate Field-Effect Devices Functionalized With

Dendrimer/Carbon-Nanotube Multilayers

F.V. Gasparyan , A. Poghossian ' , S.A. Vitusevich , M.V. Petrychuk , V.A. Sydoruk , A.V. Surmalyan , J.R. Siqueira , O.N. Oliveira Jr. ,

A. Offenhausser^ M.J. Schoning^'^

'Yerevan State University, 1 AlexManoogian St., 0025 Yerevan, Armenia ^Institute ofNano- and Biotechnologies, Aachen University of Applied Sciences, 5242, Jiilich, Germany

^Institute of Bio- and Nanosystems. Research Centre Jiilich, 52425 Jiilich, Germany ''Taras Shevchenko National University, 01033 Kiev, Ukraine

^Physics Institute of Sao Carlos, University of Sao Paulo, 369, Sao Carlos, Brazil

Abstract. Low-frequency noise in an electrolyte-insulator-semiconductor (EIS) structure functionalized with a multilayer of polyamidoamine (PAMAM) dendrimer and single-walled carbon

nanotubes (SWNT) is studied. The noise spectral density exhibits 1// '^ dependence with an exponential slope of / » 0.8 and 0.8 < / < 1.8 for the bare and functionalized EIS sensor, respectively. It has been observed that functionalisation of EIS structure with the PAMAM/SWNTs multilayer leads to an essential reduction of 1/f noise at frequencies below ~10 Hz.

Keywords: low-frequency noise, field-effect sensor, carbon nanotube, dendrimer.

PACS:73.21.-b;73.22.-f

INTRODUCTION

The functionahsation of field-effect devices (FED) based on an electrolyte-insulator-semiconductor (EIS) system with the nano- and biomaterials is one of the most attractive approaches for the development of (bio-)chemical sensors and biochips [1]. In addition, silicon nanowires and carbon nanotubes have recently attracted significant interest as a promising material for novel nanoscale bioelectronic devices [2]. Since FEDs are charge sensitive devices, each (bio-)chemical reaction leading to chemical or electrical changes at the insulator gate/electrolyte interface can be detected by coupling the gate with respective chemical or biological recognition elements. In order to obtain specific biosensors with a high performance, the layer-by-layer (LbL) technique has been recognized as a suitable tool for fabricating functional hybrid materials and nanostructured films, as it offers fine control over film thickness and architecture at the nanoscale [3].

While the noise has been extensively studied in MOSFETs (metal-oxide-semiconductor field-effect transistor) and related devices, so far the noise investigation



133

in electrolyte-gate FEDs has been limited [4]. Noise investigations in functionahsed FEDs are especially important in the case of an application of FEDs for the detection of biomolecules by their intrinsic molecular charge, where the sensor signal can be very small [1]. Moreover, the study of noise spectroscopy in functionalised FEDs can give additional insight into the detection mechanisms of the biosensor. In this work, low-frequency (LF) noises in a capacitive EIS structure functionahsed with polyamidoamine (PAMAM) dendrimer/SWNT (single-walled carbon nanotube) LbL multilayer has been investigated for the first time.

EXPERIMENTAL

The LbL film consisting of three bilayers of PAMAM/SWNT was prepared on an Al-p-Si-Si02- Ta205 (300 nm Al as a rear-side contact layer, 30 nm Si02 and 55 nm Ta205) chips via consecutive adsorption of positively charged PAMAM dendrimers and SWNTs (modified with carboxylic groups) from the respective solutions, following by rinsing and drying steps. The adsorption of each layer was monitored by capacitance-voltage (C-V) method using an impedance analyzer (Zahner Elektrik). Fig. 1 shows the schematic cross-section of the EIS structure functionalised with a multilayer of PAMAM/SWNT (a) as well as the chemical structures of the materials employed (b).

SWNT PAMAM

b)

FIGURE 1. Schematic of the EIS structure functionalized with a muWlayer of PAMAM/SWNT (a) and chemical structures of materials employed (b).

Shflding box

FIGURE 2. Experimental setup for noise measurements. E: battery; R: load resistance; DUT: device under test; LNA: low-noise current amplifier.

134

Noise spectra in bare and functionalized EIS sensors were measured in a pH 7 buffer solution in a frequency range from 0.03 to 100 Hz using the experimental setup shown in Fig. 2. A voltage from battery E was applied to the EIS structure via load resistance {R=10 MQ) and an Ag/AgCl hquid-junction reference electrode. The EIS sensor was coupled to a low-noise current amplifier (LNA). The LNA-output was connected to an HP 3 5670A dynamic signal analyzer that records the noise signal. The measuring cell with the EIS sensor, reference electrode and LNA were placed in a shielding box for protecting from external electromagnetic influences.


Fig. 3 shows the C-V curves for a bare p-Si-Si02-Ta205 EIS structure and an EIS sensor functionalised with 3-bilayers of PAMAM/SWNT measured in a buffer solution of pH 7 (left), and the enlarged C-V curves in the depletion range (right). The adsorption of each PAMAM or SWNT layer shifts the C-V curve along the voltage axis. The direction of potential shifts depends on the sign of the charge of the outermost layer. This indicates that the positively charged dendrimer molecules and negatively charged SWNTs may induce an interfacial potential change resulting in a change in the flat-band voltage of the EIS structure.

50

40

c 30 S o ^ 2 0 O

10

"'~""'' ~*"''**--

-

-

Ta205

PAMAM

SWNT

I \

*»

\ \ N

^ , 30.6

— 30.4

— 30.2 o

^ 29.8 Q. n! O 29.6

29.4

'A \\

\\

. PAMAM 6 '

- \ •• SWNT 1

SWNT 6 \ \

\ \

\ PAMAM 1 \ \

•• Tafis

-1.5 -1.0 -0.5 Voltage (V)

0.0 0.5 -0.40 -0.38 -0.36 -0.34 Voltage (V)

-0.32 -0.30

FIGURE 3. C-V curves for a bare p-Si-Si02-Ta205 EIS structure and an EIS sensor functionalized with 3-bilayers of PAMAM/SWNT measured in a buffer solution of pH 7 (left), and the zoomed C-V curves in the depletion range (right).

LF noise spectra were measured in different working points of the C-V curve by applying different gate voltages. The measured noise spectral density (NSD) exhibits l/f dependence with an exponential slope of 7 « 0.8 and 0.8 < 7 < 1.8 for the bare and functionalized EIS sensor, respectively. Surprisingly, at frequencies of/<10 Hz, noise-reduction effect has been recorded in a functionalized EIS structure. This effect depends on the applied gate voltage and is stronger in the accumulation regime. As an example. Fig. 4 demonstrates the gate-voltage noise power spectral density (Sy) as a function of frequency for the bare and functionalized EIS structures, recorded in the accumulation region at an applied gate voltage of -1.5 V and in pH 7 buffer solution. As can be seen, the presence of an additional PAMAM/SWNT multilayer leads to essential reduction (by the factor of up to 100) of the 1//noise in the accumulation mode in comparison to bare EIS structure.

135

^a'

^T=300 K, Buffer solution pH7

W 10 lrf Frequency (Hz)

FIGURE 4. Gate-voltage NSD (Sy) as a function of frequency for the bare (plot 1) and functionalized (plot 2) EIS structures, recorded in the accumulation region at an applied gate voltage of-1.5 V.

The theoretical modelling of the noise in functionalised FEDs is very complicated due to the presence of an organic/inorganic hybrid structure, and should include noise caused from the reference electrode, electrolyte, multilayer of PAMAM/SWNT, oxide/electrolyte and multilyer/electrolyte interfaces, and the solid-state device itself. It has been reported that 1 / / noise behaviour was observed for ISFETs (ion-sensitive field-effect transistor) down to 1 Hz, indicating that the noise is dominated by the trapping-detrapping mechanism at the Si-Si02 interface [4]. An application of a trapping-detrapping model to our structure, do not allow explain the effects observed in functionalised EIS structures. Thus, our results demonstrate that in addition to charge fluctuation mechanism in solid-state part, electrochemical processes at the Ta205/PAMAM and electrolyte/PAMAM/SWNT interfaces as well as in multilayer itself could be responsible for noise behaviour in functionalised FEDs. It is also possible to include phonon percolation processes in semiconductor (insulator)/other media interfaces to explain the low frequency noise pecuharities [5]. Dashed hues in Fig. 4 are calculated using only charge fluctuation model at the Ta205/PAMAM/SWNT/electrolyte interfaces. The PAMAM/SWNT multilayer could act as a "stabiliser" providing a more stable signal and a smaller drift [6] and as a tool for "noise suppression" in the investigated system [5].

ACKNOWLEDGMENTS

F.V. Gasparyan is grateful to German Academic Exchange Service (DAAD) for financial support.

REFERENCES

1. M.J. Schoning and A. Poghossian, Electroanalysis 18, 1893-1900 (2006). 2. J. Mannik, I. Heller, A.M. Janssens, S.G. Lemay and C. Dekker, Nana Lett. 8, 685-688 (2008). 3. J.L. Lutkenhaus and P.T. Hammond, Soft Matter 3, 804-816 (2007). 4. C.G. Jakobson and Y. Nemirovsky, IEEE Trans. Electron Dev. 46, 259-261 (1999). 5. S.V. Melkonyan, V.M. Aroutiounian, F.V. Gasparyan, H.V. Asriyan, Physica, B: Physics of

Condensed Matter 3^2, 65-70 (2006). 6. J.R. Siqueira, Jr., M.H. Abouzar, M. Backer, V. Zucolotto, A. Poghossian, O.N. Oliveira, Jr., and

M.J. Schoning, P/;>'5. Status SoUdi A 206, 462-467 (2009).

136

Low Frequency Noises Of Hydrogen Sensors On The Base Of Silicon Having Nano-Pores Layer

Z.H. Mkhitaryan, F.V. Gasparyan, A.V. Surmalyan

Yerevan State University, 1 AlexManoogian St., 0025 Yerevan, Armenia

Abstract. Sensors have sandwich stracture metal/porous silicon/crystalhne silicon/Al. The contact metal on porous silicon was from Au and from Pd. Porosity of the samples is 57% and 63%. Low frequency noises before, during and after influence of hydrogen gas flow are

studied. Noise spectra of the samples in general have Ij f^ forms with 0.7 < / < 1.

Keywords: sensor, porous silicon, frequency, noise, sensitivity.

PACS: 73.30.+y

INTRODUCTION

Design of control and monitoring systems permitted to receive information about composition of the several gaseous medium. This is very important for the environment monitoring. The basic part of those systems is the gas sensor. They are used for threshold detection of the several gas escapes and for control of the foul gas concentration at the technological processes. Extensive search for new materials, new techniques for increasing of sensitivity of semiconductor based gas sensors have great interest [1]. Porous silicon (PS) has high resistivity and therefore it is very sensitive to external influences, such as gaseous medium. PS/silicon substrates have been used for sensing applications. PS is successfully used for design and construction not only gas, as well as (bio-)chemical sensors and air humidity sensors [2]. The H2 gas sensors that can quickly and rehably detect H2 over a wide range of oxygen and moisture concentrations are not currently available. At the change of air gas composition concentration the electrical activity of the binding sites and charge states of the material surface is changed. Then electrical and noise characteristics of gas sensors are essentially changed. Since the time constants involved in the detection of several gases molecules via trapping on the surface electronic states of the PS layer are relatively large, in the order of seconds or higher, it would be expected that low frequency noise (LFN) is more critical than other types of noises in gas sensors.

EXPERIMENTAL

Sensors have sandwich structure Metal/PS/Crystalline silicon/Al. The contact metal on PS made from Au (samples 1) and from Pd (samples 2 and 3). Porosity of the samples 1 and 2 vas 57%, porosity of the samples 3 is 63%. PS layer is organized by



137

the electrochemical etching of crystalline p^-Si substrate having 0.01£2cm specific resistivity. Thickness of the PS layer is equal to 3 |im.

R H,

Shielding box

Gas Cell

* -LNA

I i>

Fourier analyzer

1/

i B

FIGURE 1. Experimental arrangement for noise measurements.

PS with Au contact

air + 0,1% H.

air alter biowoff of K

FIGURE 2. Noise spectra of the Samples 1 (PS 57%, Au contact). Current equal to 100 /JTO .

Noise measurements are carrying out at the 300K using experimental arrangement shown on the Fig. 1. Voltage from battery "E" is applied to sample across load resistance R. By the low-noise amplifier (LNA) "SR 560" signal level is amplified. After that we do Fourier analyze in the frequency range 2-100 Hz using dynamic signal analyzer "Handiscope 2". Spectrum averaging number is equal to 200. Samples, battery, gas cell and LNA are located in Shielding box for protection from external electromagnetic influences. Current-voltage characteristics (CVC) are measured in dry air and dry air+0.1% H2. In dry air CVC has nonlinear character. For the samples 2 and 3 the reverse CVC do not have saturation, but for sample 1 (ohmic contact) reverse branch is saturated. At the forward bias in air for sample 1 we can see rectification phenomenon. When we put up the sample 1 in air mixed with H2 only forward branch of CVC slowly changes: device resistance increases. For the samples 2 and 3 (non ohmic contacts) under influence of H2 both forward and reverse branches strongly changed and device resistance decreases. After 3 days ageing in dry air+0.1% H2 CVC of sensors with Pd contacts and 63% porosity become practically linear.

138

On the Figs. 2-4 LFN spectra are presented. In general that spectra have ij f forms with 0.7 < 7 < 1. Noise voltage is measured in current generator mode. At the influence of H2 LFN levels increases and at the same time its behavior changes. For all samples in air at the / > 30-40 Hz thermal noise dominates. In the medium dry air + 0.1% H2 noise spectra for samples I get I / / dependence (Fig. 2). LFN level of the samples 2 and 3 increase in H2 medium (Fig. 3, 4). Note that the longer we keep samples in the medium dry air + 0.1% H2 the higher is noise level.

0 7 PS With Pd contact

air+Hj

.ir(blowoff of H 3 ' ^ / > / ^ * V Y f / * r ^ / ^

FIGURE 3. Noise spectra of the Samples 2 (PS 57%, Pd contact). Current equal to 10 /JTO .

Pd contact 53% PS

10 100 I Hz

FIGURE 4. Noise spectra of the Samples 3 (PS 63%, Pd contact). Current equal to 50 mi •

SIMULATION MODEL AND DISCUSSION

Gold metallic contact does not create Schottky barrier on the surface PS/Au. Weak dependence of LFN from H2 concentration at the forward bias probably is conditioned by the weak changes of the hetero-barrier height at the adsorption of H2 molecules on the slow states located on the oxide/environment interface. Current weak growth is attended with relatively "slow growth" of the noises level (Fig. 2). For the samples 2

139

and 3 at the adsorption of H2 molecules surface resistance decreases for both forward and reverse bias cases. This is probably conditioned by the decreasing of the Schottky barrier height. Current growth is attended with quick growth of the noises level (Figs. 3 and 4). The adsorption of the gas molecules can be associated with decomposition or dissociation. A reducing molecule adsorbed on the sensor surface acts as a donor and give electrons to surface states. At the adsorption they migrate over nano-pores and accept on the slow acceptor states. As a bulk recombination process via additional surface states bring to the increase of the fluctuation of the charge carriers in the bulk Si and consequently bring to the increase of LFN level. It is clear that the higher is H2 concentration the more is adsorbed molecules and therefore the stronger is the changes of the potential barrier height. As height of the hetero-barrier at the H2 adsorption almost does not change the barrier does not have essential effect as compared with growth of the adsorbed and desorbed molecules effect on the formation of additional noise. After H2 discharge from the cell the noise level decreases. Two competing models have appeared in the literature to explain LFN: the McWhorter number fluctuation theory and Hooge mobility fluctuation theory. For explanation of LFN behavior in p-Si/PS based gas sensors we used "Correlated number-mobility fluctuation model" [3]. In this theory noise spectral density of the flat-band voltage

fluctuation can be described by: S^(f) = f - — , here / = —J2m''<S>g is AyC f h

McWhorter's or tunneling parameter, k is Boltzmann constant, T is the temperature, 7V , is the oxide equivalent trap density in eV'cm"^, A is the effective area of nano-pores contacting with H2 molecules, C is the cumulative capacitance, m' is the effective mass of the electron in the oxide layer, and O^ is the tunnehng barrier height seen by the electron at the interface. O^ decreases with increase of H2 concentration. For numerical calculations of S^ ( / ) for samples 2 and 3 we consider that in air at the normal conditions we have -2.7x10^' molecules without H2. Then at the air+0.I%H2 we have A ^ «2.7xI0'*^ c m l For Si-Si02 system at 300 K 7 = I0**cm"\ 7V , = lO'^eV'cm"^. Samples have cycle's forms with d = l.5 cm diameter. Assume that all pores identical (« parallelepipeds having lOnm x lOnm x 3 |im sizes). Then « «1.77 x lO'^ pores. For 57% PS porosity A^j,/^ « I2I0cm^, for 63% PS porosity A^y^^ «I338cm^. Analytical graphs S^(f) are presented on the Figs. 3 and 4 (direct lines). We can see acceptable fit between experimental and analytical graphs. Different behavior of LFN is conditioned by the contact metal type, as well as it depends on tie of H2 molecules after chemical adsorption on the PS surface binding sites. At storage of samples in H2 atmosphere the noise spectrum has I/f dependence.

REFERENCES

1. I. Schechter, M. Ben-Chorin and A. Kax,Anal. Chemistryf,!, 3727-3732 (1995). 2. V.M. Aroutiounian, Z.H. Mkhitaryan, A.A. Shatveryan, F.V. Gasparyan, M.Zh. Ghulinyan, L.

Pavesi, L.B. Kish, and C.-G. Granq\ist, IEEE Sensors Journal S, 786-790 (2008). 3. A. Hassibi, R. Navid, R.W. Dutton, and T. H. Lee, J. ofAppl. Physics 96, 1074-1082 (2004).

140

Nyquist Relation and Its Validity for Piezoelectric Ceramics Considering Temperature

Petr Sedlak, Jiri Majzner and Josef Sikula

Department of Physics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technicka 16, Brno 616 00, Czech Republic

majzner@feec. vuthr. cz

Abstract. In this paper, we focused on validity of the Nyquist relation for piezoelectric ceramics in temperatures 303 K - 393 K. The electrical impedance and noise spectral density were measured and compared for every 10 K in frequency range 100 kHz - 1 MHz. The measurements were made in thermal stable condition and under equilibrium conditions in the case of noise measurement.

Keywords: piezoelectric ceramics, noise spectral density, Nyquist relation PACS: 77.84.Dy

INTRODUCTION

This work, developed at Czech Noise Research Laboratory (CNRL), concerns the validity of Nyquist relation on piezoelectric ceramics considering the temperature. This ceramic material is widely used in non-destructive acoustic/ultrasonic methods, especially in acoustic emission (AE). This technique offers great potential due to its ability of quantitative evaluation such as crack location and crack characterization. The AE sensors are almost of piezoelectric ceramics type due to its ability of high sensitivity with wide bandwidth. The signal-to-noise ratio of these sensors depends on many factors including minimization of the own noise in piezoelectric part.

The main sources of voltage or current fluctuation in piezoelectric ceramics are thermal noise, polarisation noise and low frequency 1/f noise. Thermal noise is given by interaction of phonons with free electrons or holes and noise spectral density of voltage fluctuations is proportional to sensor resistance and temperature. Fluctuation of electrical polarisation in piezoelectric ceramics is an additional source of the voltage noise. Electrical dipoles are vibrating due to thermal energy and theirs motion create on electrodes induced electric charges, with time dependent value of the total charge. Induced electric charge fluctuates and this is the origin of voltage fluctuation. This is generation recombination noise with exponential distribution of relaxation constant. Noise 1/f originates from superposition of particular generation recombination spectra. In previous papers [1-3], we showed that noise spectral density of these fluctuations can be described by well-known Nyquist equation



141

S^=4kTR (1)

where k is Boltzmann constant, T is absolute temperature and R denotes a real part of electrical impedance. Figure 1 shows the noise spectral density of piezoceramic disc used in AE sensors. This dependence is simulation of PZ27 disc, and was calculated by our mathematical model [3], which is based on theory of finite element method and on Nyquist relation. The considered temperature is 300K.

In this work, we compare the noise measurements and calculations in frequency range from 100 kHz to 1 MHz for temperatures 303 K-393 K in order to prove the validity of Nyquist relation.

^r AE frequency range—^

10 10 frequency/ Hz

FIGURE. 1. Simulation results of the mathematical model based on Nyquist relation [3]

EXPERIMENT

The experiment was carried out on PCM 51 disc specimen of diameter 9.95 mm and thickness 3.20 mm. This material belongs to piezoelectric soft materials, which are suitable for sensing applications, and is produced by Noliac Ceramics, Inc [4].

The measurements of complex electrical impedance were made with an auto-balanced-bridge using RLC meter HP4285A (Agilent Technologies, Inc.). The band of analysis covers the frequency range from 100 kHz to 1 MHz with a resolution of IkHz. Measurement of noise spectral density frequency dependence requires amplifier and filter competent quality. Our apparatus consists of input low noise preamplifier (PA 15, 3S Sedlak), amplifier (AMP 22, 3S Sedlak) with high selective filters and 12bit A/D converter. The sampling rate was 5 MHz and the sample was measured under equilibrium conditions. For each temperature, the 1000 realization of noise measurement were made to get fine resolution of noise spectral density. The noise background level of measurement set-up was 2x10"'* V^ Hz"\

The temperature of the furnace was monitored and controlled by developed software from the PC via RS-232 interface. This software includes the proportional-integral-derivate (PID) control. The noise and impedance measurement were done for temperatures from 303 K to 393 K. The temperature was increased at 4K per minute

142

and sample was kept at each measuring temperature at least 30 minutes before measurement in order to ensure thermal equilibrium. Figure 2 and 3 shows measured dependences for T = 393 K.

200 400 600 frequency / kHz

800 1000 200 400 600 frequency / kHz

800 1000

FIGURE. 2. noise spectral density vs. frequency FIGURE. 3. real part of impedance vs. frequency


The influences of temperature on noise spectral density and real part of impedance are shown in Fig. 4, respectively Fig. 5. Only maximal value in both dependences is considered (figures 2 and 3). These values correspond to the fundamental planar resonance of the specimen. With increasing temperature, the noise spectral density is slightly increasing linearly by rate 5.55x10"^^ V^HZ'VK. TO the contrary, real impedance is inversely proportional to temperature, linearly by rate -76.84 Q/K.

3.6 'N

^ 3.4

b 3.2

3

2.8

2.6

2.4

c^ a = 5.55x10""V^ Hz"' /K

+ +

32

30

28

26

24

"K „ + a = -76.84Q/K

"""•+., "••+,„

"K,

+ " \

280 300 320 340 360 380 400 420

Temperature / K

280 300 320 340 360 380

Temperature / K

400 420

FIGURE. 4. Noise spectral density vs. temperature FIGURE. 5. Re(Z) vs. temperature

The specimen temperature seems to have little influence on the noise spectral density in temperature range 303 - 393 K. On the other hand, the thermal influence on impedance is not insignificant. The decrease of the real part of impedance is explained by thermal sensibility if certain characteristics of the piezoelectric material such as the dielectric constant [5]. In view of Nyquist relation, the decrease of real part of

143

impedance compensates the increase of the thermal noise, when the temperature increases.

We compared the measured and calculated noise spectral density for correspond temperature in percentage. Figure 6 represents this comparison. With increasing temperature, the results of measurement and calculations are more different, nevertheless this difference is insignificant. We suppose that Nyquist relation can be used, and is valid in measured range of temperature.

'fe

a = -2.91x10"^%/K

(D -7 o c (U

a; -7.5 T3 (D - 8 D) ro

I -8.5 o ^ n Q. -9 "o § -9-5 (U

^ -10 280 300 320 340 360 380 400 420

Temperature / K

FIGURE. 6. mean of percentage differences vs. temperature

CONCLUSION

This work concerns the validity of Nyquist relation on piezoelectric ceramics considering the temperature. The experiment was carried out on PCM51 specimen. The noise and impedance measurements were made for temperatures 303 K-393 K in frequency range from 100 kHz to 1 MHz. Comparing these measured and calculated noise spectral densities, we found that calculated results based on Nyquist relation gave us a good agreement with the noise measurement.

ACKNOWLEDGMENTS

This research has been supported by the Czech Ministry of Education in the frame of MSM 0021630503 Research Intention MIKROSYN New Trends in Microelecfronic System and Nanotechnologies and Grant MSMT KONTAKT ME896.

REFERENCES

1. J. Majzner et al, 1/f noise in piezoceramic, samples Proc. ofICNF2003, 2003, pp. 851-854. 2. J. Majzner et al. Noise in piezoceramics, Proc. ofICNF2007, 2007, pp. 347-350 3. P. Sedlak et al. Mathematical Model for Electrical Noise of Piezoelectric Sensor, Proc. of

ICNF2007, 2007, pp. 335-338 4. Noliac Ceramics, Inc. http://www.noliac.com/ 5. Z. Gubinyi Z et al. Journal of Electroceram 20 (2), 2007, 95-105

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Diagnostics of Forward Biased Silicon Solar Cells Using Noise Spectroscopy

R. Macku, P. Koktavy, P. Skarvada, M. Raska, P. Sadovsky

Department of Physics, Brno University of Technology, Faculty of Electrical Engineering and Communication,

Technicka 8, 616 00 Brno, CZECH REPUBLIC

xmacku05(a),stud.feec.vutbr.cz. http://www.feec.vutbr.cz

Abstract Our research is above all focused on non-destructive testing of the solar cells. We study a single-crystal silicon solar cells n^p and we don't have serious information about features of a/>« junction and impurities distribution. The main point of our study is characterization of the local defects in samples. These defects lead to live-time reduction and degradation of reliability. Flicker noise in forward biased solar cells is subject of this paper. We will discuss our measurement with Kleinpenning approaches for inhomogeneous semiconductors and we suggest the physical nature of the samples behaviour.

Keywords: Solar cell, flicker noise, shot noise, transport mechanism PACS: 73.50.Td

INTRODUCTION

In terms of cost reduction for photovoltaic systems the use of a new method for solar cells diagnostics is a very promising approach. Silicon based single junction solar cells are currently most widespread production alternative. The major advantage of using silicon as a base material is a stray procedure of extended dimension ingot production. The solar cell surface area is generally up to several hundreds centimeters square. The pn junction is very close to surface and it follows surface texturing. Due to this fact, the solar cell structure is very sensitive to surface damage and imperfections in bulk of material. Our aim is above all characterization of the bulk imperfections or inhomogeneties. The low-frequency noise measurement of a typical solar cell is investigated in this paper. During our study we proved that noise contributions of the solar cells are quite complicated. Their difficult character is probably result of large pn junction area and great number of local defects (such as impurities presence or junction thickness fluctuations resulting in local breakdowns). We want to characterize bulk imperfections; hence the sample selected for measurement (labeled K4) is only a fragment of the whole solar cell. Specimen surface area is less than 1 cm .



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EXPERIMENTAL AND SIMULATION DETAILS

In general, solar cell typical feature is development of the noise signal character for different bias voltage (here U{ is forward voltage and If is forward current). At the first sight we can see two types of noise (see fig. 1). The first one spectrum is inversely proportional to frequency and second one is inversely proportional to/^.

FIGURE 1. Power spectral noise density of forward FIGURE 2. IV curve of forward biased biased solar cell for different bias voltage. solar cell, sample K4, T = 16.7 °C.

Figure 2 depicts IV curve of the solar cell (here n is an ideality factor). We suggest, that the first apparently hnear region where « = 3.43 is a consequence of the shunt resistance, the second region (« = 2.58) is caused by generation and recombination (g-r process) of carriers. The ideality factor is essentially impressed with the distributed series resistance, [1].

Mathematic Noise Description and Simulation

Conventional theories usually used empirical Hooge equation but there is a reason for not doing so. The solar cell can not be allowed as a homogeneous semiconductor layer as assumed Hooge. In 1980 Kleinpenning and co-workers published a new approach concerning pn junctions. Equation (2) is based on the Kleinpenning derivation and the current transport mechanism in a sample. The transport mechanism is possible to determine through IV-curve measurement as shown in fig. 2. Saturation current /„ is lo(g-r)^ 2.7T0"^A. The relation between the minority carriers life time

and saturation current is given by [2]

h(g-r) = SpnenidJT^ (1)

Here S^^ is a/>« junction area [3], «i is a intrinsic concentration and (i is a depletion region width. In our case, the re has the same meaning as r that is effective carriers life time. The depletion region width is possible to obtain form CU measurement. The

146

resulting life time is re = 8.99-10" s. According to equation (1), the flicker noise in the pn junction is proportional to / . This statement has been verified by means of measurement of the power spectral noise density (center frequency 10 Hz) vs. forward current (see fig. 3).

shunt resistance is dominant

< CO CO

(7f=90mV, /=1.94-10-^A

10 Ilk

FIGURE 3. Power spectral noise density as a function FIGURE 4. Power spectral noise density as a of forward current, solar cell, quality factor 2 = 2. function of f/f (Q is filter quality factor).

In spite of the above, the curve in figure 3 does not correspond to the flicker noise for low level currents (in case of high level current does not be keep low injection conditions). Figure 4 depicts increase of the power spectral noise density (PSD) vs. bias voltage and frequency is here a parameter. Origin of noise with the Lorentzian spectrum correlates with measurement at frequency of 120 Hz (see fig. 1). So, the voltage limit for flicker noise observation is approximately Uf ~ 290 mV. As opposed to the high voltage limit we can find out the low current hmit. By the same taken, the shunt resistance R^h must be implemented into equation (1). We obtain

iI-UjR.^+h)Tj (2)

Here 7 is a constant representing current transport mechanism./„ is a saturation current and r^ is a life time. The y constant for the g-r process in the space charge region is 2/3. Simulation results are depicted in figure 5. The shunt resistance in case of the K4 solar cell sample is only R^h = 44.4 \<£l. In case of high currents the PSD of the flicker noise increase is proportional to the forward current and for the low lever currents it is proportional to S. « / **. This means that the flicker noise is

immeasurable in this region. So, what type of noise do we observe in low current region? We suggest the shot noise contribution. According to [1], pn junction shot noise is given by

:2e/„e"*^+2e/„. (3)

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Simulation of the pn junction shot noise has been made in agreement with eq. (4). The obtained results are depicted in fig. 5 and they are in compliance with experimental results (fig. 3). The Lorentzian spectrum shaped noise from figure 1 has been only briefly discussed because of scope of this paper. We only suggest that the time constant of the g-r process is r„ - 4-10" s. This type of noise may be caused by means of the same traps as a l//"noise.

/ / A

FIGURE 5. Noise simulation results.

CONCLUSION

The PSD vs. forward current curve in fig. 3 does not correspond to the flicker noise for low level currents. To deals with this problem, new type of noise sources has been added. Our model of solar cells noise is based on the flicker noise in the pn junction, the shot noise and the g-r noise. The research has showed that the flicker noise is given by carriers generation and recombination in the space charge region of the solar cell (carriers density fluctuation) and generation-recombination processes.

1 / / noise results from interaction between

ACKNOWLEDGMENTS

This research has been supported by the Grant Agency of the Czech Republic. The grant No. is GA102/09/H074 and the research project No. is VZ MSM 0021630503.

REFERENCES

1. VANDAMME, L., K., I , ALABEDRA, R., ZOMMITI, M.: 1/f noise as reliability estimation for solar cells, Solid State Electronics, vol. 26, no. 7, pp. 671-674, 1983.

2. SZE S. M.: Physics of Semiconductor Devices, John Wiley & Sons, New York, November 2006, ISBN 978-0-471-14323-9.

3. Macku, R.; Koktavy, P.; Skarvada, P.: Advanced non-destructive diagnostics of mono-crystalline silicon solar cells, WSEAS Transactions on Electronics. 2008. 4(9). p. 192 - 197. ISSN 1109-9445

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Flickering Noise Spectroscopy as a Powerful Tool for Investigation the Dynamics of the

Deformation Processes in Solids

Sergey G. Lakeev^ Nina N. Peschanskaya , Vitalii V. Shpeiizman , Pavel N. Yakushev , Alexander S. Shvedov^ Alexander S.

Smolyanskii^

" Karpov Institute of Physical Chemistry, 10 Vorontsovo Pole, Moscow 105064, Russian Federation loffe Physical Technical Institute, 26 Polytekhnicheskaya, St Petersburg 194021,

Russian Federation

Abstract. "Stick - slip" deformation in y-irradiated PolyMethylMethAcrylate (PMMA) samples has been tested by means laser Dopier interferometry. Stick - slip deformation becomes apparent both at elastic and plastic PMMA flow stages. It was observed that the transition component of the creep rate can be described in the framework of Flickering noise process. The parameters of the Flickering noise process have been determined. It was concluded that Flickering noise spectroscopy can be concerned as a useful and a powerful approach, explaining the nature of stick - flip deformation in solids.

Keywords: Polymer, Laser Dopier Interferometry, PMMA, "Stick - Slip" Deformation PACS: 05.40.-a; 07.05.Kf; 07.60.Ly; 61.82.Pv; 62.20.Hg; 62.30.+d; 81.70.Fy

INTRODUCTION

Development of new precise methods for measurement the little displacement in solids discovers new information about the deformation in polymers [1]. Employing of the laser interferometry for testing the creep of the different sohds can improve the quality of prognosis of their deformation behavior. Laser interferometry has following advantages: (i) absence of the time lag; (ii) contactless control; (iii) distance measurement; (iv) continuous record the creep parameters; (v) high measurement accuracy; (vi) absence of the standard.

It was shown [2] that employement of the laser interferometry technique for investigation of the deformation processes can obtain more precise information about one of the main creep parameters - the point of forced elasticity which separate the regions of the elastic and plastic flow stages. However creep data, observed by means of laser interferometry, may contain another feature important for understanding the nature of deformation in solids. Investigation of these one



149

requires the employment of modem information technologies and new approaches in explanation of the experimental data.

EXPERIMENTAL

A number of PMMA (Russian State Standard 17622-72) samples having cylindrical shape (6 mm in height, 3 mm in diameter) were used as a unit under deformation test. The polymer samples were y-irradiated up to 160 kGy at room temperature in vacuum (dose rate ~3 Gy/s). The deformation measurements have been carried out at room temperature in air.

Investigation of the PMMA creep properties has been studied by means of experimental plant containing the deformation machine coupled with the laser interferometer (Fig. 1). This plant has following characteristics: (i) creep rate range - from 10' to 10' m/s; (ii) deformation range - from 10' to 10' m; (iii) sensitiveness ~X/4 (where 1 is the laser light wavelength (1=0.633 ^m in our experiments)); (iv) deformation and/or creep rate measurement error - ±0.02%; (v) deformation testing time - from 1 to 10" c; (vi) sampling rate - from 1 to 10^ s'\

Principle of operation: the mirror 2 starting to move during the sample creep. At the same time the reflected laser beam (b) undergoing the Dopier frequency shift. As a result of interaction between the reflected laser light and the reference laser beam (a) light quantity AO = Oi - OQ is formed (Fig. 1). The AO intensity periodically changes with the beat frequency Av = Vi - Vo, where Vi, Vo - oscillation frequencies of the reflected and reference light waves. These changes of light quantity AO were detected by photomultipliers 4, 5.

The beat frequency Av connects with the creep rate e' by means of simple relation:

-2e. Av. a For 1=0.633 jnn the Av value changes from 0.3-10 to 0.3-10' s'

deformation e can be determined from the formula (2):

(1) Further, the

(2)

where N is the number of full waves.

CD

1L. f7:^^4kn

FIGURE 1. Optical scheme of the laser interferometer: 1 - laser; 2, 8 - mirrors; 3 - polarizer; 4, 5 - photomultipliers; 6, 7 - half-transmitting mirrors; 9 - the direction of the sample travel.

150

20U -ion euu T ime, s

FIGURE 2. Deformation interferogramme recorded during testing of the PMMA sample irradiated up to 160 kGy. The sampling rate is 10"' s. The value of compressive load is 80 MPa.

RESULTS

The common PMMA deformation dependence has been observed during creep test (Fig. 2, 3). It consists of two regions, corresponding to elastic and plastic flow stages. PMMA creep behavior changes at the deformation e ~ 10 ^m corresponding to the point of forced elasticity e* (Fig. 3). At e > e* the PMMA deformation becomes irreversible.

Sudden speeding-up of the PMMA deformation as well as creep rate has been observed during sample loading (Fig. 3). More than 20 events of the acceleration the deformation process have been indexed in time of mechanical test. Observed fluctuations of the PMMA deformation and creep rate appeared occasionally, both at elastic and at the plastic flow stage. These ones have different amplitude. Every creep rate fluctuation followed by relaxation time. It was observed that the length of relaxation time depends on the amplitude of the creep rate fluctuation. Possibly, the amplitude of the creep rate fluctuation is determined by the size of the defect region where acceleration of PMMA deformation is localized.

The phenomenon of the deformation acceleration is well-known and may reflect the "stick - slip" nature of the deformation process [3]. Possibly, the polymer deformation can localize in some centers (radiation-induced defects and so on) which provide the acceleration of the frictional flow between the polymer layers in defect regions.

20

u 10 ^*j 2

^ •

^ ' 0,6

0,2

E

0,0 0 200 400 600 SO

Time, s FIGURE 3. Changing of the PMMA sample deformation e, |xm (1) as well as creep rate e', |xm/s (2)

estimated according to (1), (2) from the Fig. 2 against the time of deformation test. The sampling rate is 10"' s. e* is the point of forced elasticity.

151

-12 -8 -4 lg((0,s')

FIGURE 4. The real part of the power-density spectrum estimated for occasional process characterizing the appearing of the creep rate fluctuations in the double logarithmic coordinates

From the temporal dependencies of PMMA deformation and creep rate (Fig. 3) the values of the observed times t, and creep rates e'tmax corresponding to the fluctuations of the creep rate were determined. Then by direct Fourier transform of these number series the power-density spectrum of the observed occasional process has been estimated (Fig. 4, where cO; ~ t;' ).

As follows from the Fig. 4, the power-density spectrum of the observed occasional process has two important properties:

S{a>)ocl/ofat(a>(ao (3)

5'(ft;) = 5'(0) atco<coo (4)

It is well-known [4] that is the features of the Fhckering noise process, which has the following parameters in our case: lg(S(0)) ~ 3.7 arb. un.; To = l/lmoo ~ 5L0 s; « = 1,80 ± 0,09 arb. un. (this estimation has got from the slope of linear dependence at co>coo (Fig. 4) by least-squares method. From the n value it was easy to determine the Hurst constant by means of the following formula [4]:

n = l + 2H^ (5)

In the case of PMMA deformation the Hurst constant value is 0,40 ± 0,05. The parameter To is a correlation time of the Flickering noise process [4]. Possibly, To can have a physical meaning as a mean relaxation time need for new fault nucleation where the deformation process can be accelerated.

Finally, we may conclude that observed stochastic part of the PMMA deformation process can be satisfactory described in the framework of the Flickering noise process.

REFERENCES

1. S. E. Vaiisberg, "Reversible radiation effects in polymers" in Radiation Chemistry of Polymers, edited by V.A. Kargin, Nauka, Moscow, 1973, pp. 376-443 (in Russian)

2. V. A. Stepanov, N. N. Peschanskaya, and V.V. Sheiizman, Durability and relaxation phenomena in solids, Nauka, Leningrad, 1984,245 p. (in Russian)

3. E. A. Brener, S. V. Malinin, and V. I. Marchenko, "Fracture and Friction: Stick-Slip Motion", (2004), http://www.kapitza.ras.ru/people/mar/BMM.pdf, accessed November 18,2004.

4. S. F. Timashev, and Yu. S. Polyakov, Fluctuation and Noise Letters 7, R15 - R47 (2007).

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Noise Measurement of Interacting Ferromagnetic Particles with High Resolution

Hall Microprobes K. Komatsu'', D. L'Hote' ' , S. Nakamae'', F. Ladieu^ V. Mosser , A.

Kerlain , M. Konczykowski", E. Dubois , V. Dupuis , and R. Perzynski

"Service de Physique de VEtat Condense (CNRS/MIPPU/URA 2464), DSM/IRAMIS/SPEC, CEA Saclay, F-91191 Gif/Yvette Cedex, France

''ITRONSAS, 76 avenue Pierre Brossolette, F-92240 Malakoff, France 'Laboratoire des Solides Irradies, Ecole Polytechnique, F-91128 Palaiseau, France atoire des Liquides loniques et Interfaces Chargees, UMR 7612 CNRS, Universite Pit

Marie Curie, - 4 place Jussieu, Boite 51, 75252 Paris Cedex 05, France

Abstract. We present our first experimental determination of the magnetic noise of a superspin-glass made of < i pico-liter frozen ferrofluid. The measurements were performed with a local magnetic field sensor based on Hall microprobes operated with the spinning current technique. The results obtained, though preliminary, qualitatively agree with the theoretical predictions of Fluctuation-Dissipation theorem (FDT) violation [1].

Keywords: Spin-glass, Superspin-glass, Magnetic noise. Fluctuation-dissipation PACS: 75.10.Nr, 75.50.Lk, 64.70.Q, 61.43.Fs, 64.70.kj, 75.50.Tt

INTRODUCTION

One of the most actively studied areas in the physics of complex systems such as spin glasses, polymers and colloids, is the dynamic correlation length that develops among interacting elements (spins, electrons, molecules, etc.) [2]. These length scales manifest themselves as various dynamically heterogeneous phenomena e.g., aging and critical behavior close to a phase transition. Recently, it has been shown theoretically that the mesoscopic out-of-equilibrium fluctuations (noise) and their relation to dissipation (fluctuation-dissipation (FD) relation) should reveal previously unknown spatial heterogeneity of the system [3] including the dynamic correlation length scales. However, there is little or no experimental reports on such local spatial-temporal correlation noise due to the extreme weakness of the thermodynamic fluctuations involved [4]. Indeed, the only trusted measurements on fluctuations in out-of-equilibrium systems were performed on 'bulk' samples, e.g. in spin-glasses [5] and in structural glasses [6], where the violation of the fluctuation-dissipation theorem (FDT) was observed.

In order to measure the fluctuations at mesoscopic scales, it is desirable to maximize the response from the individual elements of the system as well as the volume occupied by them. For this purpose, a concentrated ferrofluid superspin glass is a promising candidate. A 'ferrofluid' consists of ferromagnetic nanoparticles



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http://75.50.Lk

suspended in liquid matrix (in our case, glycerin). When the nanoparticles are sufficiently concentrated, long-range dipolar interactions among them produce spin-glass like behavior (aging, memory, etc.) at low temperatures. These systems are called 'superspin glasses' [7]. Due to their large magnetic moments, the magnetic fluctuations of a superspin glass can become accessible by a micro-meter sized high resolution magnetic field probes placed within a close vicinity of the sample [8].

Here, we report the first successful experimental attempts to measure local magnetic noise (micrometer scale) in 0.5 pico-liter of ferrofluid using a high resolution micro-Hall probe with spinning current technique [9].

EXPERIMENTAL

Hall microprobes used in our study are Quantum Well Hall Sensors (QWHS) based on (pseudomorphic) AlGaAs/lnGaAs/GaAs heterostructures [9, 10, 11]. Such carrier confinement provides a temperature independent carrier density over a wide temperature range (4 < T < 350K). The effective field sensitive Hall cross area of 1.6x1.6 im^ (Fig. 1 left panel) is located at 650 nm beneath the sensor surface. A small, 0.5 - 1 pi drop of ferrofluid made of /-Fe203 maghemite nanoparticles (diameter ~ 8.6nm, magnetic moment ~ lO"* ^IB) dispersed in glycerin (volume fraction -15%) is deposited directly on the probe surface as seen in Fig. 1, right panel. At low temperatures, the fluid (glycerin) is frozen and the only remaining magnetic degree of freedom is that of the particle magnetic moments. These moments (superspins) interact through the dipolar interaction leading to a superspin-glass transition at T^ ~ 69.5K. Detailed ferrofluid characteristics and preparation methods are found in [12]. In our previous attempts, the Hall microprobes' field resolution was hmited to 10-20x10"^ T for 40 < r < 90 K, not sufficient to measure the magnetization fluctuations of a superspin-glass. Recently Kerlain et al, reported significant field resolution improvements on 2DEG based Hall microprobes [9, 11] using the spinning current technique [13]. We have employed the same approach and achieved so far, a 10 fold field resolution improvement at 77K.

FIGURE 1. Left Panel: A Hall microprobe with a nominal cross surface area of 2x2nm^. Right panel : The same microprobe with 0.5-1.0 pi (12-15 |im in diameter)of Y-Fe203 ferrofluid deposited on top.

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Prior to measuring the magnetic fluctuation signal of ferrofluid, we characterized the Hall voltage noise spectra of a pristine Hall probe at various temperatures and applied fields. Subsequently, the ferrofluid was deposited on the probe, and cooled down to temperatures ranging from 4.2 to 80 K. Figure 2a shows the spectral magnetic field noise density (SB) as a function of frequency,/, at 60K in zero applied field with and without the ferrofluid. The spectra are rather noisy, due to the short data acquisition time (-10 minutes). This is because at 60 K, the ferrofluid is in the superspin glass state, therefore, it is necessary to perform the magnetization noise measurements 'before' the system reaches its equilibrium state. The magnetic noise of ~ 2 mG becomes apparent at frequencies below 3Hz. In the inset of Figure 2a, the squared magnetic noise (ASB^) due to fluctuations in the frozen ferrofluid sample at 60 and 77 K are depicted. ASB was estimated by subtracting the background noise of the pristine Hall sensor (black curve in Figure 2a, fitted to a power law function). According to FDT, the quantity ASB^ is related to the out-of-phase magnetic susceptibility ;^'(/) via ASB^ ~ ^'(/)//^ • ^^ order to test the applicability of FDT, we have estimated the/-dependence form oi )^'{f)lf from the ac-susceptibility data taken on a bulk sample (-1.5 |il, with the same concentration), as shown in Figure 2b. )^'{f) was found to increase as ~/^^^^ at 77 K and nearly/independent at 60 K for the frequency range of our interest (below 10 Hz). Therefore, if FD relation is obeyed, one would expect to obtain ASB^O behave as ~ 1 / / ^ ^ ^ and l//at 77 and 60K, respectively. As can be seen from the inset in Figure 2a, at 77 K where the sample is in the paramagnetic state, ASB follows the f-dependence predicted by FDT (the solid red line) as expected. At 60 K, however, ASB deviates from FDT predicted 1//tendency at frequencies below 1 Hz, hinting at a possible sign of FDT violation in the out-of-equilibrium state.

Frequency (Hz) 100 150

Temperature (K)

FIGURE 2. Left panel: Spectral magnetic field noise density as a function of frequency at 60 K in zero applied field with and without ferrofluid. The magnetic noise becomes apparent a t /< 3Hz. The inset shows the

squared magnetic noise due to ferrofluid magnetization fluctuation (see text) as a function of frequency. Right Panel: The out of phase component of ac magnetic susceptibility )^' vs. temperature of a bulk sample. The inset

depicts ;c"(/) at 60 and 77 K (see text).

155

CONCLUSION

In summary, we have observed the local magnetic noise stemming from a less than ~ 1 pi ferrofluid sample above and below T^ using a high sensitivity Hall microprobe. To the best of our knowledge, this is the first successful experimental attempt to measure the magnetization fluctuation close to the mesoscopic hmit in a (super)spin glass system. The experimental results imply that the FDT is obeyed at temperatures above Tg, where the system is in equilibrium. Below Tg, on the other hand, a sign of FDT violation has been found. Further improvements of the field resolution are under way. Although the results presented here are preliminary, the technique appears to be promising to study the FDT violation below Tg through the noise measurement.

ACKNOWLEDGMENTS

We thank Roland Tourbot for his help in the reahzation of the experimental setup. This work is supported by the RTRA-Triangle de la Physique (MicroHall).

REFERENCES

1. see for example, L. F. Cugliandolo, J. Kurchan and L. Peliti, PhysRev. E, 55, 3898 (1997). 2. P. Doussineau et al., Europhys. Lett. 46, 401 (1999); L. Bellon, et al., Europhys. Lett 51 551 (2000);

H. Mamiya et al., Phys. Rev. Lett. 82, 4332 (1999); E. L. Papadopoulou, et al., Phys. Rev. Lett. 82 173 (1999); A. Gardchareon, et al., Phys. Rev. B 67 052505 (2003).

3. H.E. Castillo et al., Phys. Rev. B 68, 134442 (2003); C. Chamon and L. F. Cugliandolo, J. Stat Mech. P07022 (2007).

4. E. Vidal Russell andN. E. Israeloff, Nature 408, 695-698 (2000). 5. D. Herisson and M. Ocio, Phys Rev Lett 88, 257202 (2002). 6. T. S. Grigera and N. E. Israeloff, Phys Rev. Lett 83, 5038 (1999); N.E. Israeloff et al., J. Non-

CrystaUine Sohds 352, 4915 (2006); L. Buisson and S. Ciliberto, PhysicaD, 204, 1 (2005). 7. P.E. Jonsson, Adv. Chem. Phys. 128, 191 (2004); P.E. Jonsson, et al, Phys Rev. B 70, 174402

(2004); D. Parker et al, Phys Rev. B 77, 104428 (2008). 8. D. L'Hote etal, J. Stat Mech: Theory andExp., P01027 (2009). 9. A. Kerlain and V. Mosser, Sensors and Actuators A 142, 528 (2008). 10. V. Mosser et al, 1997 Proc. 9th Int. Conf. on Solid-State Sensors and Actuators, June 1997

(Chicago, USA) pp. 401-404; V. Mosser et al 2003 SPIE Fluctuations and Noise Symposium, Santa Fe (NM), 1-4 June 2003, Proc SPIE 5115, 183; V. Mosser et al. Sensors and Actuators 43, 135 (1994); N. Haned and M. Missous, Sensors and Actuators A 102, 216 (2003); Vas. P. Kunets etal, IEEE Sensors J. 5, 883 (2005).

11. A. Kerlain and V. Mosser, Sensor Letters 5, 192 (2007). 12. Massart R., IEEE Trans. Magn., 17, 1247 (1981); E. Wandersmann et al, EuroPhys Lett 84, 37011

(2008); S. Nakamae et al, in press. Journal of Applied Physics (2009). 13. G. Boero et al. Sensors And Actuators A 106, 314 (2003); R.S. Popovic, 2"'' Edition, lOP

Publishing, Bristol Philadelphia (2004); J.B. Kammerer et al, Eur Phys J. Appl Phys. 36, 49 (2006); Steiner R et al.. Sensors and Actuators A 66, 167 (1998).

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Spatial Distribution Of Noise Sources In Thick-Film Resistors

A Kolek, A W Stadler, Z Zawislak

Department of Electronics Fundamentals, Rzeszow University ofTechnology, W. Pola 2, 35-959 Rzeszow, Poland.

Abstract. Experiments are reported which show that thick resistive films contain thermally activated noise sources. Their distribution within resistor volume is highly inhomogeneous and increases near resistor terminations. It occurs also that thermally activated sources of noise are highly influenced by the switching process triggered by the changes of microstructure.

Keywords: thermally activated kinetics, thick film resistors, low-frequency noise. PACS: : 72.70.+m, 72.15.-v, 85.40.Xx

INTRODUCTION

Experimental studies on thick film resistors (TFRs) show that excess low frequency noise apart from 1/f component contains contributions from thermally activated noise sources (TANSs) [1]. These noise sources most likely are located in the glassy matrix or conductive grain boundaries, and couple to resistance via the modulation of barrier heights for tunneling transitions in the conduction path [2]. In the paper several experiments are reported which prove that in TFRs:

(i) distribution of TANSs within resistive film is highly inhomogeneous, (ii) population and/or intensity of TANSs increase near resistors terminations, (iii) TANSs are highly influenced by the switching process triggered by changes of

film microstructure.

EXPERIMENT ALS

The basic experimental method used to arrive at the above conclusions was low frequency noise spectroscopy. We used the function of cross power spectral density Swxif) evaluated in frequency range I-IO"* Hz as a function of temperature T. Noise voltages V and F^ were taken from various parts of resistors which were prepared as multiterminal specimens with several side legs equally spaced along resistor length. Shape of the samples and measurement setup are shown in Fig. 1.

Resistive films were made either of lab-prepared pastes with ruthenium dioxide and lead-boro-sihcate glass (10% B2O3 15% Si02 65% PbO) as basic components. Current



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http://72.15.-v

pads and voltage side contacts to the resistors were made of commercially available pastes containing Au, Pt, Pd and Ag as basic ingredients. Resistors for measurements were manufactured in conventional process on alumina substrates: samples were screen printed through 200 mesh screen and after firing at peak temperature 800°C or 850°C or 900°C, for 10 minutes, gained the average thickness of 10-20 |j,m. Series of 8 specimens were fabricated. From each series a pair of samples with nearly matching resistance between contacts 1-7 (see figure la) at room temperature was selected for noise measurements. For each pair the measurements were made both on as-prepared devices and devices annealed at 120°C for 1000 h.

(a) (b)

FIGURE 1. (a) Shape of the samples and contacts enumeration, (b) Setup for noise measurements.

RESULTS

Feature (i) is concluded from the experiments like those reported in Fig.2, where fSwAf) is drawn versus frequency and reciprocal temperature in the form of two dimensional map. In such "noise map" TANS appears as local maximum which moves hnearly with \IT. Streaks observed in Fig. 2b originates from three TANSs with different activation energies. They appear in the noise spectrum of the whole resistor (measured between terminations 1-7) but not in the cross spectrum measured for part of the resistor extended between terminations 1-2. Thus, although the conclusion that noise sources that produce the streaks on the noise map are located somewhere between contacts 2-7 is not straightforward [3], the experiment proves that TANSs are nonuniformly distributed within the resistor.

The increase of density and/or intensity of TANSs in the film/termination interface was observed for many TFRs e.g. in the experiment reported in Fig. 3a. Noise power {SV^) in three zones of a resistor is plotted versus temperature. The zones were of the same size but were located at various distances from resistor terminations. Zone A comprises most inner sectors 3-4 and 4-5, zone B sectors 2-3 and 5-6 and zone C most outer sectors 1-2 and 6-7. As can be seen noise produced by TANSs increase significantly in zone C, nearest to the terminations. Other experiments show that also the number of TANS in next-to-termination sectors of zone C increase. An important feature of TANSs located in zone C is that they are either non-stationary or especially sensitive to microstructural changes occurring in TFR during coohng. This property is illustrated in Fig. 3b, where the results of 6 cooling-warming cycles are shown. During these cycles noise spectra were measured in the range 77-300 K for single-spacing

158

sectors 2-3, 3-4, 4-5, 5-6, two-spacing sector 3-5 and four-spacing sector 2-6. Simple sum-test shows that noise powers for the single-spacing sectors add up to give noise powers measured for larger sectors, 3-5 an 2-6. As the spectra for different sectors were measured in different experiments this result requires stationary processes at all measured sectors as well as minor changes in resistive film microstructure occurring during successive thermal cycles.

1/T, 1/K 1/T, 1/K FIGURE 2. Noise msips: JSwx(f, 1/T) measured simultaneously for (a) part of the resistor between

contacts 1-2: V = F1.7, V^ = F1.2 (b) the whole resistor: : V = V^= F1.7. Measurements were done with the bias current flowing through terminations 1-7.

150 200 7, K

300

FIGURE 3 Noise power (voltage fluctuations) in band 100-1000 Hz versus temperature measured in (a) different zones (b) different sectors of a TFR. Lines in (b) are the sum of noise powers in sectors 3-4 and 4-5 (lower) and 2-3, 3-4,4-5 and 5-6 (upper). The most upper plots refer to noise power measured

for the whole sample in six temperature cycles.

In each temperature cycle apart from the cross spectrum for a given sector also the spectrum for the whole sample (contacts 1-7) was measured. If the noise in sectors 1-2 and 6-7 had been stationary, noise power measured between terminations 1-7 in all 6 experiments would have been identical. This is not the case of Fig. 3c. Here the maximum at 150 K appears only in two of six "1-7" noise powers and is absent in the remaining four. This means that TANSs in zone C turn on/off during thermal cycles. Experiment reported in Fig. 4 shows that noise spectrum changes abruptly. A discussion provided in the paper [3] suggests that most likely origin of the switching

159

process that change intensity and energy of TANS is a redistribution of local currents driven by relaxation of mechanical stress, which in thick-film resistors appears due to the mismatch of the thermal expansion coefficients of the materials contained in resistive film, conductive terminations and the substrate.

10k

1 0 0

CAI -pi O OO ( D - ^ - ^ C3 —J OJ o - ^

CO C3

3 3 1/T, 1/K

FIGURE 4. Noise map of RUO2 (10% by volume)+glass TFR measured at terminations 2-6.

Similar measurement were performed on the large quantity of TFRs made of commercially available pastes from DuPont (DP2021, DP2031, DP2041) and ITME (Institute of Electronic Materials Technology, Warsaw, Poland) (R343, R344). Some of them contain bismuth ruthenate as conductive material. For all these resistors thermally activated noise sources with features (i)-(iii) were observed. Thus, TANSs should considered as a common property of TFRs: they appear even in manufacturer-optimized systems of compatible pastes and substrates.

We were able to conclude that: (iv) on average the number of TANSs, and magnitude of the signal they produce,

decrease with decreasing sheet resistivity, low-resistive devices of the size 2 x 15mm have at most two TANSs,

(v) decreasing firing temperature makes TANS less intensive and frequent, (vi) anneahng does not remove TANSs. The phenomenon described in the paper, i.e. TANSs modulated by some switching

process is active in temperature range of typical operation of TFRs. It becomes important factor that hmits stability and reliability of these devices especially in low temperatures where the phenomenon occurs to be more intensive.

ACKNOWLEDGMENTS

The work was supported by TURz grant No. U-737I/DS/BW.

REFERENCES

1. B. Pellegrini, R. Saletti, P. Terreni and M. Prudenziati, Phys. Rev. B, 27 1233 (1983) 2. A Kolek A W Stadler, P Ptak, Z Zawislak, K Mleczko, P Szalanski, and D Zak, J. Appl. Phys., 102

103718(2007). 3. A Kolek, A W Stadler, Z Zawislak, K Mleczko and A Dziedzic, J. Phys. D: Appl. Phys, 41 025303

(2008).

160

study of Organic Material FETs by Combined Static and Noise Measurements

Xu Yong", Takeo Minari*''", Kazuhito Tsukagoshi''''', Karlheinz Bock'*, Mooness Fadlallah", Gerard Ghibaudo", and J.A. Chroboczek"

"IMEP-LAHC IMP Grenoble, MINATEC, 3 Parvis Louis Neel, BP 257, 380016 Grenoble, France RIKEN, Advanced Studies Institute, Wako, Saitama 351-0198, Japan

"MANA, NIMS, Tsukuba, Ibaraki 305-0044, Japan Fraunhofer Inst. f. Zuverldssigkiet und Mikrointegration (IZM) Hansastrasse 2 7d, D-80686 Munich, Germany

Abstract. We studied low frequency power spectral density (PSD) of drain current, Ij, fluctuations in organic materials field effect transistors, (OMFETs), with pentacene and polytriarylamine channels and analyzed the data using parameters extracted from Id(Vg) characteristics, following a procedure developed for Si MOSFETs. We found that PSD spectra (i) vary as 1/f, (ii) show Ij" ,with 0!f=2, amplitude variation, and (iii) scale with the gate surface. That provides some elements for constructing a model for noise generation in OMFETs and for normalization of PSD data. We show that normalized noise amplitude in OMFETs can be up to lO' times higher than in their Si counterparts.

Keywords: molecular electron devices, conducting polymers, electrical noise, hopping, polaron. PACS: 85.65.+h, 72.80.Le, 72.20, 72.20, 71.38.-k

INTRODUCTION Perspectives of important applications of organic materials (OM) fuelled, for more than

four decades, intensive research on the OMs and more recently on the OM devices [1]. That effort is now focused on molecular and polymer materials having 7t-conjugated bonds [2]. We address here OM field effect transistors (OMFETs) with pentacene (P5) and polytriarylamine (PTAA) channel materials, belonging to these classes of materials.

The energy band structure of 7t-conjugated OMs resembles that of a standard semiconductor. The bands in the OMs are formed via interactions between the highest occupied molecular orbital, HOMO, and the lowest unoccupied molecular orbital, LUMO, for a large number of molecules. Conduction by extended states should predominate, but disorder induces localization, entailing transport by hopping. There is a growing consensus now that small polaron hopping [3-5] is involved in charge transport.

The OMFETs operate as thin film transistors; the source and drain electrodes are embedded in the OM, and the gate electrode is separated from the latter by a dielectric layer. Conduction in the HOMO band occurs via carrier injection into the OM through a Schottky barrier. The contact resistance, usually high, depends on the barrier and also on contact position in the structure. In most cases the energy band configuration favours the hole injection into HOMO band, and OMFETs are p-type, and show very small mobility.

Low frequency noise (LFN) in OMFETs has been addressed by several groups [6-8]. The PSD generally follows the 1/f-type dependence, often distorted by Lorentzian features. The interpretation is usually cast into the Hooge formalism, but that should be understood as a phenomenological description of the noise, rather than an evidence for the A|i model to hold.

This work has two objectives (i) demonstration that the analysis of the static characteristics developed for standard FETs [9] can be applied to OMFETs and (ii) exploration of LFN in OMFETs and its analysis in concordance with the static data. Finally, we provide some elements that may contribute to the construction of a model for LFN generation in OMFETs.



163

MEASUREMENTS AND RESULTS

The dependence of the drain-current, Id on gate voltage, Vg, were measured at constant Vd's, in linear and saturated regimes. Static, Id(Vg) and LFN were taken under point probes at ambient conditions, on both P5 and PTAA OMFETs. The samples were fabricated, respectively at RIKEN and IZM laboratories. They were stored in dry nitrogen between measurements. The mobility activation energy, Ea, was evaluated for PTAA at 235K<T<365K, from the expression |io=|Joo exp(-Ea/kT). The summary of the OMFET parameters extracted from the noise and static data is given in Table 1, benchmarked to Si MOSFETs.

Current-Voltage Characteristics and Parameter Extraction

We analyze Id(Vg) characteristics of OMFETs using a method developed for Si MOSFETs [9], involving the Y function, defined as follows,

Y = I,'yfg^ = ^l{W/i)C,//oF, x{Vg-V,), (1)

where gm is the transconductance and Ci is the gate capacitance. At strong inversion, Y is linear in Vg, allowing the extraction of Vt by extrapolation. The slope of Y(Vg),

S = ^{W/L)C,jUoVa (2) contains the intrinsic, low field mobility, |io, related to the effective mobility, |ieff,

jU^j^=jU,/l\ + ffiVg-V,)], (3)

with 6 being the mobility degradation factor. As known, lA^g) = iw/L)M^fqiVg-v,)xv,. (4)

If the access resistance RSD is comparable with channel resistance, 6 should be replaced by 6*,

0* = 0 + iW/L)M,qRsn. (5) Extraction of RSD proceeds by calculating

\/4g^ = [\ + dHVg-v,)]/s. (6)

This function is linear in Vg hence gives a precise evaluation of 0* and, subsequently, of RSD-

Low Frequency Noise Data Analysis Our analysis of the LFN data involved two widely used models, (i) A|J., the mobility fluctuation model, linking the normalized PSD with the carrier

density, n, via the Hooge parameter, ae,

SjJl/=a„/{WL-n-f), (7) and (ii) An, the carrier number fluctuation model, where

Sj^ 11/ = q'kTN,, I (WLq'f) X {g„ 11, f (8)

Nst in Eq. (8) is a surface density of noise-generating centers that is found by a fitting routine. If RSD is significant (as in the OMFETs), Eq. (8) acquires another term SR/RSD =SR(Id/VDs) • RSD can be determined from Eq.(5) and SR is evaluated as a fitting parameter. It measures the intensity of noise generation in RSD- Note that the access resistance contribution to PSD~Id , thus it gives an upswing to the data points at higher Id, and, ultimately, a rise ~Id .

164

Results and Discussion

This work deals with OMFETs having two different contact configurations, bottom contact (BC) and top contact devices (TC). In the latter the S and D electrodes are deposited atop the molecular layer. The contact resistance in TC devices is lower and explained by a metal penetration into the OM and metal clusters formation. In BC devices the OM is deposited on preformed metal electrode pattern. The Schottky barrier is then higher and mobility reduced because of defect formation [1]. Figure 1 shows typical Id(Vg) data in the linear regime (Vd= -0.5V) for a P5 (TC) device with gate dimensions W=500|am and L=50|am. The threshold voltage is obtained by finding the intersection of the line extrapolating the linear part of Y(Vg) with the Vg axis: We found Vt=-2.66V. The Y(Vg) slope gives the mobility, here 0.3cm /Vs. Then, using Eq.(5) we extract RSD- The data in Fig. 1 give RsD=33 MQ.|im (width-normalized). For an analogous BC P5 device RsD=600MQ.|im.

Applying the same procedure to PTAA OMFETs, we obtained Vt=-17V and |j.o= 6.6x10'cm/Vs. Such low mobility suggests a predominance of hop conduction in the material, expected in a polymer OM. The value of Rsd extracted in a PTAA device with W=0.6cm, and L=30|am was about 1.1x10 MQ|im. Figure 2 shows that the mobility in PTAA is activated, with Ea, found to be almost independent of Vg, and equal to 0.26eV. Similar data for P5 were reported by Meijer et al. [10] who pointed out that the activated transport is consistent with a multiphon polaron transport [4]. An alternative model involving an exponential distribution of shallow states near the HOMO band, as in amorphous Si, might also account for these observations [10].

' " ,„.

f ,™

' l - - j n r » q ^ l

'

S''^ ^ , ^ y ,

/ \ ^ \ ^

y W=500um, L=50um Vd=-0 5V

%

H„=H(„.exp(-Ea/kT)

fioo=14cmWs

-G.t..oit.s.(v) 1000/T(K-i)

FIGURE 1. P5 OMFET. Y function vs Vg, FIGURE 2. Temperature dependence of The Id(Vg) and gm(Vg) are shown in the inset. the intrinsic mobility in a PTAA OMFET. Noise PSD spectra in a TC 500x50|am P5 device, shown in Fig. 3, follow the 1/f

dependence and their intensity increases as Id . The PSD data for devices with W=500|am and L=50, 100, and 150|am, show a perfect gate surface area and current intensity scaling, as (Sid/Id )x(WL) versus IdXL/W) give a universal curve (Fig. 4).

Figures 5 and 6 provide a starting point for a discussion on the noise generation model. First, it is readily seen that in P5 the relation Sid/Id ~Id is followed throughout several orders of magnitude in Id intensity. Note that Eq. (7) suggests SH/I d ~Id , thus the A|J. model clearly does not account for the data. On the other hand, the S^/Id and constx(g^/Id) functions can be superimposed in a range of Id intensities spanning several orders of magnitude, yielding rather reasonable values of Njt. The upward swing of the noise data from (gm/Id) function at the higher Id, is explained by the appearance of the noise generated in the access resistance; in addition, the latter is stronger in BC devices, as explained. Correlation between S^/Id and (gm/Id) implies a noise generation by carrier number fluctuations (Eq 8). At this point a construction of a more elaborate model of noise generation in OMFETs would be a mere speculation. However, as the noise scales with the gate surface area, the noise generators must lie in the vicinity of the OM/dielectric interface. It is plausible that they are

165

defect-trapped polarons, with lattice distortion extending deep into the dielectric, as suggested m [5].

FIGURE 3. 1/f Spectra for SH measured in a P5 top contact device.

FIGURE 4. Noise scaling. SH(@20HZ)WL/I / data for 3 different P5 OMFETs, plotted vs IjXLAV.

W=500um, L=50um V9from+5a'to-5a',Vd=-1 OV f=20Hz

W=500um L=50jm Vg from +5V to-5V, Vd=-5 OV f=2CHz

FIGURE 5. TC P5 OMFET. S H / I / and const.(gVId) vs Ij. The constant gives Na=3.6xl0'VeVcm^

FIGURE 6. As in Fig.5, for a BC device. The constant gives Ns,=1.5xl0"/eVcm^.

TABLE 1: Comparison of parameters extracted from LFN (@20Hz) and static data for different types of FETs

W(nm) L(nm) Q(F/cm') (l,(cmWs) Rsd(ii.fim) (SIjXWXL)/Ij (cm7Hz) Na(l/eVcm')

SiHcon MOSFET

P5 TC OMFET

P5 BC OMFET

PTAA OMFET

10

500

500

60.000

0.12

50

50

30

2X10-'

5X10"*

5X10"*

1.26X10"'

577

0.34

0.29

6.6X10"'

170.7

33X10'

600X10'

l . l x io"

0.19X10"*

0.26X10'

2.24X10'

0.79X10'

6X10'

3.6X10"

1.5X10"

i x i o "

REFERENCES

1. CD. Dimitrakopoulos and D.J. Mascaro, IBM Journal of Res. and Development, 45, 11-27 (2001) A. Facchetti, Materials Today, 10, 28-37 (2007) D.A. da Silva Filho, Y. Olivier, V. Coropceanu, J.-L. Bredas, and J. Comil, "Theoretical Aspects of Charge Transport in Organic Semiconductors: a Molecular Perspective " in Organic Field Effect Transistors, Z. Bao and J. Locklin, Editors, CRC Press, 2007, pp. 1-26. D. Emin, Physical Review B, 61,14534-14553 (2000) N. Hulea, S. Fratini, H. Xie, C.L. Mulder, N.N. lossad, G. Rastelli, S. Ciuchi, and A.F. Morpurgo, Nature Materials, 5, 982-986 (2006) J. Plane and A. Francois, Phys. Rev. B, 70, 184203-1-11 (2004)

P.V. Nucliudov, S.L. Rumyantsev, M.S. Shur, D.J. Gundlach, and T.N. Jackson, J. Appl. Phys, 88, 5395-5399 (2000). L.K.J Vandamme, R. Feyaerts, G. Trefan, and C. Detcheverry. J. Appl. Phys, 91, 719-723 (2002) G. Ghibaudo, Electronics Letters, 24, 543-544 (1988).

E.J. Meijer, A. Matters, P.T. Herwig, D.M. Leeuw, and T.M. Klapwiijk, Appl. Phys. Lett., 76, 3433-3435 (2000).

166

Impact of the TiN Layer Thickness on the Low-Frequency Noise and Static Device Performance

of n-Channel MuGFETs

M. Rodrigues''^ A. Mercha", N. CoUaerf, E. Simoen", C. Claeys"'" and J. A. Martino''

"IMEC, Kapeldreef75, B-3001 Leuven, Belgium ''USP, University of Sao Paulo, Av. Prof. L. Gualberto trav. 3 n. 158, 05508-900 Sao Paulo, Brazil

'EE Depart KULeuven, Kasteelpark Arenberg 10, B-3001 Leuven, Belgium

Abstract. In this work, the impact of the TiN metal gate electrode with different thickness on the low-frequency noise of n-channel MuGFETs is investigated. Thicker TiN metal gate electrodes show a higher threshold voltage Vj, with a lower maximum transconductance and low-field mobility. At the same time, the equivalent capacitance thickness, GET, increases with a subsequent reduction of the gate leakage current. Higher number of TiN deposition cycles also showed an increase in the effective oxide trap traps density (Not). This increase of Not is more related to a larger amount of oxygen incorporated during the deposition of a thicker TiN layer, leading to a higher interfacial oxide layer (IL) thickness, than with the increase of the interface traps.

Keywords: TiN metal gate. Low-frequency noise, MuGFET PACS: 72.70.+m; 73.40.Qv

INTRODUCTION

Multiple-gate (MuGFET) devices are one of the most promising candidates for enabling the continued MOSFET scaling. In case undoped fms are employed, the threshold voltage (VT) tuning options in these devices are limited to work function (WF) engineering. Replacing the poly gate by deposited metal gate electrodes can lead to an appropriate WF [1]. Metal gate electrodes can also avoid the degradation caused by the HfSiO/polysilicon stacks of the carrier mobility, suppress the polysilicon depletion effect and reduce the dopant penetration through the gate dielectric. Titanium nitride (TiN) is being considered a promising metal gate electrode for MuGFETs. It is also known that its thickness variation can be used to tune the effective work function [2]. Therefore, the aim of the present work is to investigate the impact of different TiN metal gate electrode thicknesses on the static and LF noise performance of triple-gate n-channel MuGFET devices.



167

EXPERIMENTAL

The SOI MuGFETs under study have a 65nm of Si film on 150nm buried oxide. The gate dielectric consists of 1 nm SiOa chemical oxide (IL), onto which 2.3 nm MOCVD HfSiO is deposited. HfSiO/TiN capped with 1 OOnm poly was used as a gate electrode. For the TiN thickness, different splits were considered: 2nm (64 ALD cycles), 5nm (160 ALD cycles) and lOnm (320 ALD cycles). More process details can be found elsewhere [3]. Fig. 1 shows a TEM view of a finished device using 2nm TiN as gate electrode. On-wafer noise measurements were performed under the control of a BTA system and NoisePro control software in linear operation, for a drain voltage of VDs=50mV. The gate voltage was swept from weak to strong inversion.

FIGURE 1. Cross-section TEM of a device after complete processing.


Fig. 2a presents the input characteristics in linear operation for the different TiN metal gate electrode thicknesses. It is clear that a thinner metal gate reduces the threshold voltage (VT) [3] and increases the peak of the transconductance. Through capacitance measurements a reduction in the capacitance equivalent thickness (GET) was also observed. The values of VT and GET are presented in Table 1. The gate leakage current (Fig. 2b) confirms this decrement in GET for thinner metal gate, showing an increase with the reduction of the number of TiN deposition cycles.

nFinFETsW =25nm ; L=1^m ; 10fins nFinFETsW =25nm ; L=Vm ; 10 fins

« ID

'S ,n-

Gate Voitage (V)

(a)

0.25 0.50 0.75 Gate Voltage (V)

(b) FIGURE 2. (a) Drain current / transconductance and (b) gate leakage current versus gate voltage for n-

channel MuGFETs with different TiN metal gate thickness.

The noise spectra versus frequency (Fig. 3a) are typically of the 1/f type and the frequency exponent y below 10 kHz is approximately 0.89 for the different types of devices studied. The drain current noise spectral density (Si) versus drain current (ID) characteristic in linear operation at a frequency f=25 Hz is presented in Fig. 3b, for the different TiN metal gate electrode thicknesses. Si follows a quadratic law with the

drain current at lower gate voltages. In addition, a higher Si is observed for thicker metal gate thicknesses at low drain currents.

nFinFETsW =25nm ; L=lMm ; 10 fins nFinFETsW =25nm ; L=1^m ; 10 fins

>TiN V„s=50"lV

^ Sir X',-'- * " • - ^

V*k '^

1// ^ ^ - .

X

>, 10

0)

•5

W 1 0 ' ' 0)

o 2 10-='

0° 10' 10^ 10^ 10"

Frequency(Hz)

(a)

o 64 cycles • 160 cycles A 320 cycles

/i25Hz

A

• O

A "

O

10"' Drair

A ^s^a^

~ ' D ' • • ' '

-10^ 10"' Current [A]

(b) FIGURE 3. Noise spectral density (a) versus frequency and (b) versus drain current at 25 Hz for n-

channel MuGFETs with different metal gate thickness.

The normalized noise (SI/ID ) is represented in Fig. 4a for the different metal gate thicknesses, where a plateau in weak inversion and a roll-off at higher ID are observed. The normalized noise variation with the number of TiN deposition cycles can be seen in the inset. The thicker metal gate presents a higher normalized noise. This increase in the normalized noise, can also be observed (Fig. 4b) through the input-referred voltage noise spectral density (SvG=Si/gm ) at f=25Hz that is represented as a function of the gate voltage overdrive VGT=VGF-VT.

1E-8

1E-9

1E-10

nFlnFETs W| =

• o O

V„ ,=50mV

f^25Hz

o 64 cycles • 160 cycles A 320 cycles

25nm ; L=1pni ; lOtlns

% \ A

1E-9

E-10

E-11

o • A

nFinFETsW,

64 cycles 160 cycles 320 cycles

AAAAAAA

. o"o"5?)"<?

=25nm ; L=1pni ; lOfins

0

• o

. A A •

1E-7 1E-6 1E-5

Drain Current [A]

-0.2 0.0 0.2 0.4 0.6 0.8 1.0

Gate Voltage Overdrive [V]

(a) (b) FIGURE 4

density . (a) Normalized noise versus drain current and (b) input-referred voltage noise spectral versus gate voltage overdrive at 25 Hz for the different TiN metal gate thicknesses.

From the voltage spectral density at flatband, the effective oxide trap traps density (Not) can be derived, according to [4-5], where Svtb is SVG at flat-band condition, a is the tunneling parameter (10* cm"' for electrons in Si02)[6], q the elementary charge, kT is the thermal energy, CCET is the capacitance density corresponding with the CET. Not values are presented in Table 1 together with the low-field electron mobility (|j,n)

1/2^ extracted from the Y-function (=lD/gm ) [7].

5, q'kTN,,

Vfb " fln^^CET^J

(1)

TABLE 1.

# cycles 320 160 64

Extracted values of CET

CET [nm] 1.58 1.57 1.25

VT |x„, and Not for the n thickness studied.

V T [ V ]

0.57 0.51 0.45

-MuGFETs with different TiN metal gate

(J,„ [cmWsl 284 311 317

N„, [cm' eV-'] 3.40x10"* 1.92x10"* 1.91x10"*

A possible correlation between the low-field electron mobility and the effective trap density is represented in Fig. 5, where thicker metal gates present a higher density of traps and a reduced low-field electron mobility (that can be related with the reduced peak of the transconductance).

„5 320

64 cycles "---..... ^

"^--. "^ p

160 cycles

V„3=50mV

-,.^^>TiN

• • • • > * J

-

-" \ ^ 320 cycles

1x10" 2x10" 3x10" 4x10" 5x10"

Effective Oxide Trap Density (cm" eV')

FIGURE 5. Low-field mobility versus effective oxide trap density for the n-MuGFETs with different TiN metal gate thickness.

At the first instance, it is inferred that an increase in the TiN deposition cycles during the process steps could cause an interaction between the nitride and the high-A: dielectric, enhancing the permittivity and reducing the CET, but the opposite situation is observed. Alternatively, this phenomenon can also be related to a reaction between a higher level of O2 (generated during the ALD process) and the IL, increasing its thickness and the CET. As a result, the variation in the input-referred noise and of the transconductance can be related more with the variation caused by O2 in the equivalent capacitance than with the increase of the interface traps.

ACKNOWLEDGMENTS

M. Rodrigues and J.A. Martino would like to acknowledge the Brazilian research-funding agencies of CAPES and CNPq for the support for developing this work.

REFERENCES

1. Y. -C. Yeo, T. -J. King and C. Hu, J. Appl. Phys. 92, pp. 7266-7271 (2002). 2. K. Choi et al., ESSDERC 2005, pp. 101-104 (2005) 3.1. Ferain et al., ESSDERC, pp. 202-205 (2008) 4. E. Simoen and C. Claeys, Solid State Electron. 43, pp. 865-882 (1999) 5. G. Ghibaudo et al., Phys. Stat Sol. A. 124, pp. 571-581 (1991) 6. R. Jayaraman and C. G. Sodini, IEEE Trans. Electron Devices 36, pp. 1773-178 (1989) 7. G. Ghibaudo, Electron. Lett. 24, pp. 543-581 (1988)

170

Excess Noise in Transition Edge Sensors

E.Celasco^, F.Gatti'' and R.Eggenhoffner''

" Materials Science and Chemical Eng. Department - Politecnico di Torino, Corso Duca degli Abruzzi

24, 10129 Torino, Italy Physics Department and INFN, Universitd di Geneva, via Dodecaneso 33, 16146 Genova, Italy

'Nanoworld Institute, CIRSDNNOB, Universitd di Genova, Corso Europa 30, 16132 Genova, Italy

Abstract. The experimental excess noise observed in the power spectrum of three selected superconducting transition edge sensors is explained in terms of our correlated avalanche model. The agreement confirms that the excess noise characterized by a wide peak in the spectral frequency dependence originates from the dendritic regime in the superconducting state.

Keywords: Superconductivity, sensors, excess noise, avalanches. PACS: 81.05.-t; 85.25.-j; 74.70.-b;05.40.Ca

INTRODUCTION Among cryogenic microcalorimeters available for very sensitive detections.

Superconducting transition edge sensors (TES) are the most promising devices to detect with very high rate (100 }is) signals as tiny as, for instance, from single photon. The TES devices are designed for specific applications such as fast and sensitive cryogenic microcalorimeters or bolometers for high resolution, gamma ray. X-ray, single photon spectroscopy, dark matter research and for the detection of submillimiter as well as UV-IR radiation [1]. Essentially, a TES consists of co-deposited superconducting thin film, typically 100 }im thick, operated at temperatures across the steep transition between superconducting and normal state around 0.1 K as shown in Figure 1 and of a normal metal thin film acting as radiation absorber. In these conditions, they acts as type-II superconductors.

86 94 104 112

Temperature (mK)

FIGURE 1. Resistance versus temperature in a 80 nm Al thin film (left curve) [2] and in our 75 nm Ir thin film (right curve). The slope a=(T/R)dR/dT is of the order of 50 and 70, respectively. Arrows T^ and Tc2 show the beginning of the superconducting transition and the achievement of the macroscopic

superconducting state, respectively, obtained from the onset of slope changes.



171

http://85.25.-j

Suitable elemental or intermetallic superconductors requires a transition temperature in the range well below 1 K in order to decrease the thermal noise to low levels as the tiny phenomena to be detected. In a TES, the rapid change of the electrical resistance with temperature as in Figure 1 is used to detect particle or radiation energy.

The resolution limit is influenced by the thermal fluctuation noise of the TES, beyond the Johnson, feeding current and SQUID amplifier contributions to global noise power. However, the ultimate performance of these quantum sensors in terms of energy resolving power is mostly limited by the occurrence of an unexplained electrical noise with a broad peak at typically 1-100 kHz. Excess noise of this type

was shown to be roughly proportional to -Jcc, where a=(T/R)dR/dT is the dimensionless parameter for the transition steepness [3].

EXCESS NOISE MECHANISMS

Noise mechanisms with the feature of broad peaks were observed in the past in many bulk, thin and thick films of many families of superconducting low and high Tc materials. For instance, excess noise showing a peak in the noise spectrum was detected in MgBi thin films [4] and, previously, in a NbTi bulk [5] well below the transition temperature Tc.

The spectral peaks in the power density spectrum of superconducting samples were explained by correlated fluxon avalanches promoted by thermomagnetic instabilities [4,6]. At temperatures T«Tc, avalanches propagate through the thin film geometry with the typical features of dendritic structures. This mechanism can be associated to superconducting/normal phase separation as invoked by Cabrera in order to explain measured excess noise [2]. In the frequency behavior of the noise power spectrum, wide peaks appear superimposed to the 7/f and other noise sources.

A wide peak superimposed to the l/fsaA other noise sources was observed in many natural systems, including transport in mesoscopic Josephson junction arrays, polymer and DNA dynamics and single bacterium random walk in water. The power spectra measurements reported for the non linear dynamics of these complex systems were explained in terms of a statistical treatment of elementary events and of avalanche correlation [6].

In the present work we provide evidences of the close agreement of our avalanche model with experimental investigations in three selected TES systems. We explain the experimental peaks in the current power spectrum in terms of our statistical model in which noise is given by sequences of generahzed elementary events, by elementary events clustered in avalanches and by correlation among avalanches. The power spectrum is obtained by the following equation [6]:

where <p(co) (flat in the frequency range of experimental interest) is the power spectral function of completely independent events, p is the average number of pulses in each avalanche. To the average time period between two subsequent elementary events and

172

Vg the avalanche frequency. As discussed in ref.6, we introduced a scaling power relation between elementary pulse numbers n(x) correlated in a avalanche (i.e. the size avalanche) and the x-time interval between subsequent avalanches: n(x) °^ x^. The modulation function jf/<yj of the time-amplitude correlation entering eqn. 1 is defined as follows :

Zs((») = '^ s-l

sin[s • arctg 0)/v } f(r) (2)

where/(F) is this suitable combination of the Gamma function F:

f(r)=2r'(s+i)/r(2s+i). The main result from our generalization is that we provide a tool to calculate the

frequency dependence of the power spectrum with wide peaks. We remark that the deviations from the 1/f behavior are explained with few disposable parameters, i.e. p<po (the limit of p<p(co) when O)—>0), pTh (the mean avalanche duration), the avalanche frequency Vg (close to the peak frequency) and the scaling correlation parameter s.

RESULTS AND DISCUSSIONS

Many different types of TES devices with respect to the superconducting element, the geometry adopted for their realization and the substrate were considered in the hterature in connection with the occurrence of excess noise in the power spectral density. In some cases, the excess noise peak tums out to overcome every other noise sources, including Johnson, 1/f, phonon and thermal fluctuations noise as well as fluctuations in the superconducting parameters of the TES sensors. Although all these thermodynamic and quantum potential sources of noise have been studied in great details from several authors [1,2] for a long, the excess noise occurrence appears until now an unsolved problem.

o 103 104 10^

Frequency (Hz) 102 10^ 10*

FIGURE 2. Experimental current power density spectra (open circle), total background estimated noise (thin solid line), excess noise (open squares) are reported. Thick lines are the best-fit of excess noise data (open squares) obtained through the correlated avalanche model. The same vertical scale was adopted in the three Figures (a)-(c). Experiments in (a) are from Ullom et al. [3], in (b) from

B.Cabrera [2]. In (c) present work residts are reported.

173

In Figures 2 (a)-(c) we report the experimental power density spectra in TES with a transition near lOOmK realized by Mo/Cu on Si3N4 membrane [3], W on sihcon [2] and our laser ablated Ir on silicon with a Si3N4 layer, respectively.

The power density spectra selected for the fitting are related to TES realized with three different elemental superconducting element (Mo, W and Ir), different substrates, depositions and preparation techniques. Global noise power density levels appear of the same order of magnitude, whereas the excess noise peak is much lower in more recently prepared TES, as shown by Figure 2 (b),(c) vs. Figure 2 (a). Further, the background noise (thin solid curves in Figures 2 (a)-(c)) frequency behavior is significantly similar in the three treatments.

Following Nori's suggestion [5], if we assume that in every superconductors the duration time To of a fluxon jump is of the order of 1 }is, the average number p of elementary fluxons-events in a typical avalanche, estimated from our model parameter pTo^ turns out 225 in Mo TES prepared by UUom et al., 25 in the TES based on W by Cabrera's group and 825 in our Ir TES. These data are consistent with the result that larger avalanches need longer elapsing time between two subsequent avalanches and thus smaller peak frequencies are obtained, as indicated by the best fit Vg values 1200 Hz in Mo, 22000 Hz in W and 780 Hz in h".

The best fit scaling correlation parameter s is 1.4 in Mo, 1.1 in W and 0.7 in Ir. When s<l as in the case of Ir (s=0.7), the correlation between events is almost negligible and the spectrum approaches the Lorentzian shape as Figure 2 (c) indicates. When s>l as in W (s=l.l) and in Mo (s=1.4) the deviations from the Lorentzian spectrum are large, as shown by Figures 2 (a)-(b).

The excellent agreement of the avalanche model [thick curves in Figures 2 (a)-(c)] with the excess noise exhibited by all the TES devices explored in the present work and in references 7-8, irrespectively of their many differences, confirms that our model is suitable to describe the intrinsic physics involved in the generation of this type of noise.

REFERENCES 1) K.D.Irwin and G.C.Hilton, Cryogenic Particle Detection, C. Enss (Ed.), Topics Appl. Phys. 99, 63-

149 (2005), Springer-Verlag Berlin Heidelberg 2005 2) B.Cabrera, J.Low Temp.Phys. 151, 82 (2008) 3) J. N. Ullom, W. B. Doriese, G. C. Hilton, J. A. Beall, S. Deiker, W. D. Duncan, L. Ferreira, K. D. Irwin, C. D.

Reintsema, and L. R. Vale, Appl. Phys. Lett. 84, 4206 (2004) 4) R.Eggenhoffner, E.Celasco, V.Ferrando, M.Celasco, App/. Phys. Lett 86, 022504 (2005) 5) S.Field, J.Witt, RNori and X.Ling, Phys.Rev.Lettl-X, {1995) 1206 6) E.Celasco, M.Celasco and R.Eggenhoffner, J.Appl.Phys. 101, 054908 1-5 (2007) 7) D.BagHani, D.F. Bogorin, E.Celasco, M.Celasco, R.Eggenhoffner, L.Ferrari, M.Galeazzi, F.Gatti,

R.Vaccarone and R.VaUe, IEEE Transactions on Applied Superconductivity 2009 in press and Applied Superconductivity Conference 2008 (invited)

8) E Celasco, D BagUani, M Celasco, R Eggenhoffner, F Gatti, L Ferrari and R Valle, J.Stat.Mech. P01043 (2009)

174

Quantifying Response in a Class of Nonlinear Sensors with a Noise-Floor

A. R. Bulsara, V. In, A. Kho. P. Longhini, J. N e f f and S. Baglio, B. Ando^

'Space and Naval Warfare Systems Center San Diego, 53560 Hull Street, San Diego, CA 92152-5001, USA.

^Dipartimento cli Ingegneria Elettrica Elettronica e clei Sistemi, Univ. elegit Studi di Catania, Viale A. Doria 6, 95125 Catania, ITALY.

Abstract. Oscillations in unidirectionally coupled overdamped bistable systems are, now, known to occur

when a control parameter is swept through a critical value. Here, we use a protype device - the Coupled Core Fluxgate Magnetometer (CCFM)- as an example system and demonstrate how its response, quantified by a suitably defined non-spectral figure of merit, can be quantified in the presence of a noise floor

Keywords: overdamped nonlinear system, coupling, noise, oscillations PACS: 05.10.-a,05.40.-a,05.45.-a

Introduction

The CCFM is constructed by Mn/directionally coupling N{> 2) wound ferromagnetic cores with cyclic boundary condition, thereby leading to the dynamics,

i , = -x , + tanh(c(x, + Ax.+i^^^^+(- l ) '+ie)) . (1)

where x^{t) represents the (suitably normalized) magnetic flux at the output (i.e. in the secondary coil) of unit;', and e -C f/g is an external dc "target" magnetic flux, f/g being the energy barrier height (absent the coupling) for each of the elements (assumed identical for theoretical purposes). In the above, we have, also, reversed the orientation of successive cores so that the sign of the e term in (1) alternates; for TV odd, this guarantees that there will be two adjacent elements with the same sign of e. The dynamics of this coupled system have been detailed in [1] where it is shown that the oscillatory behavior occurs even for e = 0, however when e 7 0, the oscillation characteristics change; these changes can be exploited for signal quantification purposes, the underpinnings of the CCM. The above system (with TV = 3) has been realized in the laboratory [1] and is, currently being implemented in a cheap, lightweight, room temperature magnetic sensor.

A theoretical analysis [1] shows that the system (1) exhibits coup ling-induced oscillatory behavior with the following features: (1). The oscillations commence when the coupling coefficient exceeds a threshold value

lc = -£- Xi„f + c-' tanh-' x,.„ , (2)

CP1129, A' owe and Fluctuations, 20 International Conference (ICNF 2009) edited by M. Macucci and G. Basso

2009 American Institute of Physics 978-0-7354-0665-0/09/$25.00

175

withx.^^ = v / ( c - l) /c; note that in our convention, A < 0 so that oscillations occur for | A | > \Xc\-(2). The individual oscillations (in each elemental response) are separated in phase by 2n/N, and have period

T. = J^l^— + '- 1 (3)

which shows a characteristic dependence on the inverse square root of the bifurcation "distance" Ac — A, as well as the target signal e; these oscillations have been experimentally produced at frequencies ranging from a few Hz to high kHz. (3). Changing the target field strength also changes the period, via its influence on the threshold Xc, in addition, we immediately observe that increasing e leads to Ac going to larger (and negative) values i.e. a larger | A | is required to drive the system past its critical point. (4). Finally, it is important to observe that the elements in the ring switch sequentially between their stable steady states. This is apparent in figure 1 a which shows that during the switching interval for any of the elements, the other two elements remain in their (opposite) steady states, this gives the emergent oscillations the appearance of a "ripple" (visually, reminiscent of a soliton) that propagates around the ring.

In practice, it is usually more convenient to use the Residence Times Difference (RTD) At as a quantifier of the target signal. For a static hysteretic nonlinearity, the RTD quantifies the difference between the times spent by the magnetization state variable in each stable state. The RTD can be computed [1], from the i-th core element output signal, as

which vanishes (as expected) for e = 0. The system responsivity, defined via the derivative dAt/de, is found to increase dramatically as one approaches the critical point in the oscillatory regime; this suggests that careful tuning of the coupling parameter so that the oscillations have very low frequency, could offer significant benefits for the detection of very small target signals. Figure 1 a shows the oscillations obtained via a direct integration of the system (1); the agreement with experiment has been found to be very good [1].

The greatest sensitivity (defined via the slope of the RTD vs. e characteristic) is realized when the coupling strength is set closest to the critical value, but in this regime it can only detect a very small target field amplitude. Hence, the ability to tune the coupling to detect a range of target field strengths, must be a central feature of this device.

Quantifying the Sensor Performance; The Effect of a Noise Floor

We now introduce the "Resolution"; it is the minimum magnetic field that can be discriminated by the sensor against the background, after ambient static (homogenous)

176

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< -0.4

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1 40 60

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FIGURE 1. (Left) Time series from simulations on a CCM arrangement. A = 3,c = 3,A = —0.54,8 = 0.07. (Right) Return map of the (experimentally obtained) RTDs; each cluster corresponds to a different e and the straight line is the locus of the means.

magnetic fields have been nulled out, as described earlier. The Resolution is defined as:

R = STD{h^t)l[dh^t/de\ (5)

where Aj t represents the averaged RTD measured (in this case) at the Xj element and the denominator is simply the slope of the output-input transfer characteristic (the plot of Aj? vs. the target signal e), and represents the device responsivity. For small target signals, we expect this slope to be independent of e (i.e. A^t <x e); this is, of course, convenient for practical applications. The numerator in (5) is the standard deviation of theRTD, i.e., ^rZ3(Aj/').

We now describe how this quantity is measured, experimentally, in a CCFM consisting of TV = 3 cores with the "favored" element for measuring the RTD taken as Xj (t). We use an observation time (once the ambient static magnetic field nulling has been achieved) of O.I5 and an oscillation frequency that is adjusted (via the coupling A) to yield 15-20 cycles of the response during this observation window. Of course, the observation time can be increased, but this would depend on the circumstances of the particular application and, more importantly, on the statistics and stationarity (or lack thereof) of the ambient noise. Keeping e fixed, we compute the time-averaged RTD A^t by averaging the RTDs obtained in the observation window. The experiment is then repeated several times for the same e; each repetition yields a time averaged (over the observation window) RTD which is not necessarily the same as the others, due to fluctuations. In this way, one obtains a large number of time averaged RTDs corresponding to the fixed value of e. The quantity A^t is, then, the statistical average of these points (for the same value of e). The process is repeated for different values of e. A plot of A^t vs. e shows clusters of discrete points (each point corresponding to an average over the observation window) for each value of e. The locus of the statistical means of each cluster

177

of points then yields a straight line for small e. In figure lb, we have plotted the "return map" of the {experimentally obtained) RTDs. For a given e, each data point in a cluster represents the (window-averaged) RTD at 2 successive observation intervals each O.I5 long; thus, we generate a cluster of points corresponding to a plot (actually a residence times return map) of Aj^^^j vs. A^t„. Each cluster of points corresponds to one value of the target field e (in the absence of background noise, each cluster would collapse into a single point for that particular value of e; in this experimental sequence (figure lb), the point clusters correspond to values of e that are approximately l.OnT apart; one can use a smaller separation of e values, however this separation has been chosen for purposes of elucidation (with smaller separation the clusters tend to merge into one another). The density function of each cluster is near-gaussian, with a mean value corresponding to the averaged RTD over all the discrete points, and a standard deviation that can be computed from the observations. The locus of the mean values is the straight line. When one plots these mean RTD values as a function of e (not shown), the slope of this line (the respon-sivity, i.e. the denominator of (5) is 229.i3sT^^. In the figure, the standard deviations of the point clusters are (from left to right) 0.057315,0.054994,0.065573,0.04463^5, resulting in resolutions (calculated from (5)) of 250,240,286, l95pT respectively, resulting in a mean resolution of lAlpT for this particular realization of the sensor. The resolution is approximately constant (the deviations arise from experimental uncertainties and fluctuations) in this regime of low target signal. It is important to realize that, as e increases, the target signal becomes more easily "resolved". However, the analytic description of the response breaks down when e becomes comparable to (or exceeds) the energy barrier height of a single element (isolated) potential function; in this regime, the resolution becomes e-dependent.

DISCUSSION

We have developed a laboratory version of the CCFM that yields dynamic behavior that faithfully follows all the theoretical predictions. The (laboratory) resolution of this sensor is around 200 pT. In principle, the resolution can be improved (i.e. the numerical value decreases) by incorporating a larger number of cores; this is readily apparent when we realize that the denominator of (5) scales, linearly, as N. However, increasing the number of cores comes at the cost of increased engineering complexity, and additional onboard power (for the coupling circuitry).

REFERENCES

V. In, A. Bulsara, A. Palacios, P. Longhini, A. Kho, J. Neff; Phys. Rev. E68, 045102(R) (2003). A. Bulsara, V. In, A. Kho, P. Longhini, A. Palacios, W-J. Rappel, J. Acebron, S. Baglio, B. Ando; Phys. Rev. E70, 036103 (2004). A. Palacios, J. Aven, V. In, P Longhini, A. Kho, J. Neff, A. Bulsara; Phys. Lett. A367, 25 (2007).

178

Giant enhancement of low-frequency noise as precursor for the onset of a high-frequency

instability

p. Shiktorov*, E. Starikov*, V. Gruzinskis*, L. Varani''' and L. Reggiani**

'Semiconductor Physics Institute, A. Gostauto 11, LT 01108 Vilnius, Lithuania ^Institut d'Electronique du Sud (CNRS UMR 5214), Universite Montpellier II, 34 095 Montpellier

Cedex 5, France **Dipartimento di Ingegneria dell 'Innovazione and CNISM, Universita del Salento, Via Amesano

s/n, 73100 Lecce, Italy

Abstract. The spatio-temporal evolution of Gimn and Ryzhii current instabilities and accompanied noise behavior in «+««+ structures is investigated by the Monte Carlo particle technique. It is found that the transition from a passive-state to a generation-state is accompanied by an anomalous giant enhancement of the noise at frequencies quite lower than those corresponding to generation. By comparing various transport and noise features it is demonstrated that such a giant enhancement of low-frequency noise can be considered as an indicator for the onset of a high-frequency instability.

Keywords: High-frequency noise, Low-frequency noise. Current instabilities PACS: 72.20.Ht, 72.30.+q, 72.70.+m

INTRODUCTION

It is well known that in active semiconductor devices, the transition from a static passive-state to a dynamic generation-state is accompanied by a dramatic increase of the electronic noise that finally transforms into a regular periodic signal. In the spectral representation, such a pre-generation enhancement of noise is usually observed in the spectral band of expected regular signal generation. Typical examples are: the luminiscent enhancement in hot-carrier lasers and masers [1], the current-noise increase in n^nn^ diodes under Gunn-effect [2], the Ryzhii instability [3], etc. In both experimental and theoretical investigations (see, e.g. [1-3]), it is widely accepted that this pre-threshold increase of high-frequency noise is treated as a precursor of generation. The aim of this report is to demonstrate that in the pre-generation stage leading to the development of an instability, besides the noise enhancement within an expected generation band it can also appear a considerable increase of noise at frequencies well below those of the instability. This phenomenon can be addressed as an indicator of high frequency generation.

NUMERICAL RESULTS

To consider the above phenomenon, the spatio-temporal evolution of Gunn and Ryzhii current instabilities and accompanied low- and high-frequency noise behavior in n'^nn'^ structures is investigated through the simultaneous solution of the Boltzmann and Pois-



179

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400 600

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600 700 800

Time (ps)

900

FIGURE 1. Time behavior of the current flowing through «+««+ GaAs and InP structures in the region of threshold values for the bias U corresponding to the onset of self-osciUations due to: (a) Gunn and (b) Ryzhii instabilities. To separate the current evolutions, in the figure constant current values jo are added to each evolution. Accordingly, for biases respectively of: U = 2.6, 2.8, 3, 3.3, 3.6 and 6 V (left, from bottom to top), the jo values are respectively of: 0, 0.3, 0.7, 1.2, 2.0, and 3.6 10'' A/m^. For the right figure U = 0.036, 0.038, 0.040, 0.042, and 0.047 V (from bottom to top) correspond to the jo values respectively of: 0,0.7, 1.4, 2.1, and 2.8 10^ A/ml

son equations by the Monte Carlo particle technique [4]. For Gunn-effect originated by electron transfer from low T— to upper L— and X—valleys the simulations refer to a 0.5 — 7.5 — 0.5 urn n^nn^ GaAs structure with electron concentration n+ = 2 x 10'^ and n = 10 cm^ , which is similar to that experimentally investigated at room temperature in ref. [5]. To simulate the Ryzhii instability caused by spontaneous emissions of optical phonons at a low lattice temperature T = 10 K [3], we take a 0.02 — 2 — 0.02 ^m n^nn^ InP diode with n+ = 10'^ andn = 10'^ ctn^^.

Figure 1 presents the time evolutions of the current flowing through n'^nn'^ GaAs and InP structures at increasing values of applied voltage U (from bottom to top) in the range of the threshold region for the self-excitation of current oscillations caused by the onset of Gunn (Fig. 1 (a)) and Ryzhii (Fig. 1 (b)) instabilities. In both structures, at values of U less than the threshold ones (Uth ~ 3 and 0.043 V for (a) and (b), respectively), current fluctuations exhibit a time behavior characteristic of diffusion noise. By approaching the threshold, superimposed to the diffusive level there appear firstly relatively sharp spikes, which then begin to join into local groups of successive deviations (overshoots) following one after another. Above the threshold at U > Uth, the time sequence of the overshoots becomes practically continuous, thus forming a nearly periodic oscillating behavior of the current.

Figure 2 reports the spectral representation of current fluctuations in the high-frequency region corresponding to Fig. 1. Here one can see the well known scenario of fluctuation spectrum transformation when the system moves from a static into a dynamic state with the presence of self-oscillations. Far from the threshold at f/ < Uth, the noise spectrum is close to a Lorentzian one. Near the threshold U = Uth, an increase of noise takes place in the frequency region of future self-oscillations and their harmonics. Above the threshold U > Uth, the increase transforms into sharp resonant peaks inherent to a pronounced process of self-oscillations in the system. Accordingly, we shall show that the transition into a self-oscillating regime can also manifest itself

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FIGURE 2. Frequency dependence of the spectral density of current fluctuations in the simulated «+««+: (a) GaAs (Gunn-effect) and (b) InP (Ryzhii instability) structures at different biases.

FIGURE 3. Spectral density of current fluctuations at the frequency of 1 GHz and variance of current ;2 ; fluctuations Aj in the simulated n^nn^: (a) GaAs (Gunn-effect) and (b) InP (Ryzhii instability) structures

as function of bias.

in a low-frequency region placed well below the frequency of self-oscillations and its harmonics. This is illustrated by Fig. 3, which shows the intensity changes at low-frequency (1 GHz) when U crosses the generation threshold value. As follows from Fig. 3, a sharp overshoot of the noise intensity at low-frequency takes place in the region of threshold values U « Uth- AXU = Uth the noise intensity practically bumps up for several orders in magnitude. Then, at f/ > Uth it undergoes a significant sharp decrease which is correlated with the transition of the system into a quasi-stable region of self-oscillations. We stress that this overshoot of the low-frequency noise does not come from a redistribution of the noise intensity over the whole spectrum when the integral intensity of the noise, Afi = J^Sjj{co)dco, is conserved, as it takes place, for example, under noise upconversion [6] or when varying the length of the depletion region in Schottky-barrier diodes [7]. This is illustrated by the dashed curve in Fig. 3 which reports the dependence of the average variation of current deviations AJ^ with respect to current values constant in time. As follows from Fig. 3, the onset for a sharp increase ofAJ^ coincides with the overshoot of the low-frequency noise at f/ = Uth- The overshoot vanishes practically just after the threshold while AJ^ keeps its extra values until the self-oscillation processes are kept in the system.

181

DISCUSSION AND CONCLUSIONS

Here we discuss the general features of n'^nn'^ diode transition into the regime of high-frequency self-oscillations caused by the onset of Gunn and Ryzhii instabilities. In both cases, starting from a certain threshold value of the applied voltage U = Uth, the drift current cannot continue to keep a constant value along the whole n-region, that is the static distribution of free carriers becomes unstable. As a consequence, the processes of formation and travelling of clouds of excess charge, like domains or accumulation layers, are switched on in the n-region. When such a cloud leaves the structure through the anode n+-region, a current pulse appears in the external circuit. Just after the instability threshold, U = Uth, the appearance of current pulses has practically a spontaneous character in time and they represent in essence a random telegraph process with the power spectrum being concentrated mainly in the low-frequency region (like shot noise). With a further increase of U above the threshold Uth, the processes responsible for the time correlation are sharply switched on. At first, this leads to the formation of separate groups of current pulses following one after the other. Then, these groups try to join by forming a nearly continuous periodic sequence of current pulses corresponding to the regime of self-oscillations. Here, in contrast to the case of current pulses that are uncorrected in time, the spectral power is concentrated in resonant peaks placed at the self-oscillation frequency and its harmonics.

For the above scenario, the spectral density of the current pulses excited above the instability threshold must undergo a sharp transition from the low-frequency shot-noiselike spectrum to the high-frequency resonant noise. Accordingly, in the low-frequency region atU = Uth it should appear a bump of extra noise which then rapidly vanishes at U > Uth- From the practical point of view such a bump of low-frequency noise, when observed in experimental investigations, can be treated as a precursor for the onset of a high-frequency current self-oscillations.

ACKNOWLEDGMENTS

This work is supported, in part, by the Lithuanian State Science and Studies Fundation contract No P-01/2007.

REFERENCES

1. E. Gomikand A. A. Andronov (eds.), Opt. Quant. Electron., 23, S111-S360 (1991). 2. P. Shiktorov, V. Gruzinskis, E. Starikov, L. Reggiani, L. Varani, Phys. Rev. B, 54, 8821-8832 (1996). 3. V. Gruzinskis, P. Shiktorov, E. Starikov, Acta Physica Polonica A, 113, 947-950 (2008). 4. V. Mitin, V. Gruzinskis, P Shiktorov, E. Starikov, J. Appl. Phys., 75, 935-941 (1994). 5. V. Bareikis, J. Liberis, I. Matulioniene, A. Matulionis, P. Sakalas, IEEE Trans. Electron. Devices,

ED-41, 2050 (1994). 6. P. Shiktorov, E. Starikov, V. Gruzinskis, S. Perez, T.Gonzalez, L. Reggiani, L. Varani, J.C. Vaissiere,

Phys. Rev. B, 67, 165201 (2003). 7. P. Shiktorov, E. Starikov, V. Gruzinskis, L. Reggiani, L. Varani, J.C. Vaissiere, IEEEElectr Dev Lett,

If,, 2-4 (2005).

182

Low-Frequency Noise Characteristics of InGaAs/InAlAs Heterostructures

J. Pavelka\ N. Tanuma^ M. Tacano^ J. Sikula^ and P.H. Hander

'^Department of Physics, FEEC, Brno University ofTechnology, Technicka 8, Brno, Czech Repubhc Advanced Materials Research Center, Meisei University, Hino, Tokyo, Japan

'Department of Physics and Astronomy, University of Missouri, St Louis, USA

Abstract The low frequency noise of InGaAs/InAlAs heterostructures with various In concentration was measured in wide temperature range of 15K up to 450K and experimental characteristics compared with theoretical models of mobility fluctuations due to various scattering processes. Several p- and n-type doped lattice-matched Ino.53Gao.47As/Ino.52Alo.48As samples prepared by NTT reveal Hooge parameter a j s 1 and aj]S 2^-10'^, respectively, which is consistent with the l//energy partition fluctuations model. However, most of the n-type samples give aj] values of 4x10"^ to 3x10"^ or slightly higher in case of Ino.7Gao.3As/Ino.52Alo48As pseudomorphic structures, which is closer to the quantum 1//noise theory prediction of Hooge parameter about ajj= 10'^. Using the TLM structures noise analysis we determined, that contact noise was almost negligible.

Keywords: 1/f noise, InGaAs, HFET, MODFET, HEMT PACS: 72.70.+m, 73.40.Qw, 73.50.Td, 85.30.Tv

INTRODUCTION

The InGaAs/lnAlAs heterostructure is the basic element of modulation doped field-effect transistors (MODFETs), in which the intrinsic InGaAs channel is sandwiched between two InAlAs layers with planar doping applied in the top layer. Due to the narrow band gap of InGaAs, the excess population of free electrons diffuses across the heterojunction and high carrier concentration in the channel is reached without doping it with impurities. This lack of impurity scattering allows for very high mobility of the two-dimensional electron gas in the channel, which is hmited almost solely by the polar optical phonon scattering at the room temperature and then it is possible to study the intrinsic 1/f noise characteristics of a high purity material and their relation to the transport processes. In this paper we discuss two fundamental models of 1/f noise and compare theoretical predictions of Hooge parameter as a function of temperature dependent mobility.

According to the 1/f energy partition fluctuation model [1], mobility fluctuation is given by phonon scattering and the value of the Hooge parameter «// is given as a ratio of lattice constant d and electron mean free A. Since A = v(r) and mobility ju = e{i:)lm , where <r) is mean time between colhsions, v electron velocity and m its effective mass, assuming the equipartition law of energy in 2-D case E = kT= \l2m v we get



183

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theoretical dependence of the Hooge parameter on the mobility

d d_

'I" (1)

Another model of quantum 1// noise considers fundamental 1// fluctuation of the quantum mechanical cross sections and scattering rates, apphcable to the interaction of a charged particle and its own field [2]. It represents the lowest limit of ubiquitous 1// noise, inevitable in any system. The conventional, or incoherent quantum \lf noise theory gives Hooge parameter as

4a (Kvf Sn c

where a = e^/h^c is Sommerfeld's fine structure constant, c the speed of light in vacuum and Av is the change in the velocity of the carriers in the interaction process considered, which is given by cross-correlation formulas derived in [3] and shghtly modified in [4].

(2)


The relation between the intrinsic \lf noise and mobility and scattering processes can be resolved from their temperature dependences, measured in both n-type and p-type InxGai.xAs/Ino52Alo48As quantum well heterostructures. We used samples with either lattice matched (x = 0.53) or pseudo-morphic (x = 0.7) channel. The schematic view of samples' structure is given in the Tab. 1. All samples were grown on the semi-insulating InP substrate by MOCVD by NTT Advanced Technologies. The 5-doped MOCVD process is expected to induce less deep levels compared with MBE grown substrates. The ungated Hall elements were formed using In or unalloyed Ti/Pt/Au ohmic contacts and exposed cap layer removed by etching.

TABLE 1. Cross-sectional view of sample stracture: lattice-matched n-type (A) and p-type (C), pseudomorphic n-type (B)

A

n-Ino.53Gao.47As 2x10" cm-'20 nm

i-InP 6 nm

i-Ino.52Alo.48As 10 nm

S-doping 1-2x10'^ cm-^


i-Ino.53Gao.47As 15 nm


S.I. InP

B

n-Ino.53Gao47As 2x10" cm-Mo nm

i-InP 5 nm

i-Ino.52Alo48As 10 nm

n-Ino.52Alo.48As 5xl0"cm-^7nm

i-Ino.52Alo48As 5 nm



S.I. InP

C

p n-Ino.53Gao.47As 2x10" cm-'20 nm

i-InP 6 nm


p-Ino.52Alo.48As 2x10"* cm-'20 nm




S.I. InP

layer

cap

etch stop

barrier

charge supply

spacer

channel

buffer

substrate

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http://i-Ino.53Gao.47As


http://n-Ino.52Alo.48As





http://p-Ino.52Alo.48As




Temperature dependence of the mobility // and sheet carrier density ns was determined by the Hall voltage measurement and results are plotted in Fig.l. Theoretical analysis indicates, that mobility at 300K is limited by polar optical phonon scattering to about 11000 cm^/Vs, whereas at low temperatures below 50K mobility is restrained by alloy scattering to about 70000 cmWs for lattice-matched lno53Gao47As/lno52Alo48As. Much lower mobility value of about 25000 cm^A^s at T= lOK in pseudomorphic samples is probably caused by interface scattering due to the smaller thickness of channel and spacer layers.

80000

70000

60000

50000

40000

30000

20000

10000

0

^ \ A

— 4 ^

\ '•• '•• a)

\ ^

I optical \-phd)non| \ scattering \ '

----V---i—

I I

in

0 50 100 150 200 250 T [K]

300

180

160

140

120

100

80

60

40

20

0

i s ' ^—A—*^

f^ !

\ b)

^J_____L____

•"""'^r^'^]—

0 50 100 150 200 T [K]

18

16

14

12

10

8

6

4

2

0 250 300

FIGURE 1. a) Temperature dependence of mobility in n-type samples and theoretical limit given by optical phonon scattering (A denotes Ino.53Gao.47As and B Ino.7Gao.3As channels)

b) Temperature dependence of mobility and sheet carrier density in p-type sample

10'

10"

0 o o X

10"'

10"'

-^ , _ _ L . NTT-p ^ _ ^

Energy model

n ! Quantum model J^

y^yr' P : 100 200 300

T [K]

FIGURE 2. Temperature dependence of p- and n-InGaAs/InAlAs Hooge parameter, measured on various samples (NTT greek cross, lattice-matched T, C and pseudomorphic H).

Lines stand for 1/f energy partition fluctuation and quantum 1/f noise theories simulation

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Temperature dependence of the voltage noise spectral density was measured using NF Instruments preamplifier with battery power source, Advantest R9211A FFT analyser and Iwatani helium cryostat and Hooge parameter determined. Results are plotted in Fig. 2 together with theoretical «//(?) dependence, evaluated according to Eq. 1 and Eq.2 for both 1//noise models. The l//energy fluctuation model considering phonon scattering is in a good agreement with experimental results for both p-type and n-type samples, provided by NTT as a greek cross Hall elements with indium contacts. However, much lower values of Hooge parameter were measured on Hall bar samples prepared in our laboratory using NTT wafers with identical mobility. For lattice-matched samples an was in the range of 4x 10" to 3x 10" for temperature below 200K, where noise spectral density was l//"like. These values are close to quantum 1//noise theory prediction of an slightly lower than 10" . Similar results we obtained on samples provided by Communication Research Laboratory [5]. For pseudomorphic samples Hooge parameter was about one order higher. There was almost no difference between Indium or Ti/Pt/Au contacts. The influence of contact noise was found negligible by measurement of TLM structures of 5, 10, 25, 40 and 80|im width in comparison with 1000 |im Hall bar samples. For temperature higher than 250K generation-recombination noise with \lf spectra became dominant and from the Arrhenius plot of Lorentzian spectra time constant up to temperature 440K the value of activation energy of traps was determined A£ = 0.63eV.

CONCLUSION

Temperature dependence of Hooge parameter was measured on InGaAs/lnAlAs samples with various In concentration and mobility and compared to theoretical models of intrinsic 1//noise. For p-type samples experimental results were consistent with 1//energy partition fluctuation model, whereas much lower values of «//= 10" measured on n-type samples seem to approach quantum l//noise theory limit.

ACKNOWLEDGMENTS

This research was supported by grant GACR 102/08/0260, JSPS fellowship of the first author and project MSM0021630503.

REFERENCES

1. T. Musha and M. Tacano, PhysicaA 346, 339 (2005). 2. P. H. Handel, IEEE Tram. Electron. Dev. 41, 2023 (1994). 3. G. S. Kousik, C. M. van Vliet, G. Bosnian and P. H. Handel, Advances in Physics 34, 663 (1985). 4. T. H. Chung, P. H. Handel and J. Xu, Proc. Noise in Physical Systems 1/f Fluctuations, edited by T.

Musha and S. Sato, Kyoto, 1991, pp. 163-166. 5. K. Shinohara, Y. Yamashita, A. Endoh, K. Hikosaka, T. Matsui, T. Mimura and S. Hiyamizu, Jpn. J.

Appl Phys 41, L437 (2002).

Noise Limitations of FET-Based Biochemical Sensors

Peter H. Handef and Amanda M. Truong^

"Department of Physics and Astronomy and Center for Nanoscience University of Missouri, Saint Louis, 1 University Boulevard, Saint Louis, MO 63121, USA

Abstract. The gate surface of an FET can be painted with antibodies for certain biological agents, causing it to change its surface potential by 5VG and therefore also the channel resistance by 5Rch =K5m when agents of mass 5m are adsorbed. Here K is the effectivity of the antibody paint. The resulting channel current change competes with quantum 1/f noise generated in the channel. This competition determines the sensitivity. By choosing the right chemical activation deposit on the gate surface, a chemical sensor is obtained. In the present paper for the first time a complete, reliable, analytical formula is provided for the flicker floor of the FET-based biological and chemical sensors, allowing for their simpler optimization.

Keywords: Sensors, FET-based Biochemical sensors. Quantum 1/f noise in FET-based sensors. PACS: 82.47.Rs; 85.30.Tv

QUANTUM THEORY OF 1/F NOISE The 1/f resistance fluctuations 6R in a biased semiconductor yield voltage

fluctuations. They are caused by conventional and coherent quantum 1/f effects (Ql/fE), given by the general quantum 1/f formula [l]-[5], shown below in Eq. (3). It contains the coherence parameter s = 2N'ro where N' is the number of current carriers per unit length in the direction of current flow, and ro =e^/mc^=2.84.10"^^ cm is the classical radius of the electron.

The Coherent Ql/fE arises in a current of charged particles, from the definition of the electron as a bare particle plus a coherent state of the electromagnetic field. The expression for the power spectral density of fractional fluctuations given by the coherent quantum 1/f theory is

2/r St(f) = (1)

" TtfN The Conventional Ql/fE is a fundamental quantum fluctuation of physical cross

sections o and process rates Y. The Conventional Ql/fE is caused by the bremsstrah-lung energy and momentum losses of charged particles, when they are scattered or accelerated in any way. The spectral density is given by

S^°"''(/) = ^ ^ f — 1 (2) f 37:fN K c )

Both components of quantum 1/f noise are combined in the general quantum 1/f formula



187

S,,=—S''°'"(f) + ^-S''°''(f) (3) ^ 1+^ f 1+^ f

where the coherence parameter, s = 2N'ro, indicates to what extent the energy of the drift motion of the electrons adds up coherently or conventionally over the cross section of the current, in the Hamiltonian. The spectral density is related to the fractional fluctuations in each region using the following relation

S,,=R^-S^(,f) (4)

For VDS constant, the spectral densities of current and resistance fluctuations are related by

^ S R _ ^ S I

R^ I^ (5)

Quantum 1/f Noise in the FET Channel

The element of resistance dR along the current-carrying channel can be written dR =dy/(q}inN'), where }in is the electron mobility [6]. The quantum 1/f noise in the element of resistance dR is given from Eq. (3) by the basic quantum 1/f formula for mobility fluctuations, (dR)'^<(6dR)^>f =aH/fdN, where dN=N'dy=ntdy is the number of carriers in the element of length dy along the channel of thickness t and concentration n. The 1/N dependence of Eqs. (l)-(3) is used to define an effective Hooge parameter an = Uett shown below Eq. (8). Substituting, we obtain f<(6dR)^>f = [dRdy/q}inN']aH/dN. Multiplying on both sides with the channel current, Ich = -dV/dR, and integrating, yields

flch<(6R)'>f = -Ivs'^LdV/q^nN'"] (6)

Introducing Fermi statistics, for a GaN/AlxGai-xN doped n-channel HFET the number of carriers per unit length N' of the 2-dimensional channel is written in the form N'(y) = ZDeffVthlogjl + exp[(VG*-V(y))/Vth]}, where Deff = qm/%h is q times the effective density of states per unit area. Multiplying Eq. (6) by I/f, this yields the spectral density of VDS fluctuations in terms of device width Z, channel current I, and thermal voltage VTh=kT/q, with X=l+exp[(VG*-V)/Vth]

<(c^„,)^>,= ' ^ ' i(a^/Mj[dX/(X-l)WX] (7)

Here Xo and Xi are the values of X at the source (V=0) and at the drain (V=VDS),

while aH= aeCy) is the y-dependent quantum 1/f coefficient. In this case of FETs and HFETs of much larger width w»LDs>t, the kinetic energy

Ek of average motion with drift velocity Vd per unit length in the direction of LDS is still given by Nmvd^/2, but the magnetic energy Em per unit length in the direction of LDS is roughly proportional with the first power of w only, instead of w , and can be approximated by Em = 7t[ln(w/2LDs)]LDs[nevS/c]^/w. This indicates decoherence along the device width w. This yields a coherence ratio of s = Em/Ek ~ 7tnrotLDsln(w/2LDs),

which shows that only an effective width w = Weff about equal to LDS should be used in the calculation of the coherence parameter s in this special case; larger widths are subject to de-coherence. This favors lower, mainly conventional, quantum 1/f noise in these devices, in spite of the large values of w. It also explains for the first time why the huge widths are possible with impunity, i.e., without causing the much larger coherent quantum 1/f noise to affect the device.

In general, part of the channel may be sub-threshold (the part closer to the drain), starting at V(y) =Vsat=VG . The use of Fermi statistics becomes unavoidable, as is shown in Eq. (4). However, although Eq. (4) includes the field dependence of a and }i, it fails to reflect the physical situation correctly, because it neglects the impact ionization effect in the subthreshold section of the channel length. This effect is roughly included below in a first approximation, along with the variation of AlGaN polarization with y, by dividing the channel along its length into coherent and conventional parts.

Using the fact that the velocity of the carriers shows strong saturation in the subthreshold section, to include the impact ionization effect, we return to the variable V in Eq. (4), we split up the integral at V(y) = Vc-Vth, and write the second, subthreshold, part of the integral separately, with the number of carriers per unit channel length defined as N' = Ich/qvs, where Vs is the saturation velocity,

fi,,<(5R)->,=]'^+T"-q>^^

M'-'-Ch 1 - ^ -L'eff ' th ' DS "'"ch ' DS ' DS

:K^fI,,((5mf)^=K%,C;

1 s where a^.. = a;„„ H a^^^, and C is the coefficient in <(6m)^>^C/f, defined by

l+s l+s Eq. (8). The second term in curly brackets must be omitted when negative. Note, however, that the first integral was calculated here neglecting the variation of tteff, and the second integral was performed neglecting the variation of Vg.

With this quantum 1/f definition of C in Eq. (8), the ultimate two-sample (Allan) variance obtainable is <(dm)^>=2Cln2. This limits the achievable minimum detectable concentration of the biological agent.

DISCUSSION

Our calculation of the fundamental 1/f noise in the channel is applicable both to HFET-like and FET-like biochemical sensors. It shows that the ultimate achievable detection limit is proportional to the biological or chemical effectivity K of the substance (e.g., an antibody to a certain bacterium) that was deposited on the gate, as defined by Eq. (8). From a physical point of view, the direct action of the biological agent is on the surface potential VG of the FET-like structure. The change in VG, in

turn, acts on the carrier density and energy level structure of the 2-D electron gas in the channel, thus causing a change in the channel current I and resistance R at constant applied bias VDS- If the change 6VG in surface potential is small, we can introduce a coefficient k as the change 6VG per unit mass 6m attracted by the antibody deposit on the gate. We obtain for constant VDS,

6R/R = -6I/I = -gmSVc/I = -gmk6m/I =K6m/R, and K =-gmkR^A^Ds, (9)

where gm is the trans-conductance of the FET. The two constants K and k characterizing the effectivity of an antibody or chemically sensitive gate deposit, are thus related by the family of I/V characteristics and the implied value of gm-Multiplying Eq. (8) with I/(VDS), we finally obtain for the smallest detectable amount 6m the (Allan) two-sample variance

(6m,^ ) = 2C In 2 = ^ < (6R)^ >, =

2 ^ , . (10) 2R ln2 a,ffl,, , aco„vqVs . " ^ i '^"')0O1-

~ K V V , S qZ'Df„V^V,s le, V,s V,s

Here 0() is the step function of the preceding round bracket, zero for negative values and 1 for positive values of the bracket-

The above expression provides an analytical expression for the achievable flicker floor of biological and chemical detection, that can't be reduced by longer time averaging. Eq. (10) can also be used to optimize industrial FET-based biochemical sensors and detectors.

ACKNOWLEDGMENTS

The support of the Army Research Office and Army Research Laboratory is thankfully acknowledged.

REFERENCES

1. P.H. Handel: "Fundamental Quantum 1/f Noise in Small Semiconductor Devices", IEEE Trans, on Electr. Devices 41, 2023-2033 (1994).

2. P.H. Handel: "Coherent and Conventional Quantum 1/f Effect" Physica Status Solidi bl94. 393-409 (1996).

3. P.H. Handel: "Noise, Low Frequency", Wiley Encyclopedia of Electrical and Electronics Engineering, vol. 14, pp. 428-449, John Wiley & Sons, Inc., John G. Webster, Editor, 1999.

4. P.H. Handel, A.G. Tournier, "Nanoscale Engineering for Reducing Phase Noise in Electronic Devices," Proceedings of the IEEE Vol. 93, No. 10, October 2005, pp. 1784-1814

5. Hadis Morkog: Handbook of Nitride Semiconductors and Devices, Noise Chapter, Springer, 2007. 6. R.M. Warner Jr., and B.L. Grung: Semiconductor-Device Electronics, Holt Rinehart and Winston/

Saunders, 1991.

Noise in Green Transistors (Small Slope Switches)

Christoph Jungemann

EIT4, Bundeswehr University, 85577Neuhiherg, Germany

Abstract. Small slope switches with very small subthreshold swings have been proposed recently to overcome the IC power problem. In the case of these new devices the source/drain current controlled by the gate is either due to avalanche breakdown (IMOS) or band-to-band tunneling (TFET). The impact of these new transport mechanisms on noise is investigated. The performance of the IMOS is not very promising. Its noise is by orders of magnitude too large due to the avalanche breakdown and it consumes too much power. In the case of the TFET the noise is similar to conventional MOSFETs, but the gate/drain correlation coefficient of noise is larger.

Keywords: Noise, impact ionization transistor, tunnel transistor, silicon PACS: 72.10.Bg,72.30.+q,72.70+m

INTRODUCTION

The continuous scaling of CMOS devices has led to lower and lower supply voltages. This in turn has reduced the available voltage swing for turning off the MOSFETs. Since the minimal subthreshold swing of the classical MOSFET is limited to 60mV/dec at room temperature [1], this has led to an increase of the leakage currents by a factor often for each new technology cycle [2]. In the current semiconductor technologies a substantial part of the power consumption is due to this drain-source-leakage. New types of MOS-based devices have been developed which overcome the 60mV/dec limit. The two most prominent developments are the impact ionization transistor (IMOS) [3, 4] and the tunneling transistor (TFET) [5, 6], which are also called green transistors [7]. These devices are not based on the usual transport mechanisms. The IMOS is based on gate-controlled avalanche breakdown between drain and source and the TFET on gate-controlled band-to-band tunneling between drain and source. While the stationary and even some transient properties of these devices have been discussed (e.g. [8]), the noise properties have not yet been investigated. Therefore, the two device types are investigated by device simulation with special focus on their RF noise properties.

SIMULATION APPROACH

The noise simulations are performed with the 2D bipolar hydrodynamic model of Ga-lene III [9, 10]. All transport and noise parameters are generated by consistent full-band Monte Carlo simulations. Impact ionization is described by the local temperature model [11, 12] and band-to-band tunneling by the local field model of Kane [13]. The band-to-band tunneling rate is multiplied with a constant factor to obtain drain currents



191

FIGURE 1. Impact ionization MOSFET.

comparable to conventional MOSFETs. Convergence of the device model is checked by a simultaneous Newton solver for all equations.

IMOS

An IMOS is a gated pin-diode, in which the drain current is due to a controlled avalanche breakdown (Fig. 1 [14]). With a positive V^s the pin-diode is reverse biased and the drain current is very small as long as the bias is below the breakdown voltage. By applying a negative VGS a hole inversion channel is induced under the gate and the distance between the n-drain and the p-region is thus reduced. The decrease in the distance leads to an increase in the electric field and a decrease in the breakdown voltage. The breakdown voltage therefore depends on the gate bias. With a strong negative gate bias the breakdown voltage can be lowered below a sufficiently large drain/source bias and avalanche breakdown occurs. Since the device turn-on due to avalanche breakdown is quite abrupt, a subthreshold swing of less than 60mV/dec might result. In Fig. 2 the output characteristics of a CIMPAT-IMOS [14] is shown and the avalanche breakdown leads to an abrupt change in the drain current. Subthreshold swings of a few mVs/dec have been achieved (e.g. [3, 4]). The corresponding Fano factor of the drain current noise is shown in Fig. 3. In a conventional MOSFET the Fano factor is usually below one, whereas the Fano factor of the IMOS is at least two orders of magnitude larger than that due to avalanche breakdown and the IMOS is therefore extremely noisy. This might cause problems even in digital applications. In addition, the IMOS consumes far too much power due to the rather high supply voltage. Although improved IMOS devices have been proposed, even those require supply voltages of two or more Volts [15]. This is more than twice the supply bias of current CMOS technologies.

Variants of the IMOS are based on a controlled kink effect in SOI devices [16]. Since the turn-on is still due to impact ionization and feedback occurs, the noise is still very large and the devices might be slow [17, 18]. Further problems are caused by the rather unconventional I-V characteristics, which, for example, do not permit rail-to-rail operation [3, 14].

192

F

^

i n - -

10"* -

— _ - up

16.50 17.00

Drain/source bias [V]

FIGURE 2. Drain current versus drain/source bias with Vgs = —3.5F for a 5/JOT-CIMPAT device.

16.00 16.50 17.00 17.50

Drain/source bias [V]

FIGURE 3. Fano factor versus drain/source bias with Vgs = —3.5F for a 5/JOT-CIMPAT device.

TFET

A better candidate for a green transistor is the TFET. The device structure is very similar to the IMOS, where the gate covers the whole j9^-region (Fig. 4) and the junctions are very abrupt. Instead of avalanche breakdown a gate-controlled Zener breakdown is used. A positive gate bias leads to tunneling in the junction on the left-hand side and a negative to tunneling on the right-hand side (this can be avoided by an asymmetric gate). If the device is properly designed [19], the average subthreshold slope can be smaller than 60mV/dec at room temperature. This might allow to scale the supply bias considerably below one Volt [7]. The I-V characteristics are still unconventional and might require specifically designed circuits. The pn-structure might require an SOI-technology and electrical insulation of each individual TFET. Since the current is controlled by tunneling, noise is mostly due to tunneling and therefore strongly localized in the device. This leads to a strong correlation of the gate and drain noise. This in turn might reduce the noise figure [20]. Comparison of the TFET with a similar MOSFET (similar drain current and cutoff frequency) shows that both devices have similar noise

193

S o—i

FIGURE 4. Timnel-MOSFET (the bulk is lightly p-doped for threshold adjustment).

E o

-1.0 -0.5 0.00 0.50 1.00 1.50 2.00

Gate/source bias [V]

FIGURE 5. Input characteristics of the TFET.

properties. A clear decrease of the noise figure in the case of the TFET is not found. Thus, with respect to noise it seems that the TFET is neither better nor worse than a conventional MOSFET. On the other hand, the rather unusual device characteristics of the TFET might make it difficult to integrated this device into standard circuits.

TFETs suffer from rather small drain currents (in the above simulations the drain current was artificially increased to allow for a fair comparison with standard MOSFET s). New device concepts have been developed to increase the band-to-band tunneling rate [7, 21, 22]. In [7] it is suggested that TFETs could operate at supply voltages as low as 0.2F, which is much lower than the value of current CMOS devices. This would reduce the power consumption considerable.

CONCLUSIONS

With respect to RF noise and other aspects the IMOS is not a very promising device. Its noise is by orders of magnitude too large due to the avalanche breakdown. In the case of

194

— V B S = 2 0 V

V„s=1.5V

— V D S = 1 . 0 V

V„„=0.5V

M 0.00 -

-1.0 -0.5 0.00 0.50 1.00 1.50 2.00

Gate/source bias [V]

FIGURE 6. Imaginary part of the gate/drain-correlation factor of the TFET.

the TFET the noise is similar to conventional MOSFETs, but it has a larger correlation coefficient of the gate/drain noise. Its unusual device characteristics might prohibit its wide spread application.

ACKNOWLEDGMENTS

The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement n°216171 (NANOSIL).

REFERENCES

1. S. M. Sze, Physics of Semiconductors Devices, Wiley, New York, 1981. 2. International Roadmap Committee, p u i l i c . itrs. net (2005). 3. K. Gopalakrishnan, P. B. Griffin, and J. D. Plummer, IEEE Trans. Electron Devices, 52, 69 - 76

(2005). 4. K. Gopalakrishnan, R. Woo, C. Jimgemann, P. B. Griffin, and J. D. Plummer, IEEE Trans. Electron

Devices, 52, 77 - 84 (2005). 5. W. M. Reddick, and G. A. J. Amaratunga,^/J;?/. Phys. Lett., 67, 494-496 (1995). 6. W. Hansch, C. Fink, J. Schulze, and I. Eisele, Thin Solid Films, 369, 387 - 389 (2000). 7. C. Hu, "Green Transistor as a Solution to the IC Power Crisis," in International Conference on Solid

State abdIntegrated Circuits Technology, Beijing, 2008, p. P5. 8. C. Shen, J.-Q. Lin, E.-H. Toh, K.-F Chang, P Bai, C.-H. Heng, G. Samudra, and Y.-C. Yeo, IEEE

Tech. Dig lEDM, pp. 117-120 (2007). 9. C. Jungemann, B. Neinhiis, and B. Meinerzhagen, IEEE Trans. Electron Devices, 49, 1250-1257

(2002). 10. C. Jungemann, B. Neinhiis, S. Decker, and B. Meinerzhagen, IEEE Trans. Electron Devices, 49,

1258-1264(2002). 11. R. K. Mains, G. I. Haddad, and P A. Blakey, IEEE Trans. Electron Devices, 30, 1327-1337 (1983). 12. H. J. Peifer, B. Meinerzhagen, R. Thoma, andW. L. Engl, "Evaluation of impact ionization modeling

in the framework of hydrodynamic equations," in IEEE Tech. Dig. lEDM, 1991, pp. 131-134. 13. E. O. Kane, J. Phys. Chem. Solids, 12, 181-188 (1959).

195

14. F. Mayer, C. Le Royer, G. Le Carval, L. Clavelier, and S. Deleonibus, IEEE Trans. Electron Devices, 53,1852-1857(2006).

15. C. Onal, R. Woo, H.-Y. Koh, P. Griffin, and J. Plummer, IEEE Electron Device Lett., 30, 64-67 (2009).

16. U. Abelein, M. Bom, K. Bhuwalka, M. Schindler, M. Schlosser, T. Sulima, and 1. Eisele., IEEE Electron Device Lett., 28, 65-67 (2007).

17. W. Jin, P. C. Chan, S. K. H. Fung, and P K. Ko, IEEE Trans. Electron Devices, 46, 1180-1185 (1999).

18. C. Jungemann, B. Neinhiis, C. D. Nguyen, and B. Meinerzhagen, Impact of the Floating Body Effect on Noise in SOI Devices Investigated by Hydrodynamic Simulation, Proc. SISPAD, Munich (Germany) (2004).

19. K. K. Bhuwalka, J. Schulze, and 1. Eisele, IEEE Trans. Electron Devices, 52, 909-917 (2005). 20. T. C. Lim, R. Valentin, G. Dambrine, and F. Danneville, IEEE Electron Device Lett., 29, 118 - 121

(2008). 21. T. Krishnamohan, D. Kim, S. Raghunathan, and K. Saraswat, IEEE Tech. Dig. lEDM, pp. 947-949

(2008). 22. M. Schlosser, K. Bhuwalka, M. Sauter, T. Zilbauer, T. Sulima, and 1. Eisele, IEEE Trans. Electron

Devices, 56,100-108 (2009).

Low-Frequency Noise in Electronic Devices -Past, Present and Future

M. J. Deen and O. Marinov

Electrical and Computer Engineering, McMaster University Hamilton, Ontario, L8S 4K1, Canada (E-mail: [email protected])

Abstract The low-frequency noise (LFN) became prominently large in current small-geometry devices, and it occurs as a limiting factor for diverse applications. Considerable interest is paid to the "slow" (as compared to the operating frequency of the devices) noise. Therefore, we address the trends for LFN from an extensive analysis of data from many publications.

Keywords: Low Frequency Noise in electronic devices, LFN, flicker noise. PACS: 72.70.+m; 74.40.+k.

INTRODUCTION

The history of low-frequency noise (LFN) is both rich and old. For example, one finds publications on noise in electronic devices by Schottky in 1918 for shot noise [1] and Johnson and other researchers in 1920s [2, 3]. The noise was normally taken small enough in 1970s, although the presence of 1/f noise was noted in many physical systems [4]. However, the magnitude of 1/f noise is inversely proportional to the charge transport area [5-8], and the recent nm-scaled electronic devices are accompanied with increased impact of LFN in diverse applications such as sensor systems, portable electronics, wireless and

digital communications. Virtually all applications are affected by the LFN in some way. We address the trends for LFN in electronic devices, as seen from device downscaling projected in ITRS and from the experimental results reported in the literature.

LFN IN ITRS

Entering into the new millennium, the problems are complicated in following Moore's law of "twice per 3 years" for integration. The International Technology Roadmap for Semiconductors (ITRS) [9] follows Moore's law for gate length of MOS devices (Figure 1). However, downscaling of other devices is slowed down, e.g. the emitter width WE of bipolar transistors (BJT) saturates at 70nm. The 1/f noise in minimum-sized MOS transistors will continue increasing in future, whereas the noise in bipolar transistors perhaps will not change significantly, owing to the reduced rate of downscaling.

FIGURE 1. Minimum sizes and LFN in ITRS, with arrows

showing the evolution of the values vs. four editions of ITRS



197


LFN IN EXPERIMENTS

Almost 85 years ago in 1925), the "colored" L F N was reported by Johnson [2], known for Johnson-Nyquist white thermal noise [10, 11] (1928). Discussing the results in [2], the L F N was termed "flicker" noise in 1926 by Schottky [3], known for the white shot noise [1] (1918), and who proposed two features: quadratic relation between L F N power and D C current, written later as normalized 1/f noise, Snonn=S/DC^=KF/f, and frequency slope variation between 0 and 1/P, covering almost all L F N spectra investigated over the past.

FIGURE 2. Normalized 1/f noise at f=Af=lHz. Vacuum tubes (+), using cathode area; bipolar transistors ( • , data is aggregated), using emitter

•g area; MOS transistors (D, aggregated), using gate :i area; by using surface area (O) and cross-section o area ( • ) of nanowire and CNT devices; some III-V

semiconductor HBTs (A) and LEDs (X), using junction area. The predictions for MOS-Analog, MOS-RF and BJT and the Moore's law (right-hand axis) are from ITRS 2006. The regression line

1920 1940 1900 1980 2000 2020 2040 (-0.5dB/yCar) indicatcs thc prcdictions lu ITRS. Year

In contrast to shot and thermal noise, for which fundamental theories are known [1, 11], there is no unique explanation for the "colored" LFN. There are adjustable parameters in L F N models , e.g., an in Hooge equation Snonn=aH''(f^N) for intrinsic fluctuation of the population of N carriers [12], or Kp in the SPICE 1/f noise model. A reciprocal dependence between L F N and charge transport Area in electronic device exists, following from both Hooge equation and superposition models [13-15]. For M O S transistor, it is

o-H _ o-H 1 a,H q

oxide » Fairchild2N696(npn, Si)

: / ^ / M • / -0.5dB;year

'K-j : +

(pnp

X t i

'^t

G e ) \ +

/ "'"•"

^ / 1 Moore's Law

a ^

""

S LEDs

(AIGaAs)

' \ ^ v

/

° ^ V ^ i i * \ ^ I T R S MOS \ = • ^ ^ v i —-Analog :

V

/

• V ^ RF =

• \

•

f n

fN fn WL 2

fxWLC„JVG-2 S,

• V T 1 WL"

1 , Area

(1)

^ C o J f x W L t l n J (f/lHz)WL[lD

where Area=WL of the M O S channel (emitter area A E in BJT); n=N/Area; q is electronic

charge; kT is thermal energy; X~0.1nm for Si02 is tunneling attenuation distance in the

gate dielectric with trap density Nt~10^^ cm^^eV"^ and capacitance per area Coxi gm/In is

ratio of transconductance to D C current. Svo.nm' is unit-area input referred noise at f= lHz . Data [16] in Figure 2 show that (ECpxArea) is between

10" ' and 10"^ [im^, when the fabrication reaches maturity, implying LFNccArea"^ in electronic devices investigated last 85 years, from the vacuum tubes measured in 1925 up to tiny devices made of carbon nanotubes (CNTs). There is a min imum for LFN, (ECFxArea)>10"' |im^, and the device downscaling converts the nm-scale transistors into stochastic devices, in which even D C curve may not exist, as seen in Figure 3 for a singe CNT FET [17].

A dependence on oxides thickness is deduced for LFN. Since Cox=Sox/EOT in eq. (1), the models suggest higher L F N when the equivalent oxide thickness EOT is higher,

SnonnWLocEOT for M O S from Hooge model, (2)

SnonnWLccEOT^ for M O S from superposition model. (3)

FIGURE 3. Spiky, stochastic transfer "I-V curves" of a CNT

field-effect transistor

SnonnAE=cEOT' for BJT from tunnel-transparency model [18], (4)

0.1 1 10 100 100

IFO (BJT„ EOT (Equivalent Si02 Thickness), nm

iiili llli III

•Hii mi m

FIGURE 5. Flicker noise penalty in MOS transistors when using composite materials either in

the channel or in the gate dielectric. Nt •XSVG.

FIGURE 4. Input referred 1/f noise voltage at IHz in transistors of l|xm^ area. nMOS (•), pMOS (A),

and BJT (O). The gray lines represent log-normal distributions

shown in insets.

where eq. (4) for polysilicon emitter BJT with interfacial oxide with thickness IFO as described in [19]. The numerical data [20] from 51 publications in Figure 4 show the rates given by eqs. (3) and (4) for the noise increase by thicker oxides. We observe log-normal distributions, and a convergence at IFO and EOT ~0.8nm.

We note [21] higher noise by using diverse materials in active layers in Figure 5(a-f), now with more data [22], whereas a noise from lattice strain is unlikely in Figure 5(g-h).

SUMMARY AND OUTLOOK

The investigations of LFN have followed the development of electronic devices for more than 85 years, and the noise models have now achieved a certain level of maturity. The LFN sets barriers for device downscaling, since (AreaxKF)~10"^ \im^ during this entire research period. Consequently, nano-devices with active area of 10x10 nm^ may become stochastic. The noise variations tend to log-normal distributions, with standard deviation OdB=3-10dB (2-10 times), and one could experience problems with time-variant yield and non-reproducible errors, when integrating several giga transistors, while demonstrations of low-noise nano-devices is possible at the other end of the distribution.

Another issue with device downscaling is the increased impact of flicker noise at high frequencies. In conjunction with Moore law, the reduced rate of performance improvement by the downscaling is termed as "Law of Diminishing Returns" [23]. Among the many figures-of-merit that are related to device performance, the ratio fc/fi of the corner frequency fc between 1/f noise and white noise to transit frequency fx illustrates the increasing impact of 1/f noise at higher frequencies. For BJT, fc/fx is given by [21, 24]

X t J f klCp

AT;

(5) _ fc _ 71 A g X Kp

BJT fi q P l ' " q ' •E

where p is the current gain, Xt is transit time, CB is the base capacitance, AEXKF~5X10" ' \im^ [21], and fc/fx evolves with the increasing current density Jc needed for higher fx as illustrated Figure 6. Note the independence of Ag at high current density. For MOS transistors, the ratio fc/fx depends on operation regime [23, 25], and it can be shown that

f„ f. T MOS

27im 1 + C„,Wv„

IHzxS VG.nm^

4kT (6)

where gm is the transconductance at given biasing condition, being not larger than gm,sat in the saturation regime, Vsat cm/s is the carrier saturation velocity, Svo.nm' is the unit-area gate-referred voltage noise at IHz, and the parameter (m) depends on operation regime, being m={l ,1 .5 ,2} for linear, saturation and subthreshold regimes, respectively. Note that in M O S transistors with min imum gate length, gni/(CoxWvsat)~200nm/L>l, and fc/fi increases by downscaling. On the contrary, in large M O S transistors, fc/fx depends on the product SvG.um'^Cox, so, on biasing via SVG, but not on length L or width W of the transistor, which is size independence similar to that for BJT. Also, 1/f noise models for M O S transistors predict an increase of SVG with bias, resulting in an increase of fc/fi, as shown by lines in Figure 6. At present, the limited experimental data [25-29], while being in the predicted range at low biases, do not confirm the increase of fc/fi at high biases.

Gate Voltage Overdrive V,

FIGURE 6. Ratio fc/fx. The patterned area separates MOS and BJTs. Data for BJT and HBT from several publications [21]. (O) 0.8nm nMOS [25], ( • ) CSSjim SOI nMOS and pMOS [27], ( • )RF nMOS L=0.13nm W=72nm [26], (A) 0.13nm nMOS and pMOS [28], ( • ) 0.09nm nMOS [29]. The lines are calculated according to eq. (6) for a virtual MOS transistor L=30nm, Cox=3nF/cm^ n=300cm^/Vs, Vsat=10'cm/s, Nt=3xlO"cm"'eV"' and X=0.1nm for An fluctuation, Coulomb screening Hco=3 x 1 O^cm/Vs for Ajic fluctuation, corresponding to Hooge parameter aH=kTXNt(n/ncof=7.8xlO"^, 9=3V"' for phonon or roughness scattering (as=q9/nCox=5.3xlO"'^Vs) for Aji fluctuation, and then all components were combined An-Ajic-Aji. Note the linear scale for VG<VT.

Collector Current Density J^, mA/pm^

REFERENCES 1. W. Schottky, Ann. Physik, 362, 541, 1918. 2. J. Johnson, Phys. Rev., 26, 71, 1925. 3. W. Schottky, Phys. Rev., 28, 74, 1926. 4. R. Voss, Freq. Control Symp., 33, 40, 1979. 5. R. Jones, Proc. IRF, 47, 1481, 1959. 6. N. Mantena, et al, Flectr. Lett., 5, 607, 1969. 7. S. Hsu, SSF, 13, 1451, 1970. 8. A. Boomard, etal, SSC, 10, 542, 1975. 9. ITRS, http://public.itrs.net, editions 2001-2008. 10. J. Johnson, Phys. Rev., 32, 97, 1928. 11. H. Nyquist, Phys. Rev., 32, 110, 1928. 12. F. Hooge, Phys. Lett. A, 29, 139, 1969. 13. A. McWhorter, Semi. Surf Phys., 207, 1957. 14. O. Jantsch, TFD, 34, 1100, 1987. 15. G. Ghibaudo, et al, PSS (a), 124, 571, 1991. 16. Data from 48 publications for vacuum tubes,

MOS and BJTs, III-V semiconductor HBTs, optical noise from light emitting diodes, and carbon nanotube (CNT) and nanowire (NW) devices. Data is aggregated by years for CNT and NW, for BJT after 1980, and for MOS after 2000.

17. F. Liu, et al, APL, 86, 163102, 2005.

18. H. Markus, et al, TFD, 42, 720, 1995. 19. M. J. Deen, et al, IFF Proc, 151, 125, 2004. 20. O. Marinov, et al, www.RDIF.ca, 1, 75, 2007. 21. M. Deen, et al, AIP Conf Proc, 780, 3, 2005. 22. Data (a) SiGeC pMOS transistors, N I increases

with carbon content. Data (b) and (d) Si n- and pMOS transistors, Nt increases with nitridation of gate oxide. Data (c) nMOS, N, is higher for high-k dielectrics. Data (e) Nt increases with complexity of MOS structures, from Si channel to SiGe, SOI and high-k dielectrics. Data (f) Nt variations by annealing of SiGe pMOS. Data (g) and (h) nMOS without and with strained lattice.

23. L. Vandamme, et al, TFD, 55, 3070, 2008. 24. J. Tang, et al, TMTT, 50, 2467, 2002. 25. A. Amaud, et al, TCS, 51, 1909, 2004. 26. C.-C. Ho, et al, FDL, 26, 258, 2005. 27. D. Binkley, et al, TNS, 51, 3788, 2004. 28. V. Re, et al, TNS, 53, 1599, 2006. 29. M. Manghisoni, et al, NIMPR A, 572, 368,

2007.

200

http://public.itrs.net

http://www.RDIF.ca

Suppression of Random Telegraph Signal Noise in small-area MOSFETs under switched gate

and substrate bias conditions Nicola ZanoUa*, Domagoj Siprak^, Marc Tiebout**, Peter Baumgartner^,

Enrico Sangiorgi* and Claudio Fiegna*

*ARCES-DEIS, University of Bologna and IU.NET, Via Venezia 260,1-47023 Cesena, Italy Infineon Technologies AG, Am Campeon 1-12, D-85779 Neubiberg, Germany

**Infineon Technologies Austria AG, Siemensstr 2, A-9500 Villach, Austria

Abstract. In this work we focus on the impact of substrate bias on the random telegraph signal (RTS) noise in small-area MOSFETs operating under switched bias conditions. Our results clearly prove that when a MOSFET is switched between an ON- and OFF-state, the application of forward substrate bias during the OFF-state modulates the mean trap emission and capture times and determines more than one decade reduction of the RTS-noise power.

Keywords: MOSFET, RTS noise, switched bias, substrate bias PACS: 85.30.Tv, 73.50.Td, 72.20.Jv

INTRODUCTION

Low-frequency noise (LFN) strongly degrades the performance of CMOS analog and RF non-linear circuits, such as VCO's, due to the up-conversion of the noise spectrum leading to phase-noise. In small-area MOSFETs LFN is dominated by individual traps located at the Si-dielectric interface or inside the gate dielectric [1] that, by changing their occupation and charge state, produce RTS noise, i.e. switching of the drain current (ID) between two discrete levels. In large-area transistors, the superposition of the effects of different individual traps leads to flicker (1/f) noise. In circuits using small-area MOSFETs RTS noise is becoming a severe problem. For example, CMOS image sensors are affected by RTS noise that may influence the pixel read noise floor [2]. A reduction of LFN achieved by switching the gate bias between an ON-state and an OFF-state (SB), compared to a constant bias condition (CB), has been reported in the past [3]. More recently, [4] showed that the apphcation of a forward substrate bias (FSB) to a MOSFET under SB conditions effectively suppresses the 1/f noise.

MEASUREMENT METHODOLOGY AND MEASURED DEVICES

NMOSFETs with 2.2 nm thick nitrided gate oxide, gate poly length L=0.1 jUm and widths W=0.4, 0.75 jUm were manufactured in a 0.13 jUm technology [5]. RTS noise has been measured in both time and frequency domain using a differential set-up [4] (Fig. 1 (a)). Time domain analysis of RTS noise under 50% duty-cycle SB is performed using the technique adopted in [6] with switching frequency /jw=10 kHz. Measurements are



201

http://IU.NET

http://85.30.Tv

VBS (IP)

VBS (OP)

OV

VBS_OFF

(a) (b)

FIGURE 1. (a) Experimental differential set-up and (b) waveforms for switched bias measurements: substrate bias can be switched in-phase (IP) and 180° out-of-phase (OP) with respect to the gate bias.

performed for V D S = 1 0 V, under several gate and substrate bias conditions which may be relevant for analog apphcations: constant gate (CB) and substrate bias; switched gate (SB) and constant substrate bias, in-phase (IP) switched gate and substrate bias; 180° out-of-phase (OP) switched gate and substrate bias (Fig. 1 (b)).

RTS NOISE

The drain current affected by an RTS switches between two values corresponding to two different charging states of the trap. Considering an acceptor-type trap: it is negatively charged when occupied (low current) and neutral when empty (high current). RTS noise features a Lorentzian power spectral density (PSD) and a noise power (P) given by:

PSD = P = (1) (Te + Tc)[(Te ^ + Zc ^)^ + {2nffY {Ze + rc){%

where: Tg and TC are the mean emission and capture times, respectively, and AIo is the amplitude of current fluctuation. For a given trap AID is insensitive to bias conditions in our experiments. P features a maximum for Te=Tc', according to the Shockley-Read-Hall theory, the maximum occurs at EF=ET, where Ej is the trap energy.


The dependence of RTS noise on biasing schemes (Fig. 1 (b)) has been analyzed for several small-area devices for which it is possible to detect the RTS noise due to a single trap.

In all the analysis the PSD under CB conditions is divided by a factor of 4 in order to be compared to the SB measurements accounting for the intrinsic 6 dB attenuation due to the ON-OFF modulation of the drain current with 50% duty cycle [7].

Figures 2 and 3 report the results obtained for two representative traps. Figures 2 (a) and (b) show that, compared to CB conditions, switching the gate between an ON- and

202

10 10 10 Frequency [Hz]

(a)

10 10 10 Frequency [Hz]

(b)

FIGURE 2. RTS PSD for: CB; SB; gate- and substrate-SB (OP case). The CB case is divided by 4 in order to have a fair comparison with the ON-OFF modidated SB cases. Forward substrate bias appUed OP with respect to the gate bias reduces the low-frequency plateau of the PSD. (a) Trap 1 and (b) Trap 2.

an OFF-state (VGS_ON and VGS_OFF) only marginally affects the PSD; furthermore, in the case of Trap 1 (Fig. 2 (a)), since the selected VGS_ON is significantly larger than the VGS value corresponding to the maximum-P condition, SB can lead to a slight increase of P and of the low-frequency plateau of the RTS PSD; application of a forward (positive) substrate bias (FSB) during the OFF-state (OP gate- and substrate-SB) significantly reduces RTS noise in both Trap 1 and Trap 2. On the other hand, when a FSB is applied during the ON-state (IP gate- and substrate-SB, not shown) RTS is hardly affected. Figures 3 (a) and (b) report the dependence of RTS PSD on VBS_OFF in the case of OP gate- and substrate-SB, showing that a reduction of RTS noise occurs only for FSB. In particular, compared to Trap 1 (Fig. 3 (a)), for Trap 2 (Fig. 3 (b)) a different sensitivity on VBS_OFF can be noticed.

The analysis of extracted tg and tg under OP gate- and substrate-SB (upper part of Figs. 4 (a) and (b)) shows that the application of a FSB during the OFF-state significantly reduces tg and slightly increases Tc, thus enhancing the difference between the two time constants and suppressing P (lower part of Figs. 4 (a) and (b)). These results may be explained as follows: when the gate is biased at VGS_OFF=^ V, the forward substrate

10

£io-" — <^ -18 " 10 I. Q ^ 1/)

10

Y = -0 3 Y OP rr^ 1

Bs_oFF IraD 1 Y = 0 7 5 Y Y = 0 Y

GS_ON " • ' - ' * *GS_OFF f =10 kHz

0.25 Y OP • / ' " ' f •"'»V*%ifiaWf^^>t5

V,S0FF = -» ' tV0P V „ " „ „ = OY Trap 2

10 10 10 10 Frequency [Hz]

(a)

10 10 10* 10" 10" Frequency [Hz]

(b)

FIGURE 3. RTS PSD under gate- and substrate-SB (OP case) for different values of the substrate bias VBS_OPP applied during the OFF-state. Increasing VBS_OFF the PSD plateau at low-frequency decreases, (a) Trap 1 and (b) Trap 2.

203

FIGURE 4. Tc, Te and RTS-noise power P under gate- and substrate-SB (OP case) as a function of substrate bias VBS_OFF applied during the OFF-state. Increasing VBS_OFF "^e decreases affecting also the noise power that shows a strong reduction, (a) Trap 1 and (b) Trap 2.

bias tends to drive the MOS system towards accumulation; therefore, we conclude that the FSB -induced suppression of noise could be related to a transient accumulation of the silicon at the oxide interface leading to very large recombination rate of trapped carriers with accumulated holes during the OFF-state. Therefore, enhanced difference between Tg and Zc explains the suppression of RTS-noise power contributed by noisiest traps (i.e. those featuring Tg « Zc in the ON-state).

CONCLUSION

In this work we have analyzed RTS noise in small-area MOSFETs operating under switched bias conditions. We have studied the impact of the substrate bias on the noise. A more than one decade reduction of the RTS-noise power can be achieved if a forward substrate bias is applied out-of-phase with respect to the gate pulse, therefore, when the transistor is in the OFF-state. This method could be useful to reduce the RTS noise in circuit using small-area MOSFETs and therefore affected by such a noise.

REFERENCES

M. J. Uren, D. J. Day, and M. J. Kirton, "1/f and random telegraph noise in siUcon metal-oxide-semiconductor field-effect transistors," Appi. Phys. Lett., 47 (11), pp. 1195-1197,1985. Y. DegerU, F Lavemhe, P. Magnan, and J. A. Farre, "Analysis and reduction of signal readout circuitry temporal noise in CMOS image sensors for low-light levels," IEEE Trans. Electron Devices, 47 (5), pp. 949-962, 2000. 1. Bloom and Y. Nemirovsky, "1/f noise reduction of metal-oxide-semiconductor transistor by cycling from inversion to accumulation," Appi. Phys. Lett., 58 (15), pp. 1664-1666,1991. D. Siprak, N. ZanoUa, M. Tiebout, P. Baumgartner, and C. Fiegna, "Reduction of low-frequency noise in MOSFETs under switched gate and substrate bias," in Proc. ESSDERC 2008, pp. 266-269. T. Schiml et al., "A 0.13 lira CMOS platform with Cu/Low-k interconnects for system on chip applications," in Proc. Symp. on VLSI Technology 2001, pp.101-102. J. S. Kolhatkar, Ph.D. dissertation, University of Twente, Enschede, The Netherlands, 2005. A. P. van der Wei, E. A. M. Klumperink, S. L. J. Gierkink, R. F Wassenaar, and H. WaUinga, "MOSFET 1/f noise measurement under switched bias conditions," IEEE Electron Device Lett., 21 (1), pp. 43-46, 2000.

204

RTS in Submicron MOSFETs: Lateral Field Effect and Active Trap Position

J. Sikula", V. Sedlakova", M. Chvataf, J. Pavelka", M. Tacano^ M. Toita'

" Czech Noise Research Laboratory, Brno University of Technology, Technicka 8, 616 00Brno, Czech Republic, ([email protected])

Meisei University, Hino, Tokyo, Japan 'Asahi Kasei Microsystems, Nobeoka, Miyazaki, Japan

Abstract Experiments were carried out for n-channel devices, processed in a 0.3 |lm spacerless CMOS technology. The investigated devices have a gate oxide thickness of 6 nm and the effective interface area is AQ = 1.5 |lm . The RTS measurements were performed for constant gate voltage, where the drain current was changed by varying the drain voltage. The capture time constant increases with increasing drain current The model explaining the experimentally observed capture time constant dependence on the lateral electric field and the trap position is given. From the dependence of the capture time constant % on the drain current we can calculate x-coordinate of the trap position. Electron concentration in the channel decreases linearly from the source to the drain contact. Diffusion current component is independent on the x-coordinate and it is equal to the drift current component for the low electric field. Lateral component of the electric field intensity is inhomogeneous in the channel and it has a minimum value near the source contact and increases with the distance from the source to the drain. It reaches maximum value near the drain electrode.

Keywords: RTS noise, 1/f noise, MOSFET. PACS: 72.70.+m, 73.40.Qv, 73.50.Td, 85.30.Tv

INTRODUCTION

A systematic analysis of two levels RTS signal was made to obtain the information on the capture and emission processes as a fimction of gate voltage, drain current and temperature for the low and high lateral electric field [1-3]. In the submicron technology (for channel length less than 0.3 |am) the application of the drain voltage IV results in the high lateral electric field, which exceeds about 5 times the silicon critical field. The electron temperature is then higher than the lattice one and the field dependent electron mobility must be considered. Due to the small gate area and the low traps concentration we were able to activate one trap only and then two levels signal was observed in the time domain.

We suppose that there are oxygen vacancies and interstitials creating energy localized states in the Si02 gate insulating layer and on the interface Si-Si02. G-R stochastic exchange of electrons exists between the channel and the interface Si-Si02 localized states, and this is a source of 1/f noise. It is supposed that there exists electron exchange between the channel and the trap localized in the gate insulating layer at about Inm distance from the channel. These quantum transitions are sources of RTS noise. We shall try to improve the model describing



205


the electron quantum transition between the channel conduction band and the trap localized in the gate insulating layer.

EXPERIMENTAL

Experiments were carried out for n-channel devices, processed in a 0.3 |am spacerless CMOS technology. The investigated devices have a gate oxide thickness of 6 nm and the effective interface area is AQ = 1.5 |am . The RTS measurements were performed for constant gate voltage, where the drain current was changed by varying the drain voltage. The capture time constant increases with increasing drain current as is shown in Fig. 1.

1.4

1.0

N-Mos N51Ug07T260.ep2

N-MOS N51 RTS noise Ug = 0 .7V T = 2 6 0 K

y=a/(1-bx) a=1.13, b=0.330

' ^ • ^ ^ ^ •

C J^

^ T

^

./

•

I, /UA

FIGURE 1. Capture T and emission T time constants of Si MOSFET N51, measured for fixed gate voltage UQ= 0.7V as a function of drain current/^ at temperature 260 K

We will give a model explaining the experimentally observed capture time constant dependence on the lateral electric field and the trap position. Proposed model is based on these assumptions: (i) the drain voltage is so low that electric field intensity is lower than the critical one E„ = 0.7 MV/m and electron mobility does not depend on the lateral electric field, (ii) electron concentration in given point r (x, y, z) in the channel is described by

«(r) = «„ exp[-y5f/(r)] (1)

where no is electron concentration at source, yS is constant and U(f) is voltage in given point r in the channel. Electron concentration decreases with increasing distance from the source and minimum value of electron concentration is near the drain. Here are two variables: electron concentration «(r) and its gradient V«(r) then drift and diffusion current components must be considered. Drift current density is given by

/ drift -ejU^n(r)WU(f) (2)

Where e is electron charge and /4 is electron mobility. And diffusion current density can be expressed using (1) as

y. dif -e/5D^n{r)WU{r) (3)

Where D„ is diffusion constant. The relation between drift and diffusion current densities/ / / = H^I{jiD^) we get from (2) and (3). In the special case near the

206

thermodynamic equilibrium for P = Po= e/kT and using Einstein relation that /4 =PoD„ we l i a v e / ^ „ ^ , / ^ ^ = l .

Total drain current density in one dimensional case is given by

J J = -ejU^yn{x)dU{x) I dx (4)

Where y = 1 + {JiD^ )l jl^. Then we have the drain current

/^=GoJiexp[-y5f/(x)]fiff/(x)/fifx (5)

Where L is the channel length and Go is the channel conductance in the vicinity of the source given by Go = (Ae/u„no)/L.

Voltage U(x) vs. given point x in the channel could be calculated form (5) as

u{x)=Un(\i(\-pi,^iG,r)) (6)

Where ^ = x/L is relative distance from the beginning of the channel. The dependence of U(x) on ^is shown in Fig. 2.

The relation between the electron density n(x) and the drain current Id follows from (1) and (6). After the integration we have

«(x) = «„ exp[-y5f/(x)] = «„(!- pi.^lG.r) (7)

Electron concentration n(x) in the given point x in the channel decreases with the increasing drain current Id and this function is linear (see Fig. 3).

a = = 0.9 /

0 . 7 / ^

0.5, :

IS-—— 0.1

a = 0

^ ;:: — , , o.i

^ \ ^

a = 0 .9^

0.4 0.6 0.4 0.6

I;

FIGURE 2. Voltage U(x) vs. £, for different value FIGURE 3. Electron concentration n(x)/no vs. £, of a =pid/GoY and P= Po for different value of a=pi/Goyand P= Po

Drain current Id as a function of drain voltage Ud for fixed gate voltage UQ = 0.7V at temperature 298 K is shown in Fig. 4. From (5) it follows that the drain current is an exponential function of drain voftage for fixed gate voltage

Id= Idoa-eM-PUd)) (8)

Fitting the measured characteristic we have for sample N-MOS N51 the drain current Ido = 5.24 laA, P= 22.5 V"' and Gor= 1.18 x 10"'' S for the temperature T = 298 K and gate voltage f/G=0.7 V.

207

Capture time constant T of Si MOSFETs measured for fixed gate voltage Ug as a function of drain current can be expressed by

T^=Tj{\-IJ^IGj) = Tj{\-bI,) (9)

Where r„ = 1 /aVj,«o, and b = P^IG^y = ^^^.

Fitting the dependence of T on the drain current (see Fig. 5) we have b = 0.113 and we can calculate the active trap position in the channel ^= 0.67 for the temperature 298 K. We have evaluated also the characteristics measured for this sample at T= 260 K (see Fig. 1). In this case h = 0.330 and I^ determined from VA characteristic is 2.28 |J.A. Active trap position is ^= 0.75 for this temperature.

Id=

;U= 0.08

U

U(1-exp(-

5.21 E-6A

0.12

/ V

PUJ |3=22.8 V

0.16

FIGURE 4. Drain current Ij as a function of drain voltage U^ for fixed gate voltage UQ= 0.7 V at temperature 298 K

N-MOS N51 RTS noise UG = 0 . 7 V T = 298 K

y=a/(1 -bx) a=114, b=0.129

• >

T

1 2 3 4 5

FIGURE 5. Tc and T^ time constants of Si MOSFET N51, measured for fixed gate voltage UG= 0.7 V as a function of /^ for r = 298 K

CONCLUSION

Experiments were carried out for n-channel devices, processed in a 0.3 |am spacerless CMOS technology. We give a model explaining the experimentally observed capture time constant dependence on the lateral electric field. From the dependence of the capture time constant T on the drain current we can calculate x-coordinate of the active trap position. Electron concentration in the channel decreases linearly from the source to the drain contact. Diffusion current component is independent on the x-coordinate and it is equal to the drift current component for the low electric field. Lateral component of the electric field intensity is inhomogeneous in the channel and it has a minimum value near the source contact.

ACKNOWLEDGMENTS

We are grateful to Asahi Kasei Microsystems for supplying us the MOS samples. This research has been supported by the grants GACR 102/09/1920, GACR 102/08/0260, and under project MSM 0021630503.

REFERENCES

1. M. Toita et al., Proc. of the 18 Forum of Science and Techn. of Fluctuations, 2003 Tokyo, p. 63 2. M.J.Kirton and M.J.Uren, Advances in Physics, 38(1989) 367-468, No. 4 3. G. Ghibaudo, O. Roux, J. Brini, Phys Stat. Sol (a), 127, 281 (1991).

208

Low Frequency Noise Performance of Advanced Si and Ge CMOS Technologies

C. Claeys''' , A. Mercha'' and E. Simoen"

"IMEC, Kapeldreef75, B-3001 Leuven, Belgium EE Depart, KULeuven, Kasteelpark Arenberg 10, B-3001 Leuven, Belgium

Abstract. Pushing the CMOS device performance to its limits is based on the implementation of advanced process modules, new materials and/or alternative gate concepts. Several strained or non-strained Si and Ge-based substrates are being explored. All these efforts have also an impact on the low-frequency noise behavior of the devices and circuits.

Keywords: 1/f noise, germanium, CMOS, silicon passivation PACS: 72.70.+m; 73.40.Qv

INTRODUCTION

The technology scaling to continue Moore's law necessitates the use of new materials and the introduction of a large variety of advanced processing modules, such as e.g. high-K gate stacks, implementation of capping layers, fully sihcided and/or metal gates, strain engineering techniques, etc.

Beside the technological innovations one can further make use of new and/or alternative gate architectures enlarging the standard single gate transistor approach towards double and multi-gate devices based on FinFET, MuGFET, Gate-all-Around (GAA) or silicon nanowire concepts. A final parameter of choice is the starting substrate whereby standard Czochralski or Float Zone Si can be replaced by Sihcon-on-Insulator or one of the so-called high-mobility substrates based on SiGe, strained Si, Ge or even strained Ge. In addition to the potential of Ge-based technologies, there is worldwide research going on towards the monolithic integration of Ge and III-V on Si. Therefore, there exist a large number of possible future devices in order to stay in hue with the ITRS roadmap. For most of these devices, there is only very hmited or no information available on their low-frequency (LF) noise behavior. This review will highlight some general trends, illustrated by experimental noise data on devices fabricated in state-of-the-art CMOS technologies.

Advanced Materials and Process Modules

A few years ago the authors have reviewed in detail the LF noise performance of SOI based technologies [1]. The noise assessment of deep submicron CMOS process modules was the subject of reviews in 2004 [2] and 2006 [3], while more recently



209

attention was given to the impact of strain engineering on the low frequency noise characteristics [4]. This paper gives an update of the noise characterization of advanced Si and Ge processing technologies. The impact of important technological parameters, i.e., new materials and process modules will be addressed.

For 45 nm and below technologies the standard Si02 or SiON gate dielectric has to be replaced by a high-K gate stack. A large variety of gate dielectrics have been studied, including rare earths and lanthanides. Hf-based gate stacks are receiving much industrial interest. However, not only the gate dielectric itself but also the interfacial Si02 layer (IL) has an impact on the noise performance. The latter is illustrated in Fig. 1, giving the Hooge parameter versus gate voltage overdrive for an Hf02 dielectric with 0.4 and 0.8 nm IL [5]. The noise level is higher when the high-K dielectric is closer to the interface, i.e., for thinner interfacial layers.

10 Jim X 1 nm p-MOSFETs 10 m X 1 m n-MOSFETs 10"'

s 10"':

"0.0 0.2 0.4 0.6 0.8 1.0

VOS-VT(V)

8 10" ;

10'

I I I I I I I I I I I I I I I

a 0.8 nm

I I I I I I I I I I I I I I I 0.4 0.8 |V,3-V, I (V)

1.2

Figure 1. Hooge parameter versus gate voltage overdrive for two different interfacial Si02 thicknesses for HfDs n- (EOT - 0.92 and 1.44 nm) and p-MOSFETs (EOT = 1.31 and 1.35 nm), respectively.

Other important gate stack parameters are the use of a capping layer on the high-K dielectric, as shown in Fig. 2a for a LaO cap on top of an HfSiON layer [6], and the implementation of a metal gate (Fig. 2b) [7]. It can be noticed that a LaO cap can reduce the noise. Figure 2b indicates that FUSI on SiON lowers the noise level compared to the polysilicon gate reference. Recently, the impact of the metal layer thickness has been studied in detail for HfSiO/TiN stacks in MuGFETs [8]. A thicker metal gate increases the trap density and therefore also the noise. The study of the noise behavior has to take into account the whole gate-stack, i.e., thickness of the different layers (interfacial layer, high-K dielectric, capping layer), the different interfaces, and the nature of the gate electrode, in addition to the processing conditions (temperature, ambient, anneals, etc). Improved noise models for dual layers gate stacks are under development [9]. To boost up the device performance, strain engineering based on either a global approach using high-mobility substrates or the implementation of so-called processing-induced stressors has become common practice for 90 nm and below CMOS technologies. The global approach generally results in biaxial stress in the transport plane, while local stressors generate uniaxial stress in the channel direction. The impact of different strain engineering approaches on the gate stack quality and its

210

reliability, including hot carrier performance, negative bias temperature instabilities, time dependent dielectric breakdown and radiation hardness has been reviewed by the authors [10]. The impact of strain engineering technique will be further discussed.

10 nmx0.15 urn nMOSFETs 10 nmxO.25 fxm n-MOSFETs

0.2 0.4 0.6 0.8 Gate Voltage (V)

Figure 2a. Input-referred voltage noise spectral density at 25 Hz versus gate voltage for 10 |xmx0.15 |xm n-MOSFETs.

• °2''

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 .

FUSI+,10cycl

* - * FUSl

V =0.05 V DS

f=10Hz

Gate Voltage Overdrive (V)

Figure 2b. Input-referred noise spectral density .SvG at 10 Hz versus gate voltage overdrive (FDS=0.05 V) for a 10 |xmx0.25 |xm n-MOSFET corresponding with a poly silicon gate (A), aNiSi FUSI gate (•) and a NiSi FUSI gate plus 10 cycles Hf02 on top of a 1.5 nm SiON gate dielectric (o).

Relying on the lattice mismatch between Si and Ge, it is possible to grow tensile strained Si on a relaxed SiGe buffer (referred to as a virtual substrate). Figure 3a shows the input-referred noise .S'VG as a function of the gate voltage overdrive FGT [ 11 ] • For strained Si devices fabricated on the thin VS a clear plateau is observed in .S'VG at low FGT, indicating that number fluctuations associated with traps are at the origin of the noise. The .S'VG increase at higher gate overdrive bias is most likely due to correlated mobility fluctuations.

A general trend of the impact of process-induced strain approaches on the noise performance is given in Fig. 3b [10]. The used acronyms mean tensile (t) or compressive (c) contact etch stop layer (CESL), and stress memorization technique (SMT). The figure allows concluding that it is not possible to distinguish an intrinsic stress effect on the 1// noise performance. A recent study of the 1/f noise under external mechanical stress has revealed that the relative change in noise spectral density is in good approximation given by 4AID/ID [12] and is mainly associated with the strain-induced change in mobility.

Embedded recessed SiGe (compressive strain) or SiC (tensile strain) source/drain regions not only reduce the series resistance but also enhance the drive current of p-and n-channel MOSFETs, respectively. It has been shown that the application of embedded SiGe S/Ds does not degrade the I// noise spectral density as long as no strain relaxation occurs. This is illustrated in Fig. 4a, where for 30% SiGe the noise is similar as for standard Si devices [13]. For 40% the noise increases. In the case of I or 1.5% Si:C S/Ds, used for the n-MOSFETs, there is no impact on the noise performance, as shown in Fig. 4b.

211

Global strain engineering can be combined with a local technique, as shown in Fig. 5 for standard SOI and strained SOI (sSOI) combined or not with Selective Epitaxially Grown (SEG) S/Ds [14]. This has no pronounced impact on the noise performance.

f = 25 Hz VDS = 50 mV L X w = 0.5 X 10 Mm'

0.2 0.4 0.6 0.8

Gate voltage overdrive VQT (V)

SIVTT

p M O S *

-

V =50 mV D5

f = 2 5 Hz

SIVTT+C-CESL

•

,

•

No strain

O

nMOS O

O

C-CESL

•

O

t-CESL -

SIVTT+t-CESL

-40 -30 -20 -10 0 10 20 Nomalized Change in iVIaximum Transconductance (%)

Figure 3a. Input-referred noise spectral density of Figure 3b. Trend between the 1//" noise (oxide strained Si devices measured at 25 Hz. The devices trap density) and strain magnitude for p- and n-are Z X W= 0.5 •^- 10 jim^ and FDS=0.05 V. MOSFETs with or without strain engineering.

10^mx0.25^m pMOSFET

10"° 10"° 10" Drain Current (A)

10 nmxO.3 Jim nMOSFETs

Figure 4a. Normalized noise spectral density vs drain current in linear operation for a 10 |xmx0.25 |xm p-MOSFET with 30 or 40% SiGe embedded source/drain regions. The etch depth was 40 nm.

-0.4 -0.2 0 0.2 0.4 0.6 0.8 Gate Voltage Overdrive (V)

Figure 4b. Input-referred noise spectral density vss gate voltage overdrive at_/^25 Hz, showing no difference between reference n-MOSFETs and those with embedded 1 or 1.5% Si:C S/D regions.

I ,.. nFinFETT=300Kf=10l(Hz O SOI

SOI+SEG n SOI+SEG+CESL „ sSOl e SSOI+CESL

10* 10"' 10*

lo(A)

10'

10'

10"

Figure 5. Normalized drain current noise spectral density versus drain current at_/^10 kHz and at room temperature for different devices. The solid line indicates the (Gm/ o)^ parameter for the SEG+CESL+SOI device.

212

There is also much research effort devoted to the use of Ge-based technologies to enhance the performance of both n- and p-channel devices [15]. Initial investigations pointed out that the noise of Ge p^n junctions depends on the well-doping and on the use of a Ni-germanidation, as shown in Fig. 6a for large area diodes [16]. The germanidation steps lowers the noise associated with the series resistance [17]. For large perimeter diodes the noise may be dominated by surface traps and void formation in the perimeter region. Similar as for Si devices, a correlation has been found between the low-frequency noise and the mobility, as indicated in Fig. 6b. [18]. A very critical parameter from a noise viewpoint is the gate stack formation, whereby the thickness of the Si passivation layer has to be optimized [19].

,-17

- M - NiGe - • - NiGe-Low Well Dose - i - No NiGe - • - No NiGe-Low Well Dose Diode Perimeter, 10300|jm f=1 H I

10"' 10"' Forward Current (|jA/|jm)

•

• •

Qe MS p-MOSFETs L=lMmV\^10Mm Vj_=-50mV T=300K f=25Hz

• •

1.8x1(f 2.0x10

mobility \i (cm A/s)

Figure 6a. Current noise power spectral density at 1 Hz for a large-perimeter p -n junction belonging to different processing splits: NiGe low well dose (o), no NiGe low well dose (•); NiGe high well dose (n); no NiGe high well dose (x).

Figure 6b. Correlation between Syg at 25 Hz and the mobility |x for L=l |xm Ge p-MOSFETs.

The Ge technology also offers possibilities to come to a monolithic integration of Ge and III-V devices on a standard Si substrate, forming the basis for System-on-Chip (SoC) apphcations.

CONCLUSION

The implementation of new materials and advanced process modules require in-depth noise investigations in order to determine whether or not low frequency noise may become a show stopper for future advanced and emerging technologies. This may surely be the case for technologies relying on approaches based on e.g. nanowires, carbon nanotubes, graphene, spintronics, quantum devices, polymer electronics, etc.

213

ACKNOWLEDGEMENTS

The authors want to thank A. Akheyar, R. Agaiby, F. Crupi, G. Eneman, G. Giusi, W. Guo, N. Lukyanshikova, R. Todi, L. Yan for helpful discussions and the use of some co-authored results.

REFERENCES

1. E. Simoen, A. Mercha, C. Claeys andN. Lukyanchikova, Solid-State Electron. 51, 148 (2007). 2. C. Claeys and E. Simoen, J. Electrochem. Soc. 151, G307 (2004). 3. C. Claeys and E. Simoen, in Proc. Semiconductor Technology - ISTC2006, Ed. M. Yang, The

Electrochem. Soc. Ser. PV 2006-06, p. 242 (2006). 4. E. Simoen and C. Claeys, in Proceedings 8 Int. Conf. on Solid-State and Integrated Circuits

Technology -ICSICT2006, Eds T.-A. Tang, G.-P. Ru and Y.-L. Jiang, The IEEE 06EX1294, p. 120 (2006).

5. F. Crupi, P. Srinivasan, P. Magnone, E. Simoen, C. Pace, D. Misra and C. Claeys, IEEE Electron Device Lett. 11, 688 (2006).

6. F. Crupi, P. Magnone, E. Simoen, A. Mercha, L. Pantisano. G. Gisusi, C. Pace and C. Claeys, in Proc. Symposium on Advanced Gate Stack, Source/Drain and Channel Engineering for Si-Based CMOS: New Materials, Processes and Equipment, San Francisco, May, 2007 (in press)

7. P. Srinivasan, E. Simoen, R. Singanamalla, H.Y. Yu, C. Claeys and D. Misra, Solid-State Electron. 50, 992 (2006).

8. M. Rodrigues, A. Mercha, N. Collaert, E. Simoen, C. Claeys and J.A. Martino, these proceedings. 9. B. Min, S.P. Devireddy, Z. Celik-Butler, A. Shanware, L. Colombo, K. Green, J.J. Chambers, M.R.

Visokay and A.L.P. Rotondaro, IEEE Trans. Electron Dev. 53, 1459 (2006). 10. C. Claeys, E. Simoen, S. Put, G. Giusi and F. Crupi, Solid-State Electron. 52, 1115 (2008). 11. L. Yan, E. Simoen, S.H. Olsen. E. Escobedo-Cosin, C. Claeys and A.G. O'Neill, submitted to IEEE

Trans. Electron Dev. 12. J.-S. Lim, A. Acosta, S.E. Thompson, G. Bosman, E. Simoen and T. Nishida, J. Appl. Phys. (in

press). 13. E. Simoen, P. Verheyen, A. Shickova, A. Hikavyy, R. Loo, C. Claeys, V. Machkaoutsan, P.

Tomasini, S.G. Thomas, G. Groeseneken and H. Maes, Proc. ULIS2007, p. 75 (2007). 14. W. Guo, B. Cretu, J.-M. Routoure, R. Carin, E. Simoen, A. Mercha, N. Collaert, S. Put and C.

Claeys, Solid-State Electron. 52, 1889 (2008). 15. C. Claeys and E. Simoen, eds, "Germanium Based Technologies: From Materials to Devices",

Elsevier, 2007. 16. E. Simoen, S. Sonde, C. Claeys, A. Satta, B. De Jaeger, R. Todi and M. Meuris, J. Electrochem.

Soc. 155, H145 (2008) 17. R.M. Todi, S. Sonde, E. Simoen, C. Claeys and K.B. Sundaram, Appl Phys. Lett. 90, 043501 (2007) 18. W. Guo, G. Nicholas, B. Kaczer, R.M. Todi, B. De Jaeger, C. Claeys, A. Mercha, E. Simoen, B.

Cretu, J.-M. Routoure and R. Carim, IEEE Electron Dev. Lett. 28, 288 (2007). 19.E. Simoen, A. Firrincieh, F.E. Leys, R. Loo, B. De Jaeger, J. Mitard and C. Claeys, These

Proceedings.

214

Plasmonic noise in semiconductor layered structures

J.-F. Millithaler, L. Reggiani

Dipartimento cli Ingegneria dell 'Innovazione and CNISM, Universita del Salento, Via Amesano s/n, 73100 Lecce, Italy

Abstract. We investigate plasmonic noise in ungated and gated n-Ino.53Gao.47As nanochannels at room-temperature, in the absence and in the presence of an external bias by Monte Carlo simulations. The results are in agreement with analytical models but exhibit new behaviours associated with the realistic microscopic models of the system under interest. Generation and detection of microwave in the TeraHertz region should profit of present investigation.

Keywords: Classical Monte Carlo simulations PACS: 02.70.Uu, 72.30.+q, 72.80.Ey, 73.50.Mx

1. INTRODUCTION

Generation and detection of electromagnetic radiation in the TeraHertz (THz) domain is a subject in fast development because of its potential applications in different branches of advanced technologies, such as broad-band communications, high-resolution spectroscopy, medical and biological imaging, security, etc [1]. As a consequence, the realization of solid-state devices operating in the THz domain at room-temperature and with compact, powerful, and tunable characteristics is a mandatory issue. To this purpose, one of the most promising strategies lies in the plasmonic approach, which exploits the plasma frequency associated with long range Coulomb interaction of charge carriers.

In bulk semiconductors, the plasma frequency is given by the simple expression :

(1) niomeoemat

with HQ^ the three dimensional (3D) average carrier concentration, nio and tn the free and effective electron masses, respectively, and eo, Smat the vacuum permittivity and the relative dielectric constant of the bulk material, respectively. For carrier concentrations of about 10'^ cm^^ the plasma frequency is in the THz range for most materials.

For semiconductor layers embedded in an external dielectric and of width W sufficiently small (nano-meter range) to justify the in-plane approximation for the solution of the Poisson equation, it was found that the plasma frequency and its higher harmonics are given by [2]

1 I p2n^O]^ J-2D ungated ^ J_ e H^ K

CPn29, tloise andFluctuattons, 20* International Conference (ICNF 2009) edited by M. Macucci and G. Basso


215


http://02.70.Uu

where HQ^ is the average two dimensional (2D) carrier concentration, ediei the relative dielectric constant of the external dielectric, k the wavevector, / = 1,2,3... with the fundamental mode (1=1) given by A: = 7t/2L with L the length of the layer (implying that A = In/k with A the plasma wavelength) for boundary conditions corresponding to zero ac potential at the source and zero current at the drain contact, respectively. We notice that the 2D plasma frequency is dispersive, i.e. fp^ = fp^{k), and depends on the relative dielectric constant of the external dielectric.

For the case of a gated channel, within the gradual channel approximation the 2D electron gas was found to behave as the support of 2D plasma waves with fundamental frequency [2]

J2D gated Jp

1 n^e^nl^d 2K V AmomeoedieiL'^ 0)

where d is the gate to channel distance. Through these plasma oscillations, nanometric High Electron Mobility Transistors (HEMT) provided experimental evidence as emitter and/or detector of electromagnetic radiation in the THz range [4].

This work investigates the same semiconductor structures from a microscopic point of view, thus testing the predictions of the analytical approaches and provides more physical insight into the problem. To this purpose, we consider an n-type Ino.53Gao.47As layer within an ungated and a gated configuration at room temperature, and investigate the plasma frequency characteristics by analyzing the frequency spectrum of voltage fluctuations obtained from a Monte Carlo simulator coupled with a 2D Poissson solver.

2. THEORY AND RESULTS

Theoretical calculations are carried out by using an ensemble Monte Carlo simulator self-consistently coupled with a two dimensional (2D) Poisson solver and in the presence of an external applied voltage as already detailed in Ref. [4].

^ ' TtW

FIGURE 1. Schematic of the DUTs (not in scale) studied within the Monte Carlo simulation. The free charge is present only in the bar made by an n-type Ino.53Gao.47 As of length L along the x direction and thickness W along the z direction.

The Structures of the device under test (DUT), which is depicted in Fig. 1, represents a simplified version of an ungated and gated transistor channel. For the case of the ungated case, the bar is surrounded by a perfect dielectric (here taken as the vacuum) 10 ^m wide in the upper and lower region of the bar, where the 2D Poisson equation in

216

http://Ino.53Gao.47As

http://Ino.53Gao.47

the xz-plane is solved to account for the fringing of the external electric field [5]. The third dimension is used to relate the number of simulated carriers with the 3D carrier concentration. The time and space discretizations take typical values of 0.2 - 1 fs for the time step, 0.1 - 5 nm for the spatial scale of the bar and 500 nm for the spatial scale of the dielectric. In average, there are about 80 carriers inside a mesh of the bar, which are found to provide a reliable solution of the Poisson equation. The extracted potential V{t) is taken at the center of the bar while the fluctuations of current I{t) are taken at the second contact of the DUT (see Fig. 1). The contacts are realized by an infinite reservoir of thermalized electrons at electrochemical potentials differing by eU with U the applied voltage [3]. By evaluating the fluctuations of the voltage (current) around the steady value, the spectral density of these quantities is obtained from the corresponding correlation function. The characteristic peaks and the cut-off exhibited by the spectra are analyzed as detailed in [3]. We remark that, at low applied voltages, the DUT exhibits an Ohmic behaviour in the current voltage characteristics. At high applied voltages, the DUT exhibits a saturation behavior of the current.

0.1 1 10 100

Frequency CTHz)

FIGURE 2. (a) Typical autocorrelation function of voltage fluctuations and (b) associated spectral density of the DUT for a thickness W = 100 nm with n^^ = 10^* cm^^ and Z = 0.1 /jm at room temperature.

To illustrate the results of the simulator. Fig. 2 reports a typical correlation function and the associated spectrum of voltage fluctuations, Sv{f), normalized to its zero frequency value of the DUT for the case of L = 0.1 ^ m, ff = 100 nm, HQ^ = lO'^ cm^^ in the absence of an applied voltage. The results of simulations (dotted curve) compare favourably with those obtained by the 3D impedance equivalent circuit (continuous curve) [6]:

M/l = I (4)

with/the frequency, Tp = 4.28-10^''* s, andTd= 1.81 •10^''* s the plasma and dielectric relaxation times corresponding to the simulated bulk material. Here, the plasma peak is well evidenced by the good qualitative fit between numerical and theoretical results which validates the numerical approach. We notice that the relevance of the agreement mostly refers to the position of the plasma peak, since the amplitude of the peak is found to depend on the time window used for the Fourier transform of the correlation function. Furthermore, simulations evidence a cut-off decay as /^^, which is reminiscent of the presence of scattering mechanisms, instead of the sharper f^'^ predicted by the equivalent circuit model. Simulations performed for the case of current fluctuations do

217

not exhibit the plasma peak, but a simple Lorentzian decay at the collision frequency in close agreement with analytical expectations.

1 10

Frequency (THz)

0.1 1 10

Frequency CTHz)

FIGURE 3. Spectral density of voltage fluctuations normalized to the static value of the DUT for different values of the layer length, a fixed thickness W = \ nm and a free electron densities «Q 10' cm (a), andfor different values of concentration, afixedlengthZ = 0.1/jm and width rr = 100nm(b).

The results concerning the voltage spectral density under thermal equilibrium conditions are summarized in Fig. 3 (a). Here, the normalized volatge spectral densities are reported for the case ofW=\ nm, n\^ = 10'^ cm^^ at different lengths. Results show the onset of a peak in the spectrum for lengths above about 50 nm, thus at the start up of the diffusive transport-regime. By contrast, for lengths shorter than 50 nm all spectra are flat and exibit a beginning of cut-off at about 5 Thz, which is practically independent of the layer width. Remarkably, the shorter the layer length the less pronounced is the cut-off region. Figure 3 (b) reports the set of voltage spectral density at different carrier concentrations of the bulk material. The 3D plasma peak is resolved down to n -° = 10'^ cm^^ and the peak frequency is found to be in good agreement with the value predicted by Eq. (1). For n < 10 cm ^ the plasma peak is no longer resolved because the plasma time becomes comparable or shorter than the dielectric relaxation time.

" o

: X

> t

t

<

<

(bl . > o o

<

300 SOO 700

Length (nm)

10

Width (nm)

FIGURE 4. (a) Plasma frequencies of the DUT for electron density «Q^ = lO^'' and 10^* cm^^ in the diffusive transport-regime. Symbols refer to numerical simulations, curves to the analytical expression in Eq. (2) with k = n/L. (b) 2D plasma frequency normalized to the 3D value of the DUT as a function of the channel width for an electron density «Q^ = 10^* -:-10^* cm^^. The continuous line refers to the 3D of Eq. (1), the shaded region refers to the 2D of Eq. (2) covering the two cases of Z = 0.1-1-1 fim, the dashed bars refers to Monte Carlo results covering the same two cases and include the numerical uncertainty estimated within 20 %.

Figure 4 (a) reports the plasma frequencies a function of the channel length in the diffusive transport-regime for electron density n^^ = lO'^ and lO'^ cm^^. The comparison with the fundamental frequency of the analitycal model with k = n/L (see the continuous curves) is found to be satisfactory. Figure 4 (b) presents the values of the plasma

218

peak of the 2D case normalized to that of the 3D plasma peak as function of the layer width. Here we report the normalized plasma frequencies obtained from simulations for lengths covering the range of value 0.1 - 1 m and for 3D carrier concentrations in the range lO'^ - lO'^ cm^^ as bars. We also report the normalized theoretical values predicted by Eq (1) (horizontal continuous line) and Eq (2) with / = 1 and k = n/L (shaded region). The agreement between numerical results (bars) and analytical (shaded region) expectations is within numerical uncertainty, and thus considered to be satisfactory. We notice that the results evidence the constraint fjf < fj^ associated with the intrinsic characteristic of numerical simulations.

0.1 1 10

Frequency (THz)

0.1 1 10

Frequency (THz)

FIGURE 5. Voltage and current spectral densities of the DUT under thermal equilibrium conditions (a) and (c), respectively, and in the presence of a bias V = 1 V (b) and (d), respectively.

Figure 5 compares the noise spectra of voltage fluctuations with those of current fluctuations at thermal equilibrium (Fig. 5 (a) and (b)), and in the presence of an applied voltage (1 V) sufficiently high for the onset of Gunn instabilities (Fig. 5 (c) and (d)). Simulations are performed for the case of 1 nm. „3£) 10 cm ^ and a layer length L = 1 ^m. The results show that the presence of the Gunn instabilities suppresses totally the plasma peak of the voltage spectal density and is responsible of a sharp peak in both the voltage and current spectra at the Gunn domain transit time frequency of about 0.1 THz. We remark that the voltage spectrum at equilibrium evidences a second peak at the 3D plasma frequency, while the current spectrum at 1 V exhibits higher harmonics, up to four over the fundamental of the Gunn transit-frequency.

According to the analytical theory [2], the application of a gate voltage on the 2D channel should modify the internal concentration and thus the associate plasmonic frequency. Figure 6 shows the spectral densities of voltage fluctuations inside the channel of the gated structure represented in Fig. 1. In Figure 6 (a) the carrier density is fixed to

10 nm, the gate to channel distance niD . 10 cm ^, the width of the channel is is fif = 20 nm, the gate voltage is f/e = 1 V and the drain voltage is L/Q = 0.01 V. Here, only two different lengths of the sample are represented to accomodate the reader. For both lengths the spectra evidence a characteristic spike at 10 THz with is linked to the 3D plasma. Then, a second peak at 4 THz for L = 0.5 ^m and at 1 THz is found for

219

L= 1.5 jim. This particular frequency is moving as the square root of the inverse of the

(b) U D = 0 . 1 V U D = O . S V Un=1.0V

Frequency (THz)

0.1 1 10

Frequency (THz)

FIGURE 6. Spectral density of voltage fluctuations normalized to the static value of the gated DUT. The gate voltage is fixed to 1 V and «g^ = IQl** cm^^ (a) Uo = O.OlV. (b) i = 1 tm.

length and can be associated with the 2D plasma. In Fig. 6 (b) we have fixed the length to L= I jim and we have made varying the drain voltage. For a vanishing potential, the spectrum evidences as previously one peak at 10 THz, related to the 3D plasma, and a second peak at 2 THz, related to the 2D plasma. In analogy with the ungated channel and the results of Fig 5, at increasing the drain voltage the plasma peaks are replaced by a smaller frequency, 0.1 THz at the highest voltage applied, and are related to GUNN domains.

3. CONCLUSION

By studying voltage and current fluctuations with Monte Carlo simulations, we have investigated the evolution of plasmonic noise in ungated and gated nano channels of Ino.53Gao.47As. For the ungated case, in the absence of a bias, the results showed a complex scenario only partly in agreement with the 2D analytical model. In the presence of a bias we have observed a transition between the plasma and Gunn frequencies which is not explained by the analytical theory. The current spectral density does not evidence the plasma peak, as expected, but at high voltages it evidences peaks at the Gunn transit-frequency in analogy with the case of the voltage spectral density. For the case of the biased gated structure, the spectra evidence two peaks. One at the 3D plasma frequency and another that depends on the length of the chanel, as predicted by the analytical theory.

REFERENCES

D. L. WoUard, E. R. Brown, M. Pepper, and M. Kemp, Proc. IEEE 93(10), 1722 (2005). M. Dyakonov and M. S. Shur, Appl. Phys. Lett. 87, 111501 (2005). J.-F. Millithaler, L. Reggiani, J. Pousset, G. Sabatini, L. Varani, C. Palermo, J. Mateos, T. Gonzalez, S. Perez, and D. Pardo J. Phys.: Condens. Matter 20 384210 (2008). J. Lusakowski, W. Knap, N. Dyakonova, L. Varani, J. Mateos, T. Gonzalez, Y. Roelens, S. BoUaert, A. Cappy, and K. Karpierz, J. Appl. Phys. 97, 064307 (2005). M. Shur and V. Ryzhii, International Journal of High Speed Electronics and Systems 13 575 (2003). L. Varani and L. Reggiani, RivistaNuovo Cimento, bf 17, 1 (1994).

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http://Ino.53Gao.47

Shot noise suppression in p-n junctions due to carrier recombination

I. A. Maione, G. Fiori, L. Guidi, G. Basso, M. Macucci and B. Pellegrini

Dipartimento di Ingegneria dell'Informazione: Elettronica, Informatica, Telecomunicazioni Universitcl di Pisa

Via Caruso 16,1-56122 Pisa, Italy

Abstract. We have investigated shot noise suppression as a function of bias current in gallium arsenide and siUcon p-n junctions, focusing on the effect of generation-recombination phenomena. The availability of the cross-correlation technique and of ultra-low-noise ampUfiers has allowed us to significantly extend the range of bias values for which residts were available in the literature. We have then developed a numerical model, based on the Monte Carlo method, which provides a qualitative explanation of the observed noise behavior, and, with the adjustment of a fitting parameter, exhibits satisfactory agreement with the experimental results.

Keywords: Shot noise, recombination, p-n junctions.

INTRODUCTION

Shot noise in p-n junctions should in principle be close to the ideal value predicted by Schottky's theorem [1]. As however demonstrated both theoretically [2, 3, 4] and experimentally [5], when a large contribution to the current is due to charge recombination in the depletion region, shot noise is significantly suppressed.

Previous experimental results [5] have been obtained at low temperature and for relatively large current values (> 10^^ A), but a suppression of shot noise can in principle be observed, even at room temperature and at lower currents.

In this work, we present the results of measurements performed at room temperature on three different p-n junctions by means of the cross-correlation technique, which has allowed us to investigate noise levels well below those already studied in the literature. In particular, we have focused our attention on a GaAs diode and a commercial silicon diode (1N4007) with ideality factor approximately equal to 1.6, and on a silicon diode with an ideality factor equal to 1 over many current decades [8]. The experimental data have then been compared with results obtained from numerical simulations based on a specifically devised Monte Carlo procedure.

MEASUREMENTS AND NUMERICAL MODEL

We have performed an investigation of the behavior of the Fano factor over a wide range of current values, including the low injection regime; this has been made possible as a result of a particular implementation of the cross-correlation technique, a detailed description of which can be found in [6]. In general, the cross-correlation approach consists in using two different and independent amphfication channels and then computing the



221

cross-correlation of the outputs of the two channels, in order to suppress the components of the amplifier noise that are uncorrelated.

In our specific implementation, the device under test is connected in series between the inputs of the two amplifiers, with the aim of minimizing contributions for the equivalent input current noise sources.

In order to quahtatively explain the noise generating mechanism in non-ideal p-n junctions, we have developed a simple numerical model based on Monte Carlo simulations. In particular, let us consider a single story, lasting ttot seconds. At each time step At, one can observe three possible events in correspondence of the anode : 1) a thermionic electron reaches the anode; 2) a recombined electron arrives; 3) no electron arrival occurs.

From a numerical point of view, this can be modelled as follows: at each time step, a random variable x with uniform distribution is generated. If 0 < x < XA (A interval) a thermionic emission has occurred, if XA < x < xg (B interval) a recombination is attempted, while in the case of XB < x < 1 (C interval) no carrier emission happens (inset of Fig. 1).

Let us discuss in detail case 2), which refers to an electron with the following history: the electron is injected into the conduction band; it recombines via a trap in the valence band; it reaches the anode. We impose the further constraint that the trap can be occupied by only one electron at a time. For each recombination event, another random variable T, with mean exponential distribution and with value equal to T, is extracted: T is the time the traps involved in the recombination process will be occupied.

For what concerns instead the power spectral density S(0) at low frequency, it is computed by means of Milatz's theorem [7]. In particular, if Mot is the sum of the thermionic and recombination events occurring during ttou S{0) reads:

S(0) = ^va r{Mot} , (1) Hot

where varjA tot} is the variance of Mot computed over a record of 10000 stories. A key issue is the definition of the A and B intervals which specify the relative

weight of the thermionic and recombination components, respectively, that make up the total current IF- The thermionic component ID is due to the electrons which manage to overcome the barrier at the junction, and the recombination current IR is due to electrons which recombine with holes via recombination centers [9]. Such components are defined by the expressions

ID = loD

IR = loR

exp ( ^ ) - 1 (2)

where q is the elementary charge, kg is the Boltzmann constant, T is the temperature, and V is the bias voltage, while IQD and IQR represent the diffusion and recombination saturation currents, respectively.

IQD and IQR can be extracted from the I-V characteristic, by means of a least mean square fitting procedure. Once obtained, we can compute the ratio between the recombination current and the thermionic current R{IF) = IR/ID- Since ID is unambiguously determined by XA {ID = qxA/At), XB — XA can be expressed as XB—XA= RlD^t/q, obtaining the widths of the intervals at different injection regimes. The quantity XA can then be

222

<

10-2 10-4 10-^ 10-'^ 10-^ 10-8 10-9

10-10 1 0 - " 10-12 10-13

— I 1 1 1 1 —

Quasi-ideal diode n = l

1N4007 diode n=1.61 ^ GaAs diode

n=1.65

1 1

A B C

0 Xyl X B I X

-0.1 0.1 0.2 0.3 0.4 Voltage (V)

0.5 0.6

FIGURE 1. I-V characteristics of the GaAs diode, the 1N4007, and the quasi-ideal diodes. The ideality factors n are indicated. In the inset, the event selection mechanism used to perform simidations is illustrated.

considered as the independent variable of the problem, while xg can be computed as a function of XA'• XB = XA{1 + R)•

RESULTS

We have applied the above described numerical model to three different diodes: a GaAs diode, a commercial silicon 1N4007 diode, and a quasi-ideal sihcon diode manufactured by ST Microelectronics [8], whose I — V characteristics are shown in Fig.l. As can be seen, the three diodes have different ideality factors n.

In Figs.2(a) and (b), the measured (empty triangles) and computed (sohd triangles) values for the Fano factor are shown. The values of the Fano factor in the region of low bias voltage (V < ksT/q) are computed taking into account the thermal component, with the correction formula

2ql coth

^spe 2kBTj

(3)

where Sspe is the measured shot noise value at the current /. The only fitting parameter used in the simulation is represented by the mean capture

time (T), i.e. 9 ps for the GaAs diode and 3 ps for the 1N4007 diode. We observe the expected full shot noise for the quasi-ideal diode, while shot noise is suppressed in the other cases. As can be seen, the implemented models manage to quantitatively reproduces experimental results for the 1N4007 diode, while some discrepancy is present for

223

L i p

1.05 -

1 -

t3 0.95 -

o 0.9 -i^

^ 0.85 -

0.8 -

0.75 L 10-1

experimental data -numerical results -

1N4007 diode

Current (A) 10-" 10-Current (A)

FIGURE 2. Simulation and experimental data for the 1N4007 (a) and the GaAs (b) diode. In (b) shot noise experimental data are shown for the quasi-ideal diode. The solid line represents a Fano factor of one. All the measurements were performed at room temperature.

the GaAs diode. However, also in this case, the quahtative behavior of the Fano factor is recovered by the very simple statistical process that has been proposed.

CONCLUSIONS

Accurate measurements of the shot noise suppression factor have been performed over 4 decades of current on three different types of p-n junctions: a GaAs diode, a commercial 1N4007 diode, and a quasi-ideal silicon diode. Except for the case of the quasi-ideal diode, a suppression of shot noise depending on the bias current has been observed, confirming previous results and extending them to a wider current range. Such a suppression has been attributed to carrier recombination phenomena, which have been modeled by means of a simple Monte Carlo approach.

The results of the simulations, which depend on a single fitting parameter, do reproduce the quahtative behavior of the experimental data and are also in satisfactory quantitative agreement. Further work is needed to refine the model with the inclusion of direct recombination mechanisms and of the dependence of the trap occupation time on the bias voltage.

REFERENCES

1. W. Schottky, Ann. Phys. (Leipzig) 57, 541 (1918). 2. P. O. Lauritzen, IEEE Trans. Electron Devices 15, 770 (1968). 3. K. M. Van Vliet, IEEE Trans. Electron Devices 32, 1236 (1976). 4. J. A. Jimenez Tejada, A. Godoy, A. Palma, and P Cartujo, J. Appl. Phys. 90, 3998 (2001). 5. T. E. Wade, A. Van Der Ziel, Solid State Electronics 19, 909 (1976). 6. B. Pellegrini, M. Macucci, G. Basso, in Advanced Experimental Methods for Noise Research in

Nanoscale Electronic Devices , vol. 151, p. 203 (2004). 7. A. van der Ziel, A'oise in SoUd State Devices and Circuits, Wiley, New York, p. 16, 1986. 8. G. F CerofoUni and M. L. Polignano, J. Appl. Phys. 55, 579 (1984). 9. R. N. Hall, Phys. Rev 87, 835 (1952).

224

1/F Noise In Si Delta-Doped Schottky Diodes

Arkady V. Yakimov, Alexey V. Klyuev, Evgeny I. Shmelev" and Arkady V. Murel, Vladimir I. Shashkin''

"Lobachevsky State University, Gagarin Avenue 23, Nizhniy Novgorod 603950, Russia Fax: +7-831-4656416; E-mail:[email protected]

''Institute for Physics ofMicrostructures, Russian Academy of Sciences, Nizhniy Novgorod 603950, Russia

Abstract. The model of Schottky diode with S-doping is suggested. This one is aimed for the determination of technological areas of the diode, which are responsible for the 1/f noise. Series resistance Rt of base and contacts, and the possible leakage Ikai are taken into account. Parameters of the diode are defined from the analysis of the current-voltage characteristic. For an explanation of experimental data the model of fluctuations in the charge of non-compensated donors in 5-layer of Schottky junction (AA^,- model) is suggested. The analysis of the 1/f noise spectrum allows assuming that, in investigated diodes, on 10' atoms of main impurity there are 1-10 atoms of extraneous impurity the ionization energy of which may stochastically be modulated.

Keywords: Schottky Diode, Delta Doping, Current-Voltage Characteristic, 1/F Noise. PACS: 61.72.Ss; 72.70.+m; 73.30.+y

INTRODUCTION

The diode with Schottky barrier is the perspective nonlinear element used for the detection of microwave radiation. The decrease of effective height of the barrier yields the decrease of differential resistance of the diode; that allows producing the detector operating without external bias. The decrease of the effective height is reached by the increase of tunnel transparency near to the top of the potential barrier reached by strong non-uniform doping (5-doping) of semiconductor at the contact with metal.

The detailed description of investigated diodes is presented in papers [1-4]. Here we offer the model of the low-barrier Schottky diode, focused on the reveal of technological areas responsible for 1/f noise. Dependence of the noise voltage spectrum of the diode on the current is analyzed. The experimental data treatment, within frames of the model of fluctuations in the charge state of external impurity atoms, is offered. The main idea is the modification of ref [5], that is movable (bistable) defects are the source of 1/f noise.

MODEL OF THE DIODE

The equivalent circuit diagram of Schottky diode with 5-doping is shown in Fig. 1. This model is suggested for the determination of technological areas of the diode,



225


which are responsible for the 1/f noise. Series resistance Rb of the base and contacts, and possible leakage current Ikak are taken into account. Equivalent parameters of the diode are defined from the analysis of the current-voltage characteristic.

R leak

t>h '-leak

-* » Rh Schottky

barrier FIGURE 1. Equivalent circuit diagram of the diode.

The investigated diodes were manufactured on common technology and differ by the initial (at / =0) differential resistance i co, varying from 6-10' Ohm to 400 Ohm. The leakage is negligible practically in all diodes, but there is nonlinear resistance i?i.

Dependence of current/c on voltage VD supplied to Schottky barrier is described by relation:

/ ^ = / , • exp exp

Here J] is non-ideality factor; a = d/D, d - distance from 5-layer to metal, D -width of Mott barrier ( a ~0.04); VT = kT/q - thermal potential determined by Boltzmann constant k, absolute temperature T, and elementary charge q. Characteristic current/^ is equal:

/^= A**T^exA kT

where ^ ** is Richardson modified constant; Ao zero current:

effective height of the barrier at

Ao= (O - q'N^dle]-{l- dID).

This one is determined by height O of barrier at metal-semiconductor interface, surface concentration 7V of donor impurity atoms, and dielectric permeability s of semiconductor.

226

SPECTRUM OF THE NOISE VOLTAGE

Spectrum Sv of the noise voltage on the diode was measured at frequencies from a few hertz up to 20 kHz, at different currents / through the diode. The typical family of spectra (for the diode withi?flo=600 Ohm) is shown in Fig. 2.

10 1 5 . , V V H Z N F F T = 2 0 4 8

Nsp=488

Af=11.7Hz

/ , H z 1—I I I iiiii 1—I I I iiiii 1—I I i i i i i i 1—I I i i i i i i

10' 10^ 10^ 10^ 10^

FIGURE 2. Family of noise voltage spectra at different currents through the diode.

The example of the dependence of the spectrum on the current, in the region of 1/f noise, is shown in Fig. 3.

1 0 " i 5 ^ , V ' / H z

10-^S

10-'N

m'U

10 •15

< AA/,-model

I, A —1—I I I I I I I I 1—I I I I I I I I 1—I I I I I I I I 1—I I I Mill

10' 10- lo-' 10- 10' FIGURE 3. Dependence of the voltage noise spectrum on the current at frequency 12 Hz.

For an explanation of experimental data the model of fluctuations in the charge of non-compensated donors in 5-layer of Schottky junction (ATV - model) is suggested.

227

ANs- MODEL

Atoms of main impurity (Si) are ionized. However atoms of extraneous impurity (oxygen, hydrogen, etc.) may exist in 5-

layer. It is assumed that each atom has a few metastable states separated by rather low potential barrier [5]. Thermo-activated transitions between states yield the stochastic modulation of ionization energy of the impurity. This modulation may be treated as fluctuation ATV in the effective concentration of donor impurity in 5-layer.

That means, in Equation (3) value N^, should be replaced on (7V +A7V ). Thus, the effective height Ao of the barrier at zero current is subjected to fluctuations. Going through Equation (2) to Equation (1) we can determine the noise in total current ID, thus, in the noise voltage on the diode.

The analysis of the 1/f noise spectrum allows assuming that, in investigated diodes, on 10*' atoms of main impurity there are 1-10 atoms of extraneous impurity, the ionization energy of which may stochastically be modulated.

ACKNOWLEDGMENTS

Presented research was carried out in Nizhniy Novgorod State University in frames of the Priority National Project "Education". This investigation was supported by the State contract JSo6039r/8473 (PYSlC-08-3).

REFERENCES

1. V.I. Shashkin, V. L. Vaks, V. M. Danil'tsev, A. V. Maslovsky, A. V. Murel, S. D. Nikiforov, O. I. Khrykin, and Yu. I. Chechenin, Radiophysics and Quantum Electronics 48, 485-490 (2005).

2. V. I. Shashkin, Yu. A. Drjagin, V. R. Zakamov, S. V. Krivov, L. M. Kukin, A. V. Murel, and Y. I. Chechenin, Int J. Infrared and Millimeter Waves 28, 945-952 (2007).

3. V. I. Shashkin, A. V. Murel, V. M. Danil'tsev, and O. I. Khrykin, Semiconductors 36, 505-510 (2002).

4. V. I. Shashkin, and A. V. Murel, Semiconductors 3^, 554-559 (2004). 5. V.B.Orlov, and A.V.Yakimov, PhysicaB 162, 13-20 (1990).

228

Noise Enhanced THz Rectification Tuned by Geometry in Planar Asymmetric Nanodiodes

I. Ifiiguez-de-la-Torre,''H. Rodilla,''J. Mateos/D. Pardo/A. M. Song and T. Gonzalez"

"Dpto. de Fisica Aplicada, Universidad de Salamanca, Pza. de la Merced s/n, 37008 Salamanca, Spain ineering, Universii United Kingdom

School of Electrical and Electronic Engineering, University of Manchester, Manchester M60 IQD,

Abstract In this work we explore a high frequency collective phenomenon present in an asymmetric nanodiode. Charge fluctuations in the space-charge regions around the channel become visible in the noise spectra at a characteristic frequency. Though dealing with a specific noise mechanism, by virtue of the particular geometry of the device, this phenomenon enhances the DC response of the diode to AC signals, originating a THz resonance at a frequency that can be tuned by the geometry. Taking place at room temperature, this effect can be exploited to design somewhat frequency-selective detectors in the THz domain. The use of different narrow bandgap materials (like InGaAs, InAs or InSb) in the channel and the influence of the operation temperature are analyzed in order to improve the resonance characteristics.

Keywords: Monte Carlo simulation. Electronic Noise, THz detector devices, Nanoscale Diodes PACS: 02.70.Uu, 73.40.-c, 07.50.Hp, 85.35.-p

INTRODUCTION AND OVERVIEW

In modem nanoelectronics great efforts are being put to develop low-cost semiconductor devices covering the so-called THz gap. This shadowy frequency range, sandwiched between microwaves and infrared, has a variety of high impact potential applications, such as biomedical imaging and military or security tools. A recently developed planar asymmetric non-hnear nanodevice, so called self-switching diode (SSD) [1], seems to be a potential candidate for a new class of THz detectors and emitters. SSDs are planar diodes fabricated with just one lithographic step by simply etching L-shaped insulating grooves onto a semiconductor layer, as shown in Fig. 1(a). This particular geometry [top-view scheme, xy plane, shown in Fig. 1(b)] originates a strongly nonlinear rectifying/-F characteristic [2]. The planar architecture allows obtaining a parasitic capacitance between contacts substantially lower than in vertical devices of the same size, and the feasibility to integrate many SSDs in parallel without the need of interconnects overcomes the high-impedance problems typical of nanodevices. These two features, combined with the use of fast III-V materials, lead to an excellent frequency performance. Operation of SSDs based on InGaAs channels as detectors up to 110 GHz at 300 K [3] and 1-2 THz at low temperatures [4] has been experimentally confirmed.



229

http://02.70.Uu

http://73.40.-c

FIGURE 1. (a) 3D schematic view of an SSD including the details of the heterostructure used in the fabrication, (b) Top-view geometry {xy plane) considered in the 2D MC simulations. Different values of

Wy are investigated. Lc=250 nm, Wc=50 nm, Wi,=5 nm, £^=£^=1 andZ,a55=175 nm.

In this work, by means of Monte Carlo (MC) simulations, we show that a high frequency collective phenomenon present in SSDs, which leads to a tunable-by-geometry peak in the current noise spectra, couples to the DC response of the device, thus originating a THz resonance in the rectification of AC signals. First, SSDs with InGaAs as channel material are studied to identify the influence of the geometry and the link between noise and AC to DC rectification [5]. Then, we show that the amplitude and quality factor of the resonance can be improved by using narrow bandgap semiconductors with very small effective mass (like InAs and InSb) and by decreasing the operation temperature.

M O D E L

In our analysis we make use of a MC simulator self-consistently coupled with a 2D Poisson solver successfully employed in previous works for the study of SSDs [2,5,6] and different types of InGaAs based nanodevices [7]. We perform 2D simulations only in the channel [Fig. 1(b)]. The influence of the fixed charges present in other layers is accounted for by means of a "background doping" (7VDft=10 ^ m" ), and a negative charge density is considered at the boundaries of the insulating trenches to take into account the depletion originated by surface states. More details can be found in [7].

RESULTS AND DISCCUSION

InGaAs SSDs: Noise and Rectification

Initially we consider an SSD consisting of an lno53Gao47As channel with the 2D topology and dimensions shown in Fig. 1(b), and the influence of the width of the vertical trenches W^ is analyzed. We calculate two quantities at room temperature: the current noise spectral density Si(f) at equilibrium [6], and the average DC current in response to harmonic voltage signals V=Vosin{2Kft) of increasing frequency/ applied between the contacts [8]. Figure 2 allows the comparison of both quantities calculated for several values of W^. The rectified DC current (left axis) exhibits a pronounced peak just before the decay in the response. The peak frequency is shifted to higher values for wider vertical trenches. In Si(f) two peaks are observed: the one at higher frequency (around 3 THz, independently of W^) is due to 3D plasma oscillations [6], while the lower-frequency peak (1-2 THz) exhibits a dependence on Wy quite similar to the peak in the rectified DC current. This similarity may indicate that we are observing the same microscopic phenomenon displayed in different macroscopic quantities.

230

W=5nm

W^20 nm

M/,/=50 nm

o O Q

o

3.0 ^ ' E

CM

< CO

2.0 °

Frequency (THz)

FIGURE 2. Comparison between the current noise spectrum (right axis) at equilibrium and the rectified DC current (left axis) when the amplitude of the AC excitation is Fo=0.25 V, for different W^.

The microscopic information provided by the MC simulations has allowed us to identify a common origin for the peak at lower frequency in the noise spectrum and the resonance in the rectification: the dynamics of the carriers reflected back at both sidewalls of the vertical trenches. Electrons experience a deceleration/acceleration process at the potential barrier created by the surface charge at the interfaces of the vertical trenches, which originates charge density fluctuations with a characteristic frequency closely connected to plasma oscillations [9]. This is a noise mechanism, in principle negative, which is coupled to the terminals by the capacitance of the vertical trench and leads to the peak observed at lower frequency in the noise spectra. However, due to the singular geometry of the SSD, these collective charge fluctuations become also visible in the AC to DC rectification as a result of the coupling to the channel via the horizontal trench, enhancing the DC responsivity in the form of a resonance. As observed, the resonance can be tuned in the THz range by means of the geometry of the diodes, shifting to higher frequencies for wider vertical trenches, what can be quite useful for detection apphcations. We have also found that the resonance shifts to lower frequencies when the permittivity of the vertical trench e^ is increased [5], suggesting a strong influence of the associated capacitor Cv=ev/ v-

SSDs Based on InAs and InSb Channels. Temperature Effect

In this section we focus on the influence of the material constituent of the channel and on the operation temperature. Two narrow bandgap semiconductors, InAs and InSb, have been considered in the channel of the SSD (with the same geometry) to compare with the InGaAs case, by performing the same type of simulations. Figure 3(a) shows the results obtained for the noise spectra and the rectified DC current. As observed, the amplitude and quality factor of the resonance in the rectification are notably improved. For example, in the case of InSb the DC current at the resonance takes a value 12 times higher than that at low frequency. The more ballistic character of transport, achieved by the use of both higher mobility materials, enhances the observed phenomenon and shifts it to higher frequencies, what is quite interesting

231

from the point of view of applications. Concerning the noise, the 3D plasma peak moves to higher frequencies due to the lower effective mass of InAs and InSb, scahng

exactly according to the formula f^^ = 4^ob'^^I'^^^"''^ ' where Nob, e, and m* are the electron density, charge, and effective mass, respectively, and s is the permittivity of channel material. The results also confirm that the low frequency peak in the noise spectra is hnked to a plasma effect, since it scales in the same way as the 3D peak. Finally, Fig. 3(b) shows that a decrease of temperature from 300 to 77 K in InGaAs SSDs implies a quite significant enhancement of the resonance, thus improving the sensitivity of detection, again due to a superior mobility. Interestingly, the frequencies of both peaks in the noise spectra remain unaltered, since temperature is not involved in the plasma frequency.

12 -InGaAs InAs

E" 10 InSb

5

Frequency (THz) Frequency (THz)

FIGURE 3. Comparison of the current noise spectra at equilibrium (right axis) and the rectified DC current (left axis) when the amplitude of the AC excitation is Fo=0.15 V, for W^=5 nm.

(a) Influence of the semiconductor in the channel: InGaAs, InAs and InSb. (b) Influence of the operation temperature in the case of the InGaAs SSD.

ACKNOWLEDGMENTS

This work has been partially supported by the Direccion General de Investigacion (MEC, Spain) and FEDER through the project TEC2007-6I259/MIC and by the Consejeria de Educacion of the Junta de Castilla y Leon (Spain) through the project SA019A08.

REFERENCES 1. A. M. Song, M. Missous, P. Omling, A. R. Peaker, L. Samuelson, and W. Seifert, Appl. Phys. Lett.

83, 1881 (2003). 2. J. Mateos, B. G. Vasallo, D. Pardo and T. Gonzalez, Appl. Phys. Lett 86, 212103 (2005). 3. C. Balocco, A. M. Song, M. Aberg, A. Forchel, T. Gonzalez, J. Mateos, I. Maximov, M. Missous, A.

A. Rezazadeh, J. Saijets, L. Samuelson, D. Wallin, K. Williams, L. Worshech and H. Q. Xu, Nano Letters S, 1423(2005).

4. C. Balocco, H. Halsall, N. Q. Vinh, and A. M. Song, J. Phys.: Condens. Matter 20, 384203 (2008). 5. I. Wiguez-de-la-Torre, J. Mateos, D. Pardo, A. M. Song and T. GonzaXez, Appl. Phys. Lett. 94, to be

published (2009). 6. I. Miguez-de-la-Torre, J. Mateos, D. Pardo and T. Gonzalez, J. Appl Phys 103, 024502 (2008). 7. J. Mateos, B. G. Vasallo, D. Pardo, T. Gonzalez, J. S. Galloo, Y. Roelens, S. BoUaert, and A. Cappy,

Nanotechnology 14, 117 (2003). 8. K. Y. Xu, X. F. Lu, A. M. Song and G. Wang, J. Appl Phys 103, 113708 (2008). 9. M. Trippe, G. Bosman and A. Van der Ziel, IEEE Trans. Microwave Theory Tech. 34, 1183, (1986).

232

Analysis Of Noise Characteristics And Noise Generation In SubTHz And THz Frequency

Ranges

V.L.Vaks'', A.N.Panin", S.I.Pripolzin", E.A.Sobakinskaya'', D.G.Paveliev

'^Institute for Physics ofMicrostructures of Russian Academy of Science, GSP-105, Nizhny Novgorod, Russia.

Nizhny Novgorod State University, Nizhny Novgorod, Russia

Abstract. Noise characteristics of subTHz and THz devices (mixers, detectors) based on Schottky diodes and quantum superlattice diodes (SL-diodes) are presented. The test bench for measurements of device's characteristics is elaborated. Noise generator of THz frequency range based on SL-diodes multiplier and mm noise generator is also demonstrated.

Keywords: THz range, mixer, multiplier, noise generator, superlattice, Schottky diodes. PACS: 85.30.De, 85.35.Be.

INTRODUCTION

At present great efforts are being devoted to elaboration of sources and receivers of subTHz and THz ranges. The key point in this problem is noise parameters of devices (mixers, detectors, multiphers) used in a set up. Now the Schottky diodes are widely applied in various devices of THz frequency range. However, for a successful operation in this region, it is necessary to increase cut off frequency of Schottky diodes that has proven quite difficult due to fundamental restrictions. Furthermore, use of superlattice structures are expected to be more effective for frequency transformation and detection, since the lower values of inertness and parasitic capacitances and presence of negative differential conductivity (up to 1 THz) on the volt-ampere characteristic. Thus, the first part of our report concerns comparison of noise characteristics of devices, based on Schottky diodes and superlattice diodes. The second part deals with problem of noise generation in subTHz and THz frequency ranges. Many of applications, including super-high-speed data communications (up to few hundreds Mbit/s), require ultra-wide band sources, which can be obtained with employment of noise generators.

Measurement of Noise Characteristics

For analysis of the noise parameters in subTHz and THz frequency ranges we elaborated a test bench (Fig.l), which includes high-precise and stable sources of radiation. Some results of measurements for Schottky diodes and SL- diodes are presented in Fig.2-3.



233

Noise generator

Mm range synthesizer

Attenuator

I I I I L_-t_JL

Signal commutation and

filtering circuit

Measuring block n

I

DAC

Control device

IMHz generator

Low-pass filter

Digital voltmeter

F I G U R E 1. Test bench for analysis of noise parameters of various devices.

500 1000

I, mkA

F I G U R E 2. Noise parameters of SL-diodes: N E P - noise equivalent power, R- differential resistance.

234

NEP

10 w* HI' '/= «, kOhm

i l

1

\ ^ ! .

500 lOOD 1500

I, mkA

LDQD IJXQ

I, mkA

FIGURE 3. Noise parameters of Schottky diodes: NEP- noise equivalent power, R- differential resistance.

The results of our experiments show that noise parameters of Schottky and superlattice diodes are comparable for small signal. In the case of large signal superlattice diode is more stable.

Generation of Noise Signal

Noise generator up to 1.2 THz is elaborated with the help of BWO electronic beam modulation by comparatively low-frequency (relative to carrier frequency) regular and noise signals. The point is that frequency-modulated signal has dense spectrum at large modulation indexes, (3, even if modulation is done by monochromatic signal (especially non-monochromatic one). In this case the frequency modulated signal may be written as:

0} = C0„ + Aco'^d^COSnQj:

where coo is a reference frequency, Aco is a noise band, Q is a interval between spectral hues of a frequency-modulated signal. If low-frequency noise signal is additionally supplied, then the reference and lateral frequencies are broadened by noise signal frequency band Aco. If a condition Aco> Q is valid, the spectrum of modulated signal is continuous. The band of microwave noise is given by

Af-- -^^BWO^O

where uo is an amplitude of periodic signal, SBWO is a transconductance. The value of SBWO for the majority of BWO (in mm wavelength) is in the range between 30 and 80 MHzA^. Block-diagram of experimental set-up is shown in Fig.4.

235

Power supply

Low-frequency noise generator

FIGURE 4. Block-diagram of experimental set-up.

For further advancing in noise generation at higher frequencies we used frequency multiphcation of mm noise generator (BWO or Gunn generator, working in noise mode). Spectrum of a noise signal obtained by multiphcation of Gunn generator's frequency (89-117 GHz) with SL-multipher is presented in Fig.4.

T = 300 K

CD

O 1000 h Q.

Q -i U I I I I I I I I I I I I I I I I I LJj I I L

200 600 1000 frequency (GHz)

FIGURE 5. Noise signal from Gunn generator.

ACKNOWLEDGMENTS

This work is financially supported by ISTC 3174, CRDF R U C 2 - 2 8 6 7 - N N - 0 7.

236

High Frequency Noise in GaN HEMTs

J. Mateos, S. Perez, D. Pardo and T. Gonzalez

Dpto. Fisica Aplicada, Universidad de Salamanca, Plaza de la Merced s/n, 3 7008 Salamanca, Spain e-mail: [email protected], phone: +34 923 294400 fax: +34 923 294584

Abstract. In this work, by means of Monte Carlo simulations, we study the intrinsic high frequency noise of a 250 nm gate AlGaN/GaN HEMT and its temperature dependence. A consistent self-heating model is used for the correct calculation of the I-V curves, showing that both the drain conductance and fransconductance are reduced due to the high dissipated power. In the case of noise, the influence of temperature is not so strong, both on the intrinsic P, R and C parameters and the exfrinsic minimum noise figure and noise resistance, since the lower level of current fluctuations found for higher T is counteracted by a poorer dynamic performance.

Keywords: GaN HEMTs, Monte Carlo simulation. PACS: 85.30.De, 85.30.Tv.

INTRODUCTION

In spite of the extremely high speed of InP and GaAs based HEMTs, the use of narrow bandgap semiconductors is restrained to low power applications, since impact ionization limits the working voltages. In fact, HEMT technology has not been widely exploited for high power, high frequency applications due to the low value of the bandgap of the traditionally used III-V semiconductors (in the range 0.7-1.5 eV). The wide bandgap of GaN, 3.42 eV, together with a moderately high mobility (as compared to Si) of about 1500 cm^A's, allows for high temperature operation (up to 300°C) and high power amplification at microwave and millimetre-wave frequencies [1]. Another important advantage of AlGaN/GaN HEMTs for high power operation is the spontaneous polarization charge appearing at the heterojunction, which provides extremely high electron concentrations in the channel without the need for any doped layer. As a consequence, the development of GaN HEMTs has a strong practical interest for mobile telecommunications, since they can provide some advantages over the technology commonly used in base station amplifiers, where high power and linearity are required. A low level of noise is also necessary for such high frequency applications.

MONTE CARLO SIMULATION

We have developed an ensemble Monte Carlo simulator self-consistently coupled with a 2D Poisson solver for the analysis of AlGaN/GaN HEMTs. In this work special attention is paid to the study of the noise performance. The key influence of surface polarization charges P at the AlGaN/GaN interface (which leads to an enhanced electron accumulation in the channel) is taken into account. The presence of a surface charge density a at the top of the AlGaN layer, appearing as a result of polarization charges partially compensated by charge



237


trapped at surface states, is also considered. Due to the high delivered power, self-heating effects are crucial in GaN HEMTs and must be carefully considered in simulations. To this end, a self-consistent method making use of the thermal resistance of the transistors, R,/,, has been implemented.

The lattice temperature of the device r^a is continuously updated in the simulation by using the calculated value of//) according to the expression Tiatt=300K.+R,i,-lD-VDs- By using this technique, the static output characteristics of a 250 nm-gate Alo,27Gao,73N/GaN HEMT with the geometry shown in Fig. 1(a) have been calculated. Its layer structure consists of a 25 nm AlGaN barrier with 27% Al on the top of a 500 nm GaN layer. For the simulations we have usedP=12.46xlO'^ cm" (taken from [2]) and a=-5.0xlO'^ cm"^ so that «^=7.46xl0'^ cm"^ in the range of values found experimentally.

RESULTS

When analyzing the influence of self-heating, in Fig. 1(b) it can be observed that, as compared with the simulations made at constant temperature, the ID-VDS curves are very similar in the triode region and differ mainly in the saturation region for high currents, saturation being more pronounced when dissipated power is high and large R,/, is considered (even providing a negative slope).

This behavior reflects in the transconductance curves shown in Fig. 2, the maximum of g„ decreases and the curve widens as Tiatt increases (at the side of high Vgs values). The effect of self-heating in the rest of elements of the small signal equivalent circuit of the transistors is significant only in the case of g„ and Cgs (both decreasing when increasing Tiatt). This dependence is more pronounced in the case of g„, thus leading also to a reduction of the cutoff frequency/c (calculated as gJlnCgs) when the temperature increases.

2000

FIGURE 1. (a) Geometry of the simulated AlGaN/GaN HEMT and (b) intrinsic output characteristics at T=300 K compared with those obtained with the self-consistent temperature model with iJft=10xlO''

andiJa=20xlO-'K/(W/m). Top curve Fe5=-2.5V, AFe5=0.5V.

238

- V„,=5.0V ^ ^ ^ -

- J -^r— R „=10x10"'K/(W/m) "

P — • — R „=20x10"'K/(W/m) :

/ Experimental result -

-4.0 -3.0

=(V) -2.0 -1.0 0.0

FIGURE 2. Intrinsic transconductance at T=300 K compared with that obtained with the self-consistent temperature model with iJft=10xlO"' andiJft=20xlO"' K/(W/m), for (a) FD5=5 V and (b) FD5=10 V. The

dashed lines represent the experimental maximum transconductance measured in real devices.

P, R and C are intrinsic parameters that include the influence of both the current fluctuations and the intrinsic dynamic response of the device (by means of the intrinsic Y parameters). P=SM/4KBTiatt\Y2i\ represents the noise in the drain current associated to electron velocity fluctuations, R=Sjg\Y2i\ /4KBTiatt\Yu\^ the noise at the gate induced by charge fluctuations in the channel, and C=-j S^g^/ {SigS^y^ the cross-correlation between both. Being KB the Boltzmann constant and Sjj, Sjg and 5/ / the spectral densities of drain and gate current fluctuations, and their cross-correlation, respectively.

The values of the intrinsic P, R and C noise parameters have been calculated as a function of temperature (in this case with constant values of r^a). In Fig. 3 we can observe that the drain and gate current noise (represented by P and R, respectively) decreases for high temperature, but mainly under high gate bias, far from the optimum low noise conditions (near pinch off). Regarding the extrinsic noise parameters (minimum noise figure, F„,„, associated gain, Gass, noise resistance, R„ and optimum reflection coefficient. Top,), they have been calculated by adding the contributions of the source, gate and drain resistances, Rs, RG and Rp, respectively, to the simulated intrinsic noise. We consider Rs=l.O Q-mm, RG=2.6 Q/mm, and Rj)=2.0 Q-mm, temperature independent as first approach.

The extrinsic noise parameters are plotted in Fig. 3 (d)-(h), showing that the influence of temperature on F„,„ is not very significant under low bias conditions (since the decrease P and R obtained with higher Thtt is compensated by a lower fc). In order to link the behaviour of F„,„ to that of P, R and C, the approximate formula for the intrinsic minimum noise figure F™„=l+2 (f/f,) [PR{l-C^)Y'\ This formula indicates that F„,„ increases with P and R, while the drain-gate correlation reduces the total noise. As observed in the figure, the optimum low-noise conditions (minimum value of F„,„) are achieved for low current. These conditions change with temperature, shifting to higher Vgs as Tiatt increases, but providing a similar F„,„, around 1.4 dB for 10 GHz and 3.0 dB for 26 GHz. The most significant effect of temperature is the reduction of Gass with the increase of Tiatt due to the decrease of g„, while R„ and Top, remain almost unchanged.

The fact that the extrinsic resistances are considered to be temperature independent could also contribute to the observed behaviour. If the values of extrinsic resistances raise with temperature (as expected in reality), the noise would increase.

239

6

' 4

2

0 0.5 0.4 0.3 0.2 0.1 0.0

0.8

0.6

0.4

0.2

0.0 -6.0 -4.0 -2.0 0.0 2.0

FIGURE 3. Noise parameters: Intrinsic (a) P, (b) R, (c) C, and extrinsic (d) F,„,„, (e) G^,, (f) R„ and (g) magnitude and (h) phase of r„p, as a function of VQS for F/)5=10 V and several T/^tt- The results in (d)-(h) are shown for two frequencies (10 y 26 GHz).

ACKNOWLEDGMENTS We want to acknowledge the support of the Spanish Ministerio de Defensa and the

KORRIGAN initiative [3]. This work has been partially supported by the Direccion General

de Investigacion (MEC) and FEDER (project TEC2007-61259/MIC) and by the Consejeria de

Educacion of the JCyL (project SA019A08).

REFERENCES

M.A. Wakejima, K. Matsunaga, Y. Okamoto, Y. Ando, T. Nakayama and H. Miyamoto, Electronics Letters 41, 1371-1372(2005). V. Fiorentini, F. Bernardini, andO. Ambacher, ^;)p/. Phys Lett 80, 1204-1206 (2002). See for example: G. Gauthier and F. Reptin, "KORRIGAN: Development of GaN HEMT Technology in Europe." 2006 Digests of the CS Mantech Conference, 49-51, 2006.

240

Current Collapse and Deep Levels of AlGaN/GaN Heterostructures monitored

by LFN Measurements M. TACANO", N. TANUMA", S. YAGI^ H. OKUMURA^

T. MATSUI', J. SIKULA'' "Meisei University, Hino, Tokyo, 191-8506 JAPAN, ""Power Electronics Center, AIST, Tsukuba, JAPAN

'MM Device Section, NICT, Koganei, Tokyo, JAPAN, ''FEEC, Brno University of Technology, Technicka 8, 61600, Brno, CZ

Abstract: The correlation between the current collapse in the IV characteristics of AlGaN/GaN HFETs and the traps monitored through the unpassivated and SiN-passivation processes of the AlGaN/GaN heterostructures by the low frequency noise measurements is reported: the noise level of Ei(47 meV) trap decreased by 10 dBA/VHz by the SiN passivation process together with the current collapse recovery in IV curves, while £2(131 meV) and £3(235 meV) levels became apparent after SiN passivation, indicating the remarkable suppression of the El trap by the passivation. The commercially available AlGaAs/GaAs LED heads for the page and FAX printers found several deep levels introduced during the contact formation processes, which could not be assigned by the DLTS measurements. Keywords: AlGaN/GaN HFETs, AlGaAs/GaAs LED Arrays, Low Frequency Noise, Arrenius plots without DLTS PACS:70, 62.25 .De

Introduction

Best performances of AlGaN/GaN high-electron-mobility transistors (HEMTs) are obtained by growing the heterostructure on the lattice matched SiC substrates, and improved device structures obtained 550 W pulse output at 3.5 GHz, and the via hole drain structure reahzed the breakdown voltage of over 10 KV for power devices, approaching GaN device properties to those of the vacuum power devices, the cutoff frequency of 190 GHz and gm of 420 mS/mm with the 60 nm gate length was also obtained. The growth technology of wide gap semiconductors, however, still needs years to put these promising materials for the practical use. The drain current collapse is the serious problem in the AlGaN/GaN heterostructures. This is induced by the traps within the AlGaN and GaN layers, and can easily be monitored by the LFN measurement without using the DLTS measurements. We have the generation-recombination noise at the corresponding time constant in addition to the intrinsic \lf noise level in the noise spectroscopy. We report here the simultaneous deep level assignments of the AlGaN/GaN heterostructures by the LFN measurement with the drain current collapses in the HFET drain currents. LFN measurement technology is also apphed to an AlGaAs/GaAs PNPN junction LED, to which the DLTS measurements cannot be applied to monitor the deep level traps in the too much complicated structures.



241

Experiments and Results

1) AlGaN/GaN Heterostructure HFET The layer structure was prepared for a high-breakdown-voltage AlGaN/GaN metal-insulator-semiconductor HEMT (MIS-HEMT), consisting of a 4-|j,m-thick undoped GaN layer and a 15-nm-thick undoped Alo25Gao75N barrier layer on a 2-inch c-face sapphire substrate by the metal organic chemical vapor deposition (MOCVD). The sheet carrier mobility and the density were 480 cmWs and 1.3 X lo'^ /cm^ at room temperature, respectively. The Ohmic electrodes were formed at the drain and source by depositing Ti (25 nm thick), Al (100 nm), Ni (40 nm) and Au (50 nm) by an electron-beam deposition and annealing by the rapid thermal annealing at 700°C for 120s in N2 gas. The minimum specific contact resistance was 2.6 X 10" Q-cm . The gate length and source-drain distance were 2 and 6 |j,m, respectively, formed Ni (25 nm) and Au (500 nm) deposition. A 120-nm-thick SiN layer was deposited on the completed device for surface passivation by ECR sputtering. The current collapse in the device the I-V characteristics were evaluated after applying the drain bias of Vd = 10 V. Figure 1 shows both the un-passivated drain current of 184 mA/mm (dashed hues) and that of the passivated devices 264 mA/mm (solid hues) at the drain and the gate voltages of Vd =10 V and Fg =1 V, respectively. SiN film is known to reduce the surface state density, resulting in the drain current recovery in AlGaN/GaN HEMTs. The LFN measurements were done on the same wafer as the HEMTs and Hall elements. The device was mounted in a cryostat with the minimum noise level of -245 dBA/VHz up to 100 KHz. Figure 2 shows temperature dependence of noise density Si at different frequencies for the unpassivated (a) and passivated (b) devices. The peak in Fig. 2(a) corresponds to the generation-recombination (G-R) noise by the electron trap, Ei=47 meV Fig. 2(b) indicates -12 dBA/VHz suppession by the SiN passivation and those by the traps E2=131 meV and E3=235 meV becomes apparent. The Arrhenius plots of G-R noise at Ei, E2 and E3 are shown in Fig. 3, indicating the El trap the main source of the current collapse in AlGaN/GaN HFETs.

2) AlGaAs/GaAs Heterostructure LED Array The semiconductors GaAs and AlGaAs/GaAs are the fore-runners of GaN, and great many works on the deep levels are studied, too, both by the DLTS and LFN measurements[13]. Many of the deep levels in AlGaAs/GaAs LEDs, LDs and HEMTs are now deleted to have sufficient life times of the order of 10^ hrs as the practical devices, and various new applications are devised. One of the stacked LED array became commercially available as the hght source for the page/FAX printers. This hght source array can make more compact hght system compared with those made by the laser diode, suitable for the compact size color page printers or FAX printers. Figure 4 shows the schematic diagram and its expected IV curves of a light emitting thyristor, during the on-state of which the light is emitted and led to the collecting lens. Each of this thyristor is arrayed to make the LED printer The LFN measurements of

242

the anode noise can determine the deep levels existing within these complicated device stmcture, which cannot be done by the conventional DLTS measurements, i.e., most powerful way in this case. Two kinds of substrates are compared by the LFN measurements in Fig. 5. We knowthat similar deep levels are induced in these substrates, independent of the wafer preparations. Well prepared GaAlAs/GaAs HEMT has no deep levels, and observed deep levels might be introduced in the Ohmic contacting process.

Conclusion

1) AlGaN/GaN Heterostmcture HFET Low frequency noise measurements both before and after the SiN passivation of AlGaN/GaN HFETs made it possible to trace the reduction of deep level(47 meV) trap density corresponds to the dramatic decrease of the drain current collapse. This passivation also revealed another two deep levels E2=131 meV and E3=235 meV in addition to the reduction of the trap density of the deep level Ei=47 meV These traps need to be diminished by improving the substrate growth technology. We need also to make enhanced temperature measurement hopefully up to 200 C so that we can assign another huge trap just above the room temperature, which could affect much on the drain current at room temperature.

2) AlGaAs/GaAs Heterostmcture Thyristor

Light emitting thyristor is a kind of new practical application of AlGaAs/GaAs heterostmcture devices, and is now commercially available as the page printer head LED. Low frequency noise measurements of this complicated stmcture made it possible to assign the deep levels induced within the stmcture. These levels were observed independent of the substrate preparation and of the substrate lots, indicating the deep level introduction through the Ohmic contacting processes, just as typically observed in AllnAs/lnGaAs heterostmctures. Improvements of the contacting processes must be procecuted to diminish the deep levels. The thin film passivation on AlGaAs surface could improve the rehability of the device.

Low Frequency noise measurements have shown a valuable tool to assign the deep level trap energy and density of hard electronics devices made of SiC or GaN as well as those made of the shallow band gap devices like InGaAs and InAs as well as those of the conventional devices like GaAs and Si, and of the comphcated device stmctures.

243

Fig. 1. Drain IV characteristics of AlGaN/GaN HFET before (dashed) and after(solid) SiN passivation.

Temperature (K)

Fig. 2a. Temperature dependences of current noise characteristics for HFET without SiN passivation at 5 mA.

100 200 Temperature (K)

Fig. 2b. Temperature dependences of current noise characteristics for HFET with SiN passivation at 5 mA.

S 10

E3 235 TOV

i

2 f

E2

d" ^ 7

Ji

t 8 g-' ? Si'S

131 meV

t ,

.?•'' ' '

-47 meV _

•°'

5 10 15 20

iooo/r(K"')

Fig. 3 Arrhenius plots of the fluctuation time constant, r, for AlGaN/GaN HFET.

A L l f C r (anode>

p-AIGaAs AiyCr/AuGe/Cr

n-AIGaAs p-AIGaAs n-AIGaAs

n-GaAs_sub. i-GND

N

N

1 Fig. 4 Schematic diagram of light emitting thyristor

Sebstrate 1 1 . ,,, Sebstrate 2

C

• f=1Hz • 10Hz "

100Hz 1kHz 10kHz

^vKmKts^KtDm^ts^^"

100 150 200 250

Temperature [K ]

100 150 200 250

Temperature [ K ]

Fig. 5 LFN measurements of AlGaAs/GaAs light emitting thyristor

244

Optimal 2DEG Density for Plasmon-Assisted Ultrafast Decay of Hot Phonons

E. Sermuksnis, J. Liberis, and A. Matulionis

Fluctuation Research Laboratory, Semiconductor Physics Institute, Vilnius 01108, Lithuania

Abstract. Fluctuation technique is used to measure hot-phonon lifetime and its dependence on electron density and excitation level in GaN-based two-dimensional electron gas (2DEG) channels. The results are compared with those obtained by subpicosecond optical techniques. The observed non-monotonous dependence of hot-phonon lifetime on electron density is explained by resonant hot-phonon-plasmon interaction. The optimal electron density for the fastest hot-phonon decay is estimated.

Keywords: microwave noise, two-dimensional electron gas, high electric field, hot phonons, GaN-based channels. PACS: 72.20.Ht, 72.70.4-m, 72.80.Eyt

INTRODUCTION

High electron mobility transistors (HEMTs) based on GaN channels are promising for high-power applications at microwave frequencies. High electron mobility, high density of two-dimensional electron gas (2DEG), and high drift velocity at high electric fields are in the wish list for a better high-frequency performance.

It is well known, that high-field electron drift velocity measured at a high 2DEG density is lower than that achieved in semi-insulated undoped GaN (pin diode) [1]. The velocity is limited, in part, by accumulation of non-equilibrium LO phonons (hot phonons) [2]. As a result, an ultrafast decay of hot phonons is desired for an advanced HEMT technology. Fluctuation technique seems to be the best for measuring dependence of hot-phonon lifetime on 2DEG density and hot-phonon mode occupancy. The paper reports on experimental investigation of the optimal conditions for the ultrafast decay of hot phonons.

RESULTS

The pump-probe Raman scattering experiment on bulk GaN revealed a monotonous decrease of the hot-phonon lifetime as the carrier density increases (Fig. 1, bullets [3]). The dependence was interpreted in terms of LO-phonon-plasmon interaction (solid curve [4]). However, similar Raman data are not available for a high-density 2DEG, and the fluctuation technique is used most often for estimating the hot-phonon lifetime in the 2DEG channels of interest for microwave electronics (pentagon [2], diamond



245

[5]). The pioneering result (diamond [5]) is confirmed by optical pump-probe LO-phonon-assisted intersubband absorption experiment (black triangle [6]). A reasonable agreement with the data on bulk GaN is obtained if the electron density per unit volume is estimated as the 2DEG density divided by the width of the quantum well at the Fermi energy (Fig. 1).

0.1

- • — » • 293 K

GaN model [4] • GaN [3] O AIGaN/GaN [5] A AIGaN/GaN [6] O AIGaN/AIN/GaN [2]

10^ 10" 10" 10" Carrier density (cm')

10"

FIGURE 1. Hot-phonon lifetime at different carrier density for GaN (bullets [3]) and GaN-based 2DEG channels: Alo.isGao.gsN/GaN, 5x10^^ cm" (diamond [5] and black triangle [6]), Alo.33Gao.67N/AlN/GaN, lxlO"cm"^ (pentagon [2]). Solid curve stands for plasmon-assisted model [4].

'oT LL

E 0)

c 0

0

X

1

0.5

ni

* present paper A [6] 0 [5,9]

- D [8] 0 [2] 0 [7]

GaN 2DEG

* ^ -

00 r s ^ A •• >o o ^ o ^ " n « > o ; o O ^ ^^^ :

0 00 00 OOo Q?3C(-, ^%-

* 1 "

0.01 0.1 1

Occupancy of hot-phonon modes

FIGURE 2. Hot-phonon lifetime as a function of hot-phonon mode occupancy for GaN 2DEG channels: lno,i8Alo,82N/AlN/GaN, 1.2xl0"cm"^ (squares [8]); Alo.isGao.gsN/GaN, SxlO^ cm" (diamonds [5,9] and black triangle [6]); Alo,33Gao,67N/AlN/GaN, lxlO"cm"^ (pentagons [2]); Alo.22Gao,78N/GaN/AlN/GaN, SxlO^ cm" (circles [7]), lno,i8Alo.82N/AlN/GaN, 1.5xlO"cm"^ (stars, present paper).

246

http://Alo.33Gao.67N/AlN/GaN

Stars in Fig. 2 show an essential decrease of the hfetime at a high occupancy of hot-phonon modes (the lattice-matched InAlN/AlN/GaN heterostructures were grown and processed at Virginia Commonwealth University). On the other hand, at a low occupancy, a non-monotonous dependence on the 2DEG density is resolved (Fig. 3, symbols, solid curve).

2.0

1.5

1.0

^ 0.5

0.0

Fluctuations GaN 2DEG * InAIN/AIN/GaN present paper O AIGaN/GaN [5] O AIGaN/AIN/GaN [2] O AIGaN/GaN/AIN/GaN [7] D InAIN/AIN/GaN [8]

IR absorption A AIGaN/GaN [6]

0.01 0.1

2DEG density (10^ cm''

FIGURE 3. Hot-phonon lifetime measured at a low-moderate occupancy as a function of 2DEG density for GaN-based channels: lno.i8Alo.82N/AlN/GaN, 1.2xl0"cm"^ (square [8]); Alo.15Gao.85N/GaN, 5xlO"cm"^ (diamond [5] and black triangle [6]); Alo.33Gao.67N/AlN/GaN, lxlO"cm"^ (pentagon [2]); Alo.22Gao.78N/GaN/AlN/GaN, SxlO^ cm" (buUet [7]); lno.i8Alo.82N/AlN/GaN, 1.5xlO"cm"^ (star, present paper). Solid curve guides the eye.

>

10 10" 10'° 10"

Electron density (cm""

10"

FIGURE 4. Phonon and plasmon energies against electron density [10] for bulk GaN (solid curves, coupling neglected) and coupled modes (dashed curves).

247

http://lno.i8Alo.82N/AlN/GaN

http://Alo.15Gao.85N/GaN

http://Alo.33Gao.67N/AlN/GaN

http://Alo.22Gao.78N/GaN/AlN/GaN

http://lno.i8Alo.82N/AlN/GaN

The observed non-monotonous dependence can be interpreted in terms of LO-phonon-plasmon resonance (Fig. 3, solid curve): the strongest LO-phonon-plasmon interaction and the fastest decay of hot phonons are expected when the plasmon energy approaches the LO-phonon energy. In bulk GaN, the resonance electron density is close to 10^' cm" (Fig. 4). Since there is no universal relation for the plasmon energy in a 2DEG channel, the resonance 2DEG density is estimated from the experiment: -7x10^^ cm" (Fig. 3).

The LO-phonon-plasmon resonance can explain, in part, why the hot-phonon lifetime decreases as the hot-phonon mode occupancy increases (Fig. 2, squares, stars). The essential decrease is observed in the channels with high 2DEG density. According to the resonance model, the plasmon energy exceeds the LO-phonon energy if the occupancy is low. However, at a given 2DEG density, the electron density per unit volume changes with the electron temperature responsible for the increase in the hot-phonon mode occupancy. Thus, the associated plasmon energy decreases and the LO-phonon-plasmon resonance condition can be satisfied when the electrons and the phonons become hot. In other words, the resonance 2DEG density is higher, when the hot-electron temperature is higher.

CONCLUSIONS

The optimal 2DEG density for the fastest decay of hot phonons at not too high electric fields is estimated from the experimental investigation of hot-electron fluctuations. The hot-phonon lifetime depends on the hot-phonon mode occupancy as well.

REFERENCES

1, M. Wraback, H. Shen, J. C. Carrano, C. J. Collins, J. C. Campbell, R. D. Dupuis, M. J. Schurman and 1, T, Ferguson, Appl, Phys, Lett, 79, 1303-1305 (2001),

2, A, Matulionis, Phys, Status SoUdi A 203, 2313-2325 (2006), 3, K, T, Tsen, J, G, Kiang, D, K, Ferry and H, Morko?, Appl, Phys, Lett, 89, 112111-1-3 (2006), 4, A, Dyson and B, K, Ridley, J, Appl, Phys, 103,114507-1-4 (2008), 5, A, Matulionis, J, Liberis, 1, Matulioniene, M, Ramonas, L, F, Eastman, J, R, Shealy, V, Tilak and A,

Vertiatchikh, Phys, Rev, B 68, 035338-1-7 (2003), 6, Z, Wang, K, Reimann, M, Woerner, T, Elsaesser, D, Hofstetter, J, Hwang, W, J, Schaff and L, F,

Eastman, Phys, Rev, Lett,, 94, 037403-1-4 (2005), 7, E, Sermuksnis, J, Liberis and A, MatuUonis, Lithuanian J, Phys, 47, 491-498 (2007), 8, A, Matulionis, J, Liberis, E, Sermuksnis, J, Xie, J, H, Leach, M, Wu and H, Morkog, Semicond, Sci,

Technol, 23, 075048-1-6 (2008), 9, A, Matulionis, J, Liberis, L, Ardaravicius, L, F, Eastman, J, R, Shealy and A, Vertiatchikh,

Semicond, Sci, Technol, 19, S421-S423 (2004), 10, A, Matulionis and 1, Matulioniene, "Accumulation of hot phonons in GaN and related structures" in

Gallium Nitride Materials and Devices II, edited by H, Morkog and C, W, Litton, Proc SPIE 6473, 64730P-1-15, 2007,

248

A New 1/f Noise Model for Multi-Stack Gate Dielectric MOSFETs

Zeynep ^elik-Butler

University of Texas Arlington, Electrical Engineering Dept., NanoFab, P. O. Box 19072, Arlington, TX, 76013, USA

Abstract. A new Multi-Stack Unified Noise (MSUN) model based on correlated number-mobility fluctuations theory is developed to model 1/f noise in MOSFETs with multi-layered gate dielectrics. In this new model, the trap density profile takes into account the effects of energy and spatial distribution as well as the multilayer structure of the gate-stack. Correlated number and mobility fluctuation was experimentally verified as the dominant mechanism for 1/f noise, with no contribution from remote phonon scattering to the observed fluctuations. For validation of the model, experimental noise data is obtained in the 78-350K range on MOSFETs with various gate stack compositions from different processing conditions. The test samples included poly-crystalline silicon and metal gated MOSFETs with Hf02 and HfSiON high-K gate dielectrics. Process variations consisting of interfacial layers of different thicknesses were incorporated to obtain different equivalent oxide thicknesses (EOTs) as well as wide ranging channel interface properties. The MSUN model was shown to accurately predict and model the 1/f noise exhibited by multi-stack gate dielectric MOSFETs.

Keywords: High K dielectric, MOSFET, 1/f noise, multi-stack gate dielectric PACS: PACS #: 77.55.+f Dielectric thin films; 73.50.Td: Noise process and phenomena; 73.40.QV: Metal-insulator-semiconductor structure.

INTRODUCTION

Noise issues continue to be point of focus in the International Technology Road Map for Semiconductors (ITRS), especially when pertaining to novel gate dielectric materials and structures in MOSFETs. As novel gate stacks are introduced into the technology with higher dielectric constants to allow thicker gate dielectric layers for the same value of gate capacitance, although the problem of gate leakage is minimized, other issues are raised that need to be measured, analyzed, and modeled. Among these is the increased 1/f noise due to the introduction of novel and poorly understood high-dielectric constant (high-K) materials into the gate and additional interfaces resulting from these multi-layers.

The most widely used 1/f noise model for MOSFETs is the so-called Unified Flicker Noise Model introduced by the Berkeley group [1]. This model attributes the noise to the trapping/de-trapping of the channel carriers by traps in the gate dielectric that are assumed to be uniformly distributed in the energy gap and with respect to location. While the uniform trap density assumption may be reasonable for non-composite gate dielectrics, it certainly is not for the multi-layered gate dielectric stacks. In addition, high- K incorporation affects some of the fundamental parameters



249

NtlLO

/

exp(-yiLz) X exp[-yHK(z-TiL)]

X exp(-yiLTiL)

\ -\

\ \

\

of MOSFET devices, of which the carrier mobility degradation [2,3] and the low frequency noise behavior are of particular interest [4,5,6,7]. In this context, the Unified Flicker Noise Model shows significant discrepancies when applied to the analysis of noise in devices with high- K gate stacks.

Consequently, we propose modifications to include the non-uniformity in trap profile of high-K (HK) and interfacial layer (IL) materials. In this new model, the trap density profile is assumed to vary exponentially (Fig. 1) - (a) with respect to the trap energy level referred to the Si substrate intrinsic Fermi level and (b) with the location of the trap in the dielectric layer with reference to the corresponding HK/IL and IL/Si interfaces. There is an additional non-uniformity in the trap density encountered by the tunneling electrons, due to the combined effect of the non-uniform energy distribution of the traps and the band bending in the gate dielectric caused by the applied gate bias.

I HK \ \

N«

FIGURE 1. The trap density values at any location in the gate dielectric stack are referenced with respect to the values at the intrinsic Fermi level and the surface of the corresponding dielectric as depicted by Nîo and NÎKO in the diagram.

NOISE MODEL AND VERIFICATION

Based on the carrier number-correlated mobility fluctuations theory, considering an infinitesimal channel length Ax, the drain current noise spectral density Sn can be written as [1]

5V(/) ^ A

/ . -(-

I '-(^sc/êff) ÂN (x,f)Axclx (1)

WAx Nix)

Here, L and W are the channel length and width, respectively; jUeff is the carrier mobility; a^^ is the Coulomb scattering coefficient;/is the frequency; and N(x) is the

inversion layer charge density at distance x from the source. S^ (x, / ) is the power

spectral density of the mean square fluctuations in the trapped charge carriers over the area WAx. It can be written as the summation of generation-recombination fluctuations

250

(Lorentzians) due to traps in the interfacial and high-k layers with the interaction occurring through constant energy tunneling of carriers [8,9]:

TÎL (2 )

'S'AW {x,f) = AkTWAx

J NuLiEf„,z) 2^2 -dz 1 + a'-rfj (z)

T +T

J NtHK(.Ef„,z) • HK iz) l+0)^Tlf:{z)

dz

(2)

where, T is the temperature, THK and Tn are dielectric thicknesses, co=27rf, and r i s the

tunneling time constant for the charge carriers inside the dielectric expressed as

r = roexp(?^) using the WKB approximation. For IL, 0<Z<TIL, the trap density

becomes NtjL(E,z) = Ntjuoexp\^jL(E-E,)+(qAjiVgjL/TjL)z+7]jLz] and for HK,

NtHK(.E,z) = NtHKo ^W^HK(.E-E,)+{qXHKVgHKlTHKy+VHK^'^^ TIL<Z<THK+TIL,. Here,

z is the distance into the dielectric stack, z'=z-Tji, N^J^Q is the trap density at Ei and

at the Si/IL interface, Njfjf^Q is the trap density at Ei and at the IL/HK interface, it, is

the intrinsic Fermi level at Si/IL interface, ^ represents the modeling parameter that

defines the energy dependence of the traps, rj is the modehng parameter for the

spatial distribution of the traps, A is the fitting parameter for the band bending term.

Then, [Sj^) for HK/IL MOSFETs is derived from the first principles using the above

trap density and tunnehng considerations to arrive at the new MSUN Model:

^///)= AkTU

WU

Here, a - •.\l{fi,,4N(x)\ PHK

N, tHKO' ,exp[^HK(Efir-E,)]

THKÔHK (A,yHK+v„K)'r„. " ( 2 ^ ' }+(/}H,yHK+'lHK)l7Hr.

I 1 ^HK I'^HK

\+U, •HK

dx (3)

PIL=1ÎLITIL- The integration

variable is COTJJ}^ ='^HK- The contribution from the IL is neglected since this has been

shown to be small compared to the noise coming from HK. [8, 9] Experimental noise data was obtained in the 78-350K range on MOSFETs with

various gate stack compositions from different processing conditions (Table 1). The data supports the vahdity of the correlated number-mobility fluctuations theory for HK/IL MOSFETs at all temperatures. From the temperature independence of the normalized noise (Fig. 2), it can be concluded that there is no additional noise component resulting from phonon scattering in HK/IL MOSFETs. Fig. 3 illustrates the frequency exponent d of 1// spectrum to extract /ifff^, rjfff^. It can be shown that

S = [l + {J3HKVHK + VHK IYHK )\, where ^^^ = qA^^ / V , KK = ^ " ^ J V „ /T,,.

and Fj; is the channel voltage. Noting that 1/f noise in 1-IOOHz is due to HK, the five parameter set (Njfjf Q, ^fjf^ ,

^HK' VHK and jUcQ) can accurately predict the noise as indicated in Figs. 4 and 5.

251

Fig. 6 plots the average effective dielectric trap density obtained from the Unified Model and indicates an order of magnitude variation with temperature that is inconsistent with the underlying assumption of uniform trap density. On the other hand, with proper choice of ^ HK • X 'HK-and?]ffj^ in our new model,

a single value of N^fff^Q at

all temperatures can model the inherent non-uniform trap profile.

:;; 10"' I

~~ 10-

tn

J 10-

o 10"'

-23

-24

-25

-26

- © - 1.65nm (HfO^/ RCA SiO^) - • - 1.33nm (HfSiON/SiON) :

- 0 - 1.85nm (HfO^/ SRPO SiO^) - • - 1.46nm (HfSiON/SiON)

- * -1 .24nm(HfSiON/SiON) - • - 1 . 6 6 n m (HfSiON/SiON) ;

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

50 100 150 200 250

Temperature (K)

300 350

FIGURE 2. Normalized current noise spectral density showed no noticeable dependence on temperature, irrespective of equivalent oxide thickness or process.

Gate Material

Poly Si Poly Si Poly Si Poly Si

Metal (TaSiN) Metal (TaSiN)

Metal (TiN) Metal (TiN) Metal (TiN)

TABLE 1. Description of the high-K MOSFETs used

High-K Layer

HfSiON(3.0nm) HfSiON(3.0nm) HfSiON(3.0nm) HfSiON(3.0nm)

HfOj (2.7 nm) HfOj (2.7 nm)

HfSiON/10%SiO2(2.0 HfSiON/10%SiO2(2.0

HfSiO,: (2.0nm)

nm) nm)

Interfacial Layer

SiON (0.8 nm) SiON(l.Onm) SiON (1.5 nm) SiON (1.8 nm) SiO2(1.0nm) SiO2(1.0nm)

---

m this study

EOT (nm) 1.24 1.33 1.46 1.66 1.65 1.85 1.06 1.03 1.17

Process

----

RCA Clean SRPO

Plasma Nitridation Thermal Nitridation

No Nitridation

CONCLUSION

The MSUN model is shown to accurately model the 1/f noise exhibited by multi-stack gate dielectric MOSFETs with several HK/IL compositions, with different oxidation and nitridation processes and metal and poly gates. The model is being incorporated into compact models like PSP.

ACKNOWLEDGMENTS

This work is partially supported by Semiconductor Research Corporation under contract 2004-VJ-1193. We would like to thank Texas Instruments and Freescale Semiconductor for supplying the test MOSFETs. I would like to express my appreciation to Tanvir Morshed, Siva Devireddy and Shahriar Rahman..

252

1.4

B

c 0) c o Q. X 0)

1.2

1

0.8

0.6

0.4

0.2

= 1.2 0)

^ 0.8

0.6

0.4

0.2

0

EOT=1.24nm

^HK=^HK=-'-"eV-^

Tl =-7.99x10^ cm "

+ EOT=1.46nm

^HK=^HK=-°-455eV-

Tl =-4.67x10 cm"

<fa o

EOT=1.33nm

0

^K=^HK=-° -4056eV

Tl^^=-5.38x10^cm"^

o EOT=1.66nm

0 0 o 0 o o

^HK=^HK=-°-947eV-

Ti^i^=-3.53x10^cm"^

0.4 0.6 0.8 1.2 0.6 0.8 1.2 1.4

V (V) g

FIGURE 3. The frequency exponent yfor 1-lOOHz vs. gate bias for poly-gated HfSiON/SiON MOSFETs. A straight line fit is made to the data from which Tjff]^ , /i}jK ^''^ extracted.

T=172K EOT=1.24nm V =0.96V N =13x10^^cm"^eV"^ tHKO

Frequency (Hz) FIGURE 4. The calculated current noise spectral density is compared to the data for HfSiON/SiON MOSFETs with different interfacial layer thicknesses and in the temperature range of 172K-300K.

253

< n - 1 5 10

-N-10'' I

'k •D

10" ^

-1 R

10^''

C

• T=225K-230K

• ^

• ^ ^ ^ ^

•

T=300K . • -

*y^^f0'^

»_ -*—"• 1 ^

c—* * " • 1.24nm (HfSiON/SiON) '•

U 1.33nm (HfSiON/SiON) \

• 1.46nm (HfSiON/SiON)

A 1.85nm (HfO / SRPO SiO ) 2 2

• 1.65nm (HfO^/RCASiO^) "

A _ _ A _ - # _ _ _ _

fc-^^ 5- ^—2 K^^tr" ' —* " r-~" * ~»




A 1.85nm (HfO/SRPO SiO ) : 2 2 •

T 1.65nm (HfO / RCA SiO ) 2 2 .

) 0.2 0.4 0.6 0.8

Figure 6. Temperature dependence of trap density values extracted using the original Unified Model (solid) and that at mid-gap energy using the newly developed MSUN Model (open). The latter shows much less variation for both EOTs at all expected.

temperatures as

(Vg-V) (V)

. - 10 ° • >

1 10 ^ >

g 10 ' Q

Q .

^ 10^'

5

1 • "»HK

FIGURE 5. The Viinc

dependence of the MSUN Model was verified by comparing drain current noise spectral density calculated at 1 Hz to the experimental values, over the bias range of moderate to strong inversion.

(EOT=1.85imHf02/SRPOSi02) i

(EOT=1.65im HfO^/RCASiOj)

A Nj (EOT=1.85nm HfO^/SRPO SiO^)

• N| (EOT=1.65nmHfO,/RCASiO,)

^ A A A ^

' n n • g n H

A

r 0 100 150 200

g 0 H B 0 Ei

• 1 • • , • • • * A, A

A

250 300 350 4( )0 Temperature

REFERENCES

1. K. K. Hung, P. K. Ko, C. Hu, Y. C. Cheng, IEEE Trans Electron Dev 37, 654 (1990). 2. R. Chau, S. Datta, M. Doczy, B. Doyle, J. Kavalieros, M. Metz, IEEE Elect Dev Lett 25, 408 (2004). 3. A. L. P. Rotondaro, et al, IEEE Elect Dev Lett 23, 603 (2002). 4. B.Min,etal., IEEE Trans Electron Dev 51, 1679(2004). 5. Y. Yasuda, C. Uu, lEDMTech Dig 2006; 277. 6. B. Min, et a\.. Proc SPIE-Noise and Fluctuations 2005; 31. 7. E. Simoen, et al., IEEE Trans Electron Dev 51, 780 (2004). 8. S. V. Devireddy, et al.. Microelectronics Reliability 49, 103 (2009). 9. T. Morshed, et al., IEEElEDM, p. 561 , 2007.

254

Low Frequency Noise in High-k Dielectric MOSFETs. How Far From the Channel Are We

Probing the Traps?

S. Rumyantsev'' ' , C. Young", G. Bersuker", and M. Shur"

'^Rensselaer Polytechnic Institute, Troy NY 12180-3590 lojfe Institute, 194021 St. Petersburg, Russia

'SEMATECH, Inc., Austin, Texas 78741, USA

Abstract. The low frequency noise dependencies on the thickness of the high-k dielectric and aging time for n-channel Si MOSFETs revealed the presence of the generation-recombination noise in HfD2(4nm)/Si02(lnm) dielectric stack devices. Assuming that traps responsible for the generation recombination noise are located at the Hf02/Si02 interface (i.e. at the distance of ~lnm from the channel), we extracted the value of the capture cross section of those traps ~10" ''cm^, which is typical for repulsive centers. A constant gate voltage stress led to the increase of the noise at high frequencies and decrease of the characteristic time constant of the generation-recombination noise.

Keywords: High-k, MOSFET, capture cross section, generation-recombination, noise PACS: 85.30.-Z, 85.30.Tv, 72.70.+m

INTRODUCTION

Dielectrics with a high dielectric constant (high-k dielectrics) are supposed to replace silicon dioxide in the next generation of CMOS technology in order to decrease the gate leakage current in submicron devices. Hafnium oxide (HFO2) is one of the prime high-k dielectric candidates because of its high dielectric constant (>20) and relatively high band-gap (~5.8eV). However, all known high-k dielectrics (including Hf02) suffer from high density of traps that causes the mobility degradation, temperature shift of the transistor parameters, instabilities, and high level of the low frequency noise (see [1,2] and references therein). The low frequency noise dependencies on the interfacial layer thickness, gate metallization, and hot electron degradation for different types of high-k dielectrics have been studied in multiple papers [2-6]. Most of these pubhcations reported that the noise spectra exhibit small deviations from the pure 1/f noise. In the present work, we studied the noise in Hf02/Si02 stacks transistors with different Hf02 thicknesses. The generation recombination noise found in the devices with the thickest Hf02 layer (4nm) allowed us to make a conclusion about the location and capture cross section of traps responsible for noise.



255


The analysis of low frequency noise in MOSFETs is usually based on the McWhorter model [7] or so called unified models, which account for correlated mobility fluctuations [8]. In the framework of these models, the cause of the 1//"noise is the exchange of carriers due to their tunneling between the channel and traps in the gate dielectric. The lower is the noise frequency, the further away from the channel are the noise-causing traps in the oxide. Therefore, noise properties carry information not only on the overall density of traps responsible for noise but also on the location of these traps in the oxide. In particular, traps causing the generation recombination noise have to be of the same type (i.e. have the same energy position and the capture cross section) and should locate in the oxide at some well-defined distance from the channel.

Transistors were fabricated using a gate-first integration flow with the standard source/drain activation. Hf02 was deposited by atomic layer deposition on the thermally grown Si02 films. The Hf02 thickness ranged from 2 nm to 4 nm. A TiN film was employed as a metal electrode.

Figures 1 and 2 show the noise spectra and the dependence of noise on the drain current at constant drain voltage for different transistors with Hf02/Si02 stacks and for the control transistor with the Si02 dielectric without Hf02. As seen, both the amplitude and shape of the spectra depend on the Hf02 thickness for the transistors with the same Si02 interfacial layer. The transistors with the thickest Hf02 layer (4nm) demonstrated significant contribution of the generation-recombination (GR) noise (Fig.l). The increase of the noise with the increase of the high-k dielectric thickness has also been found in [6]. One of the possible reasons for this effect is that thicker high-k films stronger affect the quality of interfacial layer by generating higher density of 0-vacancies there [9]. As seen from Fig.l, the slope of the noise spectrum for the 2nm Hf02 stack is close to \lf, while the noise spectrum for the 3nm Hf02 stack shows a deviation that might be related to a higher defect density in Si02 closer to Si substrate, and the noise spectrum for the 4 nm Hf02 sample exhibits the GR noise that might be related to an additional increase of the trap density in Si02 closer to high-k.

10- ' ' -

10-^-

10-5-

10-10-

1 0 - " -

10-12-

10-13-

1nmSi02/4nm Hf02 (Vg-V,)=0.2V

^ ' X L IniTi SiOj/Snm HfOj

"Xw x/ ^V X S ^

1nmSi02/2nm Hf02 ^ S w l U

10" 10' 10^ 10^ 10* 10°

Frequency f, Hz Drain current L, A

FIGURE 1. Noise spectra for different gate stack configurations at the same gate voltage.

FIGURE 2. Dependence of the noise on the drain current for different gate stack configurations.

256

Figure 3 shows the dependence of the characteristic time of the GR noise x=l/27rfo on the gate voltage overdrive {Vg - Vt) (for the Hf02 (4nm)/Si02 (1 nm) gate stack). The characteristic frequencies, fo, were extracted as the position of the maxima on

versus frequency dependencies (see inset in Fig.3). As mentioned above, the tunneling noise mechanism links the GR noise to the traps locahzed at a certain well-defined distance away from the channel. We may speculate, based on the discussion on the spectral density slopes in Fig.l, that the observed GR noise is likely dominated by the high density of the high-k-induced traps located in Si02 layer near the Hf02/Si02 interface, i.e. about 1 nm away from the channel. Since the characteristic tunneling length in Si02 is about lO'^cm, it means that electrons should tunnel ten characteristic tunneling lengths in order to be captured. This assumption agrees with the results published in ref [10], where sihcon nano-crystals were incorporated in the oxide at the predefined distances of 1.2nm, 1.5nm and 2nm from the channel. The gate stacks with the nanocrystals located at the 1.2nm range resulted in GR noise within the same frequency range as was observed in our samples suggesting similar trap distance from the channel (assuming that the trap capture cross-section values are similar). The data on l//-like noise in high-k MOSFETs reported in ref [2] also indicates that both interfacial layer and high-k dielectric might contribute to noise, i.e. the traps can be probed within the depth of at least ten characteristic tunneling lengths in the frequency range of interest.

As seen from Fig. 3, the characteristic time x decreases with the increase of the gate voltage overdrive {Vg-Vt). The characteristic time of the GR noise is defined as

T"'=Te"'+Xc"' where Xe is the emission time and Xc is the capture time. Since Xc decreases and Xe increases with the increase of the gate voltage, we assume that capture time dominates in the overall characteristic time of the GR noise.

The hue in Fig.3 shows the characteristic time x= Xc calculated as Xc=i/c!WvV (here ny=n/d is the electron volume concentration, « is the electron sheet concentration, d is the channel thickness, v is the Fermi velocity, and a is the effective capture cross section (o=5xlO"^°cm^ as extracted by fitting). The channel thickness was calculated as d=E(/Fm, where F^ is the surface electric field in the channel, and Eg is the lowest energy electron level in the triangular quantum well. As seen, the model describes the overall trend of this dependence.

The extracted small value of the effective capture cross section indicates that the traps act as repulsive centers. (Similar or smaller values of the capture cross section are often reported in the noise measurements [11] although existence of such negatively charged centers have not been independently confirmed).

Effect of the constant gate voltage stress (~ 3V) on the current-voltage and noise characteristics has been studied on the same transistors with the Hf02(4nm)/Si02(lnm) dielectric stack. The stress led to the increase of the gate leakage current and to the decrease of the characteristic time r (see Fig.4). While at low frequencies (1-lOHz), the noise amplitude was virtually independent of stress, the noise increased at higher frequencies.

257

10-'

10--

10-

10--

! • V =1 BV J •1. 10° 10 ' 10^ 10° lO"* 10^

(Vg-v,), V

(V-V,)=0.1V

10" 10" 10"' 10° 10 10^ lO-* 10"* 10^ 10°

]ing time t, s

FIGURE 3. Dependence of the characteristic time x FIGURE 4. Dependence of the characteristic of the GR noise on the gate vohage overdrive (Vg- time x of the GR noise on the stress time for Vt). Line shows the time x calculated as d/on,v different gate voltage overdrives

(a=5xlO-^"cm^).

In conclusion, the noise level in high-k/metal gate n-channel MOSFETs is shown to depend on the thickness of the high-k dielectric. The GR noise was found in Hf02(4nin)/Si02(lnin) dielectric stack devices, and, therefore, our data suggest that the traps at the distance of up to ~lnin from the substrate might be probed at the low frequency noise measurements. A very small extracted capture cross section of those traps ~10"^°cm^ points to their repulsive nature. A constant voltage stress resulted in higher noise level at high frequencies and decrease of the GR noise characteristic time constant.

REFERENCES

L C D . Young, D. Heh, S. V. Nadkarni, R. Choi, J. J. Peterson, J. Bamett, B. H. Lee, G. Bersuker, IEEE Trans, on Device and Material Reliability, 6, 123 (2006). 2. T. H. Morshed, S.P. Devireddy, Z. Celik-Butler, A. Shanware, K. Green, J. J. Chambers, M. R. Visokay, L. Colombo, Solid-State Electronics 52, 711 (2008). 3. E. Simoen, A. Mercha, L. Pantisano, C. Claeys, and E. Young, IEEE Trans on El. Dev. 51„780 (2004) 4. F. Crupi, P. Srinivasan, P. Magnone, E. Simoen, C. Pace, D. Misra, and C. Claeys, IEEEE. Dev. Lett, 27, 688 (2006) 5. B. Min, S. P. Devireddy,, Z. (Jelik-Butler, A. Shanware, L. Colombo, K. Green, J. J. Chambers, M. R. Visokay, and A. L. P. Rotondaro, IEEE Trans on El Dev 53, 1459 (2006) 6. H. D. Xiong, D. Heh, M. Gurfmkel, Q. Li, Y. Shapira, C. Richter, G. Bersuker, R. Choi, and J. S. Suehh, Microelectronic Eng. 84 2230(2007) 7. A. L. McWhorter, Proc. of the Conf on the Phys. Semicond. Surf, 1956, Philadelphia, pp. 207- 29 8. R. Jayaraman, C. G. Sodini, IEEE Trans, on Electron Devices 36, 1773 (1989). 9. G. Bersuker, P. S. Lysaght, C. S. Park, J. Barnett, C. D. Young, P. D. Kirsch, R. Choi, B. H. Lee, B. Foran, K. van Benthem, S. J. Pennycook, P. M. Lenahan and J. T. Ryan, J. Appl Phys 100, 094108 (2006) 10. S. Ferraton, j . Zimmermann, L. Montes, G. Ghibaudo, J. Brini, J. Gurgul, and J. A. Chtoboczek, Proceedings ofSPIE Vol. 5470, 2004, p.560. 11. M. E. Levinshtein and S. L. Rumyantsev, Semicond Sci. Technol. 9, 1183 (1994)

258

1/f noise in 0.12 |Lim P-MOSFETs with High-k and metal gate fabricated in a Si Process Line on

200 mm GeOI Wafers

J. Gyani^, F. Martinez\ S. Soliveres\ M. Valenza\ C. Le Royer'' and E. Augendre''

'lES - UNIVERSITE MONTPELLIER II - UMR CNRS 5214 Place E. Bataillon, 34095 Montpellier Cedex 5, France

CEA-LETI Minatec- 17, rue des Martyrs, 38054 Grenoble Cedex 9, France

[email protected]

This paper presents an experimental analysis of the noise measurements performed in germanium on insidator (GeOI) 0.12 jam PMOS transistors. The front gate stack is composed of a Si02/Hf02 material with a TIN metal gate electrode. The residt is an aggressively reduced eqinvalent oxide thickness (EOT) of 1.8 nm. The slow oxide trap density of the front gate oxide is Nt(Ep„) = 1.2 10 cm eV and is comparable to values for nitrided oxides on Si bidk. These residts are of importance for the future development of GeOI technologies.

Keywords: pMOS, germanium, noise, high-k. PACS: 73.50.Td, 74.40.4-k

INTRODUCTION

To fulfill the ITRS roadmap requirements for sub-32 nm MOSFETs, the carrier mobility will have to be improved by a factor 2 with respect to Sihcon. Thanks to the development of high-K dielectrics and the improved transport properties in Germanium (Ge) with regards to Silicon, Ge MOSFETs with a high-K gate stack on Germanium-On-Insulator (GeOI) have become a very attractive candidate for ultra-scaled MOS technologies.

In this contribution, we present low-frequency drain current noise investigations performed on HfOi-gated pMOS transistors, (with low EOT value). The studied devices are elaborated on 200 mm GeOI wafers with a 60-80nm thick Ge active layer obtained using SmartCut^'^ technology.



259


EXPERIMENT

Low-frequency drain current noise investigations are performed on TiN/HfOi GeOI pMOSFET transistors with low equivalent oxide thickness (EOT) value. The studied devices (Figure 1) are elaborated on 200 mm GeOI wafers with a 60-80nm thick Ge active layer obtained using Smart Cut^'^ technology. The devices were fabricated using GeOI transistor processes similar to those reported in [1], with thin Si capping passivation (~2.5nm) between Ge and HfOi.

PolySi 4nm HfO,

TiNIOnm /

SiO/Si

Ge ~80nm

BOx 200nm

FIGURE 1. Schematics (left) and TEM image (right) of GeOI PMOSFETs.

The front gate stack exhibits an EOT of 1.8 nm. The buried oxide is a 200nm thick SiOi layer and is used as a back gate for experimental purposes (to suppress the back channel conduction). The devices under test for noise analysis have a width W=l|im and a length L between 0.12 to 5 |im. A complete current-voltage characterization using an Agilent 4156C semiconductor parameter analyzer was performed before noise measurements. Low-frequency noise measurements were performed using a HP89410A dynamic signal analyzer loaded by a high sensitivity current/voltage converter EG&G 5182.


Typical DC drain current evolutions versus front gate voltage VFG and their associated transconductance gm are reported figure 2 for various back gate biases VBG at VDS=-50

mV. For VFG > 0 V and VBG < 30 V the observed current is due to the back interface transistor which is normally ON due to the unintentionally doped p-Ge channel. By applying a back gate voltage of -I-60V to the bottom of the wafer, the rear interface transistor is switched off and the measured current is only due to the front interface transistor. The threshold voltage of the front interface transistor is between 0.2V and 0.5 V and is a function of gate length. From these ID/VFG characteristics, the Y function method is applied [2] and hole mobility is extracted. We obtain |io = 280 cmW.s. In order to characterize the front gate interface, a back gate voltage (-1-60 V) is applied during noise measurements. The oxide traps involved with noise fluctuations will therefore be located in the front gate.

260

i VBQ=

! 10V ' '

\

'

// li • /20V / /

V^V / /

60V

VFG (V)

VpG (V)

FIGURE 2. Typical variation of the front gate a) ID/VFG characteristics, b) gm/ VFG , with respect to the substrate bias VBG; VDs=-50mV, W=l|im, L=0.75 |im.

Noise measurements were performed with VFG varying between -0.5 V and +1.2 V, with VDS=-50 mV and VDS=VFG-VT (saturation regime) and VBG at +60V and +30V. The comparison of noise measurements performed at VBG=+60 V and +30 V showed that for these biases we do not observe any influence of back gate interface. Figure 3.a shows typical drain current noise spectrum measured at VDS=-50 mV on transistors biased from weak to strong inversion. For frequencies lower than 100 Hz we observe a power-law dependence 1/f with 1<Y<1.3. For frequencies higher than 100 Hz we observe 1/f noise with 0.8<Y<1. Figure 3.b shows the variation of the power factor y at high and low frequencies as a function of VFG- This increase of the power factor y for lower frequencies indicates that the trap density increases when moving away from the Si02/Si/Ge interface to Hf02/Si02 the interface. The higher noise level measured at 1 Hz compared to those extrapolated from 100 Hz indicates a higher trap density at the Hf02/Si02 interface. As shown figure 3.b, the excess slow oxide traps located at the Hf02/Si02 interface are active at strong inversion.

1e-18

le -m.

1e-20

1e-21

1e-22

\l\ y=1.3

"

"'WBj^ y=0.9 J

1-

[ \

i \ f \ F \

• f<1 OOHz 0 f>1 OOHz

•

•

Frequency (Hz) VpQ (V)

FIGURE 3. Slope variation of the measured 1/f spectra (left) and Variation of the power factor y at high and low frequencies as a function of VFG (right).

Figures 4.a and 4.b show typical drain current power spectral density extrapolated at 1 Hz versus drain current for the front side transistor operation, at VDS=-50 mV and VDS=VFG-VT, respectively. In both cases, the drain current 1/f noise follows the carrier number fluctuation model (AN model) [3,4], allowing the extraction of the trap density

261

at the SiOi/Si/Ge interface. From both investigations, a trap density Nt(EFn) = 1.2x10 cm" eV" is extracted. These values are comparable to those reported in literature [5]. Ref. [5] refers to PMOSFETs with a nitrided oxide on Si bulk, and much better than those reported by Srinivasan et al [6], who extracted trap densities of 1.4x10^° cm" eV" on p-MOSFETs Hf02 and metal gate on GeOI substrates. Under the voltage biases studied, we have not observed any influence of access resistance noise.

Model • Experimental data

ID (A) 1e-9 1e-8 1e-7 1e-6 1e-5 1e-4 ie^5 1e-4

ID (A)

FIGURE 4. Power spectral densities measured at 1 Hz, with VBG at +30V and +60V (left) and Comparison between the power spectral density measured at 1 Hz and the AN model (right).

ACKNOWLEDGMENTS

The authors thank French OSEO organization for financial support.

REFERENCES

1. C. Le Royer et al, Solid-State Electronics 52, 1285-1290 (2008) 2. G. Ghibaudo In : Haddara H., editor. Kluwer Academic Publishers; 1995 (chapter 1). 3. G. Ghibaudo, Solid-State Electron , 30,1037, (1987). 4. M. Valenza et al, lEE Proc-Circuits Devices Syst., 151, 102-110, (2004). 5. E. Simoen et al, Appl. Phys. Letters, 85, pp. 1057-1059 (2004). 6. P. Srinivasan et al. Material science and semiconductor processings 9, 721-726 (2006)

262

Low Frequency Noise Degradation in 45 nm High-k nMOSFETs due to Hot Carrier and

Constant Voltage Stress

M. Shahriar Rahman, Zeynep ^elik-Butler, M.A. Quevedo-Lopez*, Ajit Shanware , and Luigi Colombo

University of Texas at Arlington, Electrical Engineering Dept, NanoFab, P.O. Box 19072, Arlington, TX76019,USA

^Texas Instruments Incorporated, Dallas, TX 75243, USA

Abstract. Hafnium based materials are the leading candidates to replace conventional SiON as the gate dielectric in complementary metal-oxide-semiconductor devices. Hot carrier and constant voltage stress induced i//" noise behavior is presented for HfSiON nMOSFETs. The additional low-frequency noise introduced through stressing was evaluated on nMOSFETs with TiN metal gate and HfSiON gate dielectric. Nitridation of HfSiO was achieved either by high temperature thermal nitridation or by relatively lower temperature plasma nitridation. The difference in stress induced noise behavior is attributed to the nitrogen profile across high-k/Si interface and bulk of high-k gate oxide caused by different nitridation techniques.

Keywords: 1/f noise, high-k, MOSFET, HfSiON, plasma nitridation, thermal nitridation. PACS: 77.55.+f Dielectric thin films; 73.50.Td: Noise process and phenomena; 73.40.QV: Metal-insulator-semiconductor structures. 77.22Ch: Permittivity (Dielectric function).

INTRODUCTION

To reduce the excessive leakage current through the conventional gate oxide along with the high stand by power dissipation, planar CMOS technology is in the process of integrating novel high-k gate materials into Si MOSFETs. Hafnium (Hf) based dielectrics, especially HfSiON exhibits attractive properties among different high dielectric constant (high-k) gate materials, such as relatively high dielectric constant, stability in contact with silicon and no Hf interdiffusion into Si substrate [1,2,3]. Low frequency noise (LFN) poses a major limitation to analog and RF ICs as the device dimensions are scaled down. Therefore, it is essential to characterize the LFN not only for fresh devices but also during the lifetime of the transistor. In this paper. Hot Carrier Stress (HCS) and Constant Voltage Stress (CVS) induced noise degradation in plasma and thermally nitrided HfSiON and HfSiO devices are presented.

EXPERIMENT

As a gate dielectric, 2nm HfSiO (10% Si02) was deposited using an optimized atomic layer deposition (ALD) process that includes optimized interface between



263

high-k and Si substrate. Similar nitrogen (N) concentrations were incorporated in ALD HfSiO by plasma and thermal nitridation. For plasma nitrided samples, the N content was increased by increasing the nitridation time, while N content in thermally nitrided films was increased by increasing the temperature of the NH3 anneal. A custom made noise measurement setup with dc biasing circuitry, an EG&G PAR 113 preamplifier and HP 3562 dynamic signal analyzer was used for 1/f noise measurements. A constant drain bias of 50 mV (hnear region of operation) was applied at the drain and the gate overdrive (Vg-Vt) was varied from 0.1 to 0.7 V. A semiconductor parameter analyzer Agilent 4155C was used for DC characterization to extract the threshold voltage (Vth), transconductance (gm), conductance (gd) and subthreshold slope (S). Transistors with WxL= 10x0.1 ^m^ were used for hot-carrier and positive constant-voltage stressing, i/f noise and dc characteristics were taken before and after applying the stress for 1000s

RESULTS and DISCUSSION

For HCS experiment, worst degradation condition Vg=Vd was used, whereas for CVS the vertical field was fixed at 10 MV/cm for all transistors to achieve comparable stressing conditions. The thermally nitrided device showed the most degradation in its saturation drain cunent {Id,sat), maximum transconductance (gm,max) and threshold voltage (Vth) compared to the plasma nitrided and pure HfSiO samples (Table 1). The higher degradation in thermally and plasma nitrided HfSiON is due to the presence of nitrogen. It has already been reported that the nitrogen profile in HfON and HfSiON gate dielectric layers has a profound effect on the positive bias temperature instability induced by constant voltage stressing as measured by the amount of AVm and Agm^max [4]. Forward and reverse mode (interchanging source and drain terminal) noise behavior after HCS is shown in Fig. 1. The increment in noise in higher frequency range for plasma nitrided sample implies the increased trap density close to the high-k/Si interface is more dominant due to HCS than thermally nitrided and pure HfSiO samples. Whereas the noise increment in the lower frequency range indicates to the traps further into the gate dielectric. CVS was applied at the gate terminal with source, substrate and drain terminal grounded for 1000s. Fig. 2 shows the drain voltage noise power spectral density at 0.3V gate overdrive for three different samples. Slight increase in noise due to CVS for plasma nitrided HfSiON and HfSiO devices is observed. However, there is an order of noise magnitude increment for thermally nitrided sample and this phenomenon is attributed to the electrons filling the existing traps and increased charge damages. For the thermally nitrided sample, due to high thermal budget during nitridation, nitrogen concentration at the interface is increased. This nitrogen increment at the HK/Si interface as well as damage near HK/metal gate stack due to CVS causes higher noise. Moreover, oxygen vacancies are also likely candidates for intrinsic electron traps in devices and threshold voltage instability [5]. Atomic nitrogen is added to the system during plasma nitridation. This results in lowering of the defect levels and making those less active as electron traps due to the nitrogen incorporation into the oxygen vacancy site. On the other hand, thermal nitridation results in higher concentration of protons due to anneal at high temperature

264

in NH3 ambient. This causes an increase in positive charge centers and noise due to Coulomb scattering from these centers [6]. Fig. 3 shows the DC characteristics after HCS (forward and reverse mode) and CVS induced degradation. Higher degradation in gm,max duc to HCS indicates significant interface degradation. Trapping of charges at the bulk of the gate oxide for thermal nitrided sample is responsible for highest shift in Vth. Although transconductance characteristics shift with CVS in plasma nitridation and pure HfSiO sample, there is apparently no degradation in gm^max • This is attributed

TABLE 1. Stress induced % degradation oig„„,ax, îôjand v,i, for different samples

Type of Stress

Plasma Nitrided Snuimix ^d.sat * th

Percent Degradation Thermally Nitrided

Snuimix ^d.sat * th

No Nitridation Snuimix ^d.sat * th

Constant Voltage 0.27% 1.72% 5.07% 5.54% 10.6% 6.95% 0.62% 3.85% 2.89% (CVS)

^".TS.Ô?'^'^ 3.62% 4.19% 7.46% 5.8% 7.31% 10.76% 1.14% 1.69% 2.89% (riCcij

Hot Carrier -Reverse Mode 3.8% 4.75% 4.76% 5.1% 6.43% 10.76% 2.18% 2.72% 2.89%

Measurement (HCSR)

T>

10 100 1 1" 1""

Frequency (Hz) Frequency (Hz)

FIGURE 1. Drain voltage noise power spectral FIGURE 2. Noise characteristics of two density at 7^=50 mV and (Vg-V^=Q3 V after differently nitrided HfSiON nMOSFETs along applying hot carrier stress for 1000s on lOxO.ljim^ with HfSiO device. Fi=50mV and (Vg-V^=0.3 nMOSFET. HCS: Hot carrier stressing. HCSR: Hot V. Gate area is 10 x 0.1 jim^. Constant voltage carrier stressing, reverse mode measurement stress of lOMV/cm is applied at the gate

terminal for 1000s.

265

to the increase in electron trapping rather than increase in interface defect density. On the other hand, degradation in gm^max in thermally nitrided sample is due to pile up of N and increased Si-N bonds at the interface due to the high thermal budget during nitridation.

CONCLUSION

Various stress induced noise degradation behavior are summarized for plasma and thermal nitrided HfSiON and HfSiO devices. It is found that the LFN of deep sub-micron HfSiON nMOSFET devices can degrade by orders of magnitude over the lifetime of circuit due to hot carrier and constant voltage stress, most likely due to increased dielectric traps in nitrided sample. This problem could be alleviated to a large extent by using plasma nitridation rather than thermal. Plasma nitrided devices showed better noise performance than thermal nitrided devices, though thermal nitrided devices scaled equivalent oxide thickness (1.03nm) more than plasma nitrided devices (1.06nm).

2.510

2 10'

1.510'

110 '

510''

010°

210 '

1.510'

110 '

510''

010°

210 '

1.510'

110 '

510"'

010°

—•— - e -

^ / 1

'

Fresh

CVS

HCS ^

HCSR 1

/ • ( ^ • S Plasma

Thermal _^tfi

^ None

0.004

0.0035 0.003

0.0025

0.002

0.0015

0.001

0.0005

0.004

0.003 3

0.002

0.001

0.0035

0.003

0.0025

0.002

0.0015

0.001

0.0005

0

• V ( V )

FIGURE 3. Drain current and transconductance characteristics of three different nMOSFETs after hot carrier and constant vohage stress measured in forward and reverse modes. Gate area is 10 x 0.1 jim^. Plasma and thermally nitrided gate dielectrics to obtain HfSiON are compared to pure HfSiO (none).

ACKNOWLEDGMENTS

This work is partially supported by SRC under contract 2004-VJ-l 193.

REFERENCES

1. M. R. Visokay, etal,Appl- Phys.LettersSO, 3183 (2002). 2. K. Sekine, et al, lEDM Tech. Dig., 2003, pp.103-106. 3. M. A. Quevedo-Lopez et al., inlEDMTech. Dig. 2005, pp. 425-428. 4. C. Choi, et al., in VLSISymp. Tech. Dig, 2004, pp. 214-215. 5. J. L. Gavartin, etal, Appl. Phys. Letters, 89, 082908 (2006). 6. M. S. Rahman, et al., J. Appl. Phys., 103, 033706 (2008).

266

The consequence of continuous current branching on current-noise spectra in field-effect

and high-electron mobility transistors

E. Starikov*, P. Shiktorov*, V. Gruzinskis*, L. Varani^ H. Marinchio''' and L. Reggiani**

'Semiconductor Physics Institute, A. Gostauto 11, LT 01108 Vilnius, Lithuania ^Institut d'Electronique du Sud (CNRS UMR 5214), Universite Montpellier II, 34 095 Montpellier

Cedex 5, France **Dipartimento di Ingegneria dell 'Innovazione and CNISM, Universita del Salento, Via Arnesano

s/n, 73100 Lecce, Italy

Abstract. The problems related with the intrinsic noise in FET/HEMT channels induced by continuous branching of the total current between channel and gate are considered in the framework of a simple analytical model and its predictions on the current-noise spectra. Main branching-induced effects such as the appearance of an additional noise related to the excitation of plasma waves, its dependence on FET/HEMT embedding circuits, interference properties, etc. are analysed.

Keywords: High-frequency noise. High-electron mobility transistors. Plasma waves PACS: 72.20.Ht, 72.30.+q, 72.70.+m

INTRODUCTION

Electron transport and noise analysis in discrete-element circuits is based on the conservation law of total current (Kirchhoff's law), which in the general case writes:

hotdS = 0 (1) s

where jtot = SSQ^ + f"-^* is the local total current-density consisting of the displacement and conduction (drift) components, and S is a surface surrounding some volume of interest. In the case of two-terminal devices (resistors or diodes), a one dimensional treatment of carrier transport, which supposes that j'^"-^*||E||x, is usually used. In this case the conservation law reduces to the scalar relation:

j^[S{x)Mx,t)] = 0 (2)

which corresponds to the current flow inside a tube with cross-section S{x). Direct consequences of such a formulation are: (i) the validity of Ramo-Shockley theorem for the induced current in the external circuit; (ii) the duality (equivalence) representation of the intrinsic electronic noise in terms of Norton and Thevenin noise generators, etc. In going from two- to three-terminal (transistors) devices the local parallelism of j"^"-^* and E inside the device is in general violated, so that a one dimensional approximation no



267

longer applies. This implies the branching of total-current tubes, so that Eq. (2) becomes invalid at the branching areas where in accordance with Eq. (1) total current conservation is formulated as zero-sum rule for incoming/outcoming currents for a certain volume bounded by some closed surface S. Such a situation is typical in field-effect transistor (FET) and high-electron mobility transistor (HEMT) structures. Here, the conduction current mainly flows along a conducting channel and the governing direction of the local electric field does not coincide with the current flow due to the presence of a gate. In essence, under the gate action, the whole channel becomes practically a continuous region of total-current branching between the source-drain and channel-gate directions. The aim of this report is to elucidate noise phenomena induced by such a branching.

ANALYTICAL MODEL

Following ref. [1], we shall describe the self-consistent distribution of charge density p^^(x) and potential ^(x) in FETs/HEMTs channels by using the following system of equations which includes: (i) the charge conservation law

|p^^W + ^if^^(x) + ^ £ ^ = 0 (3)

and, (ii) the one-dimensional (ID) approximation for the two-dimensional (2D) Poisson equation

^c^^ix) + -^[Ug- <p(x)] = - | p ^ ^ ( x ) (4)

which takes into account the gate influence (second term in l.h.s. of Eq. (4)) on the potential distribution. Here, Ug is the gate potential, d{x) is the local gate-to-channel distance, 5 is the channel width, EC and e^ are the dielectric constants in the channel and the gate-to-channel spacer, respectively.

To close Eqs. (3) and (4) we use the hydrodynamic approach to describe the drift components of the currents along and transverse to channel, Jc {x) and jg"^\x), respectively, on the basis of the foUowing assumptions:

(i) charge transport between the channel and the gate is absent, i.e. jg"^*{x) = 0, and

(ii) the drift current along the channel Jc {x) = en^^{x)v{x) is determined by the free electron stream with concentration n^^ which flows upon the homogeneous donor background with concentration A - and where the velocity of the electron flow is described as:

d d ^ v + — dt ax

V e

2 m*^^ ' + evD—n^^{x) = -vv+f (5)

where m* is the carrier effective mass, v is the velocity relaxation rate, D is the longitudinal diffusion coefficient, / is the Langevin force with a spectral density 5-correlated in space, S/f = AkbTv/m* (with kb the Boltzmann constant and T the bath temperature), that describes the thermal source of fluctuations in the channel. The system of Eqs. (3)-(5) is closed, and, as shown in refs. [1,2], it allows us, on one hand, to obtain the spectral

268

densities of current/voltage fluctuations at HEMT terminals as:

l-L „3£)| S^^{m) n

0 \G^{(o,xo)Y-Sffdxo (6)

where L is the channel length, G^{(i),xo) is the spectral representation of the response function of the quantity ^ = J,U to a local perturbation at x = XQ induced by the Langevin force / . On the other hand, by linearizing Eqs. (3)-(5) we can construct an analytical model which describes transient characteristics, impedance/admittance matrixes, Norton and Thevenin generators of diffusion noise, etc.

In the most interesting case of common-source configuration, where the leading role is played by current variations in the source-drain (SD) and source-gate (SG) circuits, A/j and AJg, respectively, the fluctuations of currents and potentials in SD and SG circuits can be presented in a vector form as:

AJ = AX,

AU: AUd

Here, the explicit linearized dependence of AJ on AU and / takes the form:

AJ = 7AU+ / G/(to)/(xo)fifxo 0

(7)

(8)

where

ecSo Oil PchfiL-l

ico + v a shjiL

chPL chPL-l

- 1 (9)

is the matrix of HEMTs/FETs admittance under the common-source operation, and

G/(«,xo) = Sceo-colfi ch^xo/sh^L

5/1/3 (f-xo)/c/i/3 (10)

1(0+ v

is the response function in SD and SG circuits which determines in accordance with Eq. (6) the Thevenin generator of noise, (Dp = \/e^n/[ecEom*] is the volume plasma frequency of electrons in the channel, and a = cOp/[cOp + ico{ico + v)], ji^ = ?^^[ico{ico +

ico{ico-v)l{col- v))],A:

Z=7" ' =

\/{.£d/£c)/d5. By considering the impedance matrix,

-1 ft);

£C£Q-. 1(0 V a

shjiL chjiL+l

1 chPL

ch/iL-l

Eq. (8) can be rewritten as:

AU = ZAJ-

where

Gu{(o,xo)f{xo)dxo

(11)

(12)

Gc/(ft),xo) = -ZG/(ft),xo) = a (13) •c/i/3(f-xo)/c/i/3f" shli{L-xo)/shjiL

is the voltage response-function in SD and SG circuits, which determines the Norton generator of noise in accordance with Eq. (6).

269

FIGURE 1. Time response of currents respectively at source, drain, and gate terminals calculated by the hydrodynamic approach for a local perturbation introduced by the Langevin force at two different points xo = 0.05 fim (a) and XQ = 0.45 fim (b) of the channel. As boundary conditions we take Ug = 0 and Usd = 0. Other parameters are: channel length L = 1000 nm, thickness 5 = 15 nm, channel to gate

^^. Kinetic parameters of electrons in the distance d = 15 nm, donor concentration N^^ = 8 x 1 0 channel correspond to Ino.siGaoAl^s at room temperature, namely: v = 3 x 10 s and m* = 0.048OTO

MAIN BRANCHING-INDUCED EFFECTS

Time delay of response. As an example. Fig. 1 presents the time response of currents at the source, drain and gate terminals, respectively, AJs, A/j and AJg, caused by a local perturbation appeared at time moment t = 0 in point xo = 0.05 (just near to the source) and 0.45 jim (near to the channel centrum). When a perturbation occurs near to one of the channel boundaries (source or drain, see Fig. 1 (a)), the current response at the perturbed terminal appears practically immediately, while at the opposite terminal it appears with a time delay of about 1 ps. In the case when the perturbation takes place near to the channel centrum (see Fig. 1 (b)), the delay of the current response appears at all the three terminals. Independently of the initial perturbation location, the temporal evolution of the Alg response tends to be synchronized with either the Alg or the A/j, thus fulfilling the zero-sum rule, i.e. A/j — AJs + AJg = 0. Such a delay leads to the appearance of a series of resonant peaks, i.e. to an oscillatory behavior of noise spectra.

Excitation of plasma waves. Figure 2 (a) presents the spectral density of current fluctuations in SD and SG circuits calculated at constant voltage operation (AUg = 0, AUd = 0). Oscillations in the noise spectra are related with the resonant excitation of spatial modes of plasma waves in the dielectric layer separating the channel from the gate. Here, the resonant frequencies can be determined by the denominator of the response functions of currents in Eq. (10), which finally gives [1]:

(oL{k) = (Op ^{XL/nf + k^

0,1,2,. 1,3,5,.

,i=SD ,i=SG

(14)

The fuU set of resonances (A: = 0, 1,2, ...) corresponding to the excitation of standing waves appears for fluctuations in the SD circuit only, while in the SG circuit one observes only odd spatial modes (k= 1, 3,5, ...). As follows from Eq. (14), the spectrum of excited plasma modes exhibits an upper limit at A: oo given by the 3D plasma frequency (Dp. Note that the widely used gradual channel approximation of Poisson equation [3]

270

FIGURE 2. Spectral density of current fluctuations: (a) in SD and SG circuits calculated with Usd = 0 and Ug = 0, and (b) in an SD circuit for a short (MJg = 0) and open (A.^ = 0) SG external circuit (curves 1 and 2, respectively).

(obtained when the second spatial derivative in Eq. (4) is omitted) gives an infinite equidistant spectrum of Dyakonov-Shur plasma waves which corresponds to the limit k « XL/n in our Eq. (14).

Interference of currents in the gate. The absence of even modes in the spectrum of current fluctuations in the SG circuit (see Eq. (14) and Fig. 2 (a)) is related to the interference of local channel-to-gate currents, since, due to continuous branching, their spatial sum gives a zero value to the SG current for even modes. Indeed, a non-zero contribution is given only by those spatially excited modes which symmetry center coincides with the center of the gate. Thus, even if a mode is excited its contribution can be "invisible" due to symmetry restrictions for certain circuits.

Influence of embedding circuits on the noise spectrum. Another situation can be realized when, due to some symmetry restrictions (introduced, for example, by embedding circuits), a mode cannot be excited at all. Let us demonstrate that a change of the operation regime of one of the adjoint circuit (e.g. SG or SD) leads to a corresponding change of the current fluctuation spectrum. Figure 2 (b) reports noise spectra of current fluctuations Sjj in the SD circuit in two cases, namely: when the SG circuit is short (Af/g = 0), or open (A/g = 0). As follows from Fig. 2 (b), under the transition from the former case to the latter one, the fluctuation spectrum Sjj looses the contribution of the odd harmonics of plasma excitations, thus only the even (k = 2,4,6,...) harmonic contribution remains in the spectrum. Indeed, when the SG circuit is open, so that A/g = 0, spatially-different partial contributions of the channel-to-gate current must fully compensate each other to provide zero current at the gate. This condition can be satisfied only for even modes with the inversion centrum in the channel center. Thus, such an influence of the operation mode of one of adjoint circuits on the noise spectrum in the other one destroys the property of the Thevenin and Norton generators to describe the intrinsic noise in embedding circuits independently of their properties.

Excess noise due to plasma excitations. Figure 3 (a) demonstrates the appearance of an excess noise in the current fluctuation spectra caused by the excitation of plasma waves. For the ungated channel, (d -^ °°), we obtain the usual Lorentzian spectrum. With the decrease of the channel-to-gate distance d there appears an excess noise at frequencies below the 3D plasma value ( / = cOp/ln « 10 THz). With the decrease of

271

E 10"'

d :10 nm 100 nm

1000 nm 10000 nm

f (THz) Y

FIGURE 3. (a) Spectral density of current fluctuations in an SD external circuit calculated with U^d = 0 and Ug = 0 at different values of the distance d between the channel and the gate, and (b) integral intensity of current noise in an SD circuit vs the dimensionless parameter y = LX r^Lj-JWd.

d, the frequency region involved into the excess noise expands into the low-frequency region. This is accompanied by the growth of the noise intensity inside the excess noise region. Figure 3 (b) shows the integral intensity of current noise in an SG circuit, hfi = J^Sjj{a))da), as a function of dimensionless parameter j = L'k ^^ Ll^fbd determined by ratio between the transverse channel-to-gate capacitance and the longitudinal capacitance of the channel-under-gate region. As follows from Fig. 3 (a), when 7 < 1 (plasma excitations are absent) Kp- keeps a constant value. When 7 > 1 there takes place an exponential growth of Kp- which, in accordance with Fig. 3 (a), is correlated with the increase of the number of plasma modes excited between the channel and the gate.

ACKNOWLEDGMENTS

This work is supported, in part, by the Lithuanian State Science and Studies Fundation contract No P-01/2007.

REFERENCES

1. p. Shiktorov et al., J. Stat. Mech. 01, 01047 (2009). 2. P. Shiktorov et al., Rev. Nuovo Cimento 24, 1 (2001). 3. M. Dyakonov and M. Shur, IEEE Trans. Electron Devices 43, 1640 (1996).

272

Impurity Dispersion and Low-Frequency Noise in Nanoscale MOS Transistors

O. Marinov and M. J. Deen

Electrical and Computer Engineering McMaster University, Hamilton, Ontario, L8S 4K1, Canada

Email: [email protected]

Abstract. To remedy small-geometry effects in nanoscale MOS transistors, e.g. drain induced barrier lowering, one uses S-doping layers. The statistical variation of the atom's positions in ion-implanted S-doping layers is denoted as "frozen noise", and is correlated with the low frequency noise. The temporal accumulation of variance from "frozen noise" produces 1/f noise.

Keywords: Low Frequency Noise in MOS Transistors, LFN, flicker noise. PACS: 72.70.+m; 74.40.+k.

10-

20-

30-

40-

50-

60-

Ls„E=12nm : Z = -

^ ^ '-GATE=22nm - •

l-GATE=45nm ^ ^ _ _ ^ -

-

--..^^^ -Ls„E=90nm ^

^ -"^"^

0

20

40

60

80

100

120

140

160

180

200

220

10'''' lO''^ lO'"^ lO''^ lO'ls

Impurity Atom Density, 1/cm^

O 0) h-

CM

' - ' II E 55 P r,

II o •

li

FIGURE 1. Idealized S-doping profiles with Poisson distributions and

peak impurity concentration of lO'* cm"'. The left-hand axis is in nano

meters, and right-hand axis is in number of atomic distances.

INTRODUCTION

In MOS transistors of minimum gate length Lmm-lOOnm, 5-doping layers are implanted to remedy the drain induced barrier lowering (DIBL) [1]. The implantation depth tdep is reduced with Lmm, Figure 1, according to [2],

(1) ^ # «*dep "^ *dep'

where tj is the depth of source and drain junctions, and tox is the gate oxide thickness. Assuming tjoctiep and tox cannot be downscaled further, then eq. (1) implies that tdepOcLmm-

With a Poisson distribution for the implanted atoms [3, 4, 5], the positions of the atoms are shown in Figure 2 by dots. The lines represent depletion profiles, given by (l/tdep)^~Z(l/tia)^ for the depletion depth tdep, where tia is the implantation depth of individual atoms. Decreasing LGATE from 90nm to 12nm, the standard deviation (Jr=(odep/tdep) increases from 7% to 20%, causing local current variations

Id lo / Io l X C I V T / V T « -d tdep/ tdep ^ C^dep/tdep = ^ r ' (2)

because tdep modulates the threshold voltage VT [6]. Since the variation in tdep is due to fabrication, then o, is fixed in the device, thus one has a "frozen noise".



273


The "frozen noise" can be studied by Monte-Carlo simulations, statistical evaluation of identical samples, and theoretical extrapolation toward small devices. The Monte-Carlo simulations provide insights on forthcoming problems from device downscaling, e.g. dispersions in threshold voltage [4, 5, 7], carrier velocity [8], and noise margin [9]. Monte-Carlo simulations are feasible for high-frequency noise [10, 11, 12, 13, 14, 15], while for a low frequency of IHz, the computational volume is in the range of 10^' FLOPs (floating point operations), or about a month for supercomputers holding record rates of 1-1.1 PFLOPs/s [16]. The statistics of identical samples accesses only existing devices, and the populations of devices and measurements are also large [17]. Therefore, we seek the dependences between the "frozen noise" and temporal noise by extrapolation toward nanoscale devices.

1/F NOISE FROM ACCUMULATION OF VARIANCE

The statistical calculations for standard deviations of the spot and average depletion depths are shown in Figure 3 at the top and in the bottom. The spot "frozen noise" is proportional to the prediction in ITRS for low-frequency noise, implying that the spatial non-uniformity in nanoscale transistors is rephcated as temporal low-frequency noise. We now derive 1/f noise by assumption of step-accumulation of variance, based on the so-called "innovation variance" [18] originally developed by N. Wiener [19].

Recall the normalized standard deviation Or of the "frozen noise" from eq. (2). When a charge carrier traverses the device for a transit time Tt=KxAt in K steps, it also probes at each step and accumulates the variations in the structure, and at exiting the device

K-l

<

12nm •-GATE'

a«20%

l-GATE=22nm a«14%

4 + k=l

(3)

a«10%

•-GATE'

a«7% =90nm

Position along Channel, Atomic Distances

FIGURE 2. Dispersion of S-doping

where Ck are regression coefficients that describe the transfer of Or into variance (ok) at each probing instance k=l,...,K, e.g. position in MOS transistor channel for the "frozen noise", or scattering event for mobility. Eq. (3) is a relation in the so-called "irmovation variance" statistics [18, 20, 21], introduced in [22] by first order (one-step) auto-regression in a random sequence {XI,X2, . . . ,XK}, with a random variation AXK

from average <XK>

AXK = XK - (X) = Ai^„^ + (cK-i /cK )AXK-I , (4)

where Amnov is uncorrelated random "irmovation" added to X at step K, Amnov has variance (omnov) , and (omnov) is added at step K to the previous variance (OK-I)^ for AXK-1. Then, by also comparing to eq. (3),

+ (CK-I/CK) 2 „ 2 CT^C^K , w i t h ((Jinnov)^=(Or)^(CK)^ a n d C=CK=Ck ( 5 )

274

because dAJdt=0, (7)

Minimum Device Area, |jm^

FIGURE 3. Spot and average standard deviations of doping

atom locations compared to the ITRS prediction for 1/f noise in

MOS and bipolar transistors

As the carriers traverse the sample, for an observation time (t), the variance (ot)^ at the device terminals is

cj2=c72c2K^ = c 7 2 c 2 ^ ^ = (c72cVAt)t = At, (6) T + Zit T +

the quantities in the brackets are device constants, and the derivative of eq. (6) is

where we have also changed the time (t) with the reciprocal variable frequency f=l/t, with the meaning that twice longer time of observation gathers Vi lower frequency in the spectrum of the variance (ot)^. The variance and the power spectrum density S(f) are proportional, ((Jt) ocS(f). Substituting this relation in eq. (7), we get eS/ef=-S/f with solution ln(S)=-ln(f), from the finite difference of which we obtain the expression for the noise related to accumulation of variance in the sample as

s(fl = !.^s(f) = « ^ , S(fo) f f

where the reference frequency fo can be chosen

(8)

arbitrary, and S(fo) is the power spectrum density at fo. Noticeably, the statistical accumulation of variance produces 1/f noise, owing to the ability of the matter

to probe and forward variance. In fact, a common mechanism that causes fluctuations with 1/f power spectrum in the nature [23] is the statistical accumulation of variance.

Note that the 1/f noise from variance accumulation is statistical in origin, and not contradicting with the superposition of fluctuations probed instantly. The derivation considers hnear system of first order. Non-linear and high order effects may lead to stochastic resonance or bistability (random telegraph signal). The derivation provides qualitative result for 1/f noise behavior. The value of S(fo) in eq. (8) has to be obtained by an alternative method, not cancelling the quantities in the brackets of eq. (6).

Choosing fo=lHz in eq. (8), and since Or is normahzed standard deviation, then the equation is for normalized flicker noise and the expression for the parameter Kp is

s(f) = Kr

X DC At

1 second (9)

where X is a quantity of interest, e.g. current, XDC is the average of X, and the rate of variance accumulation cVAt-Kp/or^-O.OOl s"' can be deduced from the trend lines through the diamonds and triangles in Figure 3 for (or ) and Kp in RF MOS transistors.

DISCUSSION AND SUMMARY

The scaling rules established last 40-50 years are questionable for devices of area less than lOOnmxlOOnm. Consider the data from ITRS [1] for MOS-RF, given in

275

Figure 4 for minimum-sized transistors L=W=Ln,in, with (A) for gate referred voltage and with (A) for Kp. Shown by (-), the number of carriers (n) is low, <100 for Lmin=90nm down to few for Ln,in=12nm. By the increasing "frozen noise" Or ( • ) at low Lmm, the Hooge parameter aH=KFxn (O) decreases unreahstically. For the An noise model, the equivalent oxide thickness (EOT) decreases from 2.2nm for Lmin=90nm to 0.9nm for Ln,in=14nm, Cox {^) is in the range 16-39fF/|im^, resulting in nearly constant oxide trap density Nt~10 cm ^nm ^eV"' (O). However, for the minimum-sized transistor, assuming high-k dielectric with permittivity 4esi02, the number of acting traps around Fermi level, (•)=Ntx4EOTxLn,in^x4kT, becomes a

small fraction of one trap, which is unrealistic condition for generating 1/f noise. On the other hand, taking the values for the frozen noise G/ ( • ) from ion implantation, the substitution in eq. (9) yields nearly constant value cVAt-Kp/or'-O.OOl s"' (D) for the variance accumulation rate.

We conclude that a temporal accumulation of variance can explain the 1/f noise in nm-scaled MOS transistors, in which the extrapolation of other models meets with difficulties. The variance accumulation model provides relation between spatial non-uniformities caused by device fabrication, called here "frozen noise", to temporal low-frequency 1/f noise, a relation which is qualitatively observed very often [24]. Indeed, the variance accumulation is well established for phase-frequency noise, by the method known as Allan variance [25, 26, 27].

/ * " ] Accum. Rate c'/At=Kp/cj^', 1/s |

m^eV

IZ Decreases at lower L „

KF=f«S„o«(g „ / lo ) ' , f o r W L = L „ i „ » w i t h g „ / l p=13 V-' at ( V 3 - V T ) = 0 . 1 V

I 10-3 10-2

Channel Area=WL=L„in'

FIGURE 4. Several quantities by MOS transistor downscaling, related to An, Aji and variance accumulation models for 1/f noise

REFERENCES

1. ITRS, http://public.itrs.net. 2. J. Brews, et al, EDL, 1, 2, 1980. 3. V. De, etal, Symp. VLSI, 198, 1996. 4. H.-S. Wong, etal, MReliab., 38, 1447, 1998. 5. P. Stolk, et al, TED, 45, 1960, 1998. 6. C. Wann, et al, TED, 43, 1742, 1996. 7. S. Toriyama, et al, Phys. E, 19, 44, 2003. 8. C. Alexander, et al, TNanotech, 4, 339, 2005. 9. K. Samsudin, et al, SSE, 50, 86, 2006. 10. P. Tien, etal, JAP, 27, 1067, 1956. 11. V. Gruzinskis, et al, SST, 6, 602, 1991. 12. T. Gonzalez, et al, TED, 42, 991, 1995. 13. J. Mateos, etal, SSE, 42, 79, 1998. 14. R. Rengel, etal, SST, 16, 939, 2001.

15. B. Vasallo, et al, TED, 55, 1535, 2008. 16. Top 500, 32"'' list, www.top500.org, 2008. 17. S. Springer, et al, TED, 53, 2168, 2006. 18. T. Pukkila, et al, Biometrika, 72, 317, 1985. 19. N. Wiener, MIT Press, Mass., 1970. 20. H. Davis, et al, JASA, 63, 141, 1968. 21. A. Walden, Tr Signal Proc, 43, 181, 1995. 22. L. Froeb, "Computational Economics and

Finance ", ed. Varian, Springer, 305, 1996. 23. R. Voss, Freq. Cont. Symp., 33rd, 40, 1979. 24. L. Vandamme, TED, 41, 2176, 1994. 25. D. Allan, Proc. IEEE, 54, 221, 1966. 26. F. Vemotte, etal, TIMeas., 42, 968, 1993. 27. R. Navid, et al, SSC, 40, 630, 2005.

276


http://www.top500.org

Impact of Advanced Gate Stack Engineering On Low Frequency Noise Performances of Planar

Bulk CMOS transistors

A. Mercha'', H. Okawa , A. Akheyar", E. Simoen", T. Nakabayashi , T. Y. Hoffmann*

"IMEC, Panasonic, 'Infineon, IMEC, Kapeldreef75, B-3001 Leitven, Belgium

Abstract. This paper discusses on the impact of gate stack engineering on the low-frequency noise performance of state-of-the-art deep submicron planar CMOS technologies. Focus is on the scaling of the Equivalent Oxide Thickness (EOT) in high-k gate oxides in combination with metal gates, requiring the implementation of cap layers. As will be shown, different trends in the LF noise can be observed, indicating that LF noise optimization is a complex interplay between the different gate stack components.

Keywords: metal gate. Low-frequency noise, CMOS, high-k dielectrics PACS: 72.70.+m - 73.40.Qv

INTRODUCTION

The international effort through the ITRS presented a roadmap on the best estimates of introduction time, at the production level, of successive generations of leading technology nodes and the R&D needs. For the 45nm node and below, it will become necessary to introduce revolutionary changes in the materials, process modules and device architectures [1-2]. The classic scahng of the 1/f noise with the gate oxide thickness has first been affected with the introduction of heavily nitrided gate oxides. Due to the high trapping density, the 1/f noise in high-k dielectrics is even higher than in SiON, although a lot of improvement can be obtained through process optimization. Though progress has been made in the areas of high-k gate dielectrics and metal gate systems, several issues remain (EOT scaling, threshold voltage control, mobility degradation...) [3].

Figure 1 illustrates the level of complexity that needs to be tackled in order to achieve acceptable results with scaled gate dielectrics and metal gates. Interfacial interactions due to subsequent thermal processing (between the high-k gate dielectric and the sihcon channel as well as between the high-k material and the gate electrode) may impose the introduction of capping layers. In addition, it has been shown that the thickness of the metal gate, for example, TiN seriously impacts both the static and LF noise performance of CMOS transistors [4-5].



277

^

r e ETC w i l l

p&ly-Si or diffii^oi. metal gate froui gi

high-K Jayei" Kueial

ouMiffuii

interface eng neerini

IBBCtlOtl iw 'pDly

^ ^ r - a o a po t tdsp iseps

silicm

rTigrpm [ ^ p j n i n ^

FIGURE 1. Factors that influence metal electrodes properties. The diffusion of dopants, intermixing of films and growth of interfacial layers are key issues affecting the stability and controllability of high-k films.

The same capping layers can be also used to tune the work function [6-7]. These modules for the gate insulator and gate electrode need to be co-developed to achieve expected performances for both digital and analog performances. In addition, so-called strain engineering and the implementation of high-mobility channel materials can also impact the density of interface and oxide traps in the high-k stack. The focus in this work, however, is on the factors described in Fig. 1.


The various options are compared in a trend chart shown in figure 2. In this figure 2, the reference hue represents the data from ITRS roadmap, uncorrected for the introduction of high-k dielectrics. It can be seen that while initial results on 1/f noise in high-k gate stacks were about 2 decades higher than the reference, progress in dielectric annealing, improved metal gate deposition, and the use of Al-based or La-based capping oxides has significantly reduced the 1/f noise. This achievement has been obtained through different steps.

The dependence of the interfacial oxide inserted between the substrate and the high-k layer (Hf02) has been studied and the main conclusion was that a thick interface would be needed to fully screen the impact of the high-k layer (stars) [8-9]. At the same time, it has been demonstrated that bringing the high-k layer closer to the silicon substrate not only increases the I/f noise magnitude, but may also change the dominant mechanism from carrier number (An) to mobility fluctuations (A|j,) [10]. This transition could point to a more pronounced impact of local non-uniformities in work function on the carrier transport.

So the interface engineering alone could not be the solution as it has to be around Inm or below for the technology node of interest. A lower trap density high-k dielectric (HfSiON or amorphous Hf02) is then needed for a fixed interfacial layer of 0.8nm. In addition, post deposition anneahng in nitrogen of fluorine may be helpful in lowering the oxide trap density and LF noise [II].

278

0) Q U)

(fl

0 c

•D u

£

3

c

•o u

X

F =.

'S = Ifl

1-III 11. (fl

o s £

[Hf] lOOte A ^ T j i , , =1.5nm

HfSiON — U - * Hf02

ITRS LitteratLire HfOj and reducing interfaoial layer thickness T^

Tint =0.8nm + HfSION and reducing Hf content [Hf] HfSiON TiN metal electrode thickness Tg HfSION and TaCNO + capping layer

1.2nm(60%Hf)cap

1.5nm(60%Hf)cap

1.8nm(60%Hf)cap

1.8nm no cap

1.8nm cap PNA

1.8nm(40%Hf)cap

1.8nm<80%Hf)cap

Equivalent Oxide Thickness EOT (A)

FIGURE 2. Overview of the input referred noise voltage spectral density at 1 Hz, normalized over the area, for different gate stack options (high-k thickness, presence or not of a capping layer, content of Hf and annealing).

The noise properties are strongly dependent on the Hf content. The Hf content is certainly a knob on which the noise level can be adjusted, but it is also in conjunction of other elements like the permittivity of the resulting oxide, the thickness of the high k layer (Fig. 3), the interaction with the metal gate or the capping layer that can be inserted on top of the main high-k layer to control the threshold voltage (dipole capping layer-metal electrode). However, most of the options can suffer from limited scahng capability, such that it is today still challenging to define the options for dielectrics with EOT around Inm and below. Another issue is that in the case of gate stack first, its robustness against further thermal budget, necessary for the implementation of strain-inducing cap or Source/Drain layers should be guaranteed, as this may compromise the 1/f noise [12].

10'

E 3 i 10

-1—I—1—I—1—I—1—I—1—I—1—I—1—I—1— .100%

'" Hf content [%]

• 100% • 70%

Different passivat ion process

10"

11.5 12.0 12.5 13.0 13.5 14.0 14.5 150 15.5

EOT

Lg = 1(im W=10(im |Vos l=100mV

1.42nm < EOT <1j51nm capping layer

E 0 T - 1 . 3 n m D06 I . S n m n o c a p no capping layer

• DID 1.2nm (60%Hf)cap D15 1.5nm (60%Hflcap D20 1.8nm (60%Hf)cap

10 ' 10" 10 '

Drain current JIDSI (A)

10 '

FIGURE 3. Figures illustrating the impact of the Hf content and the passivation process [3] and the high-k layer thickness.

279

CONCLUSIONS

The impact of different options for gate stack processing in deep submicron CMOS transistors on the LF noise performance has been demonstrated and discussed. It is shown that simultaneous EOT scaling and 1/f noise scaling is difficult, yet not impossible The combination of a metal gate and cap layers may certainly help in improving the high-k gate stack integrity and defectiveness, and, hence, reduce the LF noise.

REFERENCES

1. P. Wambacq, A. Mercha, K. Scheir, et al.; Advanced planar bulk and multigate CMOS technology: analog-circuit benchmarking up to mm-wave frequencies; IEEE ISSCC 2008, p. 528-9, 633, 2008.

2. M. Fulde, A. Mercha, C. Gustin, et al.;.; Analog design challenges and trade-offs using emerging materials and devices; ESSCIRC 2007, p. 123-6, 2007.

3. C. Claeys, E. Simoen, A. Mercha et al.; Low-frequency noise performance of Hf02-based gate stacks; J. Electrochem . S o c , 152: 9, p i 15-23; 2005.

4. A. Mercha, R. Singanamalla, V. Subramanian et al.; The impact of ultra thin A L D TiN metal gate on low frequency noise of C M O S transistors; 19th ICNF 2007, p. 33-6, 2007.

5. M. Rodrigues, A. Mercha, N . CoUaert et al. ; Impact of the TiN layer thickness on the low-frequency noise and static device performance of n-channel MuGFETs ; these Proceedings.

6. V.S. Chang, L.-A. Ragnarsson, G. Pourtois et al. ; A Dy-capped Hf02 dielectric and TaCx-based metals enabling low-Vt single-metal-single-dieleclric gate stack ; lEDM 2007, p. 235-8 , 2007.

7. S. Kamiyama, E. Kurosawa and Y. Nara; Improving threshold voltage and device performance of gate-first HfSiON/Metal gate n-MOSFETs by an ALD La203 capping layer. J. Electrochem. Soc, 155: 6, p. H373-7, 2008.

8. E. Simoen, A. Mercha, L. Pantisano et al.; Low-frequency noise behavior of Si02-Hf02 dual-layer gate dielectric nMOSFETs with different interfacial oxide tiiickness; IEEE Trans. Eleclron Devices, 51, p. 780-4, 2004.

9. B. Min, S.P. Devireddy, Z. Qelik-Butler et al.; Impact of interfacial layer on low-frequency noise of HfSiON dielectric MOSFETs; IEEE Trans. Electron Devices, 53, p. 1459-66, 2006.

10. F. Crupi, P. Srinivasan, P. Magnone et al.; Impact of the interfacial layer on the low-frequency noise (1/f) behavior of MOSFETs witii advanced gate stacks; IEEE Electron Device Lett., 27, p. 688-91, 2006.

11. P. Srinivasan, E. Simoen, Z.M. Rittersma et al.; Effect of nitridation on low-frequency (1/f) noise in n- and p-MOSFETs with metal gate/Hf02 gate dielectrics; J. Electrochem. Soc, 153, p. G819-25, 2006.

12. E. Simoen, P. Verheyen, A. Shickova et al.; On the low-frequency noise of pMOSFETs witii embedded SiGe source/ti^ain and fully silicided metal gate; IEEE Eleclron Device Lett., 28, p. 987-9, 2007.

280

Length Dependent Transition of the Dominant 1/f Noise Mechanism in Si-Passivated Ge-on-Si

pMOSFETs

E. Simoen^ A. Fimncieli^'^ F.E. Leys^ R. Loo^ B. De Jaege/, J. Mitard^ and C. Claeys"'*'

"IMEC, Kapeldreef75, B-3001 Leuven, Belgium ''EE Depart, Kasteelpark Arenberg 10, B-3001 Leuven, Belgium

Abstract. The impact of the Si passivation on the low-frequency noise of Ge-on-Si pMOSFETs is investigated. A transition from number to mobility fluctuations dominated 1/f noise is found going from shorter to longer channel transistors.

Keywords: germanium, MOSFETs, 1/f noise, silicon passivation PACS: 72.70.+m; 73.40.Qv

INTRODUCTION

Historically speaking, germanium is an important material for the study of low-frequency noise and fluctuations in semiconductor materials and devices, leading among others to the famous McWhorter number fluctuations (An) model for 1/f noise [1]. Since the early 60ties most of the device and related noise research shifted towards silicon, but in the past five years, interest in Ge MOS transistors has revived, mainly triggered by the introduction of high-k gate dielectrics in 45 nm and below technology generations [2]. Recently, successful operation of ~ 100 nm gate length Ge pMOSFETs has been reported, showing a low-field hole mobility superior to the universal curve for silicon [3]. Key to this achievement is the passivation of the Ge surface by the implementation of a thin epitaxial silicon layer, which is partly oxidized and on top of which HfOa is deposited [3]. As shown recently [4], the Si passivation details, i.e., the thickness, the deposition temperature and chemistry, have a strong impact on the static device parameters. In addition, it has been demonstrated that this interfacial layer has a pronounced effect on the low-frequency (LF) noise of Ge pMOSFETs, from which a dominance of An fluctuations has been derived [5]-[8].

Here, the LF noise of state-of-the-art Ge p-channel transistors is investigated in linear operation, in function of the device length. Based on the variation of the normalized drain current noise spectral density SI/1D^ with drain current ID, it will be shown that the dominant 1/f noise mechanism changes from carrier (An) for short devices, to mobility (A|j,) fluctuations for the longer ones. This could point to a larger



281

non-uniformity and associated carrier scattering in the passivation of the long-channel transistors.

EXPERIMENTAL

The p-channel transistors have been fabricated in Ge-on-Si substrates, using a process flow detailed elsewhere [3]. The gate stack consists of a 3 to 8 monolayers (ML) Si passivation, followed by a partial ozone oxidation to form 0.4 nm SiOa. A layer of 4 nm HfOa has been formed by Atomic Layer Deposition (ALD), which was subsequently covered by a TaN/TiN metal gate. Typical input ID-VGS characteristics in linear operation (VDS=-0.05 V) are shown in Fig. 1 for 6 ML devices. Transistors with a width W=10 |am and various lengths have been characterized. Noise was measured on wafer as described before [5]-[7].

6 ML Si passivation

-6.05 V 0.0005 0.0006

-0.9 -0.6 -0.3 Gate Voltage (V)

Figure 1. Input characteristics in linear operation (VDS=-0.05 V) for 6 ML Si-passivated Ge pMOSFETs with length 0.25, 0.5 and 1 |xm.


According to the characteristics in Figs 2 and 3, the 1/f noise at 10 Hz for the 0.25 |j,m transistors behaves according to the An model. In contrast, the long 1 |j,m pMOSFET in Figs 4 to 6 follows rather the A|j, fluctuations picture.

6 ML Si passivation

•^10"21

10" 10"" 10"® 10"^ 10"^ Drain Current (A)

Figure 2. Drain current noise spectral density versus drain current of two 6 ML Si passivated Ge pMOSFETs with L=0.25 |xm and W=10 |xm. The frequency f=10 Hz.

282

N 1 10-10

0) <fl . 1 1

0 6 10 ' '

^ 4 10-11

"5)2 10'' '

6 ML Si passivation

3 a. c 0

Vos=-0.05V

f=10Hz

0.2 -1.2 -1 -0.8-0.6-0.4-0.2 0 Gate Voltage (V)

Figure 3. Input-referred noise spectral density versus gate voltage of two 6 ML Si passivated Ge pMOSFETs with L=0.25 |xm and W=10 |xm. The frequency f=10 Hz.

6 ML Si passivation N X

CM <

+ J

V) = 10 0) Q gio

+ j

u * - i n Q.10 w 0) .•210

17

18

19

V()

21

VDS=-0.05V

f^lOHz 0.25 nm :

DO.5

€>-^ 1 |im :

10 •7 10"' _ 10"" 10"" 10" Drain Current (A)

Figure 4. Drain current noise specfral density versus drain current of 6 ML Si passivated Ge pMOSFETs with L=0.25, 0.5 and 1 |xm and W=10 |xm. The frequency f=10 Hz.

•8 ^ 10 N

^ 1 0 - 9 0) V)

6 ML Si passivation

10' 10

0)

• - -11 ralO " E 110-^2

Vj3s=-0.05V

f=10Hz

1 iim

10'' 10'^ 10'^ 10'^ Drain Current (A)

10'

Figure 5. Normalized drain current noise spectral density versus drain current of 6 ML Si passivated Ge pMOSFETs with L=0.25, 0.5 and 1 |xm and W=10 |xm. The frequency f=10 Hz.

283

N10-10

> 0) (A O

6 ML Si passivation

10' •11

0)

§•10" 12

r 1 |am

Vpg=M).05V

f=10Hz

-1.2 -1 -0.8-0.6-0.4-0.2 0 0.2 Gate Voltage (V)

Figure 6. Input-referred noise spectral density versus gate voltage of 6 ML Si passivated Ge pMOSFETs with L=0.25, 0.5 and 1 |rm and W=10 |rm. The frequency f=10 Hz.

As a result, the SVG in weak inversion does not scale with 1/L. For the lowest currents measured, the high off-state drain-to-bulk leakage current can cause an increase of the 1/f noise magnitude. As will be shown, reduction of the Si passivation thickness has a strong impact on both the DC and LF noise parameters, resulting in a higher noise magnitude and a stronger A|j, tendency with channel length.

R E F E R E N C E S

1. A.L. McWhorter, Semiconductor Surface Physics, Ed. R.H. Kingston, University of Pennsylvania Press, Philadelphia (PA), p. 207 (1957).

2. C. Claeys and E. Simoen, eds, "Germanium Based Technologies: From Materials to Devices", Elsevier, 2007.

3. G. Nicholas, B. De Jaeger, D.P. Brunco, P. Zimmerman, G. Eneman, K. Martens, M. Meuris and M.M. Heyns, IEEE Trans. Electron Devices 54, 2503 (2007).

4. J. Mitard, B. De Jaeger, F. Leys, G. Hellings, K. Martens, G. Eneman, D. Brunco, R. Loo, D. Shamiryan, T. Vandeweyer, G. Winderickx, E. Vrancken, K. De Meyer, M. Caymax, L. Pantisano, M. Meuris and M. Heyns, in lEDM Tech. Dig., p. 873 (2008).

5. P. Srinivasan, E. Simoen, B. De Jaeger, C. Claeys and D. Misra, Mater. Sci. in Semicond. Process. 9,721 (2006).

6. W. Guo, G. Nicholas, B. Kaczer, R.M. Todi, B. De Jaeger, C. Claeys, A. Mercha, E. Simoen, B. Cretu, J.-M. Routoure and R. Carin, IEEE Electron Device Lett. 28, 288 (2007).

7. W. Guo, B. Cretu, J.-M. Routoure, R. Carin, A. Mercha, E. Simoen and C. Claeys, in Proc. 19"' ICNF, M. Tacano, M. Yamamoto and M. Nakao, eds, AIP Proc. Vol. 922, p. 29 (2007).

8. D. Maji, F. Crupi, G. Giusi, C. Pace, E. Simoen, C. Claeys and V.R. Rao, Appl. Phys. Lett. 92, 163508-1 (2008).

284

Numerical modeling of low frequency noise in ultrathin oxide MOSFETs

F. Martinez, J. Armand, M. Valenza

lES - UNIVERSITE MONTPELLIERII- UMR CNRS 5214 Place E. Bataillon, 34095 Montpellier Cedex 5, France

Abstract. We present a numerical low frequency noise modeling related to oxide trapping/detrapping process, based on green's function formulation and its application to ultrathin oxide characterization. This model allows slow trap density profiles to be determined. The model was applied in the investigation of the validity of the flat band voltage fluctuation model in the case of thin oxides. Numerical gate current noise modeling was applied to the characterization of nitridation-induced traps. Finally, 2D model was applied in the investigation of low-frequency degradation of MOSFETs stressed by hot-carriers, and the generated slow oxide trap density profiles were deduced.

Keywords: Low frequency noise, MOSFET, Green's function, thin oxide . PACS: 72.70.+m, 73.50.Td

INTRODUCTION

The downscaling of CMOS according to the historical law of Moore is achieved by the reduction of the dimensions. The scahng rules impose very thin oxide thicknesses of the order of one nanometer. The low frequency noise is known as an accurate tool to evaluate the quality of the interface region in MOSFETs. Within the carrier number fluctuation model, bulk oxide defects can be characterized from low frequency noise measurements [1]. Nevertheless, the aggressive reduction of the dimensions has made questionable the accuracy of simple analytical models used over past 15 years. Numerical noise modehng is an alternative that allows to take into account scaling effects. Scahng down induces a large number of new physical phenomena; among them, direct tunneling current increases exponentially and leads to modifications in the normal device operation and to degradations of the device performance. This gate current induces additional noise sources.

In this paper, we present numerical low frequency noise modeling related to oxide trapping/detrapping process, based on green's functions formulation and its apphcation to ultrathin oxide characterization.

FLAT BAND VOLTAGE FLUCTUATION CONCEPT

A very popular formulation of McWhorter based models was proposed by Ghibaudo [2], who introduces the concept of an equivalent flat-band voltage fluctuation. The flat band voltage fluctuation technique has been used extensively for



285

the last 15 years to study and model the LFN in MOS transistors. The advantage is that one can extract the slow oxide trap density with 1-V and LF noise measurements, without having to model of the drain current. However, the initial formulation of the flat band voltage fluctuation was established under approximations which no longer hold true with ultrathin oxide devices. In this first section, we present a numerical LFN model for ultrathin oxide MOSFETs.

In order to describe the static behaviour of the device, a one-dimensional formulation is used, taking into account poly-depletion effects in the gate material as well as quantum-mechanical effects. The physical model is based on a self-consistent Poisson-Schrodinger numerical resolution which includes a polysihcon gate.

For the noise analysis, charge fluctuations are introduced in the oxide, and a Green's function approach is used to calculate the spatial cross-spectrum of potential fluctuations.

S,(x,x',f)= j G^(x,xJS,^Jx, ,f)G^(xX)dx, (1) Oxide

where Gv(x,Xa) is the Green's function of the potential V(x), defined as the response of the potential V a point x due to a 5 function in the oxide at a point x . The spectral density of the fluctuation of the number of occupied traps in a volume AV is given by [3]:

S (•:f\- ^<^) ft(l-ft)Nt(x)

VJx,t)-i^27rfVxf AV ^ ^ where T(X) is the trap time constant as a function of its depth, ft is the trap

occupation probability, and Nt(x) is the trap density at a point x. The power spectral density (PSD) of the inversion charge fluctuations is expressed

by taking into account the overall collection of potential fluctuations

V (f) =jjJv"(^) Sv (x,x'/) ft'(^ydxdx' (3)

where Jy* ™ is the jacobian matrix of the inversion charge Qmv with respect to the electrostatic potential V(x). Finally, we calculate the equivalent flat-band voltage fluctuations that induce the same inversion charge fluctuation than the one evaluated by Green's functions.

S ff) ^-«"(^) = Tf^7Tf7^ ;7T^ ^ ^ * 8n(x)=F(x).8V^ (4)

F(x).F(x)dx.dx A comparison of the classical formulation SwB* (f) and the Green's function

approach to the flat-band voltage PSD SwB "'™(f) was carried out. Results are presented in Fig. 1, showing that for a given trap density, the new flat band voltage PSD SvFB "'™(f) is smaller than the classical SvFB* (f) for ultrathin oxides.

The classical formulation leads to an overestimation of the flat band voltage PSD in the case of ultrathin oxide devices. This result shows that the classical model is no longer suitable for characterization of advanced MOSFETs devices. The oxide trap density vs. equivalent oxide thickness is usually used as figure of merit of a process. This model has been used to correct the data for thin oxides, showing that trap density continues to increase even for ultrathin nitrided oxides (Fig. 2).

286

FIGURE 1 : Classical flat-band voltage fluctuations PSD SvFB^ (f) and Green's function

approach flat-band voltage fluctuations PSD SvFB°"'™(f) for a metal gate at f=lHz.

020 •

^ :'N \ : :

i

•

" nitrured o

'

\

ide

' •

0

p> o

• •

^ \

•

• 0

V

o o

/S J>° Too /o L

SiO, Oxick

; [SIMOEN'99] :

[VALENZA'04]

[MARIN'03] i

Not published ;

[WANGA'01]

[BOUTCHACHA'99{

[EYAA'03]

[1.4 EOT]

[1.2 EOT] 1

[1.2 EOT] :

i

1 100 10

EOT(mn) FIGURE 2 : Figure of merit Nt vs EOT extracted

with corrections for ultrathin oxide.

GATE LEAKAGE CURRENT LF NOISE MODELING

In this section, we present a numerical model of the gate current LFN adapted to ultra-thin oxides. The aim is to relate the gate current LFN to a slow-trap density profile in the oxide. Gate leakage current has been implement in the one dimensional solver presented in the previous section. The tunnelling transmission coefficient for electrons is expressed by:

T(t]))-exp 2 X m

=^|"(x^-q(>^.-^(y))-E) (5)

where n is the reduced Planck constant, mox is the effective mass of the electron in the conduction band in the oxide, Tox is the oxide thickness, Xb is the barrier height, ys is the surface potential, y(y) is the potential at point y, and E is the energy of the tunnelhng electron. The gate leakage current is expressed as:

lG=qN„.(©C(^)T(^)f(^) (6)

Where Nmv is the inversion charge, C is the correction coefficient due reflections at the interface, and fj is the frequency impact. The spectral density of the gate current fluctuation is given as:

with S^,(x,x',f)- j G^(x,Xj)S4^^(xj,f)G^(x',Xj)dxj (8)

where J^ is the Jacobian matrix of the gate current IG with respect to the electrostatic potential y(y), Gy(x,xl) is the Green's function of the potential y(x) defined as the response of the potential y at a point x due to a unit charge in the oxide at a point xi.

Results and discussion

In this study, n-type MOSFETs with a 1.2 nm equivalent oxide thickness (EOT) targeted value and featuring a polysilicon gate were investigated. The nitridation of the dielectric films was carried-out using either a Decoupled Plasma Nitridation (DPN) or

287

a Rapid Thermal Nitridation (RTN). The devices were processed on 200 mm diameter sihcon wafers. The gate current noise was investigated on isolated nMOS transistors with source, drain and substrate tied together with VGS varying from 0.4 to 1 V.

The main feature observed is the slope of the spectra. For RTN devices, a power factor a ~ 0.8 was observed (i.e. the LFN level varies as l/f'^). For DPN devices, we observed 1/f spectra and lorentzian spectra with a cut-off frequency of about 1-2 Hz. In order to fit this experimental data, our new gate current LF noise model was used. In particular, the slow oxide trap density profile for each nitridation technique was extracted. Fig. 3 shows the results of the power spectral density simulations for different gate voltages and for both nitridation techniques.

Experimental data — Simulation

f (Hz) Frequency (Hz)

FIGURE 3 :. (a) Transistor (W/L=10 nm/0.34 nm), with an RTN oxide. The best fit is obtained with a constant trap density profile, (b) (W/L=10 nm/10 jim), with an DPN oxide. The best fit is obtained with

a Gaussian profile.

DPN RTN

STOxide Oxide/pol intejface iriteifac

0.0 0.5 1.0 1.5 2.0 2.5 Depth (nm)

FIGURE 4 :. Slow oxide trap density profiles extracted from DPN and RTN devices using gate current

low frequency noise as a function of the physical oxide thickness.

As reported on Fig. 4, the best fitting profiles are a constant profile for the RTN devices and Gaussian profile for DPN ones. These results confirm the strong correlation between low frequency noise and Nitrogen related traps [4]. For thermal nitrided gate oxides, nitrogen profiles show that the nitrogen concentration is greater at the Si02/Si interface, and constant when approaching the poly/Si02 interface. The DPN process induces a peak in the nitrogen concentration at the PolySi/Si02 interface to preserve the Si/Si02 interface. Extracted profiles are in agreement with the typical distribution of nitrogen atoms.

288

HOT ELECTRON INDUCED DEGRADATIONS

Electrical Stress and DC characterization

Throughout this work, we used NMOS transistors obtained from a 65 nm CMOS technology. The tested devices had W= 5 ^m, L=0.3 ^m and Tox = 5 nm.

The effects of a hot carrier electrical stress on the transconductance and the threshold voltage are well described in literature. [5]. The threshold voltage variation is caused by the creation of interface state and/or fixed charges in the gate oxide. The devices are stressed by Channel Hot Carrier injection with VGS = 1.75 V and VDS=4 V. In these condition (maximum of substrate current), an electron can achieve a sufficient energy for create an electron/hole pair by impact ionization. Thus, a "lucky" electron is injected into the gate oxide and create oxide or interface defects.

The interface state creation is confirmed by the subthreshold slope variation between the fresh and the stressed devices during t=240s. The effective density of interface states extracted from the subthreshold slope is proportionnal to the threshold voltage shift. Then we assume that no significant fixed charges are creating in the oxide during the hot carrier stress.

Low frequency noise results and discussion

Fig. 5 presents the low frequency noise level at 1 Hz for fresh devices (t=0) and devices stressed during a short time (10s and 30s). For fresh devices, we assume the trap density is uniform and we extract Nt=10'^ cm" eV"\ We don't observe any change in the subthreshold LFN level, whereas we observe an increase (about 3 times) of the noise level in strong inversion. This noise behavior can't be explain with classical analytical models, and may point out a non uniformity of the oxide trap profile. Fig. 6 presents the LFN level at IHz for fresh device (t=0) and devices stressed during longer times (90s and 240s). In that case, we observe an increase of the noise level in both subthreshold and strong inversion regimes (about 5 times) for the 90 s and 240 s stress times. This noise behavior can be interpreted as an increase of the average slow oxide traps in the oxide.

In order to characterize the non homogeneous oxide trap profiles induced by hot carrier stress, we have developed a two dimensional MOS model. The low frequency noise analysis is carried out using a local oxide charge fluctuation as noise sources and Green's functions approach to compute the power spectral density of the drain current fluctuations. With this model, we can take into account slow oxide traps in the oxide overlap region (oxide over the LDD), which is degraded by hot electron stress. The Fig. 7 presents the two oxide traps profiles used to explain the noise level behavior presented in Fig. 5 and Fig 6. These Gaussian profiles correspond to a short stress time and a long stress time. We have reported on Fig. 8 the stressed device noise level normalized to fresh device noise level from subthreshold to strong inversion regime and for the two trap profiles. In the case of the short stress time profile, we observe an increase of the noise only for strong inversion biases. We attribute this trend to the influence of the traps of the LDD region. A quasi constant increase of the noise level

289

is observed when computing noise with long stress time profile, which indicates that traps created in the channel by the hot carrier stress are the origin of the 1/f noise level increase. We conclude that degradation of defects related to 1/f noise level at 1 Hz (at a depth of around 2 nm in the oxide) follows the classical scheme of interface state degradation under Hot Carrier regime [5].

le-17 j . .

®

le-23 I—

le-8 le-7 le-6 le-5 le

ID (A)

FIGURE 5 : 1/f noise level @ IHz vs. ID before and after stress short stress time.

^ le-20

® , 9 le-21

^P - T ^ ^ O

le-6

ID (A)

FIGURE 6 : 1/f noise level @ IHz vs. ID before and after long stress time.

Short stress time Long stress time

FIGURE 7: Slow oxide trap density profiles used for simulation of LFN of the drain current.

FIGURE 8: Simulation of the stressed device noise level (at IHz) normahzed to fresh device noise level for from subthreshold to strong inversion regime.

REFERENCES

M. Valenza, A. Hoffmann, D. Sodini, A. Laigle, F. Martinez, D. Rigaud, "Overview of the impact of downscaling technology of 1/f noise in p-MOSFETs to 90 nm", IFF Proc-Circuits Devices Syst., Vol 151, No 2, pp 102-110, April 2004. G. Ghibaudo, "A simple derivation of Reimbold's drain current spectrum formula for flicker noise in MOSFFTs," Solid-State Electronics,vol. 30, no. 10, pp. 1037-1038, Oct. 1987. Fan-Chi Hou, Gijs Bosman, Mark F, "Simulation of trapping noise in submicron n-channel MOSFFTs, Electron Devices, IEEE Trans, on, Vol.50, Iss.3, pp :846-852. M. Marin, J.C. Vildeuil, B. Tavel, B. Duriez, F. Arnaud, M. Stolk, M. Woo ,"Can 1/f noise in MOSFFTs be reduced by gate oxide and channel optimization", ICNF 2005, September 2005. C.Guerin, V.Huard, A. Bravaix, M.Denais, "Impact of hot carrier degradation modes on I/O n MOSFFTs aging prediction", IEEE IIRW 2006, pp.63-67.

290

Low-Frequency Noise of Strained and Non-Strained n-Channel Tri-Gate FinFETs With

Different Gate Dielectrics

N. Lukyanchikova'', N. Garba/, V. Kudina'', A. Smolanka'', E. Simoen and C. Claeys'''''

"v. Lashkaryov Institute of Semiconductor Physics, Prospect Nauki 45, 03028 Kiev, Ukraine IMEC, Kapeldreef 75, B-3001 Leuven, Belgium

'KULeuven, B-3001 Leuven, Belgium

Abstract. The influence of different front gate Hf-based high-k dielectrics (HfSiON/Si02 and Hf02/Si02) on the shape of the low-frequency noise spectra for n-channel tri-gate FinFETs processed in standard silicon-on-insulator (SOI) substrates, and global Strained Si Directly On Insulator (sSOI) wafers with/without Selective Epitaxial Grown (SEG) source and drain regions is studied. For different process splits the concentration distributions of slow traps over the thickness of the gate dielectric are estimated and it is shown that these distributions depend on the dielectric type.

Keywords: Low-Frequency Noise, 1/f noise, high-k dielectric, SOI FinFET, sSOI FinFET PACS: 85.30.Tv; 73.50.Td

INTRODUCTION

High-k dielectrics are currently replacing SiOa in sub 45 nm CMOS transistors in order to reduce excessive off-state leakage current by direct tunneling through the scaled gate oxide. However, this is traded for a loss of Ion by the degradation of the channel mobility, through scattering at defects and charges in the high-k layer. This problem can be tackled by so-called mobility-engineering, which relies either on the use of channel-strain-inducing techniques or of so-called high-mobility channel materials. In combination with the multiple-gate channel architecture offered by MuGFETs, it has been shown to yield significant performance benefits. At the same time, the question is raised how other parameters like the low-frequency (LF) noise are affected by these process changes. Therefore, the LF noise of MuGFETs fabricated in Silicon-on-Insulator (SOI) and strained-SOI (sSOI) substrates with different high-k dielectric gate stacks has been studied.

EXPERIMENTAL

In the case of a HfSiON/SiOa dielectric, the gate stack consisted of 2.3 nm HfSiON (50% Hf) on top of 1 nm interfacial SiOa while for the HfOa/SiOa dielectric, it



291

consisted of 2 nm HfOa on top of 1 nm interfacial SiOa. The parameters of the devices were: Hfl„=65 nm and 55 nm for SOI and sSOI nFinFETs, respectively, WefjÔ.02^9.^7 |am, LgÔ. 15^0.9 |am, where Hfl„ is the fm height, We/f and Le/f are the effective fm width and length, respectively. The full device width Z^f was calculated by the formula Zefj^Nfl„x(2Hfl„+Weff) where Nfl„ is the number of fms. The drain current noise spectral density Si(f) was measured on wafer in the frequency rangey=0.7 Hz to 100 kHz at 0.2 V<FGF^1.5 V and ^0^=25 mV where VGF and VDS are the gate and drain voltage, respectively. To analyze the shape of the noise spectra, the frequency dependences of [fi<Si(f)] were considered.


Figure la shows the dependences [fi<Si(f)] vs./typical for the noise spectra of the sSOI devices with a HfSiON/SiOa gate dielectric. It is seen that the main feature of the spectra is the decrease of [fi<Si(f)] with decreasing/from ^(10-30) kHz down to sufficiently l ow/ This decrease can be described as \fi<Si(f)]xJ" where a=0.17^0.25 at VGF>Vth where Vth is the threshold voltage. It has also been found that in some cases a plateau portion corresponding to the l/noise shows itself at/<100 Hz. Therefore, in the devices investigated a pure 1/noise is observed only at more or less low values of /while the (1//)''where ;i^l-ctÔ.83^0.75 is typical for a wide frequency range.

SOI HfO^/SiO^

^ W'

^^

Jj

/Hf"

>^ w^ MWH

V^pV /,iiA

2 0 3 0 4 0 5 1 e 1

4 3.S9 e 24X) 5 544 0 775 2 94.7

«?

SOI HfO^/SiO V^=OAS

---'|l|~''X-

2 m 4 1 a 23.S 5 12 2S.7 e 1.5 31.S

-

10' 10° 10° 10' itf 10' 10' 10° 10 10 10 10 10 f, Hz f, Hz f, Hz

(a) (b) (c)

FIGURE 1. Drain current noise spectral density multiplied by the frequency for the sSOI nFinFETs with the HfSiON/SiOj gate dielectric where W^frl.?,! |xm, LejrO.9 |xm and Nfl„=5 (a) and for the SOI

nFinFETs with the Hf02/Si02 gate dielectric where Wgff=9.%l |xm, Lgff=OA |xm and Nfl„=l (b) and We^M |xm, Lef=0.9 |xm andAf;i„=30 (c).

It has been revealed that for the FinFETs with HfOa/SiOa gate dielectric the shapes of the spectra [/x^H/)] vs./differ significantly from those shown in Fig. la. This is demonstrated in Figs, lb and Ic where the behaviour of the spectra shown in Fig. lb appears to be typical not only for SOI devices of Weff>0.l2 \xm or LeÔ.16 |am but also for sSOI ones [1] while Fig. Ic presents the spectra typical for the SOI FinFETs of Weff=0.02 |j,m. It is seen from Fig. lb that the high-frequency plateau corresponding to the 1//noise is observed at/>400 Hz and the decrease of [/x^H/)] with decreasing/ takes place at f<AOO Hz where a=0.3 which corresponds to the (1//)°'^ noise.

292

Therefore, the "high-frequency" 1//"noise is typical for the SOI FinFETs of Weff>0.l2 \xm or Lefj<0.l6 |am and sSOI ones with a HfOa/SiOa gate dielectric at/>400 Hz. As to the very thin SOI FinFETs, it follows from Fig. 1 c that the main component of their noise spectra is the Iz/noise which is observed fromyÔ.7 Hz up toy=5 kHz.

Figure 2 demonstrates that the equivalent gate voltage noise spectral density, SVG, is independent of the overdrive gate voltage V*={VGF-Vth) in a sufficiently wide interval of V* where SrG=Si/{gmf and g^ is the transconductance. Observation of a plateau in SrG{V) is known to be typical for the McWhorter noise of a (1//)'' type where / depends on the distribution of the density of the noisy traps Not over a distance x from the Si/SiOa interface. By using the plateau values of SrG=SrG(fmeas) measured at different fmeas, we have calculated the distribution of Not over x as follows [2]: Not{x)=Not{Xmeas)=fmeasSrG(/meas)ZeffLefj(Co) ^Q kT w h e r e X=Xmeai=-^ ln(2 ;7/êai r ^ m ) ,

1=10"* cm is the tunneling parameter, rm>j=10"'° s, Co is the equivalent gate capacitance per cm , q is the electron charge, k is the Boltzmann constant and T is the temperature. The results obtained are shown in Figs. 3 and 4.

E o

OT

^ • Q l ^ ^ r - ^

| f=10Hz|

.. C> -^

>sg»- ^ R

Wî^,\i,m i- ,|ifTi

sSOl O 2.87 0.90 A 2.87 0.15 0 0.87 0.90

^ f = 3 k H z

IV^ff-lAm l-^n,Wn

sSOI+SEG O 2.87 A 2.87 • 0.87

0.90 0.16 0.90

? W^^,\im Lîf,iim

SOI+SEG e 2.87 0.90 A 2.87 0.15 • 0.87 0.90

0.1 0.2 0.4 0.5 0.3

v,v FIGURE 2. Equivalent gate voltage noise normalized for Z^ând Z^y at different gate overdrive

voltages for sSOI, sSOI+SEG and SOI+SEG nFinFETs with a HfSiON/SiOj gate dielectric measured at / = 1 0 Hz and 3 kHz; W^j^(0.87 H- 2.87) |xm, L,^(0.15 H- 0.9) |xm andAf;j„=5 and 1.

4.0 r 4.0

3.5

- ^ 3 . 0

I 2.5 % 2 . 0

•^ 1.0

0.5

0.0.

4 sSOl ,f"<

0 0.87 0.9 0 2.87 0.9 A 2.87 0.15

1

A

2|

O

0.8 1.2 1.6

X, nm

(a)

2.0

3.5

I 3.0

; 2.5 I

% 2.0

1.5 I

• 1.0

0.5

"•"o.s

'^w

J • \ o

sSOI+SEG * 0.87 0.9 e 2.87 0.9 A 2.87 0.15

SOI+SEG • 0.87 0.9 • 2.87 0.9 A 2.87 0.15

o fi HfSiON/SiO^

© i

2 1.6

X, nm

(b)

FIGURE 3. Distributions of the slow traps concentration Nî over the gate dielectric thickness for sSOI (a), sSOI+SEG and SOI+SEG (b) nFinFETs with a HfSiON/SiOj gate dielectric; W^(0.87 H- 2.87) |xm,

Leg=(0.15 H- 0.9) |xm and Nfln=5 and 1.

It is seen from Fig. 3 that in the case of the HfSiON/SiOa dielectric a linear decrease of Not with increasing x from x=0.98 nm to x=1.4 nm is observed for the sSOI, sSOI+SEG and SOI+SEG FinFETs where the gradient of the trap concentration for the sSOI devices is higher than for the sSOI+SEG and SOI+SEG ones. At the same

293

time, at 1.6 nm<x<2.1 nm this decrease slows down (Fig. 3b) or stops (Fig. 3a). It is also seen from Fig. 3b that the dependences Notix) for the sSOI+SEG and SOI+SEG devices can be approximated by one and the same line. Therefore, the global straining technique does not influence the distribution of the slow traps over the HfSiON/SiOa gate dielectric in SEG-devices.

At the same time, in the case of the HfOa/SiOa dielectric (Fig. 4) the decrease of Not with increasing x for SOI devices of Weff>0.02 |j,m and for sSOI devices takes place only atx>1.3 nm while the range of x between 1 nm and (l.l-^1.4) nm is characterized by a homogeneous distribution of Not over x. As a result, the values of Not for the HfOa/SiOa dielectric are higher than for the HfSiON/SiOa one at 1.15 nm<x<2.1 nm. It is also seen from Fig. 4a that the global straining technique does not influence the distribution of the slow traps only at sufficiently high values of x but increases the trap concentration atx<1.65 nm. Note that even a stronger influence of the global straining technique on Notix) in the case of the HfOa/SiOa dielectric is observed for the devices of Wefri^m |am (Fig. 4b).

E o

4.0

3.5

3.0

25

b 2.0

1.5

1.0

0.5

0.0 0

W^*0.87t im

rtO

(

HfO^SiOj

'O

?''n

[2^501

8 1

I V ^ ^ i . f l - i i " '

V 9.87 0.4 e 2.87 0.9 O 0.87 0.9

SOI V 9.87 0.4 O 2.87 0.9

^JLSSOJ

'a

2 1 6 2 0

4.0

3.0

2.5

% 2.0

1.5

1.0

0.5

00

<Apt

2: SOI

W„=0.02^m

HfOj/SiOj

L-\ 3 _

V 1: sSOl

T" ''""'""

5.

sSOl < 0.4 • 0.9

SOI < 0.4 • 0.9

\ ^ y

X, nm X, nm (a) (b)

FIGURE 4. Distributions of the slow traps concentration Not over the gate dielectric thickness for sSOI (curve 1) and SOI (curve 2) nFinFETs with the HfOj/SiOj gate dielectric; f % > 0.87 |xm, L^j=(OA H-

0.9) |xm and Nfl„=l (a) and Wgg=0.02 |xm, L^g^iOA H- 0.9) |xm and Nfl„=30 (b); curve 3 presents the dependence NJx) for the sSOI nFinFETs with the HfSiON/SiOj gate dielectric; f % > 0.87 |xm,

is;r(0.15 H-0.9) ^mandNf,„=l.

CONCLUSION

The essential difference in the low-frequency noise spectra has been revealed for n-channel tri-gate FinFETs with HfSiON/SiOa and HfOa/SiOa gate dielectrics. The effect is attributed to different distributions of the concentration of slow traps Not over the thickness of the gate dielectric.

REFERENCES

1. Lukyanchikova, N. Garbar, V. Kudina, A. Smolanka, E. Simoen and C. Claeys, Semiconductor Physics, Quantum Electronics & Optoelectronics 11, 203-208 (2008).

2. R. Jayaraman and C. Sodini, IEEE Trans. Electron Devices 36, 1773-1782 (1989).

294

Low-Frequency Noise Behavior in P-channel SOI FinFETs Processed With Different Strain

Techniques

W. Guo, R. Talmat, B. Cretu, J-M. Routoure, R. Carin, A. Mercha", E. Simoen' and C. Claeys"'*'

GREYC UMR 6072 CNRS / ENSICAEN / University of Caen, 6 Bd Marechal Juin, Caen, France "IMEC Kapeldreef75, B-3001 Leuven, Belgium,

''E.E. Dept. KULeuven, B-30001 Leuven, Belgium

Abstract. The aim of this paper is to investigate the low-frequency noise behavior in p-channel SOI FinFETs processed with different strain techniques. An unusual noise behavior was observed for all devices studied. This unusual noise was investigated for different applied gate voltages and different channel lengths at room temperature. The carrier number fluctuations explain the flicker noise for all devices. The different strain techniques employed have no significant impact in the noise level.

Keywords: FinFET, SOI Substrate, Strain techniques, Low-frequency noise PACS: 72.70.+m, 73.40.Qv

INTRODUCTION

The main advantages of multi-gate FinFET (MuGFET) devices include the improvement of the short channel effects, the leakage currents, the threshold voltage dopant fluctuations, and eventual higher mobility due to the undoped channels [1-4].

The transistor performances depend on the semiconductor-dielectric interface quality. The low-frequency noise analysis is one of the tools used to analyze the quality of the gate oxide. For a better understanding of the device physics a study of the low-frequency noise is required.

In this work, the low-frequency noise performance of p-channel tri-gate FinFETs is investigated. We have first evidenced an unusual noise behavior for all the tested devices. The impact of the channel length on the unusual noise behavior is analyzed at room temperature. This unusual behavior was already observed in [5] for n-channel FinFETs only in devices processed with selective epitaxial growth. Finally, we have shown that the different strain techniques used to enhance the performances of MOS devices have no significant impact on the noise level.



295

EXPERIMENTAL

The available devices are p-channel FinFETs, with gate mask length Lpm varying from O.lS im up to l^im, mask width Wpm from O.lS im to 3|^m, fixed fm height Hpin = 65nm, and having five fms in parallel. The high-k gate stack consists of TiN/TaN/HfOa/SiOa, with a measured equivalent oxide thickness (EOT) of 1.9 nm. The device fabrication details can be found in [6].

The standard FinFETs on an SOI substrate were used as a reference (noted SOI). Other analyzed structures are: FinFETs using selective epitaxial growth (SEG) in order to reduce the access resistance by increasing the height of the source and drain regions (noted SOI+SEG), as well as FinFETs using SEG combined with a CESL (contact etch stop layer) strain technique (noted SOI+SEG+CESL).

Low-frequency noise measurements were performed directly on wafer using a 2 inch "Lakeshore TTP4" prober. The devices were biased in the linear regime with an applied drain voltage VD = -20 mV. The experimental set-up sweeps a range from 0.1 Hz to 100 kHz.


An unusual noise behavior was already highlighted for the n-channel FinFETs [5] and an empirical model was proposed (equation 1) assuming two 1/f noise levels: the one with the higher level shows an unusual frequency dependence at the two corner frequencies f and fa, and it is noted by "K2 1/f noise"; the one with the lowest level is noted "Ki 1/f noise":

S y = white noise + —^ + -^ 7 -^ (equation 1) / / 1 + A 1 + /

/ / 2

In [5] it was also demonstrated that the K2 1/f noise component could be attributed to the carrier number fluctuations in the channel. Moreover, it was clearly pointed out that in the case of the n-channel FinFETs, the unusual noise behavior is observed only for the SEG devices.

Examples of the frequency normalized gate voltage spectral density in p-channel FinFETs are showed in Figure 1 for Lmask = 0.25|^m for all tested structures. Contrary to n-channel FinFET, the unusual noise behavior can be observed for all the analyzed structures. Additional Lorentzian noise components can be observed, but will be discussed further. From Figure 1, it is clear that the different strain techniques employed have no significant impact on the noise level.

Good agreement between the measured noise spectra and the empirical model proposed in [5] has been verified for all channel gate lengths for all structures studied. Figure 2 illustrates this agreement for the case of a reference device with Lmask = 0.16|^m, with different applied gate biases. We can notice that the unusual noise can be clearly observed only in the weak inversion regime. In strong inversion operation, additional noise components can "hide" the Ki 1/f noise component, as shown in the Figure 2.

296

. n

u? . . , E . , . -

r.. .

pFinFET WyL=0.15jjm«).25jjm E0T=1.9nm

Y"--*!

- " - S O I + SEG SOI + SEG+CE

mo&^

K i

LI

f i <2

10 l o o 10OO 10OOO 1O0OOO

frequency (Hz)

FIGURE 1. Frequency normalized gate voltage spectral density for Lmask = 0.25nm

FIGURE 2. Frequency normalized gate voltage spectral density for L„,jsk = 0.16\im, for different

applied gate biases

A Study of the low-frequency noise was performed for different channel lengths at fixed drain current (i.e. ID = 3|^A). In Figure 3 the gate voltage spectral density normalized by frequency and the mask gate length for the reference device is shown. We can notice that at the same drain current conditions (i.e. ID=3|^A), the unusual noise can be observed only for gate mask lengths smaller than O.T im. This result could suggest a possible access resistance contribution on the low noise spectra.

Figure 4 shows the variation of the frequency normalized K2 1/f level noise spectral density (@VD=-20mV) versus the absolute value of the applied gate voltage. We consider the hypothesis that the K2 1/f noise component could be attributed to the carrier fluctuations in the channel (i.e. as in [5]). One can notice that there is no significant difference in the noise magnitude in the weak inversion for all the FinFET structures investigated. These results show that the different strain techniques employed do not affect the noise level.

E 5

' • " -

r n

r n

•"'

E-17-

V w v ^

A ^ ^

••wv^Jtl

JM ^

rf!jP^'

f pFinFET W=0.15|jm E0T=1.9nm

'0^

-1)

&i^

L = 0.16|jm L = 0.5|jm

—• —L = 0.7|jm L = -

^„ i

PFinFET W/L=0.15^m/0.16^m E0T=1.9nm

_ T=XOK

P'„I=P',-VJ(V)

FIGURE 3. The gate voltage spectral density normalized by frequency and the mask gate length

for the reference FinFET structures at ID=3H A

FIGURE 4. The variation of the frequency normalized K2 1/f level noise spectral density

versus the applied gate voltage

It can also be observed that in weak inversion the gate voltage spectral density related to the K2 1/f level is quasi-independent on the applied gate voltage. This suggests that the carrier number fluctuations due to hole trapping in the oxide

297

dominate for all devices in weak inversion. This result may be striking since the 1/f noise in pMOS transistors can usually be explained using the Hooge model. Therefore, the carrier number fluctuations model can be used to explain the origin of the noise and extract an average value of the oxide trap density Nit (eV'cm"^) [7]. Using the effective gate length and width, this oxide trap density was found to be abou t4 -10 'W cm" for all tested devices.

CONCLUSION

An unusual noise behavior is observed for the all p-channel FinFET structures tested in the present. This striking noise behavior clearly appears only in weak and weak to strong inversion transition. In the same ID biases, it was observed for gate mask lengths smaller than 0.7|^m. These results imply a possible contribution of the access resistances on the low-frequency noise spectra. The empirical model proposed in [5] can perfectly model the unusual noise behavior. We have found that the carrier number fluctuations dominated the flicker noise for all studied FinFET structures. Further investigations are still necessary in order to validate the origin of the two 1/f noise sources.

ACKNOWLEDGMENTS

This work was accomplished in the framework of the project "Flemish Tournesol" (Project no. 1807 IRC) of the Partnerships Hubert Curien (PHC) of EGIDE.

REFERENCES

1. Hisamoto D et al. "FinFET - a self-aligned double-gate MOSFET scalable to 20 nm". IEEE Trans Electron Dev 2000;47(12):2320-5.

2. Kavalieros J, Doyle B, Datta S, Dewey G, Doczy M, Jin B, et al., "Tri-gate transistor architecture with high-k gate dielectrics, metal gates and strain engineering", VLSI Technol Symp 2006:50-1.

3. Park J-T, Colinge J-P, Diaz CH. "Pi-gate SOI MOSFET". IEEE Electron Dev Lett 2001;22(8):405. 4. Von Arnim K et al. "A low-power multi-gate FET CMOS technology with 13.9 ps inverter delay

large-scale integrated high performance digital circuits and SRAM". VLSI Technol Symp 2007:106-7.

5. Guo W et al. "Impact of strain and source/drain engineering on the low frequency noise behaviour in n-channel tri-gate FinFETs". Solid-State Electronics 52 (2008) 1889-1894

6. CoUaert N, Rooyackers R, De Keersgieter A, Leys FE, Cayrefourq I, Ghyselen B, et al. "Stress hybridization for multigate devices on supercritical strained-SOI (SC-SSOI)". IEEE Electron Dev Letter 2007;28(7):646-8.

7. McWhorter A.L. "1/f noise and germanium surface properties". Semiconductor surface physics. PA: University of Pennsylvania Press; 1957.

298

Modeling of High-Frequency Noise in III-V Double-Gate HFETs

B. G. Vasallo

Dpto. FisicaAplicada, Universidadde Salamanca, Pza. Merced, s/n, 37008 Salamanca, Spain

Abstract. In this paper, we present a review of recent results on Monte Carlo modeling of high-frequency noise in III-V four-terminal devices. In particular, a study of the noise behavior of InAlAs/InGaAs Double-Gate High Electron Mobility Transistors (DG-HEMTs), operating in common mode, and Velocity Modulation Transistors (VMT), operating in differential mode, has been performed taking as a reference a similar standard HEMT. In the DG-HEMT, the intrinsic P, R and C parameters show a modest improvement, but the extrinsic minimum noise figure NF„j„ reveals a significantly better extrinsic noise performance due to the lower resistances of the gate contact and the source and drain accesses. In the VMT, very high values of P are obtained since the transconductance is very small, while the differential-mode operation leads to extremely low values oiR.

Keywords: Double-Gate HEMT, Velocity Modulation Transistor, Monte Carlo simulations. PACS:81.05.Ea, 85.30.De.

1. INTRODUCTION

InAlAs/InGaAs High Electron Mobility Transistors (HEMTs) have revealed an excellent performance for low-noise high-frequency apphcations [1]. To further improve their behavior, alternative solutions based on an evolution of the standard HEMT design have been proposed [2-6]. Thus, the Double-Gate (DG) HEMT, a HEMT with two gates placed on each side of the conducting InGaAs channel (see Fig. 1), has been recently fabricated. The DG architecture, originally conceived for Si devices [7], has demonstrated to provide a better noise operation in comparison with similar Single-Gate (SG) transistors [6,8-10]. In particular, the DG geometry in III-V devices allows to counteract the effect of carrier injection into the buffer (since no buffer is used in the structure), and provides a better charge control, with the subsequent improvement of the pinch-off behavior (lower drain conductance) and the transconductance gm- Moreover, the lower contact resistances contribute to a better extrinsic noise performance, mainly at high frequency [6].

As well, the progress of the DG-HEMT technology allows the design and fabrication of III-V Velocity Modulation Transistors (VMTs) [II]. In VMTs [12-15], the source and drain electrodes are connected by two channels with different mobility |a, while two gates allow the control of the global channel electron density nr. Electrons can be shifted between the two channels by changing the gate voltages in differential mode, that is, a potential VGDIFF/2 is added with different sign in each of



299

the gates to a bias voltage VGOFF which adjusts the total carrier concentration in the channel. Due to the different transport properties in the two channels. ID is modified while keeping rir constant, that is, by velocity modulation. In this way, it is in principle possible to overcome the transit-time limit for high-frequency operation. Evidently, the differential-mode operation involves an inherent change in the noise properties of DG-devices.

The purpose of this work is to review recent results on Monte Carlo (MC) modeling of high-frequency noise in III-V four-terminal devices. The MC technique has been proved to be a very useful tool in the high-frequency noise analysis of SG and DG transistors [6,10,16-18].

A thorough comparison between the noise performance of an InP-based lattice-matched 100 nm-gate DG-HEMT and the corresponding standard SG structure is performed by means of a semiclassical 2D ensemble MC simulator which reproduces satisfactorily the experimental static and dynamic characteristics of DG [6] and SG [18-20] transistors, and allows predicting their noise performance. Previously to the noise analysis, the dynamic behavior of the devices must be determined. The procedure [6,18,20,21] is the same for both DG- and SG-devices, what is correct as

long as the DG-HEMT works in common mode, that is, the potential applied at both gate electrodes being identical {Vas=Vasi=Vas2)- Once calculated the intrinsic Y parameters, the intrinsic P, R and C noise parameters are evaluated taking into account the intrinsic noise sources {Sm, Sia, and SIDIG) [22]. Then, the extrinsic minimum noise figure NFmi„, the noise resistance R„, and the associated gain Gass are determined by considering both intrinsic and extrinsic noise sources [23]. On the other hand, the analysis of the VMT behavior, still controversial, requires to include the differential-mode parameters for the input potential and current, typically VIN=VGDIFF=VGSI-VGS2, IIN=(IGI-IG2)/2 [24].

In addition, the inclusion of the parasitic elements is still uncertain when dealing with the differential-mode operation, since the small signal equivalent circuit (SSEC) is not well established, so that here we just analyze the intrinsic P, R, and C parameters.

2. PHYSICAL MODEL

Z. =lQQn , 2QQnm;innnm* *

llnAIAs

QlnGaAs

llnAIAs

igg

m 500nm ,

lOnrr^^l B f f t l ' t J i ' i M ^ ^ n W M 12nm

20nm

200nm

nid Schottky layer 1 lOjs m" b-aoping g

nid Channel B

Buffer 1

SG-HEMT

DG-HEMT

2QQnm;LpQni

VMT FIGURE 1. Schematic drawing of the simulated SG-HEMT, DG-HEMT, and VMT.

For the calculations we make use of a semiclassical ensemble MC simulator self-consistently coupled with a 2D Poisson

300

solver which takes into account important physical effects, like the influence of degeneracy in the channel by using the rejection technique [18-20]. Our MC model, by providing an accurate description of the microscopic dynamics of carriers, allows predicting adequately the noise behavior. The reliability of the model is essential in four-terminal devices since, to our acknowledge, there are no available experimental results of their noise performance. The devices under study are recessed Ino 52AI0 48As/lno ssGao 47AS 100 nm-gate DG-HEMTs, and the corresponding standard SG-HEMT, taken as a reference in the comparison. The detailed topology is plotted in Fig. 1. The SG-HEMT consists of a InP substrate (not simulated), a 200 nm InAlAs buffer followed by a 20 nm-thick InGaAs channel, three layers of InAlAs (a 5 nm spacer, a 5xI0'^ cm" 5-doped layer modeled as a 5 nm layer doped at 10^' cm" and a 12 nm Schottky layer), and, finally, a 10 nm thick InGaAs cap layer (doped 6xI0'^ cm"^). In the DG device, the buffer is suppressed and substituted by a layer structure symmetrical to that at the top of the channel, and two opposite 100 nm-gate electrodes control the total electron density in the channel. As well, they control the carrier shift between the high- and low-|a channels in differential-mode operation in the VMT device. The only difference in the VMT with respect to a typical DG-HEMT is that the channel is divided into two 10 nm-thick regions: a high-|j, undoped channel and a low-|a channel with compensated doping of 7V^=7VD=10'^ cm" . The compensated doping increases the ionized impurity scattering and thus decreases the electron mobility. The calculated low-electric-field mobilities in the high-|j, and low-|a channels are -12000 CWL'/YS and -2700 cm^/Vs, respectively.

3. RESULTS

To better understand the noise performances of the devices under study, the static and dynamic behavior of SG- and DG-HEMTs (operating in common mode) must be previously analyzed [2-5]. Due to the presence of two charge-accumulation regions in the channel, the drain current ID provided by the DG-HEMT is about twice that given by the SG-transistor for the same biasing, which leads to a significant increase of g^ (see Fig. 2). However, the higher values of gm are compensated by the increased gate-

to-source capacitance Cgs, thus providing lower values of the intrinsic cut-off frequency fc, evaluated as gm I InCgs. In particular, MC simulations predict a maximum of 401 and 387 GHz for the SG- and DG-HEMT, respectively, for KDS=0.5 V. Concerning the extrinsic dynamic performance, the experimental parasitic elements have been taken into account in the SSEC for the calculation of the extrinsic ft and fmax [5]. The DG-HEMT benefits from two gates in parallel, thus reducing its gate resistance Rg to nearly half than that of the SG-HEMT. The source and drain access resistances are also much lower in the DG-HEMTs due to the higher electron density. The

500

400

300

200

100

i-

r

r

. . , . . , . 1 . . . . , , , . . , . . , , 1 , , . . .

- • — DG-HEMT

- V - SG-HEMT j ^ — — * — :

/ ^ - ^ ^ £ r r 2 i z i j

^^^^3 W^^^--K'~Trr'^-.--

00 0.1 02 03 0.4 05 0.6

FIGURE 2. MC 1D-VDS curves in the DG- and SG-HEMT. The gate vohage of the top curves is VQS= 0 V and the step of the gate bias is tiVnfr 0.1 V for both sets of curves.

301

200 400 /„(mA/mm)

FIGURE 3. MC values of (a) P, (b) R and (c) C parameters vs. ID for the 100 nm-gate DG- and SG-HEMTs. FTO=0.5 V.

simulations show a significant improvement in the maximum values of fmax- 287 GHz for the DG-HEMT compared to 226 GHz for the SG-HEMT. Regarding ft, similar values are obtained for both the DG- and SG-devices (around 200 GHz). The improvement of fmax due to the DG architecture is more pronounced than that of ft because of the reduced value of Rg and gd, which are important for fmax without much affecting the value of^.

The intrinsic noise behavior of the transistors is illustrated in Fig. 3, which presents the (a) P, (b) R and (c) C noise parameters as a function of ID for the simulated DG-and SG-HEMTs for KDS=0.5 V. The MC results fori? and C are not very accurate due to the uncertainty in the calculation of the gate noise [18], and also the value of P for very low ID, when g^ is practically zero. Moreover, the gate leakage current and its consequent shot noise, that may affect the value of R (mainly near pinch-off) is not included in our model. For a better understanding of the figure, tendency lines have been drawn. P and R, representing, respectively, the noise due to the drain and gate current fluctuations, take lower values in the DG-HEMT than in the SG-device. This happens because carriers in the DG-device are completely confined in the channel and the current fluctuations due to electrons injected into the buffer are avoided. The change between low/high horizontal velocity of electrons in the buffer/channel leads to drain current fluctuations, while the associated vertical motion generates an excess of gate current noise. Thus, the suppression of these real space transfer processes reduces both drain and gate current noise. On the other hand, C is about the same for both types of devices, since the electron dynamics inside the channel and the gate-channel couphng are similar.

The extrinsic noise performance of the common-mode devices is analyzed in terms of the parameters NFmi„ and R„, together with Gass- In Fig. 4 we present their MC

000

100

10 |

• ' '

• (b) . . . . . . . . . . . . . . . . 150 GHz 94 GHz

- O - - • - DG-HEMT; - V - - • • - SG-HEMT:

0 200 400 600 0 200 400 /o(mA/mm) /„(mA/mm)

FIGURE 4. MC values of (a) AfF„,„, (b) G^, and (c) _R„ vs. ID for the 100 nm-gate DG- and SG-HEMTs, at 94 and 150 GHz. F™=0.5 V.

302

values as a function of ID for FD^O.5 V at 94 GHz and 150 GHz for both DG- and SG-HEMTs. The minimum noise figure shows the typical U-shape, mainly due to the influence of Rn, which increases at both high (due to the increase of the drain noise, associated with the P parameter) and low (due to the decrease of the cutoff frequency) drain current. For the same biasing, NFmi„ is lower for the DG device than for the SG one. This occurs due to not only the reduction of the intrinsic drain and gate noise (lower P and R), but also the lower parasitic contact resistances. At a frequency of 94 GHz, the minimum values taken by NFmi„ are 2.1 dB (for /D=21 mA/mm) for the DG-HEMT and 2.9 dB (for /D=68 mA/mm) for the SG-device. At 150 GHz, the minimum values of NFmi„ are 2.8 dB for the DG-HEMT and 4.2 dB for the SG-device, which proves that the improvement introduced by the DG architecture is more noticeably the higher is the frequency. The extrinsic R„ and Gass are also much improved in the DG-structure, thus allowing both for a better noise matching and a higher gain at low noise conditions, and, as a consequence, a more flexible MMIC design. The well-known increase of NFmi„ and decrease of Gass with frequency is clearly observed.

Concerning the VMT, the analysis of the static, dynamic and noise behavior requires a different treatment. Carriers are shifted between the two channels by changing the gate voltages, Vai and Va2, in differential mode. In this way. ID depends on VQDIFF due to the different electron velocity in the channels while keeping the total electron density constant (see Fig. 5). When increasing VQDIFF the electron density is transferred from the low-|a to the high-|j, channel, thus increasing the drain current [11]. However, the maximum values of gm related to this type of current control, extracted from the static characteristics, are extremely low, around 120 mS/mm in the

device under analysis. As mentioned before, since the small signal

VMT equivalent circuit is still controversial, we restrict our analysis to the study of the P, R, and C parameters. They are presented as a function of ID in Fig. 6 for FDS=0.5 V. As predicted [13], SID in the VMT is about the same to that of similar common-mode devices. On the other hand, Sia is notably lower. However, due to the low values of gm, P is significantly higher than in the SG- and

FIGURE 5. MC/i,-Foi,jj,j, curves in DG-HEMTs. In contrast, the differential-mode the VMT. Var 0.5 V

E E < E

40,

30

a, 20

10

(a)

1 0 CJ-^ 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400

/„ (mA/mm) /„ (mA/mm) /„ (mA/mm)

FIGURE 6. MC values of (a) P, (b) R and (c) C parameters vs. I^ for the 100 nm-gate VMT. 1^^=0.5 V.

303

operation leads to values ofR much lower than in common mode. Since C is about the same (within the uncertainty of this quantity), the intrinsic values of NF^m result to be similar to the DG-HEMT ones.

4. CONCLUSIONS

We have presented a review of a MC-analysis of the noise behavior in InP-based four-terminal devices. The simulations confirm that the intrinsic P and R noise parameters in the DG-device (operating in common mode) are lower than in SG-HEMT due to the suppression of the current fluctuations originated by the injection of electrons into the buffer. Furthermore, the extrinsic noise behavior (in terms of NFmi„, Gass and R„) is significantly improved due not only to the better intrinsic noise performance, but also to the lower contact resistances. However, recent results on the noise performance of VMTs (operating in differential mode) show very high values of P due to the small values of gm-

ACKNOWLEDGMENTS

This work has been partially supported by the Direccion General de Investigacion, MCyT, and FEDER through the project TEC2007-61259/M1C and by the Consejeria de Educacion, JCyL, through the project SA019A08.

REFERENCES 1. S. Tiwari, Compound Semiconductor Device Physics, New York: Academic, 1992. 2. N. Wichmann, I. Duszynski, X. Wallart, S. Bollaert, and A. Cappy, IEEE Electron. Device Lett, 25 354 (2004). 3. N. Wichmann, I. Duszynski, S. Bollaert, J. Mateos, X. Wallart, and A. Cappy, in lEDM Tech Dig., 1023 (2004). 4. N. Wichmann, I. Duszynski, X. Wallart, S. Bollaert, and A. Cappy, IEEE Electron. Device Lett, 26 601 (2005). 5. B. G. Vasallo, N. Wichmann, S. Bollaert, Y. Roelens, A. Cappy, T. Gonzalez, D. Pardo, and J. Mateos, IEEE

Trans. Electron Dev., 54 2815 (2007). 6. B. G. Vasallo, N. Wichmaim, S. Bollaert, Y. Roelens, A. Cappy, T. Gonzalez, D. Pardo, and J. Mateos, IEEE

Trans. Electron Dev., 55 1535 (2008). 7. G. K. Celler, and S. Cristoloveanu, J. Appl. Phys., 93 4955 (2003). 8. A. Lazaro, and B. liiiguez, Solid-State Electronics, 50 826 (2006). 9. B. liiiguez, T. A. Fjeldly, A. Lazaro, F. Danneville, and M. J. Deen, IEEE Trans. Electron Dev., 53 2128

(2006). 10. P. Dollfus, A. Boumel, S. Galdin,-Retailleau, and J. E. Velazquez, J. Comput Electron., 5 479 (2006). 11. N. Wichmann, B. G. Vasallo, S. Bollaert, Y. Roelens, X. Wallart, A. Cappy, T. Gonzalez, D. Pardo, and J.

Mateos, ^pp/. Phys. Lett, 94 (2009, to be published). 12. H. Sakaki, Jpn. J. Appl Phys., 21 L381 (1982). 13. K. J. Webb, E. B. Cohen, and M. R. Melloch,/£•£•£• Trans. Electron Dev., 48 2701 (2001). 14. M. Prunnila, J. Ahopelto, K. Henttinen, and F. Gamiz, Appl. Phys. Lett., 85 5442 (2004). 15. C. Sampedro, F. Gamiz, A. Godoy, M. Prunila, and J. Ahopelto,^pp/. Phys. Lett, 86 202115 (2005). 16. R. Rengel, T. Gonzalez, and M. J. Martin, Fluctuations and Noise Lett, 4 L561 (2004). 17. V. M. Polyakov, andF. Schwierz, Semicond Sci. Technol, 19 S145 (2004). 18. J. Mateos, T. Gonzalez, D. Pardo, V. Hoel, and A. Cappy, IEEE Trans. Electron Dev., 47 1950 (2000). 19. J. Mateos, T. Gonzalez, D. Pardo, V. Hoel, H. Happy, and A. Cappy, IEEE Trans. Electron Dev., 47 250

(2000). 20. J. Mateos, T. Gonzalez, D. Pardo, V. Hoel, H. Happy, and A. Cwppy, Microelectt-on. Reliab., 41 73 (2001). 21. T. Gonzalez, mAD.VarAo, IEEE Trans. Electron Devices A2 605-611 (1995). 22. A. Pucel, H. A. Haus H., and H. A. Statz,^ A. Electron. Electt-on Phys., 38 195 (1974). 23. H. Rothe, and W. Dahlke, Proc. IRE, 44 811 (1956). 24. D. E. Bockelman, and W. R. Eisenstadt, IEEE Trans. Microw. Theory Tech., 43 1530 (1995).

304

Scaling Effect Of GaAs pHEMTs Small Signal And Noise Model

S. C. Huang, W. Y. Lin and Y. M. Hsin*

Department of Electrical Engineering, National Central University, No.300, JhongdaRd, Jhongli City, Taoyuan County 32001, Taiwan (R. O. C)

Phone •• +886-3-4227151 ext 34468Fax : +886-3-4255830 *E-mail :[email protected]

Abstract. In this work, small-signal and noise model with gate-width scaling of GaAs pseudomorphic high electron mobility transistors (pHEMT) are presented. The scaling effect of the model parameters are derived from an accurate small-signal and noise equivalent circuit model for the different gate widths of pHEMTs. The experimental and model results show that noise coefficients are not dependant on the gate width while devices biasing at the same current density.

Keywords: Model, noise, pHEMT,, scaling. PACS: 72.80.Ey

INTRODUCTION

Since GaAs pseudomorphic high electron mobility transistors (pHEMTs) have demonstrated excellent microwave and noise performance, they are very attractive for millimeter-wave and optoelectronic applications. In order to optimize the noise performance of pHEMTs for low noise apphcations, a scalable noise model for the pHEMTs which provides good accuracy and scalability based on the gate width is very important for monolithic microwave integrated circuits (MMIC) design.

This work presents the small-signal and noise model of 0.15 |im-gate AlGaAs/InGaAs/GaAs pHEMTs with different gate widths fabricated by WIN semiconductor. The full noise characterization requires the four noise parameters: minimum noise figure ¥„„, noise resistance R„, optimum source conductance Gopt and optimum source susceptance Bopt. We found that the noise model parameters from different gate widths of the pHEMTs at the same current density show the similar noise coefficients unless considering the deviation in the noise coefficient P. This scaling observation fit well to the measured noise characteristics.

MODEL PARAMETER EXTRACTION

The most common noise model for pHEMTs presented by Pucel et al. is used in this study. [1] Fig. I shows the noise model, which introduces two noise sources in the



305


intrinsic equivalent circuit. Both resistances contribute uncorrelated thermal noise. In Pucel et a/.'s model, the short-circuit noise currents at drain <id> and gate <ig^> are modeled by the two equations: <id> = 4kTgmPAf and <ig^> = 4kT(Q)Cgs)2RA f/gm, with <igid > = j4kTAfQ)Cg,C(RC)" P and C, where C is a complex number.

Gate

This model needs the noise coefficients R,

Drain

FIGURE 1. The schematic noise model of the pHEMT.

A small-signal model is established first. Utihzing Yang-Long and Cold-FET method, extrinsic parameters of pHEMTs can be extracted [2] [3]. Then by using matrix operation method to obtain the intrinsic parameters, which we can get all parameters to set up the small-signal equivalent model of pHEMTs. Moreover, according to the H. Hillbrand method [4], we obtain the noise correlation matrices to match the four noise parameters from measurement for extracting noise coefficients: P, R and C. Therefore, the equivalent noise model of the pHEMTs with divinable noise characterization can be estabhshed. The Fig. 2 and Fig. 3 show the noise characterization of the 8 x 75 |xm gate width pHEMT with model and measurement data, which shows an excellent agreement.

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)

FIGURE 2. The F„,„ a.nAR„ versus frequency of the 8 x 75 jim pHEMT.

306

0.00

0.02

0.04

0.06

0 08

OKI

0.12

^ . t ^

G

^ _ , — , — « ' ' ' • '

• measure

model

L

• A v / , y ^

•

. --

-

-

0.08

0.07

0.08

0.05

0.04

0.03

0.02

0.01

0.00

2 4 6 8 10 12 14 16 18 20

Frequency (GHz)

FIGURE 3. The G„ , and Bop, versus frequency of the 8 x 75 ^rn pHEMT.

EXPERIMENT AND DISCUSSION

The pHEMTs with three gate widths are investigated including 2 x 75^ 4 x 75 and 8 X 75 |im for scaling effect. Table I shows the extracted parameters for three devices including the noise coefficients. From Table I, the intrinsic parameters show the scaling effect except the time delay (r) of gm- From the results the noise correlation matrices of the intrinsic part have the scaling effect equate to the gate width ratio when the noise coefficients maintain constant. The impedance intrinsic noise correlation matrix {C^^) has the similar trend by de-embedding the extrinsic effect as shown in Fig. 4 and Fig. 5. Where noise coefficients R and C are very close each other as shown in Table I but the noise coefficient P shows the small deviation. The deviation of P is due to the failure to scahng effect which is proportional to the gate width and the extrinsic elements.

2.0x10"

0 1.6x10'°

•g 1.4x10-'°

5 1.2x10'°

C 1.0x10"'° 0

••§ 8.0x10""

g 6.0x10"" 0 11

Q 4.0x10"

2.0x10""

, — " — 2 x 7 5 11111

A —•—4x75)1111 - / \ —' -8x75)1111

• / \ A V r "

2 4 6 8 10 12 14 16 18 20

Frequency (GFIz)

FIGURE 4. The intrinsic C'^u between the pHEMTs with different siz

307

o""

i: o

O

2.0x10

1.8x10"'

1.6x10"'

1.4x10"'

1.2x10"'

1.0x10"'

8.0x10""

6.0x10""

4.0x10""

2.0x10""

0

\ \_

\

V^

2

— • —2x75fjm

— • — 4x75 fjm

— *—8x75fjm

• ' V • . ^ — ^ • • • .

-A-^

4 6 8 10 12 14 16 18 20

Frequency (GHz)

FIGURE 5. The intrinsic C'^22 between the pHEMTs with different sizes.

CONCLUSION

In this work, the noise coefficients between different gate widths pHEMTs maintain constant though the noise coefficient P shows the small deviation. Even though P has the small deviation, the similar value can be used as the initial value of the noise parameter extraction to optimize the four noise parameters. Such that it is convenient to model and avoid the complex noise extraction procedure.

TABLE 1. The parameters of pHEMTs with three jate widths for small signal am noise model.

Extrinsic elements

device

2x75 4x75 8x75

device

2x75 4x75 8x75

R.

m 4.1 2.2 1.1

c (fF) 96 189 374

R d

m 4.2 2.3 1.25

Cgd

(fF) 25 53 111

R>

m 4 2 1

Lg (PH)

155 147 130

Intrinsic elements

Cds

(fF) 42 88 176

R i

(a) 0.8 0.4 0.2

U (pH)

171 161 150

Rds

1300 420 250

L. (pH)

6 6

3.3

gm (mS) 10.7 29 56

c (fF)

90 70 55

T

(PS) 1.1 1.2 1.2

Cpd

(fF)

63 54 44

Noise coefficients

P

0.39 0.58 0.68

R

0.15 0.16 0.15

C

0.91 0.9

0.91

REFERENCES

1. R. A. Pucel, H. A. Haus, and H. Statz, "Signal and noise properties of GaAs microwave FET," Advances in Electronics & Electron Physics, vol. 38. New York: Academic Press, 1975, pp.195-265

2. L. Yang, and S. I. Long, IEEE Electron Device Lett. 7, 75-77 (1986). 3. G. Dambrine, A. Cappy, F. Heliodore, and E. Playez, IEEE Trans. Microwave Theory Tech. 36,

1151-1159(1988). 4. H. Hillbrand, and P. Russer, IEEE Trans. Circuits and Systems 23, 235-238 (1976).

308

Investigation of SiGe Heterojunction Bipolar Transistor over an Extreme

Temperature Range

A. Shimukovitch^, P. Sakalas*'^, M. Ramonas^''^, M. Schroter*'§,

C. Jungemann , W. Kraus

Fluctuation Phenomena Lab., Semiconductor Physics Institute, Gostauto 11, Vilnius, Lithuania,

*CEDIC, Dresden University of Technology, Helmholtzstrasse 18, 01069 Dresden, Germany, ^ECEDept, University of California, La Jolla, 9500 Gilman Drive, MC 0407, 92093-0407,CA,USA

EIT4, Bundeswehr University, 85577 Neubiberg, Germany, QP34-Modeling & Simulation, Telefunken Semiconductors GmbH, Theresienstrasse 2, 74072Heilbronn, Germany,

Abstract. Dc, high frequency (hf) characteristics and noise of SiGe HBTs were investigated in a wide ambient temperature (7) range from 4 K to 423 K. SiGe HBTs with low emitter concentration (LEG) and trapezoidal Ge base doping were found good candidates for cryogenic applications. Both hydrodynamic (HD) device simulation and compact model (CM) HICUM show good agreement with experimental data in the temperature range of 300 K-423 K. The collector current did not show any leakage related to electric field assisted tunneling via traps in the base. Rapid decrease of transit frequency (fj) with T is explained in terms of the carrier delay distribution. Noise figure {NFy„j„) analysis reveals that the main noise contributors are related to collector current fluctuations (shotlike noise) and thermal noise in the base at high T. Base current fluctuations related noise becomes of importance only at high injection. Simulated diffusion noise distribution shows that collector terminal electronic noise originates at the emitter-base (BE) junction but not in base-collector (BC) junction area.

Keywords: SiGe HBTs, temperature dependence, transit frequency, diffusion noise, hydrodynamic device simulation, HICUM.

INTRODUCTION SiGe heterojuntion bipolar transistor technology nowadays offers high speed transis

tors capable to operate in cryogenic environment. Compared to BJTs, SiGe HBTs are naturally suitable for cryogenic temperatures, including 4 K [2],[3], and have achieved /j-beyond 600 GHz at 45 K [1]. Careful SiGe HBT profile optimization can yield an improved hf performance at very low temperatures [2],[4],[5]. It was shown that SiGe HBTs can be used for cryo-apphcations, such as LNAs for satellite electronics [6], amplifiers for cooled Analog Digital Converters [7] and operational amplifiers working at 4 K [8]. Profile optimization suggests to lower the emitter concentration and to use a trapezoidal Ge profile with constant base doping. Along with high speed, reduced high frequency noise of cooled SiGe HBTs is an advantageous feature for LNA design. It was shown that significant improvement of NFj„j„ was achieved at cryogenic temperatures even for SiGe HBTs with conventional doping profiles [5],[11],[12],[13]. Circuit design for these apphcations requires accurate physics based compact models (CM),



309

which capture all important effects of HBTs operating within a large temperature range. It was shown in [10] that SiGe HBTs are also reliable devices for high temperature apph-cations. In this work we present experimental and modehng results of LEC SiGe HBTs at different ambient temperatures.

DUT, EXPERIMENTAL AND MODELING TOOLS LEC SiGe HBTs with an emitter area^£•0=0.5*20.3 \im^ and a peak transit frequency

/r(3ooK)=80 GHz [9] were investigated. This technology enabled an increase of base doping, resulting in reduced base resistance. The high doping reduces carrier freeze-out at cryogenic temperature (CT). Almost constant base doping along with a trapezoidal Ge composition provide already an optimum profile for SiGe HBT operation at CT without any further profile optimization as required in [4]. Details about the measurement set-up can be found elsewhere [3],[15]. HD device simulations were performed with Galene 111 [16]. The compact model (CM) HICUM/Level 2 v.2.3 [14] was compared to measured and HD data. Note, that HD simulations were performed only for available temperature range due to limitations by unreliable or unverified mobility models for deep cryogenic temperatures.

RESULTS AND DISCUSSION Measured and simulated Gummel characteristics at different T are presented in

Fig. la, b. Good agreement of CM and HD with experimental data in the range of 300 K-423 K and also between HD and CM at T=473 K is obtained. At CT the collector current density (Jc) does not show additional leakage associated with the field assisted tunnehng via trap states in the base [5], resulting in increasing nonideality. As expected, Jc at CT shows steep and shifted (in Vgg) behavior HICUM can capture Jc behavior down to 75 K. Toward higher T, Jc increases at constant Vg^, which is caused mainly by the temperature dependence of the bandgap. The transit frequency behavior is well captured by CM and HD at high 7^=300-423 K (Fig.3, a). The fj- peak decrease is caused by the decrease of the mobility in the various transistor regions (Fig.2, b). T increase from 323 K to 423 K increase overall electron delay: in the emitter ATg(%)= 127%, base ATB{%)= 181% and collector Arc(%)= 144%. Compared to peak/y of 70 GHz at 300 K, ff reaches 160 GHz at CT (Fig.3, a). The peak value of/j-exhibits a linear dependence on T (Fig.3, b), which partially can be explained by the current gain /?(7) dependence, including band-gap narrowing A£g(7) (Fig.2, a). The current gain of SiGe HBT

P ~ (N^L /PgWJexp\-—\, where N^ and P^ are doping concentrations of emitter

310

and base respectively and Lg is emitter length and Wg is base width, AEg is band-gap reduction of the base. At CT p increases due to exponential term and decreases due to enhanced recombination of the minority carriers in the base, so compensating the current gain increase [17]. Electron delay distribution in collector exhibits peak, which was also given in [5] and is associated with HD model peculiarities.

10'

10°

"\ 10-'

^ 10-'

O 10-'

10-*

10-=

c (a)

473 K y ^

!

'3 \// « A *

r ° r ° p °

r °

HICUM HD model

^ Exp. 423 K ffl Exp. 373 K A Exp. 323 K A Exp. 300 K • Exp. 75 K o Exp.4 K

^

10'

10-

10-

10-'

10'

10-'

10'

10-

(b)

r ^^

' ^ ^^ l/f '•J/^my ^,JY

' =/ /y^ * ^ * 7 1 . I ,^:

S^ # ^ n o

^ / /

DC •o

° O cP o

HICUM

» Exp. 423 K = Exp.373 K A Exp. 323 K 4 Exp.300 K Q Exp. 75 K o Exp. 4 K

0.4 0.5 0.8 1.0 i 0.8 1.0 1.2 1.4

VBE [V] (b) VBE m FIGURE 1. (a) Jc (^BE) and (b) J^ (VBE) at different T..

2.0 ^^Base423K -•-Base323K ^^Emitter423K ^»-Emitter323K -•-Collector 323K ^-Collector 423K

0.2 0.3

X[nm]

FIGURE 2. HD simulated: (a) energy band (b) total delay distribution at Fc£=l-5 V, ¥^^=0.^ V..

160F 140

120 -

100

80 -

60

40

20

O

-IX

10.00 100

(b)

200 300

T[K|

400 500

FIGURE 3. (a)/j. vs. J^ at ^c£=l-5 V, thick line: HD, solid line HICUM (b) peak/j. vs. T.

HD simulated NFjj,j„ versus JQ is in perfect agreement with experimental data for higher T range (Fig.4, a, b). NFjj,j„ analysis revealed that the main noise contributors are related to collector current fluctuations (shot-like noise) (Fig.4, a) and thermal noise in the base at higher T (Fig.4, b). At both Tbase current fluctuation related noise becomes important at high injection only. Simulated diffusion noise spectral density distribution (Fig. 5) demonstrates that electron noise origins at the BE junction but not in BC junction area.

311

•a

(4 Z

1U-9 - 1. Al ndife IIL 6'feou

TWp|f=^G|H^|||||| .irtniii T1 miiii

2. Only collector shot ' ' - ^ ^'~ ' Vr^r^-O 8 V 3. Only base Shot ^BE ^•'^ ^

4. No base resistance thern'ial noise Symbols are measured data ^ Thick solid li

-^ f III!

——..

ne HD

^__

Ll+lf

J

' 9 / - " 3 /

/

/

/

J( [mA/fim (b) J( [mA/fim

FIGURE 4. M^„;„ vs. J^, thick line: HD, solid line: HICUM. (a): 7=323 K (b): 7=423 K.

I

Electron diffusion noise

V^ =1.5V, V„=0.8V, 1GHz

O

Hole diffusion noise distribution

=1.5V, V„ =0.8V

0.10 0.15 0.20 0.25 0.30 HM 0.00 0.05 0.10 0.15 0.20 0.25 0.3C

S (a) X[nm] (b) X[nm] FIGURE 5. (a): Collector terminal electron noise density distribution (b): Base terminal hole noise density

distribution.

ACKNOWLEDGEMENTS This work was supported by DOTFIVE European project. Atmel GmbH is acknowledged for wafers.

Authors M. Ramonas, P. Sakalas and M. Schroter are thankful to DFG for financial support.

REFERENCES

[I] p. Chevalier etal., IEEE BCTM, pp. 121-124, 2008.

[2] J. D. Cressler, et al., IEEE TED, Vol. 40, No. 3, pp. 542 - 556, 1993.

[3] H. Geissler et al.. Digest IEEE ARTFG, CD-ROM, 2006.

[4] Y. Cui et al.. Digest, IEEE BCTM 9.3, 2006.

[5] J. D. Cressler, G. Niu, "Silicon-Germanium Heterojunction Bipolar Transistors", Artech House, 2003.

[6] E. Soares et al., IEEE MTT, Vol. 48, No.7, pp. 1190-1198, 2000.

[7] D. Gupta et al., IEEE Transactions on applied superconductivity, Vol.13, No.2, pp. 477483, 2003.

[8] R. Krithivasan et al., IEEE BCTM 4.1, 2006.

[9] A. Schiippen et.al., Proc. ESSDERC, Solid-State Device Research Conference, pp. 88-91, 2000.

[10] T. Chen et al., IEEE TED,Vol. 51, No. l l , pp. 1825-1823, 2004.

[II] B. Banerjee et al., IEEE TED, Vol. 52, No.4, pp. 585-593, 2005.

[12] S. Pruvost et al., IEEE EDL, Vol. 26, No.2, pp.105-108, 2005.

[13] J. D. Cressler, IEEE BCTM 15.1, pp. 248-251, 2005.

[14] M. Schroter, lEICE Transactions on Electronics, Vol. E88-C, No. 6, pp. 1098-1113, 2005.

[15] P. Sakalas et.al., IEEE TED, Vol. 56, No.2, pp. 328-336, 2009.

[16] B. Neinhus et al., VLSI Design, Vol. 8, pp. 387-391, 1998.

[17] C Arnaboldi et al., IEEE Transactions on Nuclear Science, Vol.50, No.4, pp. 921-927, 2003.

312

Experimental Analysis of Noise in CdTe Radiation Detectors

A. Andreev^ L. G^nela^ M.Raska^ J. Sikula^ and P. Moravec''

"Department of Physics, Brno University of Technology, Technicka 8, 616 00 Brno, Czech Republic Fax: +425 4114 3133, e-mail: [email protected].

Physical Institute, Charles University, Prague, Czech Republic

Abstract Noise characteristics of CdTe gamma and X-ray detectors have been carried out. Samples were prepared at Physical Institute of Charles University in Prague by traveling heater method (THM). Measurements of high-ohmic detectors with two golden contacts and low-ohmic detectors with four contacts were carried out. Two voltage contacts were used to distinguish between metal-semiconductor junction area with depleted region and homogeneous part of the sample. The resistance of high-ohmic samples is in the range from hundreds of MD up to several GD. The noise characteristics of the samples were measured in dark and with the illumination in the range of radiation from ultraviolet to infrared.

Keywords: Noise, 1/f noise, contact noise, mobility fluctuation, GR spectra, Schottky barrier PACS: 71.55.Gs, 72.70.4-m, 0707.Df, 73.304-y

INTRODUCTION

CdTe is a member of the II-IV semiconductor family. It is a material of great importance in the fields of both fundamental research and technical applications, because of its structural, optical electronic and photoelectronic properties. In the last years, single crystals of cadmium telluride (CdTe) have become useful as a nuclear radiation detector, an electrooptical modulator, an optical material in the infrared and a solar cell, whose maximum efficiency is as high as 23%. The main application of CdTe consists in high resolution detection of y-rays and X-rays 0.

Germanium and silicon is the common semiconductor materials used respectively for gamma-ray and x-ray spectrometers. However, because of their small band gaps they must be cooled and in most cases operated near liquid nitrogen temperatures (77 K) to avoid excessive thermal currents. At the other extreme, CdTe detectors operate at room temperature because of their wide gap zone Eg~1.5eV (T=300 K). CdTe has high absorption coefficient to absorb high energy radiations. That makes CdTe the most suitable material for nuclear detectors manufacturing [2-3].



313


http://71.55.Gs

NOISE

All the samples show 1/f noise as a dominant noise and some samples also show GR noise. The noise characteristic of low-ohmic sample is in Figurel and there is 1/f" noise with parameter n very close to 1 for all the values of applied voltage.

f33b8-si-voltages-300k-15.01.2009.ep

<

f/Hz

FIGURE 1. The noise spectral density of sample F33B8 and background noise. Applied voltages U = 0.19; 1.25; 13 V.

The intensity of 1/f noise is inversely proportional to the total number of the carriers in the sample (AO [4]:

S, Nf

[1]

where Si is the noise spectral density of a fluctuating current (I) developed across the terminals of a linear resistor when a current is injected into it; an is Hooge constant, it insignificantly depends on temperature and its value for CdTe a n - 210"^.

The hole concentration within the depleted region for the applied voltage U = 1.25 V is in Fig.2. The calculation according to equation 1 shows that the value of noise spectral density should have been much less than the measured value. The hole concentration distribution within the depleted region was calculated with the assumption that the hole concentration in the homogenous part of the sample p = 710^'* cm" . The total number of free carriers can be obtained by integration of the function in Figure 2. When the result is substituted in equation 1, the current noise spectral density is the same as the measured one. These calculations were carried out for all the samples and it proves that the source of excess 1/f noise is in the depleted region at metal-semiconductor junctions.

314

f33b8-pconc-1.25v-16.06.08.ep

200 300

x/nm

FIGURE 2. The holes concentration distribution within the depleted region of p-type CdTe sample F33B8 at the metal-semiconductor junction. Applied voltage U = 1.25 V.

The noise spectral density of high-ohmic sample B39D1H with no light is in Figure 3. The dominant noise at low frequencies is 1/f" noise with parameter n very close to 1. The noise spectral density of the same sample illuminated by 548 nm wavelength is in Figure 4. The dominant noise at low frequencies up to 20 Hz is 1/f" noise with parameter n close to 1. At higher frequencies and with higher voltages the GR noise is dominant.

b39d1h-nolight-20.11.08.ep

f/Hz

FIGURE 3. Sample B39D1H. No Light. Applied voltages U = 25.6 - 44 V.

315

b39d1 h - light-548nm- 20.11.08.ep

<

1 10 100

f/Hz

FIGURE 4. Sample B39D1H, Light 548 nm. Applied voltages U = 12,9 - 43 V,

CONCLUSION

The noise spectral density of 1/f noise depends on the quantity of free carriers in the sample. The analysis of low frequency noise showed that the experimental value of 1/f noise is always much higher than the theoretical value which corresponds to the total quantity of free carriers in the sample. Free carriers are distributed uniformly throughout the homogenous part of the sample. But within the depleted region there is very low concentration of free carriers The excess value of low frequency noise is caused by the low carrier concentration within the depleted region at the metal-semiconductor junction.

ACKNOWLEDGMENTS

This research has been supported by GACR No. 102/07/0113 and by the Czech Ministry of Education in the frame of MSM 0021630503 Research Intention MIKROSYN "New Trends in Microelectronic System and Nanotechnologies".

REFERENCES

1. J. Singh, Semiconductor Devices: Basic Principles. John Wiley & Sons, Inc. 2001 2. CAPPER, P. and Co. Properties of Narrow Gap Cadmium Based Compounds. INSPEC, 1994. 648 pages. 3. ZANIO, K. Semiconductors and Semimetals. Volume 13. Cadmium Telluride. Academic Press, INC. 1978. 4. HOOGE, F.N. 1/f Noise is no surface effect. Physics Letters A. 1969, v ol. 29, no 3, p. 139-140.

ISSN 0003-6951

316

Low-Frequency Noise in a 0.18 ^m Mixed-Mode CMOS Technology at Low Temperature

p. Martin", M. Cavelier' and G. Ghibaudo''

"CEA, LETI, MINATEQ F-38054 Grenoble (France) Email: Patrick.martin&,cea.fr

''IMEP, MINATEQ 3, parvis Louis Neel, F-38016 Grenoble (France)

Abstract. We report on the characterization of Low-Frequency (LF) noise in a commercial dual gate process with 1.8 V (Tox = 3.3 nm) and 3.3 V (Tox = 6.5 nm) MOSFETs at 77 K. The LF noise behavior in various MOSFETs of this process is well described physically by the correlated carrier number - mobility fluctuation model. A simplified compact model, suitable for analog circuit simulation and valid from weak to strong inversion regimes, is presented.

Keywords: MOSFET, 1/f noise, low temperature, compact modeling. PACS: 72.70.+m, 73.40.Qv, 85.30.De

INTRODUCTION

Low Frequency (LF) or flicker noise parameters are of prime importance for the design of high performance infrared image sensors. For room temperature sensors, these parameters are generally given by the foundry as part of the design kit. However, as those high performance infrared sensors must be cooled at low temperature, typically 77 K, to attain their highest sensitivity, LF noise parameters must be extracted prior to integrated circuit design as they are generally not given by the foundry [1].

This paper reports the LF noise measurements of MOSFETs done on devices of different gate lengths and widths and having different threshold voltages or oxide thicknesses. The CMOS technology is first presented. Thereafter, experimental setup is described. The third part is dedicated to the main results and modeling.

CMOS TECHNOLOGY

We characterize a mixed mode/RF CMOS process optimized for room temperature operation. In this dual gate oxide process, two kinds of MOSFET transistors are available: (1) transistors with a physical gate thickness of 3.3 nm, a minimum channel length of 0.18 im and operating at a maximum recommended voltage of 1.8 V and (2) transistors with a physical gate thickness of 6.5 nm, a minimum channel length of 0.35 i m and operating at 3.3 V. p-MOSFET are made in a NWELL. Depending on the channel doping, transistors with different threshold voltages are provided. p-MOSFET transistors may be done either with a standard threshold voltage (STD) or with a low



317

threshold voltage (LVT). n-MOSFET transistors are made either in a P-substrate (also referred as the PWELL) or in a triple well (TWELL). N-MOSFET transistors have either a standard threshold voltage (STD), a low threshold voltage (LVT) or a zero-volt threshold voltage (ZVT). As a whole, twelve MOSFET transistors are allowed for mixed analog-digital circuit design. This process is a dual gate process and uses two kinds of polysilicon gates, N^ doped polysilicon gate for n-MOSFET and P^ doped polysilicon gate for p-MOSFET. As a consequence both NMOS and PMOS are surface channel transistors. This deep submicron process has pocket implants to combat short channel effects. Finally we have to mention that this process has LDD zones to diminish hot-carrier degradation and use Shallow-Trench Isolation (STI). Low temperature characterization of this process is presented in [2].

LF NOISE MEASUREMENTS

LF noise was measured using a specific computer-controlled system with programmable biasing current amplifiers. This system allows automatic current fluctuations measurements on up to six devices inside a cryostat using low noise switches. Low-noise voltage sources are used to bias drain, gate and bulk terminals. The noise is measured, not as it is classically done in the linear regime (Vds ~ 50 mV), but in saturation (Vds = 1.8 or 3.3 V) as it will be representative of their operation in analog circuits. Prior to noise measurements, the devices are characterized in the DC mode using an Agilent 4155 parameter analyser. Important quantities, such as subthreshold slope, drain current (Id), gate transconductance (Gm), drain conductance (Gds) and bulk transconductance (Gmb) are measured. The high sensitivity of our experimental set-up allows to measure the thermal noise floor at frequency lower than 100 kHz on long channels biased in weak inversion with a current power spectral density very closed to 2/3 x 4 kT (Gm + Gmb + Gds) as expected in saturation. Different gate lengths have been under investigations from 20 | m to the minimum allowed transistor length (0.18 or 0.35 ^im), with gate widths from 20 | m to 1 |^m. Transistors were biased at different gate voltages ranging from weak inversion (Id ~ 10 pA) to strong inversion (Id ~ 1 mA). The slope of each recorded spectrum is carefully analyzed. RTS noise components are eliminated. After this treatment, the LF noise is found to follow a 1/f variation with a mean y value of 0.97 ±0.10.


Several experiments were done in order to determine which one of the flicker noise models apply to these MOSFETs, McWhorter's carrier number or Hooge's mobility fluctuation model. Figures 1 and 2 show the experimental and simulated normalized drain current power spectral densities, SId/l/, at 295 K and 77 K on standard 3.3 V n-MOSFET. The curves describe a plateau in weak inversion and agree well with the (Gm/Id) characteristic. Experimental data were fitted using either the carrier number fluctuations model predicted by McWhorter, or the model improved by Ghibaudo et

318

al. [3] by taking into account the correlated mobility fluctuations occurring at the highest currents:

Si = [l + a n C i r G ^ S , , ( f ) w i th :S , J f ) = ^ I ^ ^ A ^ ^ i ^ -d L " ox y-^ -i m Vfb \ / Vfb v •' " I T 7 T y~l2 r> 7

CJ W L C I

(1)

where Nt (Ep) is the oxide trap volume density per unit energy (eV"' cm" ) evaluated at Fermi level, X a tunnel attenuation distance (0.1 nm) and a the Coulomb scattering coefficient (V s/C). Extracted values of a for 3.3 V STD n-MOSFETs are 7x10^ at 295 K and 9x10^ at 77 K.

^ 1 E

Improved model

1E-22 1E.09

McWhorter model

NMOS 3.3 V STD PWELL 295 K f=1Hz

1E-16

1E-17

; 1E-18 'B

h 1E-19

i 1E-20

s

NMOS 3.3 V STD PWELL 77 K f=1 Hz

Improved model

McWhorther model

1E-07 1E-06 1E-05

ld *L /W(A)

1E-04 1E.03

FIGURE 1. Experimental and best fit of Sld/Id^ versus Id for n-MOSFET at 295 K.

1E-22 1E-12 1E-10 1E-08 1E.06 1E-04 1E

ld*L /W(A)

FIGURE 2. Experimental and best fit of Sl^/Id^ versus Id for n-MOSFET at 77 K.

Figures 3 and 4 show the experimental drain current power spectral densities Sid at 77 K versus Gm for 3.3 V STD n-MOSFET and p-MOSFET, respectively. This representation shows that the following simplified carrier number fluctuations model suits in a very large range of inversion for analog transistors used in the design of image sensors:

SF KFG°' 1 W L C F

(2)

where KF and EF are parameters. The extracted EF parameter is slightly different from 2, and lies between 1.7 and 2.4 on the different MOSFETs of this process. This result was previously reported on other advanced CMOS technologies [4]. This LF noise model has been introduced in the EKV3 charge-based compact model [5].

1E-30

1E-32

T, 1E-34 t/i

g 1E-36 u _i * 1E-38

1E^0

1E^2 1E

E

NMOS 3.3 V STD PWELL 77 Kf=1Hz f'

^ •OS

•m ^

1E-07

^ EF=2.09 KF=3.4E-26

1E-06 1E-05 1E-04 1E-03 1E •02

g

\

1E-30

1E-32

1E-34

1E-36

1E-38

1E^0

1E^2

1E-)4 1

:,

EF=2.08KF » v

5.2E-27 1 ^ ^

\ '--^

V^ ^^^' • %^ • ^'•

f - * ' PIVIOS3.3VSTD77Kf=1Hz

1E.07 1E-06 1E-05 1E-04 1E.03 1E.02

FIGURE 3. Experimental and best fit of WLCox^ Sid versus G^ for n-MOSFET at 77 K

FIGURE 4. Experimental and best fit of WLCox^ Sid versus G^ for p-MOSFET at 77 K

319

Figure 5 shows the calculated concentration Nt (Ep) of oxide traps on the different transistors at 295 K and 77 K. The oxide trap density typicaUy ranges from 5x10"' to 3x l0" eV"' cm" at room temperature. Most of the transistors present systematically a higher trap density at low temperature. n-MOSFET are noisier at 77 K by a factor 7 ~ 13, while p-MOSFET are only noisier by a factor 0.6 ~ 3. The increase of Nt at low temperature should be related to higher trap density as getting closer to the band edge at low temperature [6].

;

: : •

\

A

•

A

•

A

•

A

• A

• 295 K A 7 7 K

A

•TAX •TAX •r < •r < •^AX •rJ'" •rJ

FIGURE 5. Trap concentration N, (Ep) measured on different devices.

CONCLUSION

LF noise characteristics of a mixed-mode 0.18 | m CMOS technology have been investigated at low temperature. LF noise behavior in these MOSFETs is well described by the carrier number fluctuation with correlated mobility fluctuation model. A simpler model, where only the gate transconductance Gm is used, was shown to work in a wide range of operation for transistors used in image sensors operating at low temperature.

ACKOWLEDGMENTS

This work has been supported by DGA (Delegation Generale a I'Armement).

REFERENCES

1. P.Martin and M. Bucher, Proceedings of the 6 European Workshop on Low Temperature Electronics, WOLTE-6, ESTEC, Noordwijk, The Netherlands, pp. 133-136 (2004).

2. P. Martin, M. Cavelier, R. Fascio, G. Ghibaudo and M. Bucher, accepted in Cryogenics (2009). 3. G. Ghibaudo, O. Roux, Ch. Nguyen-Due, F. Balestra and J. Brini, Phys. Stat. Sol. (a), 124, 2, 571-

581 (1991). 4. P. Martin and M. Bucher, Proceedings of the 10 International Conference on Mixed Design of

Integrated Circuits and Systems, Lodz, Poland, 26-28 June 2003, pp. 89-92. 5. A. Bazigos, M. Bucher, F. Krummenacher, J.-M. Sallese, A.-S. Roy, C. Enz, "EKV3 Compact

MOSFET Model's Documentation, Version 301.02", Technical University of Crete (2008). 6. I.M. Hafez, G. Ghibaudo and F. Balestra, J. Appl. Phys., 66, 2211-2213 (1989).

320

Electronic noise in high electron-mobility transistors under photo-excitation conditions

H. Marinchio*, G. Sabatini*, L. Varani*, C. Palermo*, P. Shiktorov''', E. Starikov''', V. Gruzinskis''', P. Ziade** andZ. Kallassy**

*Institut d'Electronique clu Sucl (CNRS UMR 5214), Universite Monpellier II, Montpellier, France ^Semiconductor Physics Institute, Vilnius, Lithuania

"Lahoratoire de Physique Appliquee, Universite Lihanaise, Faculte des Sciences, Beirut Lebanon

Abstract. The hydrodynamic approach based on the carrier concentration and velocity conservation equations is used to investigate the influence of photo-excitation of plasma waves at the beating frequency of two lasers on the intrinsic extra noise in InGaAs HEMTs caused by thermally-induced plasma oscillations. It is found that, by increasing the amplitude of the photo-excitation, a significant supression of the intrinsic excess noise is observed at the beating frequency as well as at all the frequencies where plasma waves can be excited.

Keywords: High-frequency noise, high-electron mobility transistors, photo-excitation, optical beating, plasma waves PACS: 72.20.Ht, 72.30.+q, 72.70.+m

INTRODUCTION

Recent experiments devoted to the detection of photo-excited plasma waves in High Electron Mobility Transistor (HEMT) channels in the THz frequency region [1,2,3] have stimulated the consideration of the possibility to use the photo-excitation technique also for the development of room-temperature operating tunable sources and detectors of THz radiation [4]. Such devices are based on the resonant excitation of currents in the source-drain and source-gate circuits at the beating frequency of two laser sources producing carriers photo-excitation in the HEMT channel. Due to the conversion nature of such a source, one of its important characteristics is the signal-to-noise ratio at the generation frequency [5]. Up to now, the high-frequency behavior of the electronic noise in HEMT channels was considered only near thermal equilibrium without any external excitation of plasma waves [6]. Nevertheless, these investigations have demonstrated the resonant enhancement of the excess noise in the frequency region of the plasma waves excitation. Such an extra noise will determine, in essence, the signal-to-noise ratio at the generation frequency. The aim of the present investigation is to study the behavior of such an excess noise directly under photo-generation conditions.

MODEL

For this sake, the current noise behavior at the source, drain and gate terminals of an InGaAs HEMT under a photo-excitation at the beating frequency is simulated by the



321

hydrodynamic (HD) approach based on the conservation equations for carrier concentration and velocity coupled with a quasi-2D Poisson equation [3,4]. The noise spectra are calculated in the framework of the transfer impedance method by using acceleration fluctuations during scattering events as a microscopic noise source [6]. For noise calculations under photo-excitation, an additional averaging of the spectral densities over a period of the beating signal is performed. The parameters of the HEMT structure mainly correspond to those of refs. [3,4]: the channel thickness is 5 = 15 nm, the gate-to-channel distance d=l5 nm, the gated channel lengthLg = 940 nm with 30 nm ungated regions at the source and drain terminals, the channel effective donors density 7V = 8x10'^ cm^^ [3]. The velocity relaxation rate is v = 10'^ s^'. The electron-hole pair generation rate is expressed as g{t) = Go[l + cos{27tfbt)] where Go = 10^^ s^'cm^^ and fb is the beating frequency.

The simulations of noise and the photo-excitation phenomena are carried out near thermal equilibrium. Two operation modes of HEMT are considered. In the first case, to simulate a voltage driven operation, all the potentials applied to source (Us), drain (Ud) and gate (Ug) terminals are supposed to be zero. These values are used as boundary conditions for the quasi-2D Poisson equation [3,4] in a straightforward way. In the second case, for a constant total current operation at the drain terminal, the condition Ud = 0 is replaced by Jd = 0. To fulfill the latter condition, an additional differential equation for the electric field at the drain contact: -jf- = {Jd — J^""'^)/seo is solved in parallel with the HD equations and a value Ed{t) at each time step is used as a boundary condition for the quasi-2D Poisson equation at the drain terminal.

NUMERICAL RESULTS

It should be emphasized that the characteristic feature of a gated channel of HEMT is the possibility of an excitation of standing plasma waves in the channel. As a consequence, both internal fluctuations caused by carrier scattering events inside the conducting channel and external excitations (such as photo-generation, ac voltages and currents applied to HEMT terminals, etc.) will originate in spectral response series of resonant peaks at frequencies corresponding to eigen spatial modes of plasma waves.

This is illustrated by Fig. 1 which shows the spatial profiles of electron concentration calculated by the HD approach under photo-excitation for constant drain-voltage and drain-current operation modes when the beating frequency corresponds to the fundamental plasma wave frequency of 476 and 250 GHz for the first and second operation modes, respectively. As follows from Fig. 1 excited standing modes correspond, respectively, to A/2 and A/4 waves.

Figs. 2 and 3 present noise spectra calculated without and with photo-excitation (solid and dashed lines, respectively) for the above described two operation modes. Let us discuss firstly the case without photo-excitation. For the voltage-driven operation the first resonance at the fundamental frequency / ( corresponds to the standing plasma wave with A/2 = Lg where Lg is the length of the gated part of the channel. In agreement with analytical considerations [3], the intrinsic noise spectrum at source and drain terminals contains resonances at all the frequencies fk = kf{ (k = 0,1,2,3,...), while

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FIGURE 2. Spectral density of current fluctuations at the (a) source and (b) gate terminals of InGaAs HEMT operating under constant source-drain voltage Uds = 0. Calculations are performed by the HD approach without and with carrier photo-excitation at the beating frequency / / = 476 GHz (solid and dashed lines, respectively). The arrow indicates the photo-generation frequency.

only odd spatial modes with A: = 1,3,5,... manifest themselves in the spectrum of current fluctuations. For the second operation mode, which keeps the constant total current at the drain terminal Jd = 0, the first resonant peak at the fundamental frequency f(^ corresponds to plasma wave with A/4 = Lg, so that f^^ « /]f/2. As follows from Fig. 3, here both spectral densities of current and voltage fluctuations at the source/gate and drain terminals, respectively, exhibit series of resonant peaks at frequencies fk = kf[', A:=l,2,3,. . .

To calculate the noise spectra modifications due to photo-excitation under the drain-terminal voltage and current driven operation modes we use the beating frequency of the electron-hole pairs generation rate to be equal to / ( and f{', respectively. The results are presented in Figs. 2 and 3 by dashed lines. It is evident that, when the photo-excitation amplitude is sufficiently high, a significant supression of the excess noise both at the beating frequency and all the frequencies where plasma waves can be excited, is observed. In part, this can be explained by photo-excitation of the regular plasma waves (see Fig. 1) which damp effectively "irregular" plasma waves appearing due to

323

FIGURE 3. Spectral density of (a) current fluctuations at the source and (b) voltage fluctuations at the drain terminals of InGaAs HEMT operating under constant current Jd = 0 flowing through drain terminal. Calculations are performed by the HD approach without and with carrier photo-excitation at the beating frequency / ^ = 250 GHz (solid and dashed lines, respectively). The arrow indicates the photo-generation frequency.

fluctuations.

CONCLUSIONS

Summarizing, we can conclude that, when the beating frequency coincides with the fundamental frequency of plasma waves, an increase of the photo-excitation amplitude results in a significant suppression of the intrinsic excess noise both at the beating frequency and at all the frequencies where plasma waves can be excited. Such a behavior is favourable for the development of room-temperature operating tunable sources and detectors of THz radiation based on the photo-excitation technique.

ACKNOWLEDGMENTS

This work is supported, in part, by the Lithuanian State Science and Studies Fundation contract No P-01/2007, by the french DGA project DEMETER and by the french ANR project AITHER.

REFERENCES

1. Otsuji T, et al. 2004 Appl. Phys. Letters 85 2119 2. Torres J, et al. 2006 Appl. Phys. Letters 89 201101 3. Torres J, et al. 2008IEEEJ. Select. Top. Quant. Electron. 14 491 4. Shiktorov P, et al. 2009 J. Stat. Mech. 01 01047 5. Shiktorov P, et al. 2003 IEEE Trans. Electr Dev. 50 1171 6. Shiktorov P et al. 2001 Rivista Nuovo Cimento 24, ser 4, No. 9, 1-72

324

study of the Low Frequency Noise of Metallic Emitter SiGeC Heterojunction Bipolar

Transistors

F. Pascal^ J. Raoult\ C. Leyris^

' Institut d'Electronique du Sud, UMR 5214, Place Bataillon, Universite .Montpellier. II, 34095 Montpellier Cedex 5, France.

Tel: 33 4 6714 32 14; Fax : 33 4 67 54 71 34; E-mail: [email protected] ^ STMicroelectronics 850 rue Jean Monnet, 38926 Crolles cedex, France.

Abstract. In this paper, we have been investigating both static and Low-Frequency noise (LF noise) characteristics of metallic emitter Si-SiGe:C Heteroj unction Bipolar Transistors (HBTs) based on a 0.13 jim technology developed by ST Microelectronics. This study is based on a comparison between the standard mono-emitter process and the metal emitter process. Static results confirm an increase in IB with an Ic constant and an increase in the figure-of-merit fxx BVcEo. We found a decrease in the 1/f noise level for the metallic emitter HBTs. The base current noise spectral density SJB evolves quadratically with IB and Kp is found to be inversely proportional to AE. These static and noise results will be presented and discussed. In particular, we emphasize the role of the cobalt silicidation on the excess noise.

Keywords: BiCMOS, Heteroj unction Bipolar Transistor, Metallic emitter, SiGeC base. Low-frequency noise. PACS: 85.30De, 85.30Pq, 85.40Qx

INTRODUCTION

Recent technological and design developments have allowed for the fabrication of silicon—^based heterojunction bipolar transistors (HBTs) with unity current gain frequencies (fi's) and maximum oscillation frequencies (fmax) in the 250s of GHz. With these extremely high fi and fmax, SiGeC bipolar devices become very attractive for wireless and optical communications. Such advances are made possible by the increase in collector doping, that delays the Kirk effect, and enables high collector currents. A very high current gain (3 is also achieved but with a detrimental effect on collector-to-emitter breakdown voltage BVCEO- The integration of a metalhc emitter, based on the use of a very thin poly-emitter, has thus been developed\ This metallic emitter has to increase the base current IB without reducing the collector current Ic. Hence, the BVCEO is increased without affecting the fj.



325


EXPERIMENTAL Devices structure and measurement setup

We study here the static and noise behavior of standard mono-emitter process and the metal emitter process. Figures 1 and 2 show a schematic cross-section of devices with standard emitter and with metallic emitter devices respectively. The architecture of the two HBT types is a Fully Self-Ahgned double-polysilicon structure using Selective Epitaxial Growth of the SiGeC base layer. The SiGe growth is terminated by an undoped silicon capping layer separating the emitter and the base. The Si-cap layer acts as a buffer for the carbon out-diffusion mechanism\ Arsenic in-situ doped polysilicon is deposited to form the emitter. Cleaning treatment (HF clean) associated with high annealing temperatures and times results in an epitaxial regrowth of this poly-emitter. The emitter is also partially mono-crystalline. To perform metalhc emitter process, the poly-emitter thickness is reduced from 150 nm for the standard process to 40 nm, while keeping the rest of the process unchanged. Particularly, the cobalt sihcidation module is not modified in order to keep CMOS compatibility. Finally, at the end of the process, the emitter is composed of a sihcon epitaxial layer (mono-crystalline sihcon emitter) and a thin silicided poly-emitter.

1 I I Fig 1: Schematic cross-section of HBT with

standard emitter. Fig 2: Schematic cross-section of HBT with

metallic emitter.

The static characteristics (Gummel-Plots) are measured using an HP4156 semiconductor parameter analyzer. For the LF noise measurements, the devices are biased in a common-emitter configuration at a constant collector voltage of IV. Noise analysis is based on the direct measurement of the base current noise spectral densities SIB. Noise measurements are performed in the frequency range of 1 Hz-lOO kHz, using a spectrum analyzer (HP89410A) and a low-noise current-to-voltage amplifier (EG&G5182). More details about the LF noise measurement setup are proposed in ref l

In this paper, we focus on the study of the 1/f noise. As often reported in the literature for poly-emitter B JTs^ and SiGe HBTs"*, the 1/f noise component of the base current spectral densities can be identified to the SPICE model by :

Sm =K„ - 5 — , with y ~ 1, Ap and Kp are the SPICE parameters. When Ap ~ 2 this

r model is used for a direct comparison of the 1/f noise level through the unitless parameter Kp. Moreover when Kp is inversely proportional to the emitter area, most results shows that the main 1/f noise sources are located in the intrinsic E-B volume"*.

326

Static Analysis Figure 3 compares the Gummel-Plots and current gain of standard emitter and

metallic emitter HBTs. Both the standard and metallic emitter HBTs show ideal Gummel-Plots except for the low

forward emitter-base junction bias (VBE<0.4V), where a strong band to band tunnelling current is observed\ The tunnel current is reduced when the emitter is thinner, this probably means that the emitter-base junction is less abrupt and the arsenic-to-boron distance is increased. At high forward emitter-base junction bias, the base current of the metallic emitter is increased by a factor of 2 compared to the standard emitter, without degrading

the ideality and affecting the collector current. Current gain is also decreased from 2700 down to 1300. The combination of fast surface recombination at the emitter contact and a very thin neutral emitter region allows controlhng IB by varying the remaining thickness of the mono-crystalline silicon. This principle is illustrated in ref . But the gain reduction is not sufficient to significantly increase BVCEO (here +0.4V). Further investigations have shown that when lowly doped emitters are used, the base current becomes more controllable by the emitter thickness. The current gain and the breakdown voltage are therefore more significantly changed\

VBEM

Fig 3 : Gummel-Plots and current gain comparison of metallic and standard emitter 0.15x 3.7 jim^ HBTs.

LF Noise Analysis

This study compares the noise behaviour of standard HBTs (with emitter thickness of 150 nm) and metallic emitter HBTs (with emitter thickness of 40 nm). Representative noise spectra of SIB for the metalhc emitter are reported in the figure 4.

Frequency (Hij

Fig 4: Noise spectra Sm for metallic emitter with emitter area of AE =0.3x3.7 jim^.

10-^' i * « Metallic emitter A Standard emitter

10-' 10-' 10^ 10-" 10-' 10-'

Fig 5: Noise spectra SIB at 1 Hz as a function of base current IB, for standard and metallic emitter.

The excess noise is only composed of a 1/f noise component and the white noise is reached. All investigated metallic and standard emitter HBTs show a similar behavior

327

for different geometries. According to the SPICE model, the extraction of Ap and Kp parameters are undertaken. Figure 5 shows that the 1/f noise exhibits an IB^ dependence. The 1/f noise amplitude Kp for metallic emitter device is found to be inversely proportional to the emitter area. Hence, the associated figure-of-merit KB defined as KB=KF X AE (nm^) has an excellent value ~ 3x10"'° ^m^ compared to previous pubhshed data^. A noticeable reduction of the 1/f noise level (almost one decade) is observed for the metalhc emitter HBTs. This noise reduction will be discussed in the next section.

DISCUSSION

Several assumptions can be proposed to explain the significant reduction in the 1/f noise level for the metallic emitter devices. All the parameters extracted with LF noise measurements show that the 1/f noise source are located in the intrinsic emitter-base volume. Many researchers have shown that 1/f noise, in BiCMOS bipolar transistors, is caused by a fluctuation in the density of minority carriers, which is determined by a fluctuation in surface recombination^. The base current is composed of several recombination components which can fluctuate^. Generally, in modem HBT technology, the fluctuation in the recombination current at the polysilicon/monosihcon interface is proposed to explain the origin of the 1/f noise. In the case of the devices studied in this paper, the density of interface traps can be reduced by the cobalt silicidation process. Since the poly-emitter is very thin, cobalt can have a passivation action of the interface states. Moreover, this phenomenon is enhanced by the epitaxial regrowth of the polysihcon layer. An other possible explanation for the reduction in the 1/f noise level in metalhc emitter is the strong decrease in the polysihcon thickness, from 150 nm down to 40 nm. Hence, the recombination current in the polysihcon emitter, due to dangling bonds and trapping sites at the grain boundaries, can fluctuate and produce noise. However since the polysihcon is degenerately doped this resulting noise must be very low^.

CONCLUSION

We have presented static and noise results on metalhc emitter in a high-speed SiGe HBT technology. A significant reduction of 1/f noise level has been observed compared to the standard emitter devices. We have pointed out the role of the cobalt silicidation of the poly-emitter in the excess noise improvement.

REFERENCES

1 B. Barbalat, F. Judong, L. Rubaldo et al, IEEE BCTM, 8-10 octobre 2006. 2 J. Raoult, F. Pascal, C. Delseny et al, J Appl. Phys. 103, 2008. 3 M.J. Deen, J. 1. llowski, P. Yang, J. Appl. Phys. 77, 1995. 4 F. Pascal, C. Chay, M. J. Deen, and al, lEE Proc: Circuits Devices Syst. 151, 2004. 5 J. J. T. M. Donkers, T. Vanhoucke, P. Agarwal et al, lEDM Tech. Dig., 2004. 6 A. Van derZiel, X. Zhang, A. H. Pawlikiewicz, IEEE Trans. Electron Devices ED-33, 1986.

328

Intrinsic Noise Sources in a Schottky Barrier MOSFET: a Monte Carlo Analysis

Elena Pascual Corral, Raul Rengel and Maria J. Martin

Departamento de Fisica Aplicada, Universidadde Salamanca, 37008 Salamanca, Spain. email: mjmm@usales.

Abstract. This paper presents the first results of the Monte Carlo analysis of the noise performance of Schottky barrier (SB) SOI MOSFETs, focusing our attention in saturation conditions. We analyze the influence of the applied gate bias and the barrier height over the different noise parameters. The main importance of this work lies in the innovation of the study, since this is the first time in the literature that the noise performance of SB-MOSFETs is analyzed by means of a 2D Monte Carlo simulator.

Keywords: Monte Carlo simulator, Schottky Barrier MOSFET, noise performance. PACS: 05.10.Ln, 73.30.+y, 85.30.Tv, 72.70.+m

INTRODUCTION

Schottky barrier (SB) MOSFETs, where the highly doped source and drain (S/D) are replaced by metal silicide, are receiving a lot of attention nowadays. They fulfill some of the requirements exposed in the ITRS [1] due to their interesting features, such as a reduced thermal budget in the fabrication process, a low S/D parasitic resistance and an inherent scalability to gate lengths down to 10 nm due to the low resistance of the metal [2]. For these reasons, the modelling of SB-MOSFETs is a key issue in the development of metal source/drain architectures that may potentially replace conventional MOS devices.

Within this framework, an accurate modelling of principles of operation across the Schottky barrier in SB-MOSFETs has been developed in a previous work by means of a 2D EMC simulator [3]. It must be highlighted that there are very few works in the literature that deal with the modelling of SB-MOSFETs, and they are mainly devoted to the study of I-V curves. In particular, up-to-date the noise modeling of SB-MOSFETs remains unexplored. The Ensemble Monte Carlo (EMC) method is one of the best tools to develop an accurate analysis of the dynamic and noise performance [4]; its stochastic nature mimics the real, noisy movement of carriers inside the device, without defining primary noise sources or considering any approach about its physical origin.

In the present work, we have focused on the study of the noise performance of SB-MOSFETs, and the influence of the Schottky barrier height of the contacts on the intrinsic noise sources. Such study is of great relevance since the reduction of the



329

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barrier height is proposed by some groups as a main goal for SB-MOSFETs to enhance the current drive and to be competitive with regard to conventional MOSFETs [5].

SIMULATION DETAILS

The simulated structures considered (see Table 1) consist on an n-channel accumulation-mode SB-MOSFET on SOI substrate. The topology is similar to that of a conventional SOI-MOSFET, but the ohmic contacts have been replaced by Schottky junctions; moreover, there is an underlap between the gate and the source and drain contacts. Three different Schottky barrier heights have been considered. Since quantum tunnelling across the barrier is of prime interest in these devices, it has been properly implemented in our in-house EMC. The key quantity in the procedure is the quantum transmission coefficient (TC), which is determined from the self-consistent EMC potential profile by solving the Schrodinger equation using the WKB approach. This methodology of the 2D problem is detailed in [3]. An analogous procedure has been realized and properly checked for the ID study of Schottky diodes in [6].

TABLE 1. Main parameters considered in the simulated structure. Parameter

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Value 120 nm 5 nm 10 nm

2-10'^ cm-' 0.25eV//0.20eV//0.15eV

2.2 nm 400 nm

Description Gate length

Underlap length Body thickness Channel doping

Schottky barrier height Gate oxide thickness

Buried oxide thickness

In order to determine the intrinsic gate and drain noise sources in a two-port device configuration, the instantaneous values of gate and drain current were recorded during long-time simulations (typically 1 ns simulation time). From those values the correlation functions are obtained, and through the Fourier transform of these quantities the power spectral densities of gate (Sia), drain (SID) and cross-correlation (SIGID) are directly calculated.


Figure 1 (a) shows the transfer characteristic for the device with three different barrier heights {qOsn = 0.25 eV, 0.20 eV, 0.15 eV). We observe a reduction of the threshold voltage Vj (0.95 V, 0.7 V and 0.55 V respectively) and also a clear rise of the total current as the barrier height decreases, which is mainly due to the promotion of thermionic injection current at the source. In the inset of Figure 1 (a) we show the ratio lonfloff for the three values of qOsn, where we can see that although the hn current rises when reducing the barrier height, the ratio is better for larger barriers.

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FIGURE 1. (a) Transfer characteristics for SB-MOSFETs of three different barrier heights for Vj}s = 2 V. Inset: ratio I„JI„g fox the three structures for Vj^ = 2 V and VQS adjusted to provide the same current level (200 Am''), (a) Ratio of injected tunnelling current over total current at the source versus total current for Vns = 2 V. (b) Conduction band along the x-channel for Vns = 2 V and different values

of Vas to provide the same current level (200 Am"').

Our Monte Carlo simulator considers the total current as a composition of four current components: the thermionic and tunnelling injection and the thermionic and tunnelling absorption. Therefore, we are able to evaluate the contribution of each one of them to the total current. Figure 1 (b-c) shows the ratio between the tunnelling injection and the total current as a function of the total drain current lo and the conduction band properly weighted by the carrier concentration along the x distance along the channel. The ratio (Figure 1 (b)) proves that the tunnelling current is more important for the larger values of qOsn as a consequence of the narrower tunnelling path (that can be observed in Figure 1 (c)).

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In Figure 2 we present the spectral densities of gate (Sia) and drain (SID) current fluctuations as a function of the drain current. We observe that Sia increases almost linearly with the drain current and it is not strongly influenced by the barrier height. On the contrary, Sm gets significantly decreased for high current levels when the barrier height is larger. We also have plotted in the graph the value of the ideal long channel Van der Ziel's model for a MOSFET under saturation conditions [7], which shows much reduced values as compared to the MC results.

The MC simulator provides many internal quantities that can be related to the behaviour of the intrinsic noise sources. As the barrier height decreases, for the same current level there is an important rise of the electron energy and density of scatterings

331

close to the drain (see Fig. 3), where the band bending is noticeable (Fig. 1 (c)). Consequently, there is a significant velocity overshoot and even some impact ionization events take place for hot carriers approaching the forward biased drain contact. It must be taken into account that the principles of operation of the accumulation SB-MOSFET are completely different than that of a conventional MOSFET. In this way, the behaviour of the carriers in the last few nanometres of the channel seems to have an important influence on the final SJD. On the other hand, under the gate (Sio) there is a minor effect of varying the barrier height.

It can be concluded that although lower barrier heights are needed in order to obtain a better current drive, we also obtain a larger intrinsic noise. This fact has to be taken into account in the device design, assessing the requirements of the transistor for each specific application. However, the final effect on the noise circuital parameters (NFmi„, Gass) could be tempered by an enhancement in the dynamic performance when q<pBn is reduced, since a lower barrier height is expected to provide an improvement of ft and gm [8].

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ACKNOWLEDGMENTS

This work was supported by research projects MET AMOS (IST-016677) financed by the European Commission and SA010A07 from the Junta de Castilla y Leon and TEC2008-02266/TEC from the Ministerio de Ciencia e Innovacion in Spain. We would like to thank also R. Valentin from the lEMN, in France.

REFERENCES

The International Technology Roadmap for Semiconductors, 2007 Available: http://public.itrs.net. J. M. Larson, J. P. Snydsr, IEEE Trans Electron Devices, 53, 1048-1058, (2006). E. Pascual, R. Rengel and M. J. Martin, Spanish Conference on Electron Devices 2009, IEEE Conference Proceedings, in press (2009). Jacoboni C and Lugli P, "The Monte Carlo method for Semiconductor Device Simulation", McGraw-Hill, (1989). D. ConnsWyetal, IEEE Trans. On Nanotechnology, 3, 98-104, (2004). E. Pascual, R. Rengel, M. J. Martin, Semicond Set Technol, 22, 1003-1009, 2007 A. Van der Ziel, "Noise in Solid State Devices and Circuits", Wiley. (1986). R. Valentin et al, IEEE Trans Electron Devices, 55, 1192-1202, (2008).

332


Low-Frequency Noise Measurements of the Tunneling Current in Single Barrier

GaAs/AlAs/GaAs Devices

Jacek Przybytek and Michai Baj

Institute of Experimental Physics, Faculty of Physics, University of Warsaw,

Hoza 69, 00-681 Warsaw, Poland

Abstract. The experimental results of the low temperature (T = 4.2 K) low-frequency current fluctuations measurements in the single-barrier resonant tunneling GaAs/AlAs/GaAs vertical devices with Si 5-doping in the center of the 10-nm thick AlAs barrier are reported. The dimensions of the device were 200 |lm by 200 |lm. For the small bias voltages (low transmission of the barrier) there is only the shot noise with Fano factor F close to 1 observed. For higher voltages the generation-recombination-like and/or llf' noise arises and superimposes on the shot noise.

Keywords: noise processes and phenomena, tunneling, shot noise, l//noise, generation-recombination noise PACS: 72.70.+m, 73.50.Td, 73.40.Gk

INTRODUCTION

Measurements of current fluctuations in resonant tunneling devices are complementary for the averaged current-voltage characteristics and can provide more insight into the system and its electronic transport mechanisms. Time dependent fluctuations of a tunneling current reflects the temporal correlations between charge transfer events through a conductor. The deviations from the classical Poissonian full shot noise power density 2el can provide additional information about interactions between electrons inside the tunneling barrier and the mechanism of the transport (e.g. existence and number of localized states which participate in transport) [1,2]. The single-barrier resonant-tunneling device is interesting for its simplicity and many authors have investigated electronic transport in this system until now [3,4]. Incorporating impurities inside the barriers changes dramatically their properties, enables resonant transport through the barrier and makes it useful for applications. However, not only intentionally introduced impurities exist inside the barrier. Some impurities/imperfections which only shghtly influence the I{V) characteristics can manifest its existence in current noise measurements. In this paper we present the continuation of the investigations reported previously [5].



333

SAMPLE & EXPERIMENT

Single-barrier resonant tunneling GaAs/AlAs/GaAs structure with Si 5-doping (3*10^ cm" ) in the center of the 10 nm thick AlAs barrier has been grown by MBE-technique. The dimensions of the mesa structure was ca 200 [im by 200 [im. More details about the sample and its tunneling characteristics can be found in Ref 6.

The current noise measurements have been performed by means of crosscorrelation technique [7]. In this technique the signal from the sample is provided by means of two independent signal-carrying and amplifying channels to the analog-digital converter. The power signal density of the sample-related correlated signal has been numericaly calculated as a crosscorrelation spectra from two channels. The uncorrelated noise of amplifiers will not appear in crosscorrelation spectra.

The sample has been placed inside the liquid helium container. The current signal has been provided by means of low-noise coaxial cables to the inputs of two homemade, battery-driven transimpedance amplifiers with gain ranging from 10 to lO' V/A and it has been additionally amplified by two pairs of voltage amplifiers with total gain up to 10"* VA . Two low-pass Butterworth filters prevent an ahasing effect. The PC-driven AID converter and Matlab^^ software enables the numerical analysis of the signal. Sample was polarized by means of low-noise home-made bipolar voltage source applying bias between virtual grounds of the transimpedance amplifiers.


The I{V) characteristics of the tunneling device (see Fig. la) has been measured for both biasing directions in the range |f/|<l.lV at temperature T=4.2 K. Minus-sign in the biasing voltage means the current flowing through the sample from the substrate to metahc contact on the top of the mesa structure. The most important feature of this characteristics is the bump in the region 0.45<|f/|<0.75V where the tunnehng current is mediated by sihcon impurites in the center of the barrier [6]. More features can be seen in differential conductance characteristics (Fig. la) calculated numerically from I{V) characteristics. For |f/|<0.3V one can see several small resonances of the tunnehng current - these structures are reproducible, so they do not result from current fluctuations. These resonances are probably due to impurities which are placed far away from the center of the barrier opening additional channels for transport through the barrier. Current noise measurements has been performed for several bias voltages |f/|<l.lV. Typical results for power spectral density of the current fluctuations have been shown in Figure 2. All the spectra shown have higher frequency tail influenced by low-pass filter of feedback loop of transimpedance amplifiers. Especially for small currents, where higher gains of the amplifiers are necessary, it limits the frequency band of the measurement below lOOHz (for 10 and lO' V/A). The observed spectra for small currents are almost flat as expected for white shot noise (Figure 2). Increasing bias we see additional noise which superimposes on the flat shot noise. This part of the noise can have several Lorenzian components of generation-recombination noise and/or various l/f"' character with 0.2<a<l. We explain the variety of the noise spectra by

334

different transport mechanisms at different biases. Applying the bias we change the Fermi level position and its alignment to various electronic states inside the barrier.

1 0 " ^ (b)

1.0

0.9

o o 0.8

0.6

0.5

[•

1

^ — ' — ' — f ^

^.i.SM ^ ^•-•T- :

1 1 1 1

^ — ' — ' — f ^

• M M

• M

;•••: : ••

;•-»..-!•-:

: i..I..*.J*l.»...

-

-

-

-

-

-0.5 0.0 0.5 Bias voltage (V)

-1.0 -0.5 0.0 0.5 1.0 Bias voltage (V)

FIGURE 1. (a) I(V) characteristics for the sample at 7=4.2K and differential conductance dl/dU numerically calculated from I(V) characteristics; in the regions from 0.5V<|^<0.8V one can see two bumps originating from the tunneling process with the participation of the Si impurities in the barrier; (b) Fano factor determined for several bias voltages.

(a) Bias: -I.IOV -l.OOV -0.90V -0.80V -0.70V -0.60V -0.50V -0.40V -0.30V -0.20V -0,10V -0.05V

10 100 1000 Frequency (Hz)

10000 10 100 1000 Frequency (Hz)

FIGURE 2. PSD (crosscorrelation) spectra for the sample at 7=4.2K and several biasing voltages indicated on the right hand side. For lower biases the spectra are almost flat (white noise) and starting from ca |^~0.3V they reveal generation-recombination and l/f^ components.

The value of the shot noise and the Fano factor (the ratio of the measured shot noise to the full Poissonian 2el noise) has been determined from fitting the amplifier characteristics to the high frequency tail of the measured spectra, assuming that the shot noise is a white noise with PSD independent of frequency. The Fano factor shown in Fig. 2b has values between 0.55 for higher biases and 1 for lower. Differently from other results for the sample from the same wafer [5], the Fano factor close to F=0.75, expected for tunneling through strongly localized states randomly distributed in the center of the barrier [2] is observed for biasing voltages -0.7V<t/<-0.1V and 0.1V<t/<0.4V. Certain asymmetry of the Fano factor is probably related to the nonintentional, determined by growth technology, non-central position of the Si 6-layer in the barrier.

5. CONCLUSIONS

For biases |t^<500mV smaller than those for which resonant tunneling through intentionally introduced impurities is observed, we have observed Fano factors lower

335

than 1, which indicates that even below this voltages we have a variety of different transport mechanisms. Only for the lowest biasing voltages (|f/|<0.1V) the Fano factors tend to/is equal F~l. For higher biasing voltages 500mV<| f/|<700mV, where in I{V) characteristics the resonant tunnehng through the impurity states is observed, we have measured the Fano factor close to 0.7 and 0.6 for negative and positive biases, respectively. Because low frequency noise originating from the trapping electrons on impurities/imperfections superimposes on the shot noise, the measurements of the pure shot noise is difficult in the configuration of experimental setup with transimpedance amplifier, which limits the band below lOOHz for the highest gains used. However, because in our experiment Fano factor F does not exceed one, it means that in our tunneling device we have multiple uncorrelated sequential and/or parallel transport channels, each of which is governed by a full Poissonian process and a resulting transport statistics of a whole system is exclusively sub-Poissonian.

ACKNOWLEDGMENTS

Authors are very much indebted to Dr Antonella Cavanna, Dr Ulf Gennser and Dr Giancarlo Faini from Laboratoire de Photonique et Nanostructures (Marcoussis, France) for the growth and fabrication of the samples.

Work has been partially supported by the European Community project No. MTKD-CT-2005-029671 and by the Polish Ministry of Science and Higher Education project No. N N202 192534.

REFERENCES

1. Ya.M. Blanter, M. Biittiker, Physics Reports 336, 1-166 (2000). 2. Y.V. Nazarov, J. J. R. Straben, Phys Rev. B 53, 15466 (1996). 3. G. lannaccone, M. Macucci, and B. Pellegrini, Phys Rev. B 55, 4539 (1997) 4. J. Davies, P. Hyldgaard, S. Hershfield, J. Wilkins, Phys Rev B 46, 9620 (1992) 5. J. Przybytek, M. Baj, ActaPhysicaPolonica 112, 221 (2007). 6. M. Gryglas, M. Baj, B. Chenaud, B. Jouault, A. Cavanna, and G. Faini, Phys Rev. B 69, 165302

(2004). 7. G. Ferrari, M. Sampietro, Rev. of Set Instr. 73, 2717 (2002).

336

Suppression of 1/f Noise in Accumulation Mode FD-SOI MOSFETs on Si(lOO) and (110)

Surfaces W. Cheng", C. Tye , P. Gaubert'', A. Teramoto'', S. Sugawa , T. Ohmi"'"

"^New Industry Creation Hatchery Center, Tohoku University ''Graduate School of Engineering, Tohoku University

"World Premier International Research Center, Tohoku University Aza-Aoba 6-6-10, Aramaki, Aoba-Ku, Sendai, 980-8579, Japan

Abstract. In this paper, a new approach to reduce the 1/f noise levels in the MOSFETs on varied silicon orientations, such as Si(lOO) and (110) surfaces, has been carried out. We focus on the Accumulation-mode (AM) FD-SOl device structure and demonstrate that the 1/f noise levels in this AM FD-SOl MOSFETs are obviously reduced on both the Si(lOO) and (110) surfaces.

Keywords: Accumulation-mode, SOI, 1/f noise, silicon orientation.

INTRODUCTION

The noise of CMOS is dominated by 1/f noise and increases continually while scaling the transistor size and limits all the electronic devices, especially analog and RF apphcations [1-2]. To reduce 1/f noise levels, the MOS transistor fabrication processes are extensively studied, especially the high quality gate oxide and the oxide/silicon interface [3-4]. On the other hand, SOI technology and Si(llO) devices are very useful for improving the analog and RF performance [5]. In previous research, we demonstrated the MOSFETs drain current can be obviously improved by optimizing the SOI device structure introducing Accumulation-mode (AM) SOI MOS transistors. However, 1/f noise characteristics are stiU not thoroughly investigated. In this study, a new approach has been carried out to improve the CMOS performance and suppress 1/f noise levels by optimizing the SOI device structure.

EXPERIMENTAL

In this paper, the novel accumulation mode (AM) FD-SOI MOSFETs shown in Fig. 1 are fabricated on Si(lOO) and (110) surface to investigate the noise characteristics and revel the mechanisms. The SOI layers impurity concentrations (Nsub) are adjusted from lO'* to 2xl0'^cm"l The thickness of SOI layers (Tsoi) is from 40 to 50 nm. To avoid the increases of 1/f noise induced by the surface roughness at the oxide/sihcon interface, the radical oxidation has been repeated 4 times to achieve flattened interfaces with the Ra of 0.08 nm for the SOI devices shown in Fig. 2 and maintained by 5-step room temperature cleaning [6]. Gate oxides are formed by high quality



337

microwave excited radical oxidation at 400°C [7]. B^ ions (1.0x10^^ cm" ) are implanted to gate Poly-Si layer (300 nm) for AM n- and inversion mode (IM) p-MOSFETs. As^ and BF^^ (1.5x10^^ cm" ) ions are implanted to Source/Drain region for n- and p-MOSFET, respectively. Each step of device fabrication and measurement was carried out in the Fluctuation Free Facility for New Information Industry at Tohoku University.

|n+

P+ Poly

n n+l |P^ Buried Oxide

Si Substrate

n+ Poly

P P1 Buried Oxide

Si Substrate

Fig. 1 Schematic of the accumulation mode (AM) FD-SOI n-MOSFET and p-MOSFET.

(a) SOI surface initial (b) SOI (100) surface after (c) SOI (IIO) surface after Ra=0.I3nm surface flattening process surface flattening process

repeatedly 4 times repeatedly 4 times Ra=0.07nm Ra=0.08nm

Fig. 2 AFM images of the flattened SOI (100) and (110) surfaces before gate oxidation. Ra is reduced to 0.07nm on (100) and 0.08nm on (110) surface.


At the 1/f noise measurement, the drain current noise measurements were carried out using a Vector Signal Analyzer (AGILENT 894lOA) connected to a low-noise preamplifier (Princeton Applied Research 5184) with contacts directly taken on wafer. Transistors were initially biased by a modular DC source (HP 4142B) in order to find the target bias point parameters. This DC source was then replaced by an Ultra-Low Noise DC Source (SHIBASOKU PA14A1) for the final noise measurement. All transistors had low frequency 1/f noise in the measurement range from 10 to 10,000Hz. We systematically investigated the 1/f noise characteristics in both the AM and IM MOSFETs on Si(lOO) and (110) surfaces and demonstrate that AM SOI device structure is very useful to effectively reduce 1/f noise in MOS devices. Reduction of 1/f noise has been exhibited in all the MOS transistors operated at either linear regime or saturation regime for both the Si(lOO) and (110) devices. Fig. 3 shows the measured power spectral density (PSD) of n-MOSFETs drain current on Si(lOO) as a function of the measurement frequency at linear operation region with the bias of Vg-Vth=2.5 V, VD=2 V. It is observed that the 1/f noise level in AM n-MOSFET operated at the linear region on Si(lOO) is reduced about 1 order of magnitude compared with that in IM n-MOSFET. Fig. 4 shows the measured PSD of p-MOSFETs drain current on Si(lOO) as a function of the measurement frequency at hnear operation region with the bias of Vg-Vth=-2.5 V, VD=-2 V. It is observed that the 1/f noise level in AM p-MOSFET operated at the hnear region on Si(lOO) is obviously reduced compared with that in IM FD-SOl p-MOSFET and bulk p-MOSFET. Fig. 5 exhibits the measured PSD of n-MOSFETs drain current on Si(lOO) as a function of the measurement frequency at saturation operation region with the bias of 1D=0.6 mA, VD=2 V. It is observed that the 1/f noise level in AM n-MOSFET with higher SOI layer impurity concentration operated at the saturation region on Si(lOO) is reduced about 1 order of magnitude compared with that in IM n-MOSFET. However, the reduction of the 1/f noise level in the AM n-MOSFET with lower impurity concentration is almost not

338

observed. Fig. 6 shows the measured PSD of n-MOSFETs drain current at 10 Hz on Si(lOO) as a function of the drain current from 0.1 to 1 mA at saturation operation region with the bias of VD=3 V. It is observed that the 1/f noise levels have been suppressed very well at a wide range in AM n-MOSFET operated at the saturation region on Si(lOO). Especially, the reduction of the 1/f noise levels turns much remarkable, what is very useful for analog circuits, in our experiment. Fig. 7 shows the measured PSD of n-MOSFETs gate voltage on Si(llO) as a function of the measurement frequency at linear operation region with the bias of ID=0.1 mA, VD=0.05 V. 1/f noise levels of MOSFETs on Si(llO) surface are higher compared with those on Si(lOO) and are difficult to be effectively reduced. However, the 1/f noise level in AM n-MOSFET operated at the linear region on Si(llO) is obviously reduced compared with that in IM FD-SOI n-MOSFET. Fig. 8 shows the measured PSD of n-MOSFETs drain current on Si(llO) as a function of the measurement frequency at saturation operation region with the bias of ID=1 mA, VD=2 V. Suppression of 1/f noise level in AM n-MOSFETs operated at saturation region on Si(llO) has been exhibited.

Si(100) FD-SOI n-MOSFETs Linear regime

UW=10/20nm TQj =7.5nm TgQ=50nm

V^-y =2.5V

accumulation mode """ ^ - ./'N

Frequency (Hz)

Fig. 3 1/f noise level in AM n-MOSFETs operated at linear region on Si(lOO) is reduced about 1 digit compared with that in IM n-MOSFET.

ID" Frequency (Hz)

Fig. 4 1/f noise level in AM p-MOSFETs at linear region on Si(lOO) is greatly reduced compared with that in IM SOI and bulk p-MOSFETs.

Sl(100) FD-SOI n-MOSFETs Saturation regime

10 10'

Frequency (Hz)

Fig. 5 1/f noise in AM n-MOSFETs at saturation region on Si(lOO) is reduced about 1 digit compared with that in IM n-MOSFFT

10' 10*

:

"

;

Si(100)

______ m—

o

1^1 OHz

FD-SOI n

O

— • -

o

-MOSFETs

' ' ' _ • - — • o1

o o

o L/W=10/20^im

V^=3V

- inversion mode | •

accumulation mode \

1

.("^) Fig. 6 Measured drain current noise at lOHz as the function of drain current from 0.1 to 1 mA. A suppressed 1/f noise level has been observed at AM devices at wide range.

339

Si(110) FD-SOI n-WlOSFETs

Inversion-mode

Si(110) FD-SOI n-WlOSFETs

Inversion-mode

100 1000 Frequency(Hz)

10000 1° 100 1000 Frequency (Hz)

Fig. 7 1/f noise level in AM n-MOSFETs Fig. 8 Suppression of 1/f noise level in AM n-operated at linear region on Si(l 10) is reduced MOSFETs operated at saturation region on about 1 digit compared with that in IM n- Si(l 10) has been exhibited. MOSFET.

CONCLUSION

Reduction of noise is achieved not only by optimized processes but also by optimized the device structure. In this study, we demonstrate that accumulation mode SOI device is a potential candidate for the advanced analog and RF circuits in the future.

ACKNOWLEDGMENTS

This work was partly supported by the project under Grant-in-Aid for Specially Promoted Research (project No. 18002004) and the project under Grant-in-Aid for Young Scientists (A) (project No. 19686019), supported by Japanese Ministry of Education, Culture, Sports, Science and Technology.

REFERENCES

1. 1. M. Stey aert et al., IEEE International Symp. On Circuits and Systems, 2(1993) 1447 2. Y. C. Tseng et al.JEEE Trans Electron Devices, 48, 1428 (2001). 3. P. Gaubert et al., IEEE Trans. Electron Devices, 53, 851 (2006). 4. R. Kuroda et al., European Solid-State Device Research Conference, 83-86, 2008 5. A. Teramoto et al., IEEE Trans. Electron Devices, 54, 1438 (2007) 6. T. Ohmi et al., Phys D: Appl Phys., 39, Rl (2006). 7. T. Ohmi, J. Electron-chem. Soc., 143,2957 (1996).

340

Noise and Interface Density of Traps in 4H-SiC MOSFETs

S. L. Rumyantsev''' , M. E. Levinshtein , P. A. Ivanov , M. S. Shur"", J. W. Palmour', A. K. Agarwaf, B. A. Hulf, S. H. Ryu '

"Rensselaer Polytechnic Institute, Troy NY 12180-3590, USA an Academy of Sciences, 26 P oliteklmicheskaya, 194021 TREE Inc., 4600 Silicon Dr, Durham NC 27703, USA

loffe Institute of Russian Academy of Sciences, 26 P oliteklmicheskaya, 194021 St. Petersburg, Russia

Abstract. The low frequency noise was studied in 4H-SiC Metal Oxide Semiconductor Field Effect Transistors in the frequency range from 1 Hz to 100 kHz. The trap density responsible for the noise extracted using the McWhorter model increases approaching the conduction band edge, where it reaches the values up to -10^" cm"'eV"'.

Keywords: 4H-SiC, silicon carbide, trap density, MOSFET, noise PACS: 85.30.-Z, 85.30.Tv, 72.70.+m

INTRODUCTION

The first SiC power DMOSFET was reported in 1997. [1] Even since, higher and higher blocking voltages were reported for this device reaching 10 kV [2]. Radio frequency 4H-SiC MOSFETs reached cutoff frequencies of 7 GHz and 12 GHz [3,4]. When devices are used as active elements oscillators or mixers, their low frequency noise is one of the major factors determining the phase noise characteristics. Low frequency noise measurements are also a tool to study impurities, defects, interface states in semiconductor structures, and to diagnose quality and rehability of semiconductor devices. Low frequency noise in Si MOSFETs has been studied extensively in hundreds of papers (see, for examples, [5,6] and references therein). However, the low frequency noise studies of SiC field effect transistors are limited to a very few pubhcations. In the present work, we present the results of the experimental study of the low frequency noise in vertical and lateral double implanted 4H-SiC MOSFETs. More detailed description of the results is presented in [7].


The device structures under study were 1.2 kV, 10 A 4H-SiC Vertical Double Implanted MOSFETs (VDMOSFETs) and Lateral DMOSFETs (LDMOSFETs). The VDMOSFETs consisted of multiple identical elementary cells connected in parallel (Figure 1) with the total active operation area. A, of 0.10 cm , and 4.4x10"'' cm^ (for the test VDMOSFET). The LDMOSFETs with the gate lengths of 10, 25, 50, and 100 |am, had the same effective p-well doping and oxide thickness.



341

10 |jm thick n Drift Layer Epitaxy, 6 x 10'^ CITT^

N" 4H-SiC substrate

FIGURE 1. Schematic cross-sectional view of 4H-SiC DMOSFET elementary cell. Gate

length Lg=0.5|xm.

The noise spectra were close to the 1//'dependence with Y= 0.9- 1 over the entire range of frequencies, / and of Fg and Fd values for all the structures: large and test DMOSFETs, and LDMOSFETs (Figure 2). At frequency/« 1 kHz, a weak bulge in this dependence was often noticeable.

10 10 Frequency f, Hz

FIGURE 2. Frequency dependencies of spectral noise density at different gate voltages (from Fg = 1.8 V (deep sub-threshold region) to Fg » Fih (strong inversion) for large and test VDMOSFETs. Vj = O.IV. Dashed lines show the 1/f slope [7].

The dependencies of noise on Id at constant drain voltage Vd (when Id was varied by Fg) had the form, which qualitatively differed from that for conventional silicon MOSFETs. Figure 3 shows the dependencies of the relative spectral noise density S/Id^ on the drain current (at the constant drain voltage) for the large VDMOSFET, test VDMOSFET and LDMOSFET. For Si MOSFETs, as well as for many other FETs, the noise Si/Id^ decreases in strong inversion ~l//d^. At low currents, below the threshold, this dependence usually tends to saturate. As seen in Fig. 3, the Sj/h^ dependence on/^ for the SiC MOSFETs is very different.

342

test VDMOSFET

v j \ ••

^ \ VDMOSFET

V

10"' 10° 10° 10" 10° 10" 10"'

E

•a

O test VDMOSFET

0.00 0.05 0.10 0.15

(E - E l , eV

FIGURE 3. Dependencies of noise spectral density on FIGURE 4. Traps density in oxide, A'tv, as a li (at Vi = 0.1 V ) . / = 10 Hz. Arrows show the current, function oi_Ec-Ep [7]. which corresponds to the threshold voltage VQ^ at given

Over the entire range of Id, from deep sub-threshold to strong inversion, Sj/Id decreases with the current increase approximately as /d'°^. Similar dependences are often observed in amorphous and polycrystalline TFTs [8]. These noise versus current dependences can be explained by the energy dependence of the density of the oxide trap states.

Figure 4 shows dependencies of trap density TVtv on energy (£'c-£'F)_for different MOSFETs extracted based on the McWhorter model. As seen, the characteristic value of Mv atEc-EF=0.l V is about 10^' cm"^ V \ which is of the same order of magnitude as for polycrystalhne TFTs and about two to three orders of magnitude higher than corresponding values in conventional Si MOSFETs.

In reference [7], we proposed a new method for extraction of interface trap density Dit{E) based on the analysis of sub-threshold hiVg) characteristic of MOSFET and using the comparison of measured Id{Vg) characteristic with "ideal" MOS charactreristics (with Dit=0) (see Fig. 5)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

(E,-Ep), eV

FIGURE 5. At (E) extracted from saturated sub-threshold current-voltage characteristic [7].

343

As expected, the interface density of states sharply increases closer to the conduction band edge, Ec. The value of Dit « 4x10 cm " eV' at EC-EF = 0.2 eV, where E^ is the Fermi level, agrees rather well with the typical values of D^ for 4H-SiC MOSFETs extracted by alternative techniques [9,10].

CONCLUSIONS

The low effective channel mobility in SiC MOSFETs (3-7 cm^ A^ s) can be explained by a high density of localized states near the conduction band in the thin ion implanted SiC layer. The energy dependence of trap density, TVtv, was estimated from the gate voltage dependencies of noise. Mv ranges between 10^' and 10 ° cm"^ V \ which is of the same order of magnitude as in polycrystalline TFTs and about two to three orders of magnitude higher than in Si MOSFETs.

ACKNOWLEDGMENTS

This work was supported by Cree, Inc. At the loffe Institute this work was supported by Russian Foundation for Basic Research Grant No. 08-02-00010_. The work in RPI was supported by the National Science Foundation under the auspices of the I/UCRC "Connection One."

REFERENCES

1. J.N. Shenoy, J. A. Cooper, and M. R. Melloch, IEEE El. Dev. Lett. 18, 93 (1997) 2. S.-H. Ryu, S. Krishnaswami, M. Das, J. Richmond, A. Agarwal, J.W. Palmour, and J. Scofield,

IEEE Electron Device Lett 25, 556 (2004). 3. D. Alok, E. Arnold, R. Egloff, J. Barone, J. Murphy, R. Conrad, and J. Burke, IEEE Electron Device

Lett 22, 511 (imi). 4. G. 1. Gudjonsson, F. AUerstam, E. O. Sveinbjornsson, H. Hjelmgren H, P. A. Nilsson, K. Andersson,

H. Zirath, T. Rodle, and R. Jos, IEEE Trans, on Electron Devices 54, 3138 (2007). 5. T. H. Morshed, S. P. Devireddy, Z. Celik-Butler, A. Shanware, K. Green, J.J. Chambers, M.R.

Visokay, and L. Colombo, Solid-State Electronics 52, 711 (2008). 6. M. Haartman and M. Ostling, Low-Frequency Noise in Advanced MOS Devices (Analog Circuits

and Signal Processing). Springer. 2007 7. S. L. Rumyantsev, M. S. Shur, M. E. Levinshtein, P. A. Ivanov, J. W. Palmour, M. K. Das, and B.

A. Hull, J. Appl Phys 104, 094505 (2008) 8. M. Shur, Physics of semiconductor devices. Prentice Hall, Englewood Cliffs, New Jersey, 1990. 9. J. A. Cooper Jr., M. R. Melloch, R. Singh, A. Agarwal, IEEE Trans on Electron Devices 49, 658

(2002). 10.T. Endo, E. Okuno, T. Sakakibara, S. Onda, Abstracts of Intern. Conf on Silicon Carbide and

Related Materials lCSCRM-2007, Otsu, Japan, 14-19 October 2007, We-P-50

344

Monte Carlo analysis of noise spectra in InAs channels

from diffusive to ballistic regime

G. Sabatini*, H. Marinchio*, L. Varani*, C. Palermo*, J.F. Millithaler''', L.Reggianil', H. Rodilla**, T. Gonzalez**, S. Perez** and J. Mateos**

*Institut d'Electronique clu Sucl (CNRS UMR 5214), Universite Montpellier II, Place Eugene Bataillon - 34095 Montpellier Ceclex 5 - France

^Dipartimento cli Ingegneria dell 'Innovazione and CNISM, Universita del Salento, Via Arnesano s/n - 73100 Lecce - Italy

**Departamento de Fisica Aplicada, Universidad de Salamanca, Pza. Merced s/n - 37008 Salamanca - Spain

Abstract. Modem technology allows the downsizing of electronic devices deeply into the nano-metric scale thus reducing the electrons transit time between contacts. By means of Monte Carlo simulations coupled with a two-dimensional Poisson solver, we have analysed the transition from diffusive to ballistic transport in InAs channels of differents lengths. The increased number of ballistic electrons associated with the shrinking of the device length is found to progressively modify the time dependence of the autocorrelation function of current fluctuations while significant modifications of plasma oscillations appear in the autocorrelation function of voltage fluctuations inside the structure.

Keywords: Monte Carlo, Ballistic transport, InAs PACS: 73.23.Ad,72.30.+q, 72.70.+m

INTRODUCTION

The increase of the operation speed and the associated downsizing of electronic device dimensions is often obtained using high-mobility materials such as InGaAs and, more recently, InAs, employed for the fabrication of novel devices where ballistic or near-ballistic transport is expected even at room temperature. For instance, recent studies showed that High Electron-Mobility Transistors (HEMT) with InGaAs channels can be used as emitters or detectors in the THz domain [1]. On the other hand, despite its small gap, InAs has attracted recent interest for advanced electronic applications because it presents a higher mobility and higher intervalley separation than InGaAs, thus implying that its transport properties should be even better [2].

By means of Monte Carlo (MC) simulations coupled with a two-dimensional Poisson solver, we can evaluate the possibility to use this semiconductor as the active zone of ultra fast devices exploiting ballistic transport. In particular, we have calculated the correlation functions of current and voltage fluctuations in several InAs channels and discussed their microscopic behavior, in diffusive and ballistic regimes.



345

MODEL DESCRIPTION

We consider a thin semiconductor slab, with thickness ff = 50 nm and length L which can be shorter (longer) than the carrier mean free path so that ballistic (diffusive) regimes can be analysed. The carrier concentration ND is equal to 10'^ cm^^. Electrons are injected in the channel at a constant rate from contacts treated as thermal reservoirs according to the model detailed in Ref [3], which is well adapted to the case of meso-scopic conductors. The Monte Carlo model here used to simulate transport in InAs at room temperature has been previously validated by comparison with experimental data [4]. For the conduction band we have used three non-parabolic spherical valleys. The scattering mechanisms which are included are the collisions with ionized impurities, the transitions due to absorption and emission of polar and nonpolar optical phonons, the collisions with acoustic (elastic) phonons, and the intervalley scatterings. Carrier-carrier interaction and impact ionization are neglected.

BALLISTIC TRANSPORT

To asses the ballistic or diffusive character of transport, we have performed simulations of InAs channels of different lengths. Fig. 1 reports on the left the fraction of ballistic electrons as a function of the channel length at equilibrium. While for a 1 ^m-length

1 ^ 1 1 1 ^

100 90 80 70 60 50 40 30 20 10 0

D 200 400 600 Length (nm)

800

CO

_C

to O

Cf)

1000

1000

100

10 r / ' ' L /

1 ' 0 0.5

y' ^ —

'*'' .^^ / ' /— 50 nm —

100 nm —-1000 nm

1 1.5 2 2.5 3 3.5 ' Voltage (V)

FIGURE 1. Fraction of ballistic electrons at equilibrium as a function of length (left) and number of scatterings (rigth) as a function of voltage for InAs channels of different lengths.

channel transport is basically diffusive, when reducing the length, the fraction of ballistic electrons strongly increases: for instance, about one half of electrons are ballistic for a 50 nm-channel. Moreover, due to the high doping level, most of these interactions are elastic collisions with ionized impurities. On the right panel of Fig. 1 we have reported the average number of scatterings As- a carrier undergoes inside channels of different lengths as a function of the applied voltage. At equilibrium, electrons crossing the shortest channels make few scatterings while in the long channel the average number of scatterings is around 200. These numbers increase with the applied voltage showing the activation of different collision processes with characteristic energy thresholds. To better understand the behavior of these channels, we have calculated and reported in Fig. 2 also the free flight time TF = TT/NS where TT is the transit-time spent by a carrier to

346

move from one contact to the other. At equilibrium Tp is slightly shorter than 0.1 ps, i.e, it approaches TT as it must be in ballistic conditions.

^ 0.04

1.5 2 2.5

Voltage (V)

FIGURE 2. Free flight time as a function of voltage for InAs channels of different lengths.

At increasing voltages, tp decreases due to the increasing efficiency of scattering processes. The strong decrease observed around 1 V, which is more evident in short channels, is associated with the onset of intervalley transfers.

CORRELATION FUNCTIONS

To analyse the noise properties of the simulated structures, we have calculated the autocorrelation functions of current (Q) and voltage (Cr) fluctuations at equilibrium, reported in Fig. 3. For the long channel, as expected, Ci{t) decays exponentially and the time behavior can be fitted with the analytical formula

cf'ff(o = C/(o) exp (1)

where the velocity relaxation time Ty = 0.15 ps coincides with the one estimated from the ohmic mobility calculated with the MC simulation. When the length of the channel becomes progressively smaller and comparable with the carrier mean free path, deviations from the standard exponential behavior are expected. In particular, for the classical ballistic regime, the MC results are in good agreement with the Stanton and Wilkins analytical formula [5]:

CfXO '-(T) 2t

Tt^/^Tj-1 —exp T)- (2)

Here, we have taken TJ = 0.1 ps from the MC results shown. The agreement between the numerical and analytical results is evident only in the initial time region, where the correlation function Cj{t) decreases for no more than one order of magnitude. The longtime tail of Ci{t) is not reproduced by the simulation because, even in the short structure, transport is still far from being fully ballistic, and moreover the accuracy of numerical calculations is not sufficient to provide an adequate resolution.

To complete the analysis we have calculated also the autocorrelation function of voltage fluctuations taken in the middle of the channel. The results reported in Fig.

347

10

o

O 0.1 r

0.01

^^X**"^* '-

Y"

100 1000

***<•

' •" . .

nm nm

-;.:.;..•....

• - . .

•-•.

- 1 1 1 1—

100 nm — 1000 nm —

0.05 0.1 0.15 0.2

Time (ps) 0.25 0.3 0.4 0.6

Time (ps)

FIGURE 3. Equilibrium correlation functions of current (left) and voltage (right) fluctuations for InAs channels of different lengths. On the left, dots correspond to eq. (1) and crosses to the eq. (2).

3 (right) shows damped oscillations which are associated with the presence of three-dimensional plasma osciUations at a frequency o f / = 15 THz, in fuU agreement with the analytical formula co=\/e^ND/meeo. For the short channel, where partially ballistic transport is present, no significant variation of the plasma frequency is observed.

CONCLUSIONS

We have presented a Monte Carlo investigation of transport and noise in InAs channels of different lengths in the range 50 - 1000 nm. The transition from diffusive to ballistic transport is monitored by a direct calculation of the number of scatterings inside the structures. However, in spite of the high-mobility typical of InAs, even for a 50-nm channel only 50% of electrons are ballistic at room temperature and for a carrier concentration of lO'^ cm^^. This transition is found to progressively modify the autocorrelation function of current fluctuations from the standard exponential behavior to a more complicated non-exponential decay in partial agreement with analytical results of classical ballistic transport. The autocorrelation function of voltage fluctuations in the center of the channel, apart from a different damping, has been found to be not significantly affected by the size-effect within the range of lengths here studied.

REFERENCES

1. J. Lusakowski, V. Knap, N. Dyakonova, L. Varani, J. Mateos, T. Gonzalez, Y. Roelens, S. BoUaert, A. Cappy, and K. Karpierz, J. Appl. Phys. 97, 064307 (2005).

2. T. Daoud, G. Boissier, J. Dvenson, G. Sabatini, L. Varani, A. Baranov, and R. Teissier, 20th International Conference on Indium Phosphide and Related Materials, 2008. IPRM 2008. pp. 1 ^ (2008).

3. T. Gonzalez, J. Mateos, D. Pardo, L. Varani, and L. Reggiani, Semicond. Sci. Technol. 13, 714 (1998). 4. J. Mateos, T. Gonzalez, D. Pardo, V. Hoel, and A. Cappy, IEEE Trans. Electron Devices 47, 250

(2000). 5. C. J. Stanton, and J. W. Wilkins, Physica B 134, 255 (1985).

348

1/f Noise in p-Channel Screen-Grid Field Effect Transistors (SGrFETs) as a Device Evaluation

Tool

K. Fobelets^, S. L. Rumyantsev'', P. W. Ding^, and J. E. Velazquez-Perez''

"^Department of Electrical and Electronic Engineering, Imperial College London, Exhibition

Road London SW7 2BT, U.K lojfe Institute of Russian Academy of Sciences, 194021 St-Petersburg, Russia

"^Departamento de Fisica Aplicada, Universidad de Salamanca, Edificio Trilingile, Pza de la Merced s/n, E-37008 Salamanca, Spain

Abstract. Screen-Grid Field Effect Transistors (SGrFETs) are midti-gate devices with a novel gate geometry consisting of oxide wrapped metal cylinders standing perpendicidar to the current flow between source and drain. TCAD simidations show robust downscaling performance and high functionality of single devices. Experimental residts are presented on fabricated p-type SGrFETs. The fabrication process has been characterized via FIB-SEM images and via the use of low frequency noise measurements. Low frequency noise measurements give an insight into the influence of the gate cyUnders inside the channel. It is shown that the flicker noise in these devices does not increase systematically with an increase in the number of gate cylinders. This residt strengthens the conjecture that SGrFETs are suitable for low power applications with midti-gate functionality.

Keywords: MugFET, flicker noise, SOL PACS: 72.70.+m; 73.40.Qv; 85.40.Qx; 85.30.Tv

SCREEN-GRID FIELD EFFECT TRANSISTOR

The Screen-Grid Field Effect Transistor (SGrFET) was first presented at the MIEL conference in 2006 [1]. Since then TCAD studies have been carried out to study the analogue and digital performance of these devices [2]. The schematic 3D structure of a two unit cell SGrFET with two rows of gate cylinders between source and drain, is given in Fig.l. The source and drain are at the left and right side of the channel region. Current is flowing parallel to the plane of the SOI wafer in the Si body and the electric field lines from the gate voltage are perpendicular to the direction of the current flow. Thus the width of the channel is controlled by the extension of the carrier depletion caused by the gate voltage at both sides of the channel. This operation is similar to a MESFET but with the gate insulated from the channel by a gate oxide, allowing both depletion and enhancement mode operation of the device. The function of the second row of gate cylinders, near the drain, is to electrostatically shield the source from the drain potential, reducing drain-induced barrier lowering (DIBL). Increasing the current can be done by a parallel connection of unit cells. It was shown by TCAD that this



349

geometry is promising for low power, high frequency operation [2], and for fast low power switching of logic circuits with a minimum amount of SGrFETs [3]. The given gate geometry is not the only one possible. In the experimental study, we have fabricated non-self-aligned SGrFETs with 1, 2 and 3 rows of gate cylinders between source and drain in order to study the influence of the number of gate rows on the device characteristics and on the low frequency noise.

FABRICATION AND OPERATION OF THE SGrFET

Thermal SiO

Si3N4

Buried Si02

D

Channel

All Steps were performed by optical lithography except the definition of the gate holes which were defined via e-beam writing. The final layer structure of the fabricated SGrFET from top to bottom is 50 nm deposited Si3N4, 10 nm thermally grown SiOi, 200 nm p-type Si body (channel region), 200 nm buried oxide (BOX) and Si substrate. The dimensions of the devices are: source-drain distance LsD = 2}im, diameter of the gate vacuoles Lo = 200nm, and distance between holes Lmter =100 nm. The gate oxide - this is the wall of the cylindrical vacuoles - is estimated at tox = 10 nm. It is grown at the same time as the 10 nm top oxide.

Note that the SGrFET with 3 rows of gate cylinders has a source-drain distance of LSD = 3}im, thus 1 micron larger than the other samples in this comparison. The fabricated SGrFETs are not self-aligned. The gate vacuoles are filled with Al and the Ohmic contacts are also Al. No effort was made into optimizing the fabrication steps of these devices. Although the 200 nm Si body is sufficiently thick to be equivalent to PD SOI, the threshold voltage is not determined by this thickness but is determined by the distance between the gate cylinders in 1 row (Lmter). The SGrFET does not suffer from floating body effects as the cylindrical gates control the channel over the whole height of the Si body. The deposited nitride layer was designed to be 200 nm thick in order to minimize any top surface gating. However during fabrication 150 nm were accidently removed, leaving only 50 Si3N4. As a consequence, some top-gating will occur. Fig. 2 gives the SEM pictures of a fabricated SGrFET. It was noted from the FIB-SEM that the gate vacuoles were not completely filled with metal, leading to only partial gate cylinder gate action. Notwithstanding this error in the SGrFET devices, they still outperform the behavior of the MOSFETs on SOI with the same geometry fabricated on the same die. The performance parameters extracted from the DC

Figure 1. Schematic configuration of a 2-row SGrFET. Top: 3D view,

bottom: channel region only.

350

current-voltage characteristics are given in Table 1. Although the devices are by no means optimal at the moment, there is a small but consistent trend of improvement in the DC performance parameters of the MOSFET with the introduction of gate vacuoles. These measurements also serve to prove that the SGrFET geometry can be fabricated with CMOS compatible process steps.

FIGURE 2. FIB-SEM of a fabricated SGrFET at different scales. Left: complete device, middle: gate area with two gate cylinder rows and right: SEM of FIB cross section through the gate vacuoles.

TABLE 1. Comparison between MOSFET and SGrFET with different source-drain distances (LSD). The SGrFETs have a gate hole diameter of Lo=200nm. VT: threshold voltage, S: sub-threshold slope,

and DIBL: drain induced barrier lowering. Gating cylinders None (MOSFET) 2 rows (SGrFET) L S D (|.tm)

V T ( V )

S (mV/dec) DIBL (mVA^)

3 -0.80 98 144

2 -0.73 162 285

3 -0.79 93 124

2 -0.75 117 206

LOW FREQUENCY NOISE OF THE SGrFET

SGrFETs are fabricated on SOI (in this case SIMOX wafers), as a consequence the low frequency noise can be influenced by the quality of both top and bottom oxides [4]. Additionally, the gate cylinders reside inside the charmel region. This increases the Si02 surface that accounts for traps causing 1/f noise in MOSFETs. However the SGrFET operates as a MESFET, making the carriers flow in the "bulk" of the charmel rather than along the Si02 interfaces. This creates the hope that the addition of the gate cylinders in the SGrFET might not deteriorate the noise behavior of the EFT.

The low-frequency noise was measured using a shielded probe station in a frequency range from 1 Hz to 50 kHz at 300 K with the FETs in common source mode at source-drain in the linear region of the output characteristics for all gate voltages, VGS- The voltage fluctuations Sv from the load resistor RL cormected in series with the drain were analyzed by a SR770 EFT Spectrum Analyzer. The spectral noise density of the short circuit drain current fluctuations. Si, was calculated using the

expression: S; =Sy[(R^+Rf;p}/(R^Rf;p}] , where RSD is the drain-to-source differential resistance. High levels of 1/f noise were measured in all devices, and no generation-recombination was observed, hi fig. 3 the normalized current noise spectral density at f=10Hz is given as a function of current drive for devices with a different number of gate rows. The amplitude of the flicker noise in all devices is rather high.

351

indicating a high density of interface traps at the SiOi-Si interfaces. No H2 annealing was performed to reduce the trap density. It is interesting to note that the noise level, although high, does not systematically increase with an increase of the number of holes in the channel of the FET. This might indicate that in these devices the noise is mainly caused by the top and bottom SiOi surfaces rather than the SiOi surfaces surrounding the gate vacuoles. This observation in noise shows that surface gating is

still important in these structures and that for complete SGrFET behavior the thickness of the top Si3N4 layer needs to be larger than 50 nm to avoid classical MOSFET gating action in these devices. It also indicates that introducing additional gate cylinders in the FET, does not deteriorate its noise behavior, thus making the SGrFET with its peculiar gate geometry applicable in low power applications.

i.E+00

i.E-01

i.E-02

- l.E-03

l.E-04

^ l.E-05

l.E-06

l.E-07

^*^

^ ^

l.E-08 -

l.E-07 l.E-06 l.E-05 l.E-04

1(A)

Figure 3. Normalized current spectral noise density as a iunction of current drive in the different SGrFETs and the

MOSFET. Crossed: MOSFET, blue: 1 row SGrFET, green: 2 row SGrFET and red: 3 row SGrFET.

ACKNOWLEDGMENTS

K.F. acknowledges financial support from EPSRC for device fabrication (by Mir Enterprises, UK), for FIB-SEM (LCN FIB access scheme) and travel grant.

REFERENCES

K. Fobelets, P.W. Ding, and J.E. Velazquez-Perez, 25th International Conference on Microelectronics, Serbia and Montenegro 14-17 May 2006 K. Fobelets, P.W. Ding, Y. Shadrokh. and J.E. Velazquez-Perez, International Journal of High Speed Electronics and Systems (IJHSES) 18(4), Chapter 1 (2008) Y. Shadrokh, K. Fobelets, and J.E. Velazquez-Perez, Semicond. Set Technol 23(9) 095006 (9pp) (2008) E. Simoen, A. Mercha, C. Claeys and N. Lukyanchikova, SoUd State Electron. 51 16-37 (2007).

352

The Low Frequency Noise of SiC MESFETs

Jacek A. Cichosz, Alicja Konczakowska

Gdansk University of Technology Faculty of Electronics, Telecommunication and Informatics

Department of Optoelectronics and Electronic Systems 11/12 G. Narutowicza str., 80-952 Gdansk, Poland

[email protected] I

Abstract: In the paper results of low frequency noise measurements were presented. The investigations were carried out on SiC MESFET transistors type CRF24010. The system for low frequency noise measurements was constructed, equivalent noise circuit for MESFET was elaborated. From the results of noise measurement it was found that the dominant noise is the flicker noise of the drain current.

Keywords: l//noise, SiC MESFETs

PACS: 07.50.Hp, 85.30.-z

INTRODUCTION

Silicon carbide (SiC) transistor technology is quickly gaining acceptance in both military and commercial application due to the bandwidth and power characteristics of production-level devices. The aim of this paper is to present low frequency noise properties of SiC MESFET type CRF24010 produced by CREE [1]. The authors plan to apply results of noise measurements to an reliability evaluation of SiC transistors.

Cree's CRF24010 is an unmatched SiC RF power Metal-Semiconductor Field-Effect Transistors. SiC has superior properties compared to silicon or gallium arsenide, including higher breakdown voltage, higher saturated electron drift velocity, and higher thermal conductivity. SiC MESFET offer greater efficiency, greater power density, and wider bandwidths compared to Si and GaAs transistors.

In literature, the authors did not find reports about noise properties of SiC transistors.

INVESTIGATIONS

The investigations were carried out on SiC transistors type CRF24010 (MESFET -CREE). At first we had bought 3 specimens (sample 1), which in the paper are numbered No. 1, No. 2 and No. 3 and next we have bought 4 specimens (sample 2), numbered No. 4, No. 5, No. 6, No. 7.

In Fig. 1 the static characteristics ID =f(lDs) and IG =f(UG) of transistor No. 7 are presented. For transistors from sample 1 the gate current IG can be accepted as equal to zero in the range of UG from 0 to 3 V, and for sample 2 from 0 to 10 V.



353

http://85.30.-z

0.080

0.060

0.040

0.020

0.000

;=-10 —4—Ugs=-9

• •" ' i •

X :z: ^ '

^ ^ ^

- • - • — i • •

13 . ^

: ^ o

CRF2410-7 lG=f(UG)

•

/

/ /

U D M

•

(a) (b)

FIGURE 1. Characteristics for transistor No. 7 (a) - ID =f(lDs), 0 ) - IG =f(UG)

In elaborated MESFET noise model at the first moment the all sources of noise (thermal noise of the drain, gate and source resistances, thermal noise generated in the channel, flicker noise of the drain current and shot noise of gate current) were taken into consideration. It was find that some of them can be neglected. The thermal resistance of gate and source are very small ones, also the thermal noise generated in channel. Summing up, we take into account the thermal noise of the drain resistance iff^, flicker noise of the drain ij^ and shot noise of gate current / ^ . The mean-square values of noise sources represented in the noise equivalent circuit (see Fig. 2) are:

•2 -2 ( • AkT

\ _2

-'Sm J

K J'*

f -nJ

in = i'kg = ^QIG

where: gm - transconductance, Kf- flicker noise coefficient.

The gate current in investigated transistors is equal to zero or on the order of |iA (see Fig. 1). It means that the intensities of noise source il is not important. On the

base of noise measurements the contribution of ;'„

equivalent source of transistor will be compared.

and i^^ sources in e^ noise

RESULTS OF NOISE MEASUREMENTS

The noise measurement system constructed specially for CRF24010 transistor is presented in Fig. 2. The power spectral density function of voltage noise Su (/) of CRF24010 transistors in the frequency range 10 Hz ^ 20 kHz was measured for all transistors and results are presented in Fig. 3, for both sample separately.

The inherent noise of MESFET is l/fnoise type. In Fig. 4 The results of noise measurements as a function of drain current are

presented.

354

100

220J

Shield'

FIGURE 2. System for low noise measurements of SiC MESFET type CRF24010 and noise equivalent circuit

0"

0"

0"

0"

Nos. 4.5.6.7 at ld=50mA

1 — 1 - 1 + i + i ^ - i - i - n - - m i - i — I H\-\ [ — i - i i - i t i ^ - | - | - n - - n i i - | - - t i - i i i 1 1 1 1 11 1 1 1 1 1 1 111 I I 1 1 111

1 — | - ^ T K I > ^ " ^ ^ f e i i H " " m i ~ i — ^ n n

= = =t = = i = ± i ± i = = = i = i = i=VVSll5k,= = = i = i = = t y i J = = = = = =t = =l==l=l=l=t= = = l = l = |=|=| :^?rf f iE= =|=|==tl=t|: |= = =

1 1- -1-1+ 1 ^ - 1- l-l-l- - 1 ^^%i l | i | teJ- 1 ^ I-I I-I [ ~ " 1 " TIT 1 ^ ~ r r i T T i r ~^©fe&>WLn'"r

1 — 1 _ 4-14-1 ^_i_i_i_i-4-ij '^^>fe^*flmBlli = = zt = = l = ± l ± h = d=l=l=lzl: i l : l t = = =l= i m i l t » ~ ^

r — i - T i T i ^ - r r i - r - r i i r i — r n i i [ " T T i T i ^ " r r i T T i i r r i n i T

10* l[Hzl

(a)

10 = = , = = =

Nos. 1.2.3 at ld=50mA

I ' z ' a ' ^ ^ & i ^ ^ i t = d z t tizt t t i t = zi z t tizt d ± = z:

z = zt = = = ± ± t = =i = i= i = i 3 * J t = =1 = t t izt t t ± = =:

I L_LI ^ _ L LI_L U IP^L _l _ L L I J U 1

I I I I I I 11 11111 I l W t t i ^ = = =f = = = TT I= = =l = P Pl=f I=fl1= = =1 = P pHl^igfflftjq

h h-h 1 ^-i-i-nhm i-i-i-H 'MfUli l

^=i===!i^=i=riri=^|j^ (b)

FIGURE 3. The results of noise measurements for SiC MESFET type CRF24010 (a) - N o . l , No. 2, No. 3 (sample 1), (b) - N o . 4. No. 5, No. 6, No. 7

s; 10 = = ='

= = =l = 1 = T I = T T I = = T = l = l = l = f n T = = l = = f = l = f I = i n P = = | - l - - | - | - i - i - | t - | - | - | - t l - l t | - - t - l i - | - l l - | l 1 -4 -4 -1 -4 -4 -1 4 - - l - l - l - t U 4 - 1 1-14- U 4-11 1 - 4 - 4 - 1 - 4 - 4-1 4 - - I - I - M I - I 4 - 1 1 -14 I-I 4-11

-~._^^^lld=50lnA I I I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 111 = =~^f^^^~^^Ti E E E E IE lEI 3 B E E E E 3 B 3 El B E E S

3|i||||i|n^:i:^^?nf^^ = = =I = 4 = T F T T I = = T = & ^ * f i 3 n ' " % X = 1= =t =1 =f PI 4=1 P = =

E E E I E E E E E E I E E E E I E I E I 3 B E E E I E 3 B E^Pf i f f iD f tS i

| - - | - - | - | - i - - H + - | - | - | - f H i - 1 l - l i - H - H I 1 1 1 1 1 II 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 I n 1 1 1 1 1 1 1 1 1 llm 10' f[Hz]

FIGURE 4. The results of noise measurements as a function of a drain current (U= 5 V) for transistor No. 7

355

From the results of noise measurements in the form Su (f = const) = / (Ijf) for / = 100 Hz the values of/? coefficient are calculated and they are presented in Table 1.

TABLE 1. Values of p coefficient

No. of transistor

^

1

0,56

2

0,68

3

0,37 sample 1

4

0,82

5

0,86

6

0,86

7

0,79 sample 2

From the Table 1 we see that the values of y for sample 2 are similar. In Fig. 5 the Noise Scattering Pattern (NSP) [2] of measured noise of transistor

No. 7 is presented.

Samples [1:100000]

FIGURE 5. The NSP for instantaneous values of low frequency noise of transistor No. 7

From the view of NSP one can conclude that the character of noise is Gaussian.

CONCLUSIONS

The character of measured low frequency noise of transistors is a Gaussian noise. The RTS noise was not be observed.

From the results of noise measurements one can conclude that the intensity of noise is proportional to drain current (Fig. 4). It means that the participation of the thermal noise of drain resistance in e^ noise source is not important, and the flicker noise of drain is important part of noise.

ACKNOWLEDGMENTS

This work was supported by Polish Ministry of Science and Higher Education -the project No. PBZ-MEiN-6/2/2006.

REFERENCES 1. Cree Incorporated, USA http://www.cree.com/. 2. A. Konczakowska, J. Cichosz, A. Szewczyk, A New Method for RTS Noise of Semiconductor

Devices Identification, IEEE Transactions on Instrumentation and Measurements, vol. 57, No. 6. June 2008, pp. 1199-1206.

356

http://www.cree.com/

The Methods for RTS Noise Identification

Alicja Konczakowska, Barbara Stawarz-Graczyk

Gdansk University of Technology Faculty of Electronics, Telecommunication and Informatics

Department of Optoelectronics and Electronic Systems 11/12 G. Narutowicza str., 80-952 Gdansk, Poland

[email protected] I

Abstract. In the paper authors present two methods, which allows to identify the RTS noise in noise signal of semiconductor devices. The first one was elaborated to identify the RTS noise and also to estimate the number of its levels. The second one can be used to estimate all of the parameters of Gaussian and non-Gaussian components in the noise signal in a frequency domain.

Keywords: RTS noise, 1/f noise, noise identification, spectra estimating, optocouplers. PACS: 07.50.Hp, 05.40.Ca

INTRODUCTION

The Random Telegraph Signal (RTS) noise observed in noise signal of semiconductor devices is an indicator of their quality. If the device under test generates the RTS noise it means that this device is a poor quality one. That is why it is so important to eliminate the devices that generate that kind of noise.

Below, authors present two methods, which allows to identify the RTS noise in noise signal. It was assumed that noise signal can consist of a Gaussian component (1/f" noise, which dominates in low frequency noise in noise signal) and a non-Gaussian component (RTS noise) [1,2]. All results presented in the paper were collected during the low frequency noise measurements of optocouplers CNY17.

THE AUTOMATIC METHOD FOR RTS NOISE IDENTIFICATION

The first method was elaborated for the time domain and allows to identify the RTS noise in noise signal and also to estimate the number of its' levels [3]. It was based on Noise Scattering Pattern (NSP) method presented in [4].

The elaborated algorithm was written as a Matlab script. It uses 10 data samples which is sufficient number of samples to observe RTS noise. The first procedure of algorithm (afterwards the data are read from the text file) is elaborating the NSP pattern from which the special matrix G is created. The values of matrix G are the number of points appearing in a particular part of NSP pattern (in this case the number of compartments is equal to 30). It is very important to type the proper size of the matrix G, because setting too big size can cause a non-homogeneous pattern and too



357

small size can cause the appearance of two centres in one compartment. The size of the matrix G was experimentally set to 30x30.

Below, in Fig. 1 the noise signal, its NSP pattern, fragment of matrix P and graphical interpretation of matrix G is presented.

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(c) (d) FIGURE 1. (a) - the exemplary noise signal, (b) - the NSP pattern, (c) - the fragment of

matrix G created from NSP pattern, (d) - the graphical interpretation of matrix G

The next step of algorithm is searching the number of local maxima of matrix G and therefore establishing if the device generates RTS noise. For this purpose the supplementary matrix P is created, which size was experimentally set to 5x5.

In the following steps the algorithm rewrites the values from matrix G to matrix P and in each step the algorithm checks if there is the local maximum in matrix P. If the following conditions are fulfilled: the value gi+2j+2 is the maximal value, and the matrix P does not include values equal to 0, the central value of matrix P is recognized as a local maximum. Afterwards the whole matrix G is searched, the number of identified maxima is summed up and interpreted as follows: value 1 means, that in the matrix G one local maximum was found - the noise of device consists only of a Gaussian component, value 4 means, that in the matrix G four local maxima were found - the noise of device consists of a Gaussian and a non-Gaussian components (two-level RTS noise), value 9 means, that in the matrix G nine local maxima were found - the inherent noise of device consists of a Gaussian and a non-Gaussian components (three-level RTS noise).

Figure 2 shows the result for three optocouplers: whose inherent noise contains only a Gaussian noise, whose inherent noise contains two-level RTS noise, and three-level RTS noise.

358

(b)

^jJ^^^

FIGURE 2. The NSP patterns and the graphical interpretations of matrix G for noise signal with: (a) - Gaussian component, (b) - Gaussian and non-Gaussian components (two-level RTS noise),

(c) - Gaussian and non-Gaussian components (three-level RTS noise)

The presented method enables to identify also higher number of RTS noise levels than three.

THE IDENTIFICATION OF RTS NOISE PARAMETERS

The second method is based on composing estimators of two spectra, corresponding to 1/f" noise (Gaussian component) and two-level RTS noise (non-Gaussian component) and consists of two modules shortly described below [5].

The 1/r noise signal generator was designed in the Micro-Cap 9.0 program. It was elaborated as an enhanced noise macromodel of an optocoupler with the possibility of changing the value of exponent a, which was only available due to a MOSFET transistor (MTP15N06L) [6]. The most important information that the generator allows to gain are the values of following parameters: Kf - the 1/f noise level, a - the 1/f" spectrum slope.

The RTS noise signal is generated in a Lab View virtual instrument (vi) [5]. The program generates a two-level RTS noise signal with random length of impulses and random distance between them. The information available due to the generator are as follows: A - RTS noise signal amplitude, yj j s - RTS noise comer frequency, tup, tdown - the mean time the impulse remains in up and down state.

The process of composing the spectrum identical with the measured one is carried out in MS Excel. As the input data macro collects the measured spectrum and the files with Power Spectrum Densities of Gaussian component: \lf noise signal and non-Gaussian component: RTS noise signal. Afterwards, the proper graphs are elaborated. In case that the estimated spectrum is not identical with the measured one, the macro enables to correct and to set the appropriate level and slope of 1/f and the right level of the RTS noise signal.

In Fig. 3 results of two simulations are presented. The estimated parameters: Kf, a, A /RTS, tup and tdown are collected in Table 1.

359

S [nV/Hz"']

the measured spectrum

the composed spectrum

f[Hz] f[Hz]

(a) (b)

FIGURE 3. The results of simulation and measurement (a) — the Gaussian component, (b) - the Gaussian and non-Gaussian components

TABLE 1. The identification of component parameters

Device generating: Only Gaussian component Gaussian and non-Gaussian components

Gaussian component Kf [V^]

14.3-10-"'

e.i-io-"

a 1.0 1.1

Non-Gaussian component

^[Vl -

4.5-10-"

IRTS [HZ]

-250

tup [ms] -

1.27

tdom, [ms] -

1.27

The composed spectrum are identical with the measured ones. In both cases the parameters of Gaussian and non-Gaussian components were selected properly.

CONCLUSIONS

Presented in the paper automatic method for RTS noise identification allows for fast and easy recognition of non-Gaussian component in noise signal. The advantage of this method is that the user gets the results in a few seconds and without the visual observation of the noise or NSP patterns. The results of spectra composing presented on examples allow to assume that the method of component identification works properly and very precisely. The advantage of this method is also the possibility of identification of all component parameters such as Kf, a, A,fRTs, Up and tdom-

REFERENCES

A. Konczakowska at all, IEEE Transactions on Instrumentation and Measurements 57 (6), pp. 1199-1206(2008). A. Konczakowska at all. Fluctuation 'Noise Leters 6, (4), pp. L395-L401 (2006). B. Stawarz and B. KarczewsM, Zeszyty Naukowe Wydzialu ETI Politechniki Gdanskiej, 5, Seria: Technologie InformacYJne, Gdansk, tom 14, 05.2007, pp. 649-655, (in Polish). J. Cichosz and A. Szatkowski, Noise scattering patterns method for recognition of RTS noise in semiconductor components. Proc. of the 18th Intern. Conf Noise and Fluctuations ICNF'2005. Salamanca, Spain, 19-23 September 2005, pp. 673-676. B. Stawarz-Graczyk at all, Opto-Electronics Review 17 (2), 2009. B. Stawarz-Graczyk at all. Bulletin of the Polish Academy of Sciences 56 (1), pp. 59-63 (2008).

360

Modification of A. Van der Ziel relation for natural noise in diodes with non-ideality factor

of I-V characteristic tj >1

Alexey V. Klyuev, Arkady V. Yakimov and Evgeny I. Shmelev

Lobachevsky State University, Gagarin Avenue 23, Nizhniy Novgorod 603950, Russia Fax: +7-831-4656416; E-mail: [email protected]

Abstract. The spectram of natural noise injunction of the diode with non-ideahty factor of the I-V characteristic greater than one, ?/ >1, is investigated. It is shown, that the model by A. Van der Ziel is inapplicable for ?; >1 in the region of small currents (except the "ideal" case ri =1). At small currents the spectrum of noise is described by the Nyquist relation. To solve this problem we accept that the I-V characteristic is described by a few "ideal" junctions connected in serial; in case of ?; = « = 1, 2, 3,... these may be n identical junctions. The total current noise includes noise sources of all junctions. The presented result is the modification of A. Van der Ziel relation for ?; > 1.

Keywords: Natural Noise, Diodes, Non-ideality Factor, Nyquist relation. Model by A. Van der Ziel. PACS: 72.70.+m; 85.35.Be.

INTRODUCTION

For the description of spectrum of natural noise in p-n junction the model by A. Van der Ziel [1] is used. At small currents A. Van der Ziel relation is transformed in the Nyquist relation [2]. It is correct only for the diode with non-ideality factor of the I-V characteristic ri=\.

In case of the diode with non-ideality factor of the I-V characteristic greater than one, ?/ >I, the model by A. Van der Ziel is inapphcable in the region of small currents.

To solve this problem we present result of modification of A. Van der Ziel relation foxr}> 1.

NATURAL NOISE IN THE REGION OF SMALL CURRENTS

The spectrum of natural noise in junction of the diode with non-ideality factor of the I-V characteristic greater than one, rj >I, is investigated. Rather low frequencies, at which inertial properties of the junction are not essential, are considered. The I-V characteristic of the junction is as follows:

/= / , .{exp(F/ i )F , ) - l} . (1)



361


Here h is saturation current; V - supphed voltage; VT = kT/q - the thermal potential determined by Boltzmann constant k, absolute temperature T, and elementary charge

Differential resistance of the junction is

(2)

Initial (at / =0) resistance is equal to

(3)

To analyze the noise, we originally used the model by A. Van der Ziel (for the diode with "ideaf />-« junction, rj =1), see Fig. 1.

lit)

»

W D y^ I I

FIGURE 1. Equivalent circuit of the "ideal" junction.

Current noise i{t) of the junction has the spectrum

(4)

This approach yields the following relation for the spectrum Sv of the voltage noise on the junction:

S^--2q-[l^2I^)Rl. (5)

In the region of small currents, / « h, the presented result is transformed to spectrum 5*170:

S,. AkTijRj- (6)

This one contradicts the Nyquist relation (except the "ideal" case rj =1). Thus, the model by A. Van der Ziel is not apphcable when rj > I because

pecuharities of the current transfer are not taken into account.

362

MODIFICATION OF A. VAN DER ZIEL RELATION

To solve this problem we consider, at the first, the noise of the current caused by recombination of carriers in the junction depleted region, ri = 2. The I-V characteristic of the junction is modeled by two "ideal" junctions connected in serial (Fig. 2), which have identical saturation currents, hi = hi = h-

a • 3

FIGURE 2. Equivalent circuit of the junction with ?7=2.

Differential resistance RD is the sum of identical, RDI = Rm, differential resistances of separate junctions:

RD - 2-Rfli (TTT (7)

Initial (at7=0) resistance is equal to RDO = I-Vjlh-Noise sources of separate junctions are not correlated. That yields the spectrum of

total voltage noise:

S^= 2\2q-[I^ 21JR (2q/2]-(U2lM (8)

In the region of small currents, / « h, this relation is transformed to the Nyquist relation: Svo = ^kTRm.

Generalizing the obtained result, we accept that the I-V characteristic (I) is described by a few "ideal" junctions connected in serial; in case of ?/ = « = 1, 2, 3,... these may be n identical junctions. The total current noise i{f) includes noise sources of all junctions. As far as these sources are not correlated, we obtain the following relation for spectrum S of the total noise current:

^,= (29/))).(/+2/J. (9)

The presented result is the modification of relation (4) for;/ > 1. In Fig. 3 the voltage noise spectrum of Schottky diode with 5-doping at zero

current, as an illustration, is presented.

363

10 i jg

10-'

10-'

10-'

10-'

Sv, V'/Hz

n I'^lm-M ^ r

5 H) 4kTRDo h^^ig^^p.

/kHz

-r 4

T "T" 12 20 16

FIGURE 3. Voltage noise spectrum of Schottky diode (triangles); full line - spectrum 5™, calculated fromNyquist relation; daggers - t he measured spectrum of thermal noise of resistor 40 kOhm.

The diode is made in the Group by V. Shashkin [3, 4] (Institute for Physics of Microstructures, Russian Academy of Sciences, Nizhniy Novgorod). Initial resistance of the diode is RDO = 40 kOhm; non-ideality factor;/ = 1.21. It is it seen from the figure that experimental data are in satisfactory agreement with the Nyquist relation.

CONCLUSION

In this work we present result of modification of A. Van der Ziel relation for the spectrum of natural noise in p-n junction with non-ideality factor of the I-V characteristic greater than one, ?/ > I.

ACKNOWLEDGMENTS

Presented researches were carried out in Nizhniy Novgorod State University in frames of the Priority National Project "Education". This investigation was supported by the state contract JSo6039r/8473 (PYSIC-08-3).

REFERENCES

1. A. Van der Ziel, Noise: Sources, Characterization, Measurements, Prentice Hall, Englewood, Cliffs, N. J., 1970.

2. H.Nyquist, Physical Review. 32, 110-113 (1928). 3. V.l. Shashkin, V.M. Danihsev, O.l. Khrykin, A.V. Murel, Yu.l. Chechenin, A.V. Shabanov. Proc.

Int. Semicon. Dev. Res. Symp. (ISDRS 1997), Charlottseville, USA, 1997, p. 147. 4. V. Shashkin, Yu. Chechenin, V. Danil.tsev, O. Khrykin, A. Maslovsky, A. Murel, V. Vaks. Proc

23rd Int. Conf Microelectronics (MIEL 2002), Nis, Yugoslavia, 2002, p. 335.

364

Investigation of Intensity and Phase Feedback Effects on the Relative Intensity Noise in Blue

Laser Diodes under High Frequency Modulation

Jong Chang Yi, Hyung Uk Cho and Jong Dae Kim

Department of Electronic Engineering, Hongik University, Sangsu 72-1, Mapo, Seoul, 121-791, Korea

Abstract. The quantum noises in typical Blu-ray laser diodes have been modeled after the quantum Langevin formalism into three level rate equations. The required material parameters have also been extracted by using the multiband Hamiltonian for strained wurtzite crystals. The results reveal the existence of an optimum current modulation level to minimize the relative intensity noise.

Keywords: Laser diode, quantum noise, GaN, multiple quantum well. PACS: 42.55.Px, 42.50.Lc, 71.20.Nr, 73.21.Fg

INTRODUCTION

Since the InAlGaN blue laser diodes (LDs) proved to be an essential building block for the optical data storage system such as Blu-ray, the blue LDs have gone through extensive efforts to meet and/or lead the requirement for faster data access time, which are often limited by the noises in the laser diode cavities [1]. In this paper, the noise dynamics, known as the relative intensity noise (RIN), are to be analyzed regarding the 405nm AlGaInN multiple quantum well laser diodes. The modeling of such noises has been developed based on the quantum Langevin formalism [2]. The required material parameters and device parameters have also been extracted by using the multiband Hamiltonian for strained wurtzite crystals [3]. The optimum operation modes for the laser diodes have been quantitatively assessed by utilizing the high frequency modulation schemes [4].

MODELLING OF REV

The quantum noise due to the external optical feedback from the optical disc surface into the laser cavity is modeled as shown in Fig. 1.

FIGURE 1. Schematics for LD under the external optical feedback.



367

http://42.55.Px

F and R shown in the figure denote the forward and reverse traveling optical fields, rgy^t, (pen, hxt the reflection coefficient, phase retardation, and the length of the external cavity, respectively. The band structure of the blue LD to be modeled is as shown in Fig. 2. This AlGaInN SCH MQW LD structure is divided into three states for stable and accurate analysis. Each of them is referred as Gateway (G) state. Quantum well (QW) state, and Separate confinement hetero-structure (SCH) state, respectively [5, 6].

Cladding SCH - „ , SCH Cladding QW state ^ ^ °

state state state state

Cladding SCH state state

^ Gateway

»i state

QW state SCH Cladding state state

1-SCH 1-OW LsCH

FIGURE 2. The band structure of the AlGaInN Blue MQW LD.

The injected electrons and holes to the semiconductor junctions were captured in quantum well after passing from the cladding state to the SCH state and the Gateway state. Most of the carriers captured in QW were emitted via light and a few carriers are spilled out of Quantum well by thermionic emission. Each state can be represented by the charges and its corresponding relaxation time constant. Therefore, the three-level rate equation of blue LD can be expressed as following: [5].

dN, No No . Nr,

dt q -r,

dNr nS

dt

dNQ

dt

Ns

^D

NG_

^C

No

^G

_N^

'ÊSC

No N NQ

^nG ^C ÊSC

N, vfi{n,)P

^na 1 + ^ . ^ + F,,

^P.ÔM^ Up^..,P,.F, dt

d(j)

dt

\ + e„P "nQ

( av, .G{n,)

V 1 + ^ . ^ -{co-Q)+x<l)jt+F^

(1)

(2)

(3)

(4)

(5)

368

/; is the injection current, and Ns, No, NQ, P and 0 are the carrier populations of SCH, G, and QW, and the cavity photon population, respectively. The -r„'sare the radiative and nonradiative time constants for spontaneous recombination processes in each of Gateway, QW, and SCH states. TD is transit time constant, computed from charge-control theory, which relates to carriers which travels the SCH by ambipolar diffusion [6]. Tcis the ambipolar capture time constant for carriers transferring from the gateway states to the QW bound states. TECS is the corresponding time constant for the inverse process of escape via thermionic emission. TG is the time constant associated with the lifetime of carriers in the gateway states. Tp is the photon lifetime, Sp the gain suppression coefficient applicable to the photon population, y5 the spontaneous emission factor, and G(«g) the carrier-density-dependent modal gain [3, 8]. Vg, a, n„, rig, x^ is the group velocity, linewidth enhancement factor, mode refractive index, the group refractive index, the external feedback coefficient [7, 8], respectively, and i^,'s are the Langevin noise factors [2].

REV CHARACTERISTICS

The relative intensity noise (RIN) of the laser diode [7] is defined as the ratio of the laser intensity noise dP{t) to the average laser power P(?), or

^^^^,_S,ico)_\dr{5Pit^r)5Pit)Y-p2 pi

In the time domain analysis, the Langevin noise functions were implemented by random number generators with proper amplitudes [2]. Sometimes the low-frequency RIN level exceeds industry tolerance, for example, around -125 dB/Hz for optical pickup application. One way to reduce the external feedback effect is to modulate the injection current to the laser cavity up to a certain frequency [1, 4]. Figure 3 shows different driving method of a laser diode with a high frequency current superimposition regarding the modulation depth. Figure 4 shows the low-frequency RIN levels versus modulation frequency for different modulation depth. One can see from this figure that the RIN improvement due to the high frequency modulation scheme is noticeable when the modulation depth is \ .^{jBias-hh) or \ 5{lBias-hh), whereas negligible when the modulation depth is 0.5{lBias-Ith) or LOilsias-Ith)- Here, we assumed the external feedback is 1 % and the external feedback length is 1 cm and IBias is the bias current for 5 mW output from the laser diode and Ith is LD's threshold current. In Fig.4, the dashed horizontal lines indicate the -125dB/Hz requirement and the solid horizontal lines the RIN level without high frequency modulation. One can see 20dB reduction in the RIN level at 4GHz for the modulation depth of \.5(lBias-Ith)- Hence one can conclude that there exists an optimum modulation depth to minimize the RIN level by using the high frequency modulation. The three-level rate equations along with the Langevin quantum noise coefficient obtained from band calculation proved efficient in optimizing the actual operating point such as the bias current and modulation frequency into the LDs for optical pickup under external optical feedback.

369

o

3

I

(a)

P o /

V Threshold//

Mas • Bias

/

'\ A /\ /\ \/ V 1/ v Current

^ ^ In,: Modulation Ith : Threshold^

Ibias: Bias

FIGURE 3. Modulation schemes of a LD with a high frequency current superimposition: (a) /™=0.5(/6te-/ft), (b) /™=1.0(/6te-/ft).

-80

-90

-100

-110

-120

-130

-140

(b)

'•^^0 1Q0D

\

kmaiskii* l„=l-5(Ibi„-l,K)-^1

2000 3000 4000

f [MHz] m

•

•

•

5000 eoc 1000 2000 3000 4000 5000 6000

FIGURE 4. The RIN spectra of the LD vs. the modulation frequency for Zext=lcm and_/^xt=l%: (a)4=0.5(4te-/ffi) & 1.0(4to-/ffi), (b)4=1.5(4to-/ffi) & 2.0(1 uas-Ith)-

ACKNOWLEDGMENTS

The authors wish to acknowledge the support of KOSEF grant funded by MOST under RO1-2007-000-20048-0.

REFERENCES

1. G. R. Ray, A. T. Ryan, and G. P. Agrawal, Optical Engineering, 32, 739-45 (1993). 2. W. W. Chow, S. W. Koch, and M. Sargent III, Semiconductor Laser Physics, Berlin:Springer-Verlag, 1994,

pp.337-364. 3. J. C. Yi andN. T>ag\i, IEEE J. Quantum Electronics, 31, 208-18 (1995). 4. H. Kondo and N. Mori, Proc ofSPIE, 6282, WPD 5 (2006). 5. D. McDonald and R.F.O'Dowd,/£•£•£•/. Quantum Electronics, 3,1, 1927-1934(1995). 6. R. Nagarajan, M. Ishikawa, T. Fukushima, R. S. Geels, and J. E. Bowers, IEEE J. Quantum Electronics, 28,

1990-2007 (1992). 7. G. P. Agrawal and N. K. Dutta, Long-wavelength semiconductor lasers, NY:Van Nostrand Reinhold,

1987,pp.250-286. 8. J. C. Yi and H. U. Cho, Phys. Stat. Sol. (c), 4, 1617-1620 (2007).

370

Equivalent Noise Input Photon of MWIR Hg Cd Te Avalanche Photodiodes used in a

focal plane array . B. Orsal", J.Foltz ", J.Kayaian J.Arnoult

[email protected] a Research team "Bruit Optoelectronique", Institut d'Electronique du Sud (lES), CNRS JJMR 52141

University/ Montpellier 2, CC 084, Place Eugene Bataillon, F-34095 Montpellier Cedex 05, France

b Laboratory Research & Development Soc Contralco Avenue Mas Faugeres BP 23 34150 Gignac, France

Abstract. Based on past experience with MWIR HgCdTe avalanche photodiode, the solution is a combined sensor where in the same focal plane array and optical system is used for passive and active operating modes. Noise equivalent photon counting can be very low if the dark and background current are very low.

Keywords: middle wave infrared, HgCdTe, excess noise factor, noise equivalent photon counting PACS: 85.60.-qor42.79.-e http: //www. aip. org/p acs/index.html

INTRODUCTION Thank to its tunable bandgap, Mercury Cadmium Telluride (HgCdTe) is

considered as the most important modern semiconductor alloy for infrared IR applications. In a previous work, Orsal' and Leveque ^ have shown that the electron ionization rate a (E) are much higher than hole ionization rate p (E) when the average composition x < 0.5. This material is particularly suitable for Median Wave Infrared Radiations MWIR^. When the average composition x = 0.3, this semiconductor material is adapted for radiation remote sensing (3 im < X < 4,2 im) in order to determine Noise Equivalent Photon counting: NEPh for various applications ^ ''.

EXPERIMENTAL SET UP The HgCdTe Avalanche photodiode (APD) used in this study comprises the

following layers: p+ contact, p wafer-absorber, junction p/n", n via contact . The device is a front-side, illuminated, cylindrical, n-on-p photodiode that is formed around a small via in the HgCdTe as shown in reference^. The via serves the interconnect between the n side and the input to the read out circuit of focal plane area (FPA) . Another interest: the via creates an nl n" region around. The result is an region that is doped in the low lO'"* cm" range by the residual donors^. The etch damage at the surface of the via results in a thin n+ surface layer. The results is an n /n" cylindrical junction^. The devices are illuminated through an optical window situated on the P wafer. With increased reverse bias, the depletion width continues to grow until the field is high enough to generate avalanche multiplication in p/n". As the bias is further increased, the n" region multiplication becomes completely depleted. Photocarriers are generated in the wafer p region around this depletion region, it is the



371


absorption region. Photogenerated minority electrons in the p region diffuse to the multiplication layer P/N . In this device (x=0.3) only electrons are multiplied and holes are drifted to p+ contact without multiplication. Based on past experienced with MWIR HgCdTe avalanche photodiode, the solution is a combined sensor where in the same focal plane array (FPA) and optical system is used for passive and active operating modes. The (FPA) is connected to the readout integrated circuit (ROIC) and the case of switches for diode selection^ Experimental set up at 77 K (figure 1 and 2) have been shown in a previous work .

Case of switches for dkxJe selection

Speclrvim analyser

Transim|>edance amplifier

FIGURE 1. Cryostat. FIGURE 2 Case of switches for diode selection

MULTIPLICATION NOISE

The APDs are typically designed to favor generation and multiplication of the preferred minority carrier in the perimeter neutral region. In our case, the electron as described in a previous work ' . In these experiments, we use low frequency shot noise measurements versus the electrical frequency range (IHz < f < IMHz). This range correspond to electrical bandwidth Af=1 MHz used in these infrared photodiodes. The interest of this work reported here is to show the noise experimental results figures 3 and 4: multiplied dark current lobs(V), multiplied background photocurrent Iph(V, X), dark, background photocurrent multiplication coefficient Mobs(V), Mph(KV) respectively, dark noise spectral density Siohs(V) versus dark current, background photocurrent noise spectral density Siph(V,X) versus photocurrent.

White noise spectral density for various currents can be expressed by eq 1 ' : ,iy) AVn) (1) 'Ph hkiV,^)

IpHpiV,,?^)

l E - 1 1 l E - 1 0

Dark Current l .n (A)

FIGURE 3. Dark multiplication Noise Spectral FIGURE 4. Photocurrent multiplication Noise density Siobs versus dark current lobs Spectral density Sph versus photocurrent Iph

372

We can extract from the experiments, excess noise coefficients under obscurity or illumination respectively Xp^s, Xph for three devices given in Table 1:

TABLE 1: Noise experimental results Device

1 2 3

Iobsp(Vp) pA

3,95+/-0,05 4,05+/-0,05 3,92+/-0,05

iphp(Vp, A ; TiA

0,557+/-0,005 0,417+7-0,005 0,317+/-0,005

Xobs

0,076+0,015 0,077+0,015 0,073+0,015

FobsiMots)

1,41 1,41 1,39

Xph

0,048+0,015 0,016+0,015 0,039+0,015

FphiMpu) 1,24 1,07 1,19

EXCESS NOISE FACTOR We can compute excess noise factor values given in table 1 and shown in

figures 5 and 6. The value of excess noise factor depends on the ratio of ionization coefficients of holes and electrons k=|3/a as shown in reference \ This very low noise gain property is consistent with pseudo deterministic ionization process investigated by Mclntyre in a paper and given by eq 2 ' .

Fp,(Mp,) = M

FobsiMobs)= ^c

(l+(l-^).(—f f Mph

V] ^M

(2)

:M.

The amount of the excess noise factor determines the gain value when the APD noise reaches the system noise and hence the noise equivalent power PEB/^f can be expressed as a function of NEPh and integration time Tmt. It is given by eq 3:

-| l l / 2

FEB,, = ^ M,,

Af

Af WP^^F^S

NEPh IK

T:„, A (3)

where S (v x) is the primary sensitivity of APDs, F gain unity Fill Factor and

Si^ ROIC white noise spectral density. 3 1 1 1 1 1 { 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

— Fth

s M I 1

° [ I f ] I F = M°'°"

III 1' 1 1 GainM

\ \ j F(

Wl ""' F(

M)th =

M)th =

M)th =

M ^

^ 1 n — f — —

» F(M)e

• F(M)e

—F(M)th

—F(M)th

—F(M)th

IB u

p Device 1

p Device 2

p Device 3

Device 1

Device 2

Device 3

T

FIGURE 5:Dark Excess Noise Factor versus Multiplication Coefficient: Gain Mobs

FIGURE 6:Background Excess Noise Factor versus Multiplication Coefficient: Gain Mpf,

NOISE EQUIVALENT PHOTON COUNTING The noise equivalent photon performance of a focal plane array, depends on

many factors including the background photon flux, the dark current, the gate time Xmt , as well as the ROIC noise spectral density. This circuit provides all bias and clock signals for the detector. To compute NEPh " in eq 4, we use the main parameters at 77K given in table 1: primary dark current lohsp(Vp) and primary background current

373

NEPh-%Pug

\ obsp obs) '•' ' php int 0;.T;,

?^M, ' (4)

ph

at the onset of the multiplication Iphp(Vp,X), dark and background excess noise factor Fobs(Mobs), FphfMpij) respectively, Fug the unity gain mode fill factor. We can express NEPh by using eq 4 and results of Table 1:

NEPh = — — (5)

TABLE 2. APDs Design Parameters Parameters Design Targets

Wavelengh 3 |im <X < 4,2 |im Multiplication Coefficient: Gain 50 <Mpi, < 500 Excess Noise Factor for Ip^ a 4x10-'" A Fpi,(Mpi) = l,l6 Primary dark current/„(,y =(Iob/MobJ 4 x 10"' A Primary background photocurrent 1 ; , = (Ipi/Mpf) Ipf,p = 4,2. 10"'" A Integration time Ti„t tint = 10 " s Quantum efficiency Vq"^ Unity gain Fill Factor (Af=1 Mhz) F„j, = 0,95 5,-, white noise spectral density of ROIC 1,6.10" A/Hz

Noise Equivalent Power (Mpt, =100) 4,7 pW Noise Equivalent Photon Counting {Mp,, =100) NEPh = 83 photons

Summary: The excess noise factor becomes nearly independent of gain, remaining low (1< F(M) < 1,4) even as gain is increased to high levels. This low excess noise factor value is the consequence of the unique band structure of the HgCdTe material''^ which results in a near unity excess noise factor for electron injection HgCdTe APDs at VVK"*. The primary dark current and particularly background photocurrent, excess noise are essential in determining performance given by NEPh = 83 photons for integration time equal to lO'^s.

CONCLUSION

The ultimate low level of NEPh = 14 photons can be obtained for the same time integration.lt shows these APDs are very suitable in order to detect very low level middle infrared RMS power signal (0,83 pW) if the dark current is lower than 4 pA and the background photocurrent detected in the range 3 im <X < 4,2 im can be neglected for several applications: spatial detection, bioluminescence, ethylometry.

REFERENCES

1 B.Orsal,R.Alabedra,G.Lecoy,Y Flachet 5 Intern Conf on Noise in Physical Systems pp 207-211-ISBN997150397-2

2 G.Leveque, M.Nassert, D.Bertho, B. Orsal, R.Alabedra Semicond.Sci and technol.,7,1317(1993) 3 J.P.Perez, R.Alabedra, B.Orsal IEEE,TED Vol 1 N°l 1 November 2002 4 I Baker,S.Duncan,J.Copely, Proc SPIE5406,133 (2004) M. P. Brown and K. Austin, The New

Physique, Publisher City: Publisher Name, 2005, pp. 25-30. 5 J.Beck, C.Man, M Kinch, and J.Campbell, "The Mercury HgCdTe Electron Avalanche Photodiode"

Journal of Electronic materials Vol 35, N°6, 2006 6 R. J. Mclntyre, IEEE Trans. Electron. Dev. 46, 1623 (1999)

374

http://integration.lt

Noise Sources of a-Sii.xGexOy Microbolometers And Their Reduction By Forming Gas

Passivation Mukti M. Rana" and Donald P. Butler''

'^Department of Electrical and Computer Engineering, University of South Alabama, Mobile, Alabama 36688, USA

''Department of Electrical Engineering, The University of Texas at Arlington, Box 19072, Arlington, Texas 76019, USA.

Abstract. Forming gas annealing was performed on Sii.xGe^Oy microbolometers in order to reduce the electrical 1//^noise. The passivation time and temperature were optimized to obtain least possible noise voltage power spectral density (PSD). To observe the effect of forming gas annealing at different intervals of time on the noise voltage PSD of the devices, the noise voltage PSD was measured before and at the end of each interval of passivation time by tracking individual devices. The microbolometers fabricated from Sii.xGe^Oy were placed inside a rapid thermal annealing chamber in the presence of forming gas at 250 °C with different intervals of time starting from 0.5 hour to 8 hours. The value of flicker noise coefficient .STj, or an/N from the Hooge model was used as a measure of electrical l//^noise. The l//^noise is believed to originate from the defects and dangling bonds in the sensing layer. The hydrogen passivation reduces the electrical l//^noise by repairing the dangling bonds and defects. Passivation done for excess time increased the electrical l//^noise of the device slightly, indicating that the dangling bonds contribute to trap states and scattering centers for the carriers.

Keywords: Bolometer, Noise, Forming gas passivation, Silicon germanium oxide. Annealing PACS: 85.60.Gz / 85.60.Bt / 85.40.Qx

INTRODUCTION

A microbolometer is an optical sensor whose resistance changes with the absorbed radiant energy. Noise plays a key role to a bolometer's performance. Six possible noise sources affecting bolometers are Johnson noise, shot noise, generation recombination (g-r) noise, flicker noise (1//), temperature fluctuation noise, and background radiation noise and limit the microbolometer's performance [1]. The dependency of microbolometer's figures of merits on noise sources are discussed elsewhere [2]. Among these noise sources, l//-noise plays a significant role in determining the microbolometer's performance [3].

Amorphous GCxSii-xOy, a compound of conventional semiconductors, is an attractive sensing material for microbolometer with high temperature coefficient of resistance and simple deposition technique, although the presence of high 1/f-noise was found to be an issue that hmits the microbolometer performance [4, 5].

In this work, the effect of post-deposition forming gas passivation on the performance of SixGci.xOy microbolometer is investigated.



375

http://85.60.Gz

EXPERIMENTAL

The fabrication of microbolometer pixels, the forming gas passivation, and current voltage (I-V) measurement techniques are presented elsewhere [2]. Figure 1 (a) and 1 (b) show the cross sectional schematic view and a SEM micrograph of the fabricated amorphous Sii-xGcxOy microbolometers respectively. To observe the effect of forming gas annealing at different intervals of time on the noise voltage power spectral density (PSD) of the devices, the noise PSD was measured before the passivation, at the end of each interval of passivation time, tracking individual devices. Also, the crystallinity and atomic composition of the thermometer were investigated before and after forming gas passivation. Four devices namely ssl3, sltl, srb4 and slbl having resistance of 15 kQ, 70 kQ, 65 kQ and 70 kQ respectively were tested for this purpose. Figure 2 shows the schematic of noise measurement setup where the value of metal film resistor Rl was varied to obtain the bias currents (7 ) of 0.85 [j,A, 0.6 [j,A, 0.3 [j,A, 0.1 [lA and 42 nA for the microbolometer.

The \/f-noise or flicker noise can be expressed by Hooge's formula

S,. KVi

f (1)

where,/' ' is the electrical frequency with y has the value near 1 for l//^noise and 2 for g-r noise or non-stationary l//noise, ^ i s the flicker noise coefficient and is equal to a/////where an is Hooge coefficient of llf-no\sQ and N is the number of fluctuators in the sample, V^. is the dc bias voltage in which p has the ideal value of 2, and S^ is the voltage noise PSD.

p-Si Substrate

NiCr S

SiN L J Al ^ pj 2737

Ni ^ SixGei-xOy |

(a) (b) FIGURE 1. Sii.xGexOy Microbolometer (a) Cross Sectional View (b) SEM Micrograph after

fabrication.

The value of y5 was determined by plotting LOGio(-S'vy) versus LOGio(//,j using the 1, 5, and 10 Hz frequency data for four bias currents and taking their average value. The value of y was determined from the slope of Sv versus/log-log plot at different bias currents in the frequency range of 1 to 10 Hz. From the equation of trend line the average value of y was determined for four bias currents. The value of Kf was determined by using Eq. 1 at each bias current and their average was taken to determine the final value of ^ as the measure of flicker noise.

376


No XRD sharp peaks corresponding to Ge, Si, or Si-Ge were observed before or after the passivation process, although the observed sharp peaks of Ge02 and Si02 after passivation indicated that the samples turned into a least partially crystalline from an initial amorphous state during the passivation at 250 °C.

The hnear current-voltage characteristics determined before and after passivation imphed that there was no potential barrier in the metal-semiconductor junctions, eliminating the possibility of shot noise [6] and no change in electrical resistance.

TABLE 1. Effect of Passivation Time on Noise Parameters For Various Devices Passivation

Time (Min.)

SRB4 P

SLTl P

SSL3 P

SLBl P

0 30 60 120 180 240 480

5.89x10-°" 2.53x10-°" 1.19x10-'° 1.50x10-°' 7.94x10-°'

5.54x10-" 9.51x10-"

1.18 1 2.14 1 1.44 1 1.65 1 2.00 1

1.42 1 1.75 1

06 13 19 03 01 10 01

1.61xl0-°«

1.47x10-°' 4.81x10-°'

1.10x10-°" 5.03x10-"

2.85x10-" 8.76x10-"

1.50 1.39 1.70 2.35 1.30

1.52 1.31

1.12 1.18 0.97 1.15 1.01

0.96 1.01

2.27x10-°"

1.18x10-°' 1.25x10-°'

1.02x10-°" 1.66x10-°'

1.90

2.64 1.97

1.71 1.41

1.15

1.33 1.19

1.08 1.01

7.54x10-°' 2.85x10-°' 1.04x10-°" 5.69x10-°' 1.89x10-"

8.84x10-'" 2.21x10-'°

1.26 2.17 1.78 1.89 0.90

1.22 1.78

1.09 1.25 1.11 0.96 1.20

0.94 1.01

The values of Kf, y, and p for different devices at different conditions are shown in TABLE 1. The values of ^/decreased with passivation time indicating the reduction of l//-noise. For the longest passivation time (480 minutes), the values of Kf were found to increase slightly with devices srb4, sltl, and slbl. For cases like device sltl, the value of Kfis increased from 4.81 x 10"' to 1.10 x 10" as the passivation time passed from 60 minutes to 120 minutes probably because of the increase of p. The average value of y for all the devices varied between 0.96 to 1.25, indicating classic frequency dependence for the l//-noise.

Figure 3 shows the noise voltage PSD for slbl at 0.3 |iA bias current at different passivation times. Noise sources in the bolometer are mainly due to the sensing layer of Sii-xGCxOy [2]. The value of Kf depends on the quality of the crystal, and on the scattering mechanisms that determine the mobility // [6]. l//-noise can be attributed to the trapping and detrapping of carriers with a distribution of time constants [7]. The mobility on the other hand depends on the defects in the materials. From Fig. 3, the higher slope of the noise curve before and after 0.5 hours and one, two, three hours passivation time have no flat portions. They are characterized by combination of 1//-noise and g-r noise where these two noise sources dominate over Johnson noise. Their presence is because of the fluctuation of electrons between the conduction band and the traps [6]. After 4 hours and 8 hours of passivation time, the measured noise spectra displayed Johnson noise above 100 Hz, although between 1-100 Hz frequency range the spectra are still dominated by 1/f-noise. The other three devices behaved similarly.

The bonding of Si with the oxygen can have one major type of charge trap-E'-type paramagnetic defect [8-10]. This type of defect center consists of silicon dangling bonds [11]. Similar dangling bond of Ge can exist for Ge-0 or Ge-Ge bonding [12].

It can be seen from Fig. 3 that as the passivation time progressed, the electrical noise decreased through 4 hours of passivation time, clearly indicating the passivation of the defects at the grain boundaries and thus reduced the recombination centers [13-15].

377

The increase in Kf from 4 hours to 8 hours may be because of defects caused by excess hydrogen, after passivating the dangling bonds [13-14].

The increase in values of ^/for devices like srb4, sltl as the passivation time passed from 60 minutes to 120 minutes needed to be investigated further.

" M "

5 \

i 1 1 1 z

io-«

in-Q 10

1 0 - "

1 0 - "

10-^^

1 0 - "

10-^"

1 0 - "

1 0 - "

Before Passivation t After 0.5 Hour I f l o f e L ^ / After 1 Hour

^ f ^ J f l p M l ^ After 2 Hours

• X^ ^^^^Mtnil^ '^"^'^ ^ Hours i|>t. / ^ f t e r 4 Hour;

^ ^ After 8 Hours " ^Wl l i

.

-•

. -

» . | I \ -

_

10 100 1000 10"

Frequency (Hz) FIGURE 2. Noise Voltage PSD of a Device at Different Annealing Conditions with 0.3 jiA Bias

Current

ACKNOWLEDGMENTS

This material is based in part upon work supported by the National Science Foundation under grant ECS-0322900.

REFERENCES

1. E. L. Dereniak and G. D. Boreman, Infrared Detectors and Systems, New York, John Wiley & Sons, 1996.

2. M. M. Rana and D. P. Butler, Thin Solid Films 514, 355-360 (2006). 3. S. Sedky, P. Fiorini, K. Baert and R. Mertens, IEEE Trans Electron Devices 46, 675-682 (1999). 4. A. Ahmed and R. N. Tait, InfraredPhys Teclmol 46, 468-472 (2005). 5. E. Iborra, M. Clement, L. V. Herrero and J. Sangrador, J. Microelectromech. Syst 11, 322-329

(2002). 6. F. N. Hooge, IEEE Trans. Electron Devices 41, 1926-1935 (1994). 7. A. \an der Zie\, Noise in Solid State Devices and Circuits,¥iew York, John Wiley & Sons, 1986. 8. S. P. Kama, H. A. Kurtz, W. M. Shedd, R. D. Pugh and B. B. K. Singaraju, IEEE Trans Nucl Sci.

46, 1544-1552(1999). 9. A. Stirling and A. Pasquarello, Phys Rev. B 66, 245201- 245211 (2002). 10.T. Uchino, M. Takahashi and T. Yoko, Phys Rev. B 64, 081310-081314 (2001). U.S. Mukhopadhyay, P. V. Sushko, A. M. Stoneham and A. L. Shluger, Phys Rev. B 71, 235204-

235209 (2005). 12.W. L. Warren, K. Simmons-Potter, B. G. Potter Jr. and J. A. Ruffner, Appl Phys Lett 69, 1453-

1455 (1996). 13.K. L. Brower, Phys Rev. B 38, 9657-9666 (1988). 14.K. L. Brower, Appl Phys Lett 53, 508-510 (1988). 15.D. M. Fleetwood, M. J. Johnson, T. L. Meisenheimer, P. S. Winokur, W. L.Warren and S. C.

W-Aczak, IEEE Trans Nucl Sci. 44, 1810-1817 (1997).

378

Optical noise of a 1550 nm fiber laser as an underwater acoustic sensor.

B. Orsal", K. Hey Tow ", R. Vacher , D. Dureisseix^ a Research team "Bruit Optoelectronique ", Institut d'Electronique du Sud (lES), CNRS UMR 5214 / University Montpellier 2, CC 084, Place Eugene Bataillon, F-34095 Montpellier Cedex 05, France

b Societe d'etudes, de recherche et de developpement industriel et commercial (SERDIC), 348 avenue du Vert-Bois, F-34090 Montpellier, France

c Research team "Systemes Multi-contacts ", Laboratoire de Mecanique et de Genie Civil (LMGC), CNRS UMR 5508 / University Montpellier 2, CC 048, Place Eugene Bataillon, F-34095 Montpellier Cedex 05, France

Abstract. The goal of this presentation is to provide first results concerning the optical

noise of a fiber laser used as an underwater acoustic sensor: hydrophone. The main sensor characteristics are: -1): A sensitivity allowing to detect all noise levels above background sea noise (the so-called deep-sea state 0). Among other applications, one may mention: seismic risk prevention, oil prospection, ship detection, etc. -2) : An optical noise reduced to its minimal value: it is the lower bound below which no acoustic pressure variation is detectable.

We therefore present here the first results for the expected sensitivity of the acousto-optic sensors, the frequency and amplitude of optical noises induced by the fiber laser and all the devices on the optical line. These results exemplify the possible detection of signal levels as low as the deep-sea state noise 0, especially for low frequency bandwidths, from several Hertz up to several kiloHertz.

Keywords: Distributed FeedBacIi Fiber Laser, Optical Noise, Underwater, Acoustic Sensor, Deep Sea State Zero. PACS: 85.60.-qor42.79.-e

INTRODUCTION

The detection of underwater signals and noises has a large number of applications in geological observations '• . It can be used for the observation of geologic movements, associated to seismic events. Underwater acoustic testing is of primary importance for off-shore oil prospection. It can also be used to detect the motion of ships. Piezoelectric systems have been used since several decades as hydrophones. Their main qualities for this apphcation are their small volume and their mechanical simplicity, leading to robust sensor elements. However, their sensitivity is limited, of the order of ImV/Pa, when the deep-sea state noise 0 is in the uPa range .

Sensors based on optical fibers have well known advantages over conventional electromechanical sensors. They offer electrically passive operation and immunity from electromagnetic fields because the fiber is entirely made with dielectric materials. The principle of the commercial passive fiber optic hydrophones is to measure the change of length of an optical fiber which is induced by a variation of hydrostatic pressure associated with the acoustic waves. To achieve the necessary



379

sensitivity, a fiber of a few hundred meters must be used. This fiber must be supported by a mandrel producing a large deformation with hydrostatic pressure. With such hydrophones, high sensitivity and wide dynamic measurement range can be achieved allowing the detection of the deep-sea state noise 0. They have multiplexing capabilities for a quasi-distributed measurement configuration by using a single optoelectronic control unit. Remote measurement is also possible. Indeed, the very low signal attenuation (0.2 dB/km from 1.48^m up to 1.55^m) makes it possible to place the optoelectronic control unit several kilometers away from the measurement point. However, the mechanical system allowing achieving the required sensitivity is complex, which reduces the durability and makes it difficult to use the hydrophone under elevated hydrostatic pressures in deep sub-marine investigations. This also leads to an elevated cost. More recently, laser fiber hydrophones have been proposed. They have very small dimensions (diameter of 125 ^m and length varying from 5 cm up to 10 cm for a standard bare fiber)^. One of the problems to be solved is to reach the necessary sensitivity for deep sea state noise 0 detection with simple mechanical systems. Moreover, the free running relative intensity and frequency optical noise of the laser can limit the performance of deep sea detector systems '

For the first time, the dependence on relative intensity and frequency optical noise detected of distributed feedback fiber laser (DFB FL) used as sensor versus acoustic frequency is shown. We also discuss the acousto-optic sensitivity of the sensor. We show that the performances that can be reached make fiber lasers suitable for acoustic detection of seismic risks, tsunami prevention and oil prospection with large deep-sea detectors.

DISTRIBUTED FEEDBACK FIBER LASER

The studied sensor we are concerned with is a fiber-laser based deep-sea hydrophone. Its basic principle lies in the measurement of a laser emission frequency from an erbium-doped optic fiber, with two imprinted mirrors acting as an optical cavity. The length variations of the fiber, proportional to the acoustic pressure levels, induce a change in frequency of the emitted light. To allow for an accurate detection of these length variations, the fiber laser is embedded into an acoustic/mechanical device whose aim is to amplify strains arising from acoustic waves. Such a device is shown in Figure 1.

^/vV'VM 10 nm pump light | ) < ( I

. soft material •. ', _9|^ '

land-plate \ fiber laser ; hard outer-cylinder

FIGURE 1(a) Acoustic amplification after FIGURE 1 (b) DFB FL sensor

Acousto-optic sensitivity S* The deformation of the DFB fiber laser is small when a bare fiber laser is

placed directly in water. Its sensitivity can be increased by using an acoustic amplification as shown in figure 1 . Typically we have calculated that for underwater surveillance applications, an amplification of the sensitivity of about 500 - 1000 times

380

is required to approach the deep sea state zero noise level in the sea. ' The sensitivity can be expressed for the general case of an acousto-optic amplification by:

5o. = ^ = 6 . ( 0 , 7 8 . ^ ) (1)

where A is the sensitive surface area, k is the fiber stiffness, XB is the wavelength and LpL is the DFB laser length equal to 5 cm, SAO is the sensitivity parameter of the overall device. Indeed, the sensitivity SAO depends both on the DFB grating distortion due to axial strain, and to the dependence of the optical properties of the material on the strain state, due to the elasto-optic couplings . The mechanical amplifier can be made at least with two technologies: (i) for medium-range depth (up to 500 m), a suited compliant mechanism with a large amplification capability, such as the one in Figure 1 can be used (several devices are mentioned in the literature ^'\ (ii) for large-range depth (up to 5000 m), a more robust design, though less sensitive must be constructed. The mechanical device that we have designed provides the necessary amplification^.

EXPERIMENTAL SETUP

The experimental apparatus is described in Figure 2. The excitation power is produced by a laser pump located on an emission/reception station on shore. A transmission fiber, whose length may be up to several dozens of kilometers, guides the pump power (with a 1480 nm wavelength) up to the fiber lasers acting as sensors. These fiber lasers emit a backtracking beam with a different wavelength. In order to increase the detection area, several sensors may be located on the same line, provided that they all respond with different wavelengths, close to 1550 nm. Once they reach the reception station, the different signals are first separated with a wavelength demultiplexer, and second, driven to a Mach-Zehnder interferometer (MZI) shifted by using a line of 300m optical fiber that detects both the acoustic signal frequency and the optical noise generated by the fiber laser. To improve the phase detection, the two separated beams in the interferometer are modulated with an oscillator which delivers a frequency of 100 MHz and we use a phase-meter and a fast Fourier transform (FFT) analyzer for low frequency analysis of the main signal and optical noise for homodyne detection. The oscillator is connected to both the acousto-optical modulator (AOM) and phase meter to fix the phase to that of the AOM in Figure 2^'^. The MZI converts the pressure-induced wave-length shift of the radiation emitted by the DFB fiber laser, into a phase delay which is a function of the FL output wavelength shift Ak and of the optical path difference OPD = w^ L, where L is the length unbalance between the two interferometer arms.

A0 = «, .(Afc).L = 2fi .AA (2) A

A<j) = <j)^ + 8<j), where <j)^ is the phase delay related to the pressure induced wave

length shift and 8<p is the noise component associated with the signal.

381

\ AOM j \

^—1 ^ \~^ "^

1 Os dilator

PHASE METER

X

Y

F[^ Analyser

Detection unit

FIGURE 2: Experimental Setup

The optical intensity detected by two photodiodes PIN at the input of the phase-meter is in the form eq 3:

/ =/„ (l+ycos(A<l))) (3)

w h e r e / o is the mean opticalintensity, y=^S2x sss-is the visibility. The in ter ferometer max mill

must be in quadrature (multiples of JT / 2 ) to provide linear responses; hence we must use a sinusoidal phase carrier signal to carry the phase delay created in the interferometer \

The sensitivity of the interferometer to the frequency modulation by the carrier is obtained by differentiating eq 2:

A < 1 ) = -'eff^ •sm(cot)- (4)

The expression gives the following by using the Bessel functions eq 5:

I(A(t>) = I„V. Jfj (a) + 2 > J21^ (a) ms{2b mt) cos(A</))+/„y

Y(A0)

2 y / 2 i + i (a)sin((2*:+ 1)0)0

~ X(A0)

sm{A(j)) (5)

The even harmonics of the carrier are all amplitude modulated by the cosine of the phase delay while the odd harmonics are amplitude modulated by the sine of phase delay. The phase meter gives Y(A0) and X(A0) as outputs. Both signals can be connected to two channels of a FFT analyser from which the phase delay can be extracted both in time and frequency domain in order to plot frequency noise 8<j) versus acoustic frequency/ at different hydrostatic pressures is given by eq 6 .

arc tan |^ ' " (^ ,^M = ^^ cos (S^) )

(6)

382

OPTICAL SENSOR NOISE SOURCES

A noise source refers to any effect that generates a random signal which is unrelated to the acoustic signal of interest and interferes with precise measurement. In the remote interrogated optical hydrophone sensors, there are several optical noise sources that contribute significantly to the total sensor noise^. The main are: i) laser intensity noise, ii) laser frequency noise. Other noise sources such as optical shot noise, obscurity current noise, oscillator phase noise and fiber thermal noise and input polarization noise are generally less significant and will be ignored .

DFB fiber laser frequency noise: A typical RMS frequency noise S(f,X) ( HZ/VHZ). is shown in figure 3 at 1552.06 nm. The frequency noise of the laser was measured using the experimental set up described in figure 2. The frequency noise of the DFB fiber laser was found to exhibit an f " relationship where y = 0.5 for frequencies up to 1 kHz. Between 3 kHz and 10 kHz, the frequency noise spectrum detected was flat at 7Hz/V Hz at 5 kHz. These results are comparable with those given by Cranch et al .

f (0

1,0E+01

1,0E+00

1,0E +00 1,0E-K)1 1,0E-K)2 1,0E+03 1,0E-K)4

Acoustic frequency (Hz)

FIGURE 3: RMS Frequency fluctuations of a DFB FL at 1552.06 nm

Laser intensity noise: Fluctuations in the intensity of the laser contribute to the sensor noise and generate a noise photocurrent on the detection indistinguishable from the sensor phase signal by the phase-meter and FFT analyser. Measurements carried out on a single DFB FL pumped at 1480 nm with a power of 140 mW, are obtained by using the relationship given by the experimental set up^. This is characterized in terms of the relative intensity noise spectral density (RIN) eq 7 where

RIN{f,X)-. ( / .A)

(7)

Sgp is the spectral density of the optical power fluctuations and (P) is the mean optical power generated by laser near A = 1.55 im. For the case where the RIN occupies a bandwidth much narrow than the heterodyne beat frequency, RMS induced phase fluctuation is given by eq 8 :

383

6<j>^^=^RIN{f,X) (8)

A typical (50„„ RMS fluctuations is shown in Figure 4. It was found to exhibit

an f " relationship where y = 0.5 for frequencies up to 10 kHz. Our measurements have given that RIN(f,X) levels less than -110 dB/Hz between lOkHz and lOOkHz thanks to a RINpumpiaseAower than 10"' s ' .This behavior proves that sufficiently low RIN can be obtained from DFB FL with a good choice of pump lasers powered with a very low noise current source^'^. Generally, the laser frequency-induced phase noise arises from the small path imbalance present in each sensor. The frequency noise is converted into phase noise by the interferometer and is proportional to the path imbalance in the interferometer. The optical phase fluctuations shown in Figure 4, are given eq 9 by:

2nn,ffL 2mi,„L 6(t> freq •- n ff (Sk)L = -5v = .( n.S/{f,X) ) (9)

8v is due to the RMS frequency fluctuations or hne-width''. In our interferometer fiber sensor, the detected RMS phase resolution can be

degraded by both RMS intensity noise RIN(f) and Sf (f,X) RMS frequency noise fluctuations given by eq 8 and eq 9 respectively.

J , . ,

1,0E-04

1PE

"——.

^ ^ _ . __

'~"~'~-~-*K .

"'*

"^^--^ -, . ' ~*~~--^

-HDD 1,0E-rf)1 1,0E-rf)2 1,0E-rf)3 1 PE-HD4


1 - . - ? ? Freq (rad/?Hz) - . - ? ? DSSO (rad/?Hz) - ^ ?? RIN (rad/?Hz) - . - ? ? ambiant (rad/?Hz) |

FIGURE 4: Detected RMS phase fluctuations 6(|)jiaf, 6(|) gj, 6(|) xxo, ^'^omMem versus acoustic frequency.

Deep sea state Noise (DSSO) :When a hydrophone is placed in the ocean, the background acoustic noise contributes to the total noise. This has been empirically determined for the quietest sea state, referred to as deep sea state zero (DSSO) in Figure 4, such as the root mean square amplitude of the pressure fluctuations in Pa/VHz can be approximated by the Knudsen relationship eq lO ' ' :

<5Pn • dP^.W , / / o N0,85 with (5P„ =lQ-'Pal4Hz (10)

For/=/o, (5Pz555o(/; = lkHz) = 158.48 uPa/VHz; and a t / = 0.3kHz, (5Pz555o(0.3kHz) = 441 ^iPa/VHz. The frequency fluctuations due to the pressure fluctuations detected by the fiber laser are given eq 11 by:

384

^^DSSO =6Po.lO (11)

where SAO is the acousto-optic sensitivity, equal to 4 10" nm/Pa. At f = 0.3kHz 6>^DSso(0.3kHz)=6.61 10"'nm/VHz.

EVTERFEROMETRIC PHASE RESOLUTION

The detected RMS phase fluctuation due to sea state is given eq 12 by:

SO, :(5P„.10' (12)

where f is the acoustic frequency and GMZI is the gain of imbalanced interferometer given by the relationship eq 13:

G„ 2jt n,.L

dX (13)

X' with the values A = 1552 nm, «,^= 1.465, L = 300m, GMZI = 1.149 10* rad/nm. Between 3 kHz and 10 kHz, the RMS phase fluctuation 6(|)<„(,fe„, detected was flat at 5 kHz equal to 1.2 10" rad/V Hz for L= 300m. These results are comparable with those given by Cranch et al . In this case, the RMS phase noise fluctuation due to local oscillator and thermal fluctuation of imbalanced interferometer fiber respectively, are negligible ^ and are not shown in the Figure 5.


-Sao=1,5E-5(nm/P3^ =•=?? Freq (rad/?Hz) Sao = 4,0E-5 (nm/Pa^ -t-Sao=0,75E-5 (nm/Pa^ ?? ambiant for Sao = 4E-5 (rad/?Hz) |

FIGURE 5: Phase fluctuation resolution b<^ambient versus acoustic frequency at various acousto-optic sensitivity Sj^o

NOISE EQUIVALENT PRESSURE

Noise equivalent pressure SPpj^g (Pa/VHz) can be computed, it is given by the model eq 14:

<5Pp, v^v 4^llN+^'l']req (14)

In order to compare it with sea noise equivalent pressure SPQXSOCP^/VHZ). The

acoustic pressure resolution of the hydrophone can be computed for the two cases

385

limited by the sensor self noise and acoustic noise d^ooso using eql4 and eql5, respectively versus sensitivity SAO-

SPnsso=^^^ (15)

The results are plotted in Table 1 when S p g = 8<J)ODSO •

TABLE 1.. Sao(nm/Pa) Frequency (Hz) (50^^^ = S^oo^^ (rad/VHz) dPpj,^ OiPa/VHz)

L5 10"' 1 0.1 57900 3.0 10-'= 10 0.03 8690 5.0 10-'= 38 0.015 2600 7.5 10-'= 100 0.01 1150 1.010^ 300 0.005 434 1.5 10- 800 0.0032 185 4.0 10- 10000 0.0012 26

When the sensitivity is higher than 1.5 10" nm/Pa, the acoustic limit of detection is only due to the deep sea state zero (DDSO). When the sensitivity is lower than 1.0 10" nm/Pa at IHz, the limit of detection is only due to the laser noise.

CONCLUSION

In this paper, we have shown the first noise measurements (detected) of a single mode DFB FL used as an underwater hydrophone. The low frequency pressure resolution in water becomes limited by Deep Sea State zero ambient acoustics if the acousto-optic sensitivity is sufficiently high (> 1.5 10" nm/Pa). If the sensitivity is lower, then the frequency resolution is limited by self noise which is nearly equal to DFB FL frequency noise when the phase noise related to relative intensity noise is negligible because the DFB fiber laser is pumped with a 1480 nm laser with a very low RINpump < 10''^ Hz"\ This type of system can be adapted for any applications requiring networks of sensor elements to be efficiently multiplexed. In particular, for seismic surveying arrays such as those positioned on ocean floor, for instance plugged to the Deep Sea Net used by Ifremer.^

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Acoustic Sensor." Certified Project supported by POPSud OPTITEC (November 7* 2008). 3 A.Dandridge, A.B.Tveten and T.G.Giallorenzi Homodyne demodulation scheme for fiber optics

sensors using VGC IEEE J. Quantum Electron 18 1647-53 (1982). 4 D.J. Hill etal, SPIE Vol 3860, 55 (1999).D.J. Hill etal, SPIE Vol 5855, 904 (2005). 5 G.A.Cranch et P. Nash. IEEE. JLT Vol 19 N°5 May 2001 pp687-699 6 C.Chluda, M.Myara, P.Signoret, B.Orsal "Pump to Signal and Pump to ASF Noise Correlations in

Copropagative Raman Amplifiers" Fluctuations and Noise Symposium-20May 2007 Florence, Italy. 7 J.P.Tourrenc , P.Signoret, M. Myara„J.P. Perez, B.Orsal and R. Alabedra «Low frequency EM

Noise Induced Line-shape: A Theoretical and Experimental Approach)). IEEE Journal of Quantum Electronics. - Vol. 41 n° 4 pp.549-553 April 2005.

8 B.Orsal, R Vacher, D. Dureisseix "Optical noise of an 1550nm erbium doped fiber laser" pp 214-218, Second international Conference on optical Complex Systems OCS'08 March 17-20 2008, Cannes.

386

Low-Frequency Noise of Single Junction GaAs Solar Cell Structure

Jungil Lee'', Byung-Yong Yu'', Gerard Ghibaudo", Seong-Il Kim", and Ilki Han '

'^Nano Device Research Center, Korea Institute of Science and Technology, 39-1 Hawolkok Seongbuk, Seoul, Korea

Advanced Metal Research Center, Korea Institute of Science and Technology, 39-1 Hawolkok Seongbuk, Seoul, Korea

'IMEP-MINATEC, INPG-CNRS, 3 rue Parvis Louis Neel BP257, 38016 Grenoble, France

Abstract. In this paper we report the preliminary results of current-voltage and low-frequency noise measurements on GaAs single junction solar cell with and without multi-quantum well intrinsic layer. The current-voltage characteristics showed typical curves for semiconductor p-n junctions. The spectral density of low-frequency noise shows l//behavior in general. However, both structures showed Lorentzian components in addition to the 1/f behavior. Noise sources are discussed from the analysis of the low-frequency noise data and the current-voltage characteristics, with available models for noise generation.

Keywords: Solar cell. Low-frequency noise, p-n junction. Compound semiconductors. PACS: 73.20.At, 73.21.Fg, 73.50.Pz, 73.50.Td, 73.63.

INTRODUCTION

Low frequency noise measurements in semiconductor devices is a sound tool to diagnose the materials, processes and device stmctures which can provide useful information on the materials quality, defects which can affect the performance and reliability of the devices. In this study, we fabricated GaAs single junction solar cells with and without multi-quantum well (MQW) and measured the photovoltaic response, current voltage characteristics under dark condition and low-frequency noise characteristics. The current dependence of the noise density was discussed with both mobility and number fluctuation models.

EXPERIMENTAL DETAILS

The structures of GaAs single junction solar cell with and without MQW intrinsic layer were grown on «^ GaAs substrate via metalorganic chemical vapor deposition (MOCVD) at Innolume, and had p-AXGaAs window layer for reduction of recombination velocity and «-AlGaAs back surface field layer for reduction of leakage current. The multi-quantum well consisted of three InGaAs and GaAs pairs with 200 A of GaAs and 20, 40, and 60A of InGaAs (Fig. I).



387

p^-GaAs 0.01 |xm (SXIO"* cm' ) p-Alo.9Gao.1As 0.05 |xm (SXIO"* cm' ) p-GaAs 0.8 im (IXIO"' cm' ) n-GaAs 2.5 im (2X10" cm' ) n-Alo.35Gao.65As 1.0 im (3X10"' cm' ) n-GaAs 1.0 im (IXIO"' cm' ) n+-GaAs sub (2X10"^ cm' )

(a) Stripe pattern solar cell

p+-GaAs0.01 ^im(lX10'^ D-AlnoGan,As0.05 umH: p-Alo.9Gao.1As 0.05 |xm (IXIO"* cm-^)~ p-GaAs 0.6 |xm (2X10"* cm' ) GaAs 200 A Ino.2Gao.8As 20 A GaAs 200 A Ino.2Gao.8As 40 A GaAs 200 A Ino.2Gao.8As 60 A GaAs 200 A n-GaAs 1.5 im (3X10" cm' ) n-Alo.3Gao.7As 1.0 |xm (1X10"* cm' ) n-GaAs 1.0 |xm (IXIO"* cm' ) n^-GaAs sub (2X10"* cm' )

(b) Circular pattern solar cell

FIGURE 1. Sample Structures

5^m (line width) 1 95 i m (between the lines)

2.5mm

J200^m

t200^m

2.5mm

2.5mm 2.5mm

(a) Stripe pattern solar cell (b) Circular pattern solar cell

FIGURE 2. Electrode Patterns

The solar cell devices were fabricated using the grown structures. The metal contacts on the top and bottom surface of the solar cell devices were prepared by using photolithography, metal deposition, and lift-off process, sequentially. Ti/Pt/Au of 30nm/30nm/300nm ohmic metal was used for the j9-type GaAs surface and AuGe/Ni/Au of 30nm/30nm/300nm was used for the ?7-type GaAs substrate. The deposited metal structures were annealed by rapid thermal annealing process at 375 °C. Stripe patter was applied to the solar cells without MQW layer and circular pattern was applied to the devices with MQW layer. The areas of the electrodes were the same (Fig. 2).

Current-Voltage (I-V) characteristics for photovoltaic response were measured by using solar simulator at one sun condition (100 mW/cm^). The model name of the solar simulator was XES-301S of San-Ei electron company. The light source was xenon lamp (1 kW). Also I-V characteristics under dark condition were measured with Precision Semiconductor Parameter Analyzer, HP4156A. Noise measurements were performed with Dynamic Signal Analyzer, HP3562A.

388

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FIGURE 3. Current-voltage characteristics

The spectral density of low-frequency noise shows 1/f behavior in general (Fig. 4).

10 10

Frequency (Hz)

(a) Stripe pattern solar cell

10" 10"

Frequency (Hz)

(b) Circular pattern solar cell

FIGURE 4. Noise spectral current density

The conversion efficiency was about 13% in all the tested devices with and without MQW. Fig. 3(a) shows the photovoltaic response of several devices collectively. In Fig. 3(b) the results are presented and showed typical characteristics of semiconductor p-n junction with the ideality factor n of about 2.3 and the saturation current of about 10" A. One can note that the reverse leakage current is much larger in the devices with MQW layer. Noise measurements showed 1/f behavior in general. However, both structures showed Lorentzian components in addition to the 1/f behavior. For the devices without MQW layer showed Lorentzian at higher frequencies whereas for the devices with MQW the corner frequency moved to lower value as the current decreased. It means that the defect levels responsible for the Lorentzian noise are located deeper into the bandgap in the

389

devices with MQW layer. The current dependence of the spectral density of l//noise at 10 Hz is presented in Fig. 5. Both structures showed linear dependence at low current and a quadratic one at higher currents. The transition current was about 10" A which is still in the middle of linear region of /« / vs. Fplot as shown in Fig. 3(b).

According to the mobility fluctuation model [1], S^ x '^1/, where a is Hooge

parameter and r is carrier lifetime in the junction for the case of n « 2 as in our devices. At extremely high current the noise from the series resistance dominates and current dependence becomes quadratic [1]. Recently a number fluctuation model involving tail states in the space charge region [2] were developed which predicts quadratic current dependence. Since the transition current (from linear to quadratic) is in the middle of linear region in Fig. 3(b), the noise at higher current in our data can be explained by the number fluctuation model.

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(a) Stripe pattern solar cell (b) Circular pattern solar cell

FIGURE 5. Current dependence of the noise density

CONCLUSION

GaAs single junction solar cells with and without MQW were fabricated and characterized in particular the low-frequency noise characteristics. Different models for the generation of low-frequency noise are utihzed for the analysis and possible origins of noise sources are discussed. The defects may play an important role in enhancing the quantum efficiency of the solar cells.

REFERENCES

1. L. K. J. Vandamme, R. Alabedra and M. Zommiti, Solid-StateElec. 26, No. 7, 671-674 (1983). 2. A. P. Dmitriev, M. E. Levinshtein, E. N. Kolesnikova, J. W. Palmour, M. K. Das and B. A. Hull,

Semicond. Set Technol. 23, 015011 (2008).

390

Noise of Reverse Biased Solar Cells

p. Skarvada, R. Macku, P. Koktavy, M. Raska

Department of Physics, Brno University of Technology, Faculty of Electrical Engineering and Communication,

Technicka 8, 616 00 Brno, CZECH REPUBLIC

xskarv03(a>,studfeec.vutbr.cz, http://www.feec.vutbr.cz

Abstract. The non-destructive testing and analysis of single crystal silicon solar cell is the focal point of our research. The noise spectroscopy and I-V curve measurement of reverse biased/?« junction provide information that is connected with solar cell reliability and that provide for not only local defect characterization. We propose a new electric solar cell model, as a base for an enhanced noise model, which is in accordance with the experimentally obtained I-V curves. We suggest the physical nature of an unconventional behavior in reverse I-V characteristics, which is typical for solar cells without apparent local avalanche breakdowns.

Keywords: Solar cell,/?«junction, noise. PACS: 73.50.Td

INTRODUCTION

A clean energy source is an up-to-date subject and photovoltaic industry boom assists development of technology and testing methods. Solar cell cost reduction and efficiency improvement are the aim of present research. Despite of various solar cell technologies existence, the diffusion technology based silicon solar cells are stiU favorite article.

Diffusion technology based sihcon solar ceUs are characteristic by a large pn junction. There are many defect regions formatted during and after fabrication process, which can affect the function of a solar cell. The noise spectroscopy and I-V curve measurement provide information that is connected with solar cell reliability and provides for defect characterization. During the study of solar cell properties it turned out necessary to divide the characterization of solar cell noise properties into two groups depending on pn junction polarization. Only reverse biased solar cell results are presented in this paper.

MICROPLASMA NOISE OF SOLAR CELLS

Typical I-V curve of a reverse biased pn junction can be seen on Fig. 1. With increasing of the reverse voltage UR the reverse current /R is slightly rising. Avalanche multiplication of charge carriers occurs when the reverse voltage is increased enough.



391

http://feec.vutbr.cz

http://www.feec.vutbr.cz

More increasing of the reverse voltage leads to homogenous breakdown of a sample. In marked I-V curve region local avalanche breakdowns are present. Bistable microplasma noise can be measured in small range of UR, over this range the channel is permanently ionized. This type of breakdowns relates to local imperfections of the pn junction. An equivalent circuit of the pn junction introduced by Haitz [1] is depicted on Fig. 2. Ideally, the circuit is composed of the power supply U, the reverse biased ideal diode D and the load resistor R. Experimental results indicate that the capacitor C (barrier capacitance of the pn junction) can be neglected because of small sample area. The stochastic switch S, that is connected in series with the voltage source Uu and the series resistor Ru, models bistable behavior of the microplasma noise corresponding to one local region.

4 Second microplasma region

First microplasma region

FIGURE 1. Qualitative I-V curve of reverse biased /?«junction of solar cell.

FIGURE 2. Equivalent circuit of reverse biased/?«junction with microplasma region

In order to noise study the power spectral noise density has been measured. Measurement has been performed up to 13 V of the reverse voltage UR with fine voltage step of 50 mV (with view to pn junction destruction). The experimental results are depicted on Fig. 3. There are three range of the reverse voltage with microplasma noise. The measurement technique of the power spectral noise density is not quite easy because of a low-level measured signal and strong electromagnetic interference. The temperature of the solar cell sample has to be constant during measurement due to microplasma breakdown temperature dependence [2].

FIGURE 3. Power spectral noise density for various reverse voltages, sample K2, T= 26°C

392

MICROPLASMA NOISE FREE SOLAR CELLS

In the case of microplasma free samples, we observed different noise sources (see Fig. 4). A noise model of these samples is more complicated, because the noise type depends on the applied bias voltage and the structure of studied samples is inhomogeneous. The power spectral noise density measurement for various reverse voltages can be seen on Fig 4. Slope of the power spectral noise density is changing with the reverse voltage. The slope is changing from/"°^ t o / " ' ^ in the measured reverse voltage range. It should be noted, that there is unconventional behavior in I-V characteristics (see Fig. 5), which apparently relates to slope changing. I-V curves are superposition of many processes in a sample. Every of these processes depends on the temperature by different way and that is why the temperature measurement can emphasize the I-V curve deflection. The quantum tunneling effect is not the nature of deflection as indicates its temperature dependence. The I-V curve T= 16.1 °C was measured twice to exclude sample degradation. The Back contact of the sample K4 is not integral but grating type and this could be the reason of characteristics distortion.

FIGURE 4. Power spectral noise density for various reverse voltages, sample K4, T=26°C

FIGURE 5. Solar cell measured I-V curves, sample K4, S,,^ = 276,303 mm^

Electric model of reverse biased solar cells

The proposed electric model, which results from a diffusion technology process, is divided into two parts (Fig. 6). A mathematical model is defined by a transcended equation Eq. I, where /rsh + hi corresponds to the model part A and /D2 to the model part B. /si and /s2 are saturation currents of diodes Di, D2. Rsh is the shunt resistance and Rsi, Rsi relates to a metal-silica junction and silica bulk resistance. The equation can be solved using Lambert ^-function; which is inversion function to exponential function/(z) = z e ^ where z is an arbitrary complex number. Simulation results using model parameters from Tab. I are depicted on Fig. 7 together with measured data set.

U-RJ '-rsh ^ -^ D\ ^ -^ D2

R.

e(U-R,-iI)

+ L -e(U-R,J-R,2lD2)

n2kT (1)

393

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u

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l\ 1 |RS.

— I > i

Rs2

v

A B

i i D2

FIGURE 6. Solar cell electric model

-10" u„/v

FIGURE 7. Simulation and measured data

TABLE 1. Parameters of mode analysis we come

Model part A

r = 7 . 5 ° C

Model part B

r = 7 . 5 ° C

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CONCLUSION

We discovered the transistor npn structure in solar cell, created during diffusion process. This structure has not been removed sufficiently, thus close to the back contact it is forward biased pn junction. This unwanted pn junction causes the I-V curve distortion and it have connection with the power spectral noise density.

We have proposed the new electric model of solar cell that is in accordance with the referred presumption. Simulated dataset using the proposed model traces the measured data set (sample K4) well. The unconventional value of the transport mechanism coefficient «2 is probably caused by generation-recombination process affected by distributed serial resistance.

ACKNOWLEDGMENTS

This research has been supported by the Grant Agency of the Czech Republic within the framework of the project GACR 102/09/H074 "Diagnostics of material defects using the latest defectoscopic methods" and by the Czech Ministry of Education in the frame of MSM 0021630503 Research Intention MIKROSYN New Trends in Microelectronic System and Nanotechnologies. These supports were gratefully acknowledged.

REFERENCES

1. Haitz R. H.: Model for the Electrical Behaviour of Microplasma, } . Appl. Phys., vol. 35, no. 5, pp. 1370-1376, 1964.

2. Macku R., Koktavy P.: Study of Solar Cells Defects via Noise Measurement, Proceedings at 31th International Spring Seminar on Electronics Technology, Budapest, Hungary, pp. 96-100, 2008.

394

Acoustic Emission, Electroluminescence Intensity Surface Distribution and Light

Fluctuations Correlation in GaAsP/GaP and InGaN/GaN structures

Vitaliy P. Veleschuk", Olexander I. Vlasenko'', Oleg V. Lyashenko and Maxim P. Kysselyuk*

"Institute of Semiconductor Physics ofNASU, 03028, 45 Prospect Nauki, Kyiv, Ukraine ''Kyiv National University ofTaras Shevchenko, 03680, 2 Prospect Glushkova, Kyiv, Ukraine

E-mail: [email protected], [email protected]

Abstract In work it is shown, that in the GaAsP/GaP and InGaN/GaN heterostructures at current passage simultaneously with operation of acoustic emission sources take place a redistribution of electroluminescence (EL) intensity on a structure surface, are observed local (on areas of surface) and integrated fluctuations of EL intensity and current.

Keywords: Acoustic Emission, fluctuation, LED, Electroluminescence. PACS: 43.35+d; 43.50+y; 72.70+m; 73.50.TD; 78.60.Fi; 78.66.Fd

INTRODUCTION

Defect formation and fluctuation processes in heterostructures impose the certain restrictions on field of their application. The decision of these problems demand research of their physical mechanisms, dynamics of defects development, transformation of functional parameters and also their interrelation [1,2].

It is known that, to increase intensity of an electroluminescence LED most easier to do by increasing density of injection current J, but thus the following problems are shown [2]: non-uniformity distribution J on the active area of structure, including on any cross-section of a current tube (including - current crowding, [3,4]), and also -problem of significant local overheat of the active area [2,3,5].

In the LED active area the gradient AT(r)/Ar reaches lO"* 7cm, what, at take proper account the distinction of thermal expansion coefficients a, of layers and lattices constant a,, leads to making static and local thermomechanical inelastic strains in p-n junction [1-3,5]. Failure of these induced strains result in LED occurrence of chaotic acoustic radiation- to acoustic emission (AE) materials [1,2].

In this paper is analyzed the dynamics of local processes of a relaxation and defect formation in the hght-emitting InGaN/GaN and GaAsP/GaP heterostructures, induced by the direct current, leading simultaneous to change of surface distributions of the EL intensity of structure I(S), to AE, oscillation of integrated EL intensity, current fluctuations and the general degradation of electrophysical parameters of the LED.



395



EXPERIMENT

Objects of researches were low- (GaAsP/GaP) and nano- (InGaN/GaN) dimensional structures. Research of correlation AE, fluctuations of EL intensity AI and a current it was done by a technique [1,2,5]. AE signals were registered by the piezoelectric transducer of specialized acoustic-emission device AF-15; integrated / was registered by photodiodes; further these signals, and also current fluctuation were processed by ADC.

EL intensity surface distribution AI(S) (or along the chosen cross-section) received computer frame-by-frame analysis of digital video (or from original digital photo) of the structure surfaces, made by means of microscope with total zoom x98.

RESULTS

On Fig. 1 (b) it is presented I(S) of a GaAsP/GaP structures (a) on cross-section of structure along a hue Z in a spectral range of 410-700 nm. Apparently from Fig.l, degradation of intensity at AE occurrence and the active processes of a local defect formation and partial destruction along cross-section have happened disproportionately on various distances from an electrode (E on Fig.l). At increase of density of current J from 80 A/CM^ (less than threshold for AE occurrence) up to J= 140 A/cM^ (at which are observed long, but attenuated continuous and bursts AE), EL intensity near to an electrode has decreased in 1.35 times, and at distance 200 microns - more than in 2 times. At subsequent decrease of J up to 80 A/cm^ and AE termination, degradation / on the LED surface also is non-uniform. It is necessary to note, that as a whole degradation of / happens not symmetrically concerning to an electrode and individual dynamics of/ in various local areas is observed.

FIGURE 1. GaAsP/GaP LED (a) and I(S) along line LatJA

200 -100 0 / ( j i m )

(b) ; 2 - 140; 3 -

300

1 A/cM^ after AE (b)

On Fig. 2 (a) photos of surface an InGaN/GaN structure are presented at J = n o A/cm^ at the AE moment. Were observed change / in separate local areas AlfS,) on a surface of structure. Arrows I and 2 on a photo indicate local areas (St) of sharp changes 1(8,) where occur "flickers" - fluctuations of/.

On Fig. 2(b) it is presented three I(S) of a InGaN/GaN LED in a direction of an arrow L on (a) in each 40 ms. From Figure 2 follows, that during AE I(S) in local areas

396

quickly changes, and in some local areas local fluctuations Al(Si), i.e. increase-decrease I(Si) on the part of surface are observed.

1 5

.^J^

j ^ r ^

2

3 ^ 2

i \ •

"1 •

I , 155? •-•

'•

0 40 80 120 160 200 240 280 / (jim)

(a) (b) FIGURE 2. InGaN/GaN LED (a) and I(S) along arrows L (b) at sequential moments, J = 110 A/CM^.

Electroluminescence

emission

FIGURE 3. AE signal and fluctuation of the EL intensity and current of a InGaN/GaN LED.

It is necessary to note, that for these structures at J = 110 A/cm^ have been registered simultaneously with AE signals and Al(Si) also integrated (on all surface of the structure) fluctuations AI(S) and current fluctuations. Usually, as on Figure 3, following correlation between them was observed: actually to each group of AE signals there corresponds change superfluous current noise andzI/fS^.

DISCUSSION

One of problems of manufacturing and operation of these structures in which the average value of internal mechanical strains reaches ~10' Pa [1,2] is local heterogeneity of conductivity, thermal resistance, a difference of elastic modules [2], and also - a, and a;, noted above. That leads to (at current passage) to formation of temporal local thermo-mechanical strains (> 10^ - 10 Pa) and temperature gradients (10^-7-10^ °C/cm) is unpredictable (chaotically) distributed both in the structure, and on its surface and areas adjoining it.

The complex of these reasons leads to individual dynamics of degradation in local areas of the heterostructure. Integrated display of it are simultaneous with AE degradation and fluctuations [1,2] of electro-physical properties (in particular -integrated EL intensity, [2,5]), and also current noise locally-is non-uniform thermo-strained heterostructure.

397

Owing to current non-equilibrium processes there is an activation and operations of AE sources (formation and change of a defects state and areas adjoining to them) to AE signals radiation, and more intensive AE obviously correspond more intensive defect formation and to changes of energy state of a lot of defects.

Formed (changed the state) defects in active area operate as the additional centers of carrier scattering and as their tunnehng centers due to formation of additional energy levels system in the barrier area. At J exceeding AE threshold, because of activization of defect formation processes, / decreases and locally changes, thus there is a redistribution of relative I(S) on the structure surface.

Intensive generation of structural defects and also changes of their energy state, which are accompanied by AE, leads to local fluctuations zfJ; due to fast local changes of the LED resistance. Accordingly, AJi leads to fluctuations of injection of carriers in a quantum well which recombine with radiation that causes fluctuations Al(Si), and integrally - to AI(S). It is note, that temporal local occurrence (shunting/>-« junction) and breaks of conducting channels are simultaneously possible.

Fluctuations of recombination current because of redistribution recombination and tunnel components of the current, caused by rise of a leakage current are possible also at crossing p-n junction by defects (dislocations) and decrease in injection coefficient due to capture of carriers on traps at defects generation in the area of contact.

CONCLUSION

It is shown, that in the GaAsP/GaP and InGaN/GaN heterostructures acoustic emission occurrence and EL intensity fluctuations, degradation and evolution of electroluminescence intensity distribution along active area correlate in time and have the common origin. Thus during the moments of AE radiation electroluminescence intensity distribution lengthwise active area becomes non-uniform.

REFERENCES

1. O. V. Lyashenko, V. P. Veleshchuk, O. I. Vlasenko and R. G. Chuprina "Dynamics and Time Correlation of Acoustic Emission, Electrical Noises and Quantum Yield Fluctuations in Optoelectronic Devices" in Noise and Fluctuations-2007, edited by M. Tacano et al., AIP Conference Proceedings 922, American Institute of Physics, Melville, New York, 2007, pp. 216-222.

2. V. P. Veleschuk, O. I. Vlasenko, O. V. Lyashenko, etc., Ukrainian J. ofPhys. 53, 240-246 (2008). 3. A. V. Zinovchuk, O. Yu. Malyutenko, V. K. Malyutenko, etc., J. Appl. Phys. 104, 033115 (2008). 4. X. Guo and E. F. Schubert, Appl. Phys. Letters IS, 3337-3339 (2001). 5. V. P. Veleschuk, O. I. Vlasenko, O. V. Lyashenko, A. BaiduUaeva, B. K. Dauletmuratov, Physics

andChemistry ofSohdState 9, 169-174 (2008).

398

Low Frequency Noise Characteristics and Aging Processes of High Power

White Light Emitting Diodes

V. Palenskis, J. Matukas, B. Saulys, S. Pralgauskaite, V. Jonkus

Radiophysics Dep., Vilnius University, Sauletekio 9 (III), 10222 Vilnius, Lithuania phone: +370 5 2366078, fax: +370 5 2366081, e-mail: [email protected]

Abstract. Noise and operation characteristics of high-power nitride-based white hght emitting diodes have been investigated during long time device aging. Investigated LED degradation is caused by formation of leakage microchannels due generation of defects and migration of atoms at high aging current and at elevated temperature due to Joule heating in the active layer. It is shown that the most sensitive parameter to the LED degradation processes is the correlation factor between optical and electrical fluctuations.

Keywords: correlation factor, electrical noise, light emitting diode, optical noise. PACS: 05.40.Ca, 07.50.-e, 81.40.Cd, 85.60.Jb.

INTRODUCTION

It is well known that low frequency noise characterization of electronic devices is valuable method for their quality and rehability evaluation. In our earlier works [1, 2], we have shown that investigation of optical and electrical noise characteristics of various optoelectronic devices (laser diodes, photodiodes and light emission diodes) is important not only for noise level evaluation, but also it is highly sensitive and informative method for clearing up the physical processes in device structure and predicting device quality and reliability.

In this work we present the comprehensive investigation of high-power nitride-based white light emitting diodes to clear up physical processes related with accelerated aging of these LEDs.

INVESTIGATION AND RESULTS

Output hght intensity, optical (light output power) and electrical (diode terminal voltage) noise spectra, correlation factor between optical and electrical fluctuations (in the frequency range from 10 Hz to 22 kHz) dependencies on forward current have been measured for initial samples and during long time aging (about 8000 h) at maximum permissible current /max=l A. Optical output power and current-voltage characteristic changes during aging are presented in Fig. 1. Optical output power



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gradually decreases during all aging experiment and after 8000 h aging it decreases about 30 %.

6000 8000 00 0 5 10 15 2 0 25 3 0 UOJ)

FIGURE 1. Photovoltage (proportional to the LED output light power at 1=30 mA) dependency on aging time fag (a) and current voltage characteristics (b) at different aging time: 0 - before aging; 1 -after 2200 h; 2 - 2500 h; 3 - 2540 H; 4 - 2730 h; 5 - 4120 h; 6 - 6640 h; 7 - 7000 h; 8 - 8000 h.

/(mA)

FIGURE 2. Optical (>?opt) and electrical (S^i) noise spectral densities (at/=280 Hz) and correlation factor (r) dependencies on aging time at 1=30 mA (a) and correlation factor dependency on current (b) at different aging time: 0 - before aging; 1 - after 64 h; 2 - 223 h; 3 - 335 h; 4 - 951 h; 5 - 8000 h.

In the initial phase of aging (till 400 h. Fig. 2 (a)) LED electrical noise intensity slightly decreases, while optical noise level decreases about two orders, what shows that in initial stage of aging not stabile defects and impurities in the active layer migrate to more stabile position of ordered structure. Correlation factor at the first 100 h decreases from initial value ~25 % to almost zero, and increases during further aging (Fig. 2 (b)). In the aging range from 2000 h to 3000 h it is observed a strong correlation between the large changes of the leakage currents at low bias (Fig. 1 (b), curves 1-4) and noise characteristics (Fig. 2 (a)) in the normal LEDs operation range. Such large changes of leakage currents and noise characteristics show on migration and reordering of defects, formation of leakage microchannels during aging at high currents and at elevated temperature due to Joule heating in the active layer. The most sensitive parameter to the LED degradation is correlation factor between optical and electrical fluctuations: its value during aging changes in the range from 0 to 80 % (Fig. 2). Electrical noise spectra consist of Lorentzian noise component superposition, which can be approximated as l//type noise (Fig. 3). While optical noise spectra at higher frequencies have shot noise component due to incident photon shot noise and it increases with current increase.

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10' 10* f(Hz)

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FIGURE 3. Electrical (on the left column) and optical (on the right column) spectra at different forward currents: (a) before aging, (b) after 223 h aging, (c) after 1400 h aging.

In order to evaluate, what part of the low frequency electrical and optical noises are completely correlated, it is convenient to approximate the LED optical and electrical noise spectral densities as follows:

S. opt —^ + -^ + A^; a>l; 0 „ i

/ + Bo

where/is the frequency, a is between 1 and 2, the quantities Ai, A2 and 5i depend on current, the quantities A3 and B2 correspond to the white noise sources. The optical noise spectral density Sopt approximation by above presented equation is shown in Fig. 3 (c) by solid lines, the dots present the experimental results. Such

401

approximations are very useful for calculations the total variances and their components of investigated noises in order to evaluate the correlation factor

60

:4o

20

0 10"

: r^ \ \ \ "

10" 10" 10"' 10"

FIGURE 4. The correlation factor dependency on photodiode photocurrent (proportional to the emitted light power) (solid line - experimental data, dashed line - calculated considering that \lf type optical and electrical fluctuations are completely correlated.

10"'

10"'

SlO"' CO

10"'

10"' 10"' 10" 10"'

'p. C^A)

10"' 10"

FIGURE 5. The optical low frequency noise spectral density (f= 65 Hz) dependency on photodiode photocurrent (solid line - experiment, dashed line - the low frequency noise level of that part of optical noise that is completely correlated with electrical \lf type noise.

The experimental and calculated results are presented in Figs. 3 (c), 4 and 5. High correlation between optical and electrical fluctuations of investigated LEDs is determined by optical and electrical l//"type noises. It is seen (Fig. 5) that at higher output light power (that is proportional to the photodiode photocurrent /ph) experimental results differ from calculated considering that 1// type optical and electrical fluctuations are completely correlated. Therefore, the completely correlated low frequency optical and electrical noises dominate only at lower output hght power, while at higher power there appears 1//" type optical noise (a>l) that is not correlated with electrical low frequency noise.

CONCLUSIONS

Noise characteristic investigation of high power white light emitting diodes fabricated on nitride-based materials have shown that investigated LED degradation is determined by formation of leakage microchannels due generation of defects and migration of atoms at high aging current at elevated temperature due to Joule heating in the active layer. Comprehensive noise characteristic investigation cleared up physical processes that are related with investigated device aging. The observed noise characteristic features (especially correlation factor between optical and electrical fluctuations) can be used as indicator of quality and reliability of investigated devices.

REFERENCES

1. S. Pralgauskaite, V. Palenskis, J. Matukas, J. Petrulis, and G. Kurilcik, Fluctuations and Noise Lett. 7, L367-L378 (2007).

2. S. Pralgauskaite, V. Palenskis, and J. Matukas, Fluctuations and Noise Lett 4, L365-L374 (2004).

402

Study Of Radiation Spectrum Emitted From Local Regions In PN Junctions

Ondrej Krcal, Pavel Koktavy and Tomas Trcka

Brno University of Technology, Faculty of Electrical Engineering and Communication, Department of Physics, TechnickaS, 616 00 Brno, Czech Republic

E-mail: [email protected]. vuthr. cz

Abstract The microplasma discharges in PN junction local defect micro-regions are as a rule accompanied by the emission of light, as it has been reported by Chynoweth and McKay [1]. The emission of visible light can be observed on small regions on the surface of solar cells. This study deals with investigating the spectrum of emission and determines the wavelength with maximum intensity.

Keywords: microplasma, solar cell, radiation spectrum, wavelength, PN junction. PACS: 52.80._s

INTRODUCTION

The radiation from solar cell PN junctions was measured by means of an optical fiber connected to the optical input of a photomultiplier. The optical fiber is divided into two parts, their connection is realized across the removable optical filter. Every optical filter has known characteristics measured by the spectral analyzer. A 2-D image of the irradiation local regions has been created by inching the fiber by means of a computer-controlled X-Y plotter above the cell surface. It is seen that a cell of a superficial area of 100 square cm contains a number of defects, which has similar basic characteristics.

REALIZATION

One of main goals of emission measuring is to uncover the frequency spectrum, which indicates features of emission that could be obtained with difficulty by the other measurement method. In case of this study is to determine the center of maximum emission wavelength and the spectrum course. As it is known, the emission should be in the visible light spectrum. That is why the filters for the visible light were chosen for the purpose of filtering. The characteristics of filters have been measured by the differential method with using the digital spectral analyzer. Frequency throughputs of filters normalized according the filter with the highest throughput on the central frequency are in figure 1.



403


— 0.6

350

— 15

— 26

— 135

— 139

— 165

— 707

— 779

— 738

400 450 700 500 550 600 650

wavelenght[nm]

FIGURE 1. Filters throughput characteristics

750 800

From know characteristic of filters the frequency construction masks was made by the addition/subtraction algorithm, which handles band overlapping caused by the other near placed filter(s). The indexes of emitted wavelengths for the maximum pass can be easily computed by the construction masks. That is the solution for real time measuring. Frequency construction masks are determined only once that spends a lot of time at the beginning, but index estimation of emitted light for each filter is quick.

The polarization filters were tried for this purpose as they the very narrow band pass. Their unsuitable feature is very high attenuation that is the main reason, why they couldn't be used. The detector - photomultiplier is not able to catch any emission. The used belt filters covers the wavelength from 350 nm to 700 nm. The next figure 2 shows the final emission characteristics. It can be seen the maximum is on the 580 nm wavelength, it is possible to monitor a significant downward trend to the nearly zero-intensity radiation near the maximum frequency. Digressing from maximum it can be observed, the intensity is growing moderately.

The next investigation was focused on the finding out, whether all the emitted local regions have the same or similar spectrum. The number of the emitted regions was determined by the measuring without the filter, than the comparisons with the measuring with the filter have been done. The results show us the spectrum emitted

404

from solar cell surface, its maximum - characteristic wavelength. This information can nearly indicated the micro plasma discharges, what is the main goal of this study.

350 400 450 500 550 600 650

vinova delka [nm]

FIGURE 2. Spectrum of emission from solar cell surface

CONCLUSION

The radiation characteristics of light emitted from local regions of solar cell as a PN junction using photomultiplier and optical filters has been measured. The computing algorithm was designed to measure in real time based on the determination of mathematical models of filters and their indices. It was found the significant maximum of emission for the wavelength of 580nm, which corresponds to the starting assumptions as the visible light emission can be observed . Furthermore, it was found that every local region with emission of visible light has the same characteristic.

ACKNOWLEDGMENTS

This paper is based on the research supported by the Grant Agency of the Czech Repubhc, the grant No. 102/06/1551 and the project VZ MSM 0021630503.

REFERENCES

1. A. G. Chynoweth and K.G. McKay, Phys. Rev. 102,369 (1956) 2. A. G. Chynoweth and K.G. McKay, Journal of Applied Phys. 30,11,1811 (1959)

405

Using gate voltages to tune the noise properties of a mesoscopic cavity

M. Totaro*, P. Marconcini*, S. Rotter^, D. Logoteta* and M. Macucci*

*Dipartimento di Ingegneria dell'Informazione, Via Caruso 16,1-56122 PISA, Italy ^Institute for Theoretical Physics - Vienna University of Technology, Wiedner Hauptstrafle

8-10/136, A-1040 Vienna, Austria

Abstract. We propose a layout for a tunable mesoscopic cavity that allows to probe the conductance and noise properties of direct transmission channels ("noiseless scattering states"). Our numerical simulations demonstrate how the variation of different gate voltages in the cavity leads to characteristic signatures of such non-universal processes. Using reaUstic assumptions about scattering in two-dimensional heterostructures, our proposed layout should define a viable protocol for an experimental realization.

Keywords: shot noise suppression, mesoscopic cavity PACS: 72.70.4-m, 73.23.-b

INTRODUCTION

Shot noise suppression in mesoscopic cavities is an active research field to which many investigations have recently been dedicated (see [1] for an introduction to the field). On the experimental side, high precision measurements of current shot noise have become possible. These have revealed detailed device characteristics which were unaccessible by way of conventional measurements of the average current. On the theoretical side, different approaches, such as those based on Random Matrix Theory, on semiclassical formulations, and on numerical simulations, have been applied to the evaluation of the shot noise suppression factor (Fano factor).

A particularly intriguing subject in this context is represented by so-called "direct processes," i.e., events of direct transmission between the entrance and the exit constrictions of a cavity. It has been shown recently that such non-universal processes can behave like classical ("noiseless") transmission channels, thereby leading to deviations from universal values for shot noise [2].

Here we propose a cavity layout (see Fig. 1(a)) that should allow for an experimental verification of the noise properties of direct processes. In order to provide useful indications for such an envisioned experiment, we performed detailed numerical investigations based on realistic assumptions on scattering in two-dimensional semiconductor heterostructures.

In particular, we have focused on a model that, although still relatively easy to handle in terms of computation, retains the capability of providing a reahstic description of the confinement potential landscape and of the transport and noise processes.



409

0.14 5" 0.12 — 0.1 a 0.08 cvi 0.06 .= 0.04 S 0.02 1 0 S

i

1 1

0.4

0.35

0.3

0.25

0.2

0.15

0.1 0

Deflector gate scan

Depletion gate scan

/^y\y-^-~^--^--10 -20 -30 -40 -50 -

b)

-100 -200 -300 -4

Gate voltage (mV)

FIGURE 1. (a) Proposed gate layout, with a 500 x 600 nm depletion gate (top left comer) and a 1272 nm long, 130 nm wide deflector gate, placed at an angle of 45° (bottom center); (b) Effect of gate voltages on the Fano factor, for a cavity with both apertures 250 nm wide; the inset contains an enlargement of the low-gate-voltage region.

MODEL

We consider a cavity defined by means of depletion gates in a 2-dimensional electron gas (2DEG) obtained by modulation doping in a conventional GaAs/AlGaAs heterostruc-ture. For a fast but reasonable estimation of the confinement potential at the 2DEG level without solving the complete self-consistent problem (which would be too costly numerically for large parameter scans), we use a technique based on the semianalytical evaluation of the potential, with the inclusion of screening from the charge in the 2DEG [3].

This approach is derived in the approximation of linear response, assuming that the change in kinetic energy of the two-dimensional electron gas is small compared to the contribution from the electrostatic potential. With such an assumption, the 2DEG can be considered as an equipotential surface for the solution of the Poisson equation. Then the charge density induced in the 2DEG is computed from Coulomb's theorem as 5<T = —eoEr^^/^zl ^^, where eo is the vacuum permittivity, e is the relative permittivity of the semiconductor, ^ is the electrostatic potential and d is the depth of the 2DEG.

Following Ref. [3], the variation in kinetic energy is given, to first order, by

5£ = 5N_

3E

da

TZtl'-

Ttff egErd^

m dz (1)

z=d

where e is the electron charge, m is the electron effective mass, h is the reduced Planck constant, and the expression for the density of states in a two-dimensional electron gas has been used. Finally, it is sufficient to divide the variation in kinetic energy by the electron charge, in order to obtain the screened potential.

A graphic representation of a typical electrostatic potential in a cavity is presented in Fig. 1(a), obtained with the procedure described above. Such a potential is then decimated along the longitudinal direction, by coalescing all the shces that are characterized by potential values that differ along the longitudinal direction by less than a given amount. This step is important to reduce the computational time in the recursive Green's function calculation [4] which we use to determine the transmission matrix through the

410

structure. The Fano factor is finally obtained from the transmission matrix [5] by dividing the actual shot noise power spectral density by the one expected from Schottky's theorem [6]. Both quantities in this fraction are independently averaged over energy values in a window, centered around a Fermi energy of 9 meV, with a width corresponding to the bias realistically applicable between the entrance and the exit of the cavity (about 10 kT, to prevent excessive heating of the electron gas while keeping the shot noise level well above thermal noise). A temperature of T=30 mK has been assumed.

RESULTS

We have focused specifically on the action of tunable cavity openings [7] and gate voltages, the latter being located at different positions of the cavity (see Fig. 1(a)). We consider a "depletion" gate located in the upper left corner of the cavity, and a so-called "deflector" gate, located in the middle of the bottom boundary of the cavity. As being located right between the two quantum point contacts, such a "deflector" gate should be able to suppress direct processes. The "depletion" gate, on the contrary, should not influence the direct processes in a systematic way. To test this picture, we explicity study the dependence of the shot noise on both gate voltages and thereby extract interesting information on the scattering dynamics in the system.

0.23

0.21

I 0.19

« O0.17 c

"o.15

0.13

a) 0.22

-300 -200

Depletion gate "^°° voltage (mV)

-400 -300

-200 100 Deflector gate voltage (mV)

0.1

Deflector gate scan b)

Depletion gate scan

-100 -200 -300 Gate voltage (mV)

-400

FIGURE 2. (a) Fano factor as a function of the bias voltages applied to the depletion and the deflector gate for a cavity with 900 nm constrictions; (b) same as in (a), but for a constant null bias applied to one of the two gates.

We first focus on the "quantum" regime with two narrow (250 nm) cavity openings that allow for a propagation of just a few transverse modes (N ^ 3). Setting one of the two gates to zero voltage and tuning the other gate away from zero voltage, we find sligthly increased Fano factors (see Fig. 1(b)); the two gates seemingly play analogous roles here. This behavior can be well understood by considering that, in the regime of low mode numbers N propagating in the constrictions, direct processes are strongly suppressed, but symmetry considerations become very important. For this regime of low mode numbers Random Matrix Theory predicts that cavities with a left-right symmetry (i.e., with both gate voltages set to zero) have a Fano factor of F = 1/4. Asymmetric cavities (i.e., with either gate at finite voltage) are expected to lead to significantly

411

higher noise values [8]. Since our numerical results follow these predictions, we may thus conclude that the slightly increased noise values (F > 1/4) observed in Fig. 1(b) are produced by a gate-voltage induced breaking of the left-right cavity symmetry. (Earlier studies have shown that a transition from regular to chaotic scattering dynamics can be excluded here as a reason for increased noise values [9, 10].)

In the "classical" regime of high mode numbers in both cavity openings (A^« 34, for a width of 900 nm) the situation is quite different. Here we find that activating the deflector gate systematically increases the noise (from F « 0.14 to F « 0.22), both in the case of an active or an inactive depletion gate. (See Fig. 2(a) for a plot of the dependence of the Fano factor on the two gate voltages). By tuning the depletion gate bias, on the contrary, the noise properties of the cavity are changed only slightly, regardless of the deflector gate bias (the action of each single gate, when the other one is kept at zero bias, is shown in Fig. 2(b)). We emphasize that this behavior of shot noise can be directly understood by way of a classical scattering picture in which direct trajectories between the openings are disrupted by the deflector gate but are left unchanged by the depletion gate. Since the direct scattering processes are less noisy than those that explore the cavity [2], blocking direct processes by way of the deflector gate increases the noise, as observed in our numerical data.

We note parenthetically that we have also exphcitly studied the dependence of the conductance on the two gate voltages and on the cavity openings. These results (not shown due to space constraints) demonstrate an exphcit dependence of the conductance on the deflector gate voltage for wide cavity openings (as expected from the above findings).

In summary, we have numerically investigated a layout for a mesoscopic cavity which should allow for an experimental investigation of the conductance and noise properties of direct scattering processes. Our results clearly show that in the "classical" regime of many open modes direct scattering processes indeed play an important role, which, as we demonstrate, can be probed by way of tunable gate voltages. In the "quantum" regime of small mode numbers direct processes are suppressed and the symmetry properties of a cavity dominate its noise characteristics.

REFERENCES

1, Ya, M, Blanter and M, Bilttiker, Phys, Rep, 336, 1 (2000), 2, P, G, Silvestrov, M, C, Goorden, and C, W, J, Beenakker, Phys, Rev, B 67, 241301 (R) (2003), 3, J,H, Davies, LA, Larkin and E, V, Sukhorakov, J, Appl, Phys, 77, 4504 (1995), 4, F, Sols, M, Macucci, U, Ravaioli and Karl Hess, J, Appl, Phys, 66, 3892 (1989), 5, M, Bilttiker, Phys, Rev, Lett, 65, 2901 (1990), 6, W, Schottky, Ann, Phys, (Leipzig) 57, 541 (1918), 7, S, Oberholzer, E,V, Sukhonokov, and C, Schonenberger, Nature (London) 415, 765 (2002), 8, V, A, Gopar, S, Rotter, and H, Schomerus, Phys, Rev B 73, 165308 (2006), 9, P Marconcini, M, Macucci, G, lannaccone, B, Pellegrini and G, Marola, Europhys, Lett, 73, 574

(2006), 10, R Aigner, S, Rotter, J, Burgdorfer, Phys, Rev, Lett, 94, 216801 (2005),

412

High frequency shot noise of phase coherent conductors

E. Zakka-Bajjani*, J. Dufouleur*+, P. Roche*, D.C. Glattli***, A. Cavanna*, Y. Jin* and F. Portier*

*Nanoelectromcs Group, SPEC/IRAMIS/DSM (CNRS URA 2464), CEA Saclay, 91191 Gif-sur-Yvette, France

^Now at Walter Schottky Institut, Technische Universitdt Munchen, Munchen, Germany **Laboratoire Pierre Aigrain, Departement de Physique de I'Ecole Normale Superieure, 24 rue

Lhomond, 75005 Paris, France ^LPN CNRS, Route de Nozay, F-91460 Marcoussis, France

Abstract. We report on direct measurements of the electronic shot noise of a Quantum Point Contact (QPC)

at frequencies v in the range 4-8GHz. The very small energy scale used ensures energy independent transmissions of the few transmitted electronic modes and their accurate knowledge. Both the thermal energy and the QPC drain-source voltage V j are comparable to the photon energy leading to the observation of the shot noise suppression when V j < hvIe. The measurements provide the first direct comparison to the high frequency shot noise scattering theory without adjustable parameters. Furthermore, the developed method allows cross-correlation measurements and computation of higher moments of current noise. This possibility is demonstrated by the measurements of the fluctuations of the power emitted by a biased tunnel junction.

Keywords: Quantum Transport, Shot Noise, Finite Frequency PACS: 73.23.-b; 42.50.Ar; 73.50.Td

The excess quantum shot noise power of a phase coherent conductor is expected to vanish at frequency V for bias voltage |y | < hv/e and temperature T <C hv/k^ [1, 2]. Indeed, finite frequency noise detection requires that the electron energy provided by the bias is large enough to emit a quantum hv toward the detector. This prediction was first observed in a diffusive conductor [3]. However, this system was by nature unable to probe the role of the transmissions and hence unable to provide a full comparison with theory. The QPC used in the present experiments fills this gap.

A two-terminal conductor made of a QPC realized in a 2DEG (35 nm deep with 36.7 m^ V^^ s^^ mobility and 4.4 10^^ m^^ density) in GaAs/GaAlAs heterojunction is cooled at 65 mK by a dilution refrigerator and inserted between two RF transmission lines. The experimental set-up is represented in fig. 1-a. To increase the sensitivity, both contacts are connected to 50 Q. coaxial transmission lines via two quarter wave length impedance adapters, raising the effective input impedance of the detection lines to 200 Q. for frequencies between 4 GHz and 8 GHz. The transmitted signals are amplified by two cryogenic Low Noise Amplifiers, with input noise temperature 7i -2^ 5K. Two circulators, thermally anchored to the mixing chamber, isolate the sample from the current noise of the cryogenic amphfiers. After further amphfication and narrow bandpass filtering at room temperature, current fluctuations are detected using two calibrated quadratic detectors whose output voltage is proportional to the noise power.



413

200ii/50ii A/4 adapter

- ^ ^ t J T i

a

f

f

b

—(9>»y, ADCi

lo©

—(g>*.Vj ADC2

->

^sample

Sl,{t),SP, •*• r.t.

' filters

FIGURE 1. Schematic diagram of the measurement set-up.

«i

To remove the contribution from the input noise of the amplifiers, we measure the excess shot noise, i.e. ASi{v,V) = Si{v,V) — 5/(v,0). Practically, this is done by applying a 93 Hz 0-V square-wave bias voltage on the sample through the DC input of a bias Tee, and detecting the first harmonic of the square-wave noise response of the detectors using lock-in techniques. In terms of noise temperature, referred to the 50 Q. input impedance, an excess shot noise ASi{v,V) gives rise to an excess noise temperature k^ATi°^{v,V) = ASi{v,V) X ZeffZ2^^pi,/(2Zeff+ Zsampie)2 • This shows the benefit of increasing Zgff: in the high source impedance limit Zgampie > Zgff, ATjf"" is proportional to Zgff. Last, the QPC differential conductance G is measured with a 127 Hz lock-in technique.

0,20

—-, 0,15

=o °-i°

<1 0,05 "O

-J o

U

o/ / ^

^ 1

-0.5

o

\ -

C

--0.4 -0.3 -

Gate Voltage(V)

-0,50 -0,45 -0,40 -0,35 -0,30 -0,25 V,(V)

1,0

G/G„

FIGURE 2. Left: Open circles: conductance G of the sample versus gate voltage. Solid Line: Fit to the saddle point model. Dashed lines: variations of the transmissions of the first three electronic modes. Inset: variations of G with drain-source voltage V at G ~ Go/2. Right: Open circles: dASj/d{eV). Full line : theoretical prediction. The only fitting parameter is the attenuation of the microwave signal.

As the experiment is performed at zero magnetic field, G exhibits plateaus, quantized in units of Go = 2e^/h. The small 80nm width of the split gates combined with the large 14meV Fermi Energy provides large 1-D sub-band energies making conductance almost independent of bias voltage (5G/G < 6% for V < SO uV for G ^ 0.5 Go). As shown by the left part of fig. 2, each mode transmission can be obtained from a fit of the conductance with the saddle point model [4].

We then set the gate voltage to get only one half transmitted mode (G c^ 0.5 Go), and measure ASi{v,V). As shown in the left part of fig. 3, we observe a striking

414

Observation Frequency v (GHz) 0 2 4 '

E 0,(1

I ' II 1

o 4.22 GHz A 7.63 GHz

2XV,(7.63GHz

So

2 X V,(4.22 GHz

D/

/ /

-80 -60 -40 -20 0 20 40 60 80

Drain Source Voltage V (nV)

10 15 20 25

hv/e (|iV)

FIGURE 3. Left: Ar OQ ( ^ y^ measured at 4,22 GHz (open circles) and 7,63 GHz (open triangles). The dashed Unes represent the linear fits to the data, from which the threshold VQ is deduced. The solid lines represent the expected excess noise Sj{v,Ti,{V),V) — Sj{v,Ti,{0),0), using Ti,{V) due to heating effect. Right: Variations of VQ with v. The dashed lines correspond to the low frequency and high frequency limits, and the solid line is a fit to theory, with the Tg as only fitting parameter.

suppression of ASi{v,V) at low bias voltage, and that the onset of noise increases with the measurement frequency. Quantitatively, if Di stands for the transmission of the i"' mode of the QPC, the expected excess shot noise power reads [1,2]:

A5/(v,r ,y) = 2 G o ^ A ( l - A ) hv -eV hv + eV 2hv

gPihv-eV) _ I ' gPihv+eV) _ I gPhv _ I (1)

with j5 = {kBT)^^. For eV > hvMT, one gets

A5/ (v , r , y ) -2eGo^A( l -A) (V-yo ) with eVo =/JVcoth(i3/jv/2) (2) i

At low frequency, VQ characterizes the transition between thermal noise and shot noise (eVo = 2^8^), whereas in the low temperature limit, it marks the onset of photon suppressed shot noise (eVb = hv). Experimentally, Vb is determined by the intersection of the linear regression of the excess noise measured at high bias and the zero excess noise axis. Fitting Vo( v) with eq. 2 (see the right part of fig.3) gives an electronic temperature T = 12 mK, in good agreement with the fridge temperature (65 mK). We will show that heating effect can account for this small discrepancy.

The right part of fig. 2 displays dASi/d{eV), measured above VQ, at a fixed frequency of 5.65 Ghz, as a function of the sample conductance, both normahzed to GQ. The data agree qualitatively with the D{1 — D) dependence of pure shot noise, showing maxima at G = 0.5Go, and G = 1.5Go, and minima at G = Go and G = 2Go. The short QPC is responsible for the non zero minima, as when the second mode starts to transmit electrons, the first one has not reached unit transmission (see the left part of fig. 2). However, eq. 1 is not compatible with a second maximum higher than the first one. To explain this, one has to take into account the increase in electron temperature due to finite bias-voltage and energy redistribution. The sample dimensions being much larger than

415

the electron-electron energy relaxation length, but much smaller than electron-phonon energy relaxation length, there is a gradient of electronic temperature T^ from the QPC to the ohmic contacts assumed to be at the fridge temperature. Combining the dissipated power IV with the Wiedemann-Franz law, one gets [5]:

r / = 7o + ^ 7 ^ ( l + = ^ l ( ; ^ ) , (3) . 2 _ ^ 2 24 G_f 2G\ f eV Y

Tt^Gm V Gm) \2kBj

where G^ stands for the total conductance of the 2D leads, estimated from conductance measurements to be 12 mS ±20%. The increased noise temperature is then due to both shot noise and to the increased Johnson-Nyquist noise. For a lattice temperature of 65 mK and G = Go/2, T^ will increase from 69mK to 77mK as V increases from 50jUV to SOjuV. This accounts for the small discrepancy between the fridge temperature and the electron temperature deduced from the variations of Vo( v). As G increases, the effect is more important, as can be seen both in the right part of fig. 2 and eq. 3. The solid line in the right part of fig. 2 gives the average derivative with respect to V of the total expected excess noise, Si{v,V,Te{V)) — Si{v,V = 0,Te{V = 0)), using the attenuation of the signal as a free parameter. The agreement is quite satisfactory, given the accuracy of the saddle-point model description of the QPC. We find a 4.7 dB attenuation, which is in good agreement with the expected 4 ± 1 dB deduced from calibration of the various elements of the detection chain. Last, the voltage dependent electron temperature obtained from eq. 3 can also be used to evaluate Si{v,V,Te{V)) — Si{v,V = 0,Te{V = 0)) as a function of V at fixed sample conductance G = 0.5Go. The result, as shown by the solid lines of the left part of fig. 3, is in excellent agreement with experimental observations.

Other measurements of the finite frequency shot noise of a QPC have been reported, using on-chip detectors, based on the DC detection of the photocurrent of a Quantum Dot operated in the Coulomb Blockade Regime [6], or the monitoring of the charge configuration of a double-dot [7]. The advantage of the present approach is that the fluctuating currents resulting from shot noise are amplified and can be computed at room temperature, thus giving access to various moments of the distribution of the transmitted charge, and to perform cross-correlation between both sides of the detection circuit. To demonstrate this possibihty, we replace the Quantum Point Contact used above by a 500 Q. tunnel junction. The current signals reflecting the shot noise of the biased junction can then be analyzed using three alternative techniques. The first two methods are devoted to the study of electronic shot noise. First, we repeated the experiments described above, leading to similar results, albeit with a much higher signal to noise ratio due to the almost perfect impedance matching between the tunnel junction and the detection impedance. Alternatively (case b of fig. 1), they are digitized, using an AP240 Acqiris acquisition card capable of a IGsample/s sampling rate, after down conversion achieved by mixing with a suited local Microwave signal. The third method, on which we focus, is dedicated to the study of photon noise. It is a hybridization of the two first: the outputs of the two quadratic detectors are digitized, aUowing to study the auto and cross correlated fluctuations of the powers emitted in the two branches of the detection circuit. Pi ,2, following the original Hanbrury-Brown Twiss (HBT) experiment [8].

one expects the emitted photons to have the same chaotic statistics that thermal photons [9], although their basic origin -and hence the frequency dependence of the

416

emitted power- is totally different. The cross-correlated power fluctuations are positive, reflecting the bosonic nature of the emitted photons, and proportional to the products of the power emitted in both detection branches. The autocorrelated power fluctuations are enhanced by the contribution of the noise temperature of the amplifiers [10]:

Spouipoui = 2G2Avi[fe2(7;i+A7;i)2]

Spouipoui = GiG2[AVinAv2]klAT„iAT„2,

where P™' stand for the amphfied power detected at the end of the two detection branches, Avi,2 for their bandpass , and Ar„i 2 for the excess noise temperature due to shot noise. Eq. 4 shows the benefit of HBT cross-correlation to study photon statistics, as it suppresses the term due to the mixing of the input power with the input noise of the amplifier. A convenient way to get rid of the imperfectly known gains and attenuations is to normalize both the excess noise temperatures and the excess autocorrelated power fluctuations by their value at zero drain source voltage, and the cross-correlated power fluctuations by the geometric mean of the zero drain source autocorrelated power fluctuations. We start with the auto-correlated excess power fluctuations. The left part of fig. 4 represents the excess power fluctuations spectrum AS pout pout, normalized by the zero bias value Spoutpout{V = 0) as a function of AT„i/T„i, measured with Vi = 5 GHz,Avi = 180 MHZ and Vi = 7 GHz,Avi = 220 MHZ. The solid line represents the theoretical prediction:

ASpoutpout / A T 1 \ 2 AT 1 M M _ / ^^«1 \ I 2 (5)

Spout pout {V — 0) \ T„i J T„i

which agrees with the experimental data within 3%. We now turn to the normalized cross-correlated power fluctuations, measured with Vi = V2 = 5 GHz,Avi = 180 MHz,Av2 = 200 MHz. As shown by the left part of fig. 4, the cross-correlated power fluctuations are positive, showing the bosonic character of the emitted excitations. According to theory, assuming Avi C AV2 one should observe:

?poutpout(V = 0)Spoutpout(V ^ 0 )

Vr'i'l"' _ /AVi AT„i AT„2 1/2 V AV2 7;i T, n2

Although the normalized cross-correlated power fluctuations are indeed found proportional to the product of the normalized excess noise temperatures, the experimental pro-portionahty factor is 0.9 instead of 0.95, estimated from the overlap of the bandpass of the two room temperature filters and the difference in wave path length. We believe that these discrepancies can be attributed to the frequency variations of the various microwave couplings, gains, and noise temperature.

In conclusion, we have developed an experimental set-up that aUows the extraction of the noise current of mesoscopic conductors in the 4-8 GHz frequency range. This aUowed us to perform the first direct measurement of the finite frequency shot noise of the simplest mesoscopic system, a QPC[11]. The data are found in quantitative agreement with theoretical predictions. The method also allows the measurement of higher moments of the noise currents, as shown by the first determination of the statistical

417

1,0 6GHz 7GHz

-Theoretical Prediction

0,25

0,20

0,15-

0,10-

0,05

0,00

5 GHz -Theory

T

0,00 0,05 0,10 0,15 0,20 0,25

A7;i AT„,

T T '-nl '-n2

FIGURE 4. Left: Normalized auto-correlated power fluctuations vs normalized excess noise temperature, measured at 5GHz and 7 GHz. Symbols: experimental data. Right: Normalized cross-correlated power fluctuations, vs the product of excess noise temperatures. The solid line corresponds to the theoretical predictions.

properties of the photons emitted by the shot noise of a tunnel junction. This opens the way to the exploration of the interplay between the statistical properties of electrons and of the photons emitted by phase coherent conductors [9].

ACKNOWLEDGMENTS

It's a pleasure to acknowledge precious help from Q. Lemasne, P. Bertet and D. Vion with the tunnel junction fabrication. We greatly benefited from technical support from P. Jacques, D. Darson, C. Ulysse, and C. Chaleil for during the experiment design. The laboratoire Pierre Aigrain is "unite mixte de recherche" (UMR 8551) of the Ecole Normale Superieure, the CNRS and the Universities Paris 6 and Paris 7. This work was supported by the ANR contract 2e-BQT, and by the C'Nano Idf contract QPC-SinPS.

REFERENCES

10. 11.

G. B. Lesovik, Pis'maZh. fiksp. Teor. Fiz. 49, 513 (1989), [JETP Lett. 49, 592 (1989)]. S.-R. E. Yang, Solid State Commun. 81, 375 (1992). R. Schoelkopf, P. Burke, A. Kozhevnikov, D. Prober, and M. Rooks., Phys. Rev. Lett. 78, 3370 (1997). M. Bilttiker, Phys. Rev B 41, 7906 (1990). A. Kumar, L. Saminadayar, D. C. GlattU, Y. Jin, and B. Etienne, Phys. Rev Lett. 76, 2778 (1996). E. Onac, F. Balestro, L. H. W. van Beveren, U. Hartmann, Y V. Nazarov, and L. P. Kouwenhoven, Phys. Rev. Lett. 96,176601 (2006). S. Gustavsson, M. Studer, R. Leturcq, T. Ihn, K. Ensslin, D. C. DriscoU, and A. C. Gossard, Phys. Rev Lett. 99, 206804 (2007). R. H. Brown and R. Twiss, Nature 177, 27 (1956). C. W. J. Beenakker and H. Schomerus, Phys. Rev Lett. 93, 096801 (2001). C. M. Caves, Phys. Rev D 26, 1817 (1982). E. Zakka-Bajjani, J. Segala, F. Portier, P. Roche, D. C. Glattli, A. Cavanna, and Y Jin, Phys. Rev. Lett. 99, 236803 (2007).

418

Shot noise and linear conductance in a transport through quantum dot coupled to polarized leads

A. Golub

Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva, Israel

Abstract. We study the influence of polarized leads and of magnetic field on the noise power and on transport through a link which may be a quantum dot or a point junction. We suggest that such link is tuned to the local spin regime and reveals Kondo type behavior The implication of superconductivity is also analyzed in the case when one of the leads is a superconductor. Specifically, we calculate the noise power to the third order in the Kondo coupUng. With the help of fluctuation-dissipation theorem we can further define the Unear conductance as a function of the polarization and magnetic field. For dot spin operators we used their representation in terms of mixed Dirac and Majorana fermions. The important output of the derivation with both, spin polarization and magnetic field included, is the potential scattering contribution which acquires logarithmic dependence on the band width. Motivated by experiment [1] we analyze a set configuration when only one lead is polarized. The Kondo temperature is defined with the help of renormalization group equations. In particular, some of these equations follow from the invariance of the shot noise under scale transformation.

Keywords: Scale invariance, Kondo effect PACS: 72.10.Fk,72.15.Qm,73.63.Kv

INTRODUCTION

The Kondo effect was observed in electron transport and shot noise measurements of semiconductor and carbon nanotube quantum dots. The majority of works considered the unpolarized leads fabricated from normal metals or superconductors. Recently [2] it was shown that also for ferromagnetic leads the Kondo correlations in the linear conductance persist. Moreover, the Kondo effect depends on the orientation of the magnetization vector in the leads. In the parallel magnetic configuration of the ferromagnetic electrodes, the Kondo resonance sphts without magnetic fields. With applying magnetic field the sphtting is reduced and standard Kondo peak is restored [3, 4].

Here we calculate the shot noise power and liner conductance using perturbation theory to third order in potential and exchange couplings. In addition to the ordinarily discussed configurations of parallel and antiparallel orientations we also consider the case when both leads are of the same material, however, one of the lead is non-polarized. This can be relevant for short ballistic hole quantum wires [1]. Indeed, if the in-plane magnetic field is parallel to the wire, then these wires form point contacts which show the complete spin-resolved conductance. However, much stronger in-plane magnetic field perpendicular to the constriction does not affect the quantized steps.



419

HAMILTONIAN

Quantum dot can be described by the Anderson hamiltonian which reads

+ eY_dXda + Ud^.^didld^+t ^ [dXaja{0)+H.C.] a j=L,R,o

where (in standard notations) the first line represents leads hamiltonians and the Zeeman interaction of the impurity spin with magnetic field. The second line stands for the dot hamiltonian and the hybridization term. The polarized leads cause a renormahzation of bare level position. This renormahzation results in a spin-dependent contribution to e: e ^ e"<j = e — r7<j. The Anderson model is equivalent to the Kondo model in the limit of the frozen charge fluctuations. Therefore in the Kondo regime a new effective magnetic field influences the tunneling:

B^B = B + r]f-r]i=B + 5r] o

In the limit U ^ o°ior ferromagnetic leads we have Srja = {P/7t)rin{D/e). For finite U

For superconductor-QD-ferromagnetic junction in the case the superconducting gap A e the effective magnetic field acquires a form

5r],, = {P/n)rL{l + rs/A)ln{D/e)

where Ts, TL are corresponding tunneling width (Tj = Int^Vj). Here Vj = {Vji + Vj|)/2 are the densities of electronic states, D denotes the band width. The equations also include the polarization parameter P .Pj = (v^j — Vj|)/(2Vj). Thus the total hamiltonian is a sum of leads hamiltonians, polarization-dependent Zeeman interaction and the Kondo hamiltonian H = HI + HR — BS^ + HK, where for normal leads

HK= Y, «a<j(0)(^5<J,<J'+'^^'^<J<J')«a'<T'(0) aa'aa'

here i and S are the spin operators of the electrons in the leads and of the electron on the isolated level, respectively.

The superconductor-QD-ferromagnetic junction is convenient to describe in Nambu representation for conducting electrons «„! = Vai, aL = Vai (here a, a' stand for notation of the electrodes). In Nambu space we can write HL + HR = Y.ka vlai^kaT^z + ASa,RTx)y^k,a, while the Kondo hamiltonian acquires a form

iJ=L,R ^ ^ V 2

where S_ = [S, - iSy)/V2, and s)' = {YJUWI)2/V2

420

SHOT NOISE

The noise power is given by current-current correlation function

s=^-<[i{t),i{t')]+>-<i>^

To calculate linear conductance G which is a function of polarization and magnetic field, we use the fluctuation dissipation theorem. We start with the limit of zero effective magnetic field B, and explicitly separate the contributions from exchange and potential scattering. To second order in the coupling constants for a junction biased by voltage V we get

5 ' = 'êVcoH^)VLVR[{l+PLPR){Jl + Ji) + 2{l-PLPR)J^]

The third order contribution includes the cut-off energy D under the logarithm which is the hallmark of Kondo effect. The potential scattering in this case {B = 0) does not depend on D and can be neglected. Thus

^ ( ^ ) ^ ^ f - e y 4 > o c o t h - l n -

where <^0 = VLVR[2{VL+VR){1-PRPL) + VL{1-PI) + VK(1 - P | ) ]

The expression for conductance directly follows from definition G = dS/2dT. Next we calculate the conductance and noise power in the case when effective mag

netic field B > T,eV.To present the final expressions in a simple form we use the following notations:

g± = J.'<\/VL]-,lVRl,]-, gR,L = Jsy^VR^L]VR,Ll^ 8ti= Jsy^VLl^V^l, g^-î= Jp\/VL\XVî

We proceed with perturbative calculation in general case JpÔ and introduce the value

Then to the third order in the Kondo interaction the shot noise power in normal junctions with polarized leads acquires a form

eV D D S = eVcoth—{A + 2R+ln—+2R^ln — }

Here A = {g'^)^ + {gl)^ R+ = g+ig^gR + glgi), R = g-{g'^gL + glgR), B± =B±eV. From the scaling invariance of the noise we obtain the renormahzation group equation for coupling constants

^ W + {glf + 2{R-+R^)\n^^ = 0

421

The linear conductance in the same limit (B > T) takes a form

where R = R-+R+.het us consider PR = 0,PL = P. Then A = 2g'^{l+f+2Pf) and R = 2g^ (1 + Vl — P^ — 2fP). Factor / is related to the potential scattering and magnetic moment: f = 2< Sz> Jp/Jz-

In junctions with one lead as a superconductor and the other as a ferromagnetic, the noise power at T=0 shows a threshold behavior at the voltages below the superconducting gap. We can write the total shot noise power as S = S ^ + S'^^\ where

S(2) = ^eVCOth(^)VLVK Vy2 _ A2[/2 + +3/2]

2n A

In the last formula we drop terms which do not include large cut-off depending logarithm.

CONCLUSION

Using nonequilibrium perturbation theory we investigate the transport and shot noise power of a quantum dot in the Kondo regime coupled to ferromagnetic leads. We also analyze the case with one superconducting electrode, and find that the shot noise power shows the threshold behavior at the voltages below the energy gap and zero temperature.

The finite polarization of the electrodes and nonzero applied magnetic field lead to logarithmic dependence of the potential scattering on the band width. Therefore, the potential coupling plays an important in the transport and shot noise. Similar effect of potential scattering was found earher for non-polarized leads and in non-zero magnetic field [5]. However, for non-polarized leads only the non-linear part of the conductance is influenced.

We also note that the experiment [1] opens the unique possibility to have point contacts between the polarized and non-polarized leads at given magnetic field.

REFERENCES

1, Danneau R,, et al, Phys, Rev, Lett,, 100, 016403 (2008) 2, J, Martinek, et al, PRL 91, 127203 (2003) 3, A, N, Pasupathy, et al, Science 306, 86 (2004) 4, K, Hamaya, et al, AppUed Phys, Lett,, 91, 232105 (2007) 5, J, A, Appelbaum, Phys, Rev,, 91, 633 (1966)

422

Effect of localization on the Fano factor of cascaded tunnel barriers

p. Marconcini*, M. Macucci*- , G. lannaccone*-^ and B. Pellegrini*'

*Dipartimento di Ingegneria dell'Informazione, University di Pisa, Via Caruso 16,1-56122 Pisa, Italy

^CNR-IEIIT(Pisa), Via Caruso 16, 1-56122 Pisa, Italy

Abstract. We study, by means of numerical and analytical quantum mechanical calcinations, shot noise suppression in a series of tunnel barriers, finding results that strongly differ from the 1/3 limit for the Fano factor that woidd be expected from semiclassical models. The reason for the observed residts is attributed to the presence of strong localization, which in the case of just one-dimensional disorder makes it impossible to reach the diffusive transport regime.

Keywords: tunnel barriers, shot noise suppression, strong localization PACS: 72.70.+m, 73.23.-b, 73.40.Gk, 73.23.Ad

INTRODUCTION

It is well known [1] that shot noise in diffusive conductors is suppressed, with respect to the value expected from Schotfliy's theorem, by a factor 1/3. De Jong and Beenakker [2, 3] showed that the same value for the shot noise suppression factor (Fano factor) is obtained for a series of tunnel barriers, using a semiclassical model based on the Boltzmann-Langevin equation. Here we show, by means of numerical and analytical calculations, that a quantum mechanical model yields different results and, in particular, the asymptotic 1/3 limit is not achieved, and we propose an explanation for these results.

NUMERICAL RESULTS

We focus on the case of unevenly spaced barriers (see Fig. 1(a)), i.e. of one-dimensional disorder, where (differently from the case of equidistant barriers) the transport behavior is not dominated by resonances between the interbarrier regions. As a result of the absence of mode-mixing in this structure, transport along the propagation direction x can be analyzed studying separately each propagating mode. We have therefore evaluated the transmission T„ of the generical w-th mode through the device, with the scattering matrix formalism. Then we have computed the conductance G, the shot noise power spectral density S; and the Fano factor y as [4, 5]

(q is the value of the elementary charge, h is Planck's constant, / is the average current flowing through the device and V is the extemaUy apphed voltage). The considered



423

w

(a)

3 4 5 6 7 Number of barriers

(b)

FIGURE 1. (a) Sketch of the considered stracture. (b) Fano factor (averaged over 50 sets of interbarrier distances) for a series of ideal barriers, represented for three values of the transparency F. The empty symbols represent the numerical residts, while the solid symbols come from the analytical calculation.

Structures have a confinement width W of the order of some microns. In particular the values of the Fano factor and of the resistance that we show in the following have been obtained for W = 8 jum, with a Fermi energy of 9.03 meV, which corresponds to 320 modes propagating in the structure; the results are properly averaged over a range of energy qV = 40 jUeV. In order to smooth out the fluctuations resulting from interference effects, we have averaged the results obtained from 50 different sets of interbarrier distances. In Fig. 1(b) we show, as a function of the number of cascaded barriers, the values of the Fano factor achieved for three barrier transparencies, assumed to be independent of the longitudinal wave vector of the impinging electrons. It is apparent that these results (which are very close to those we have obtained with an exact description of the tunnel barriers) do not approach, increasing the number of barriers, the common 1/3 limit expected from semiclassical arguments.

STRONG LOCALIZATION

Indeed, due to the presence of just 1-D disorder, and consequently to the absence of mode-mixing, in this structure the transport problem is equivalent to a collection of intrinsically one-dimensional problems. Therefore, the locahzation length L/ is equal to the mean free path LQ and it is impossible to satisfy the condition for diffusive transport (Lo « Ld « Li, with Ld being the length of the device). The presence of localization in the considered structure is confirmed by the exponential behavior of the resistance, as a function of the number of barriers, obtained from our numerical calculations (see Fig. 2(a), where we have considered ideal barriers and averaged over 50 different interbarrier length sets). This is evident for opaque barriers, but also the nearly linear behavior observed for a series of highly transparent barriers actually represents a slowly exponential increase. Thus locahzation, which does not appear in common semiclassical models (not including phase coherence), does appear in quantum mechanical approaches and represents the key difference between the two types of theoretical analysis. In the ab-

424

0.007

0.006

0.005

0.004

0.003

1 2 3 4

- ' r=o.9

-- ^ ^ ^ S f - " " ^ " " ^

- , , ,

5 6 7 8 9 10

.—* -----

3 4 5 6 7 Number of barriers

(a)

10

0.2 0.4 0.6 O.f Occupancy

(b)

FIGURE 2. (a) Resistance (normalized with respect to the resistance quantum 12906.4 Q.) of a series of ideal barriers with F = 0.1 and 0.9, as a function of the number of barriers, (b) Distribution of the occupancy for the propagating modes in the 3rd interbarrier region of a series of six unequally spaced tunnel barriers with F = 0.1.

sence of a hypothetic mechanism leading to complete dephasing, the diffusive behavior can be recovered for a series of barriers only if strong mode-mixing is introduced in the structure, the number A of propagating modes is large, and the length of the device is such that the condition LQ < < L^ < < L/ (with L/ = A^LQ) is satisfied. The absence of mode-mixing makes it impossible also to define an occupancy depending only on energy in each interbarrier region, while this is one of the main assumptions that need to be made in semiclassical derivations. As an example, we have computed the occupancy of each propagating mode in the 3rd interbarrier region of a series of 6 unequally spaced barriers with an average transparency 0.1. This has been achieved for each mode dividing the partial density of states due to injection from the left lead by the total density of states. The distribution of the occupancies is shown in Fig. 2(b); the dispersion of these values demonstrates that it is impossible to define a unique value for the occupancy in the interbarrier region.

ANALYTICAL CALCULATIONS

Assuming ideal barriers with a transparency independent of the longitudinal wave vector of the impinging electrons, it has then been possible to find closed-form expressions for the Fano factor of the series of 2 and 3 tunnel barriers, as a function of their transparency. Indeed, from our simulations we have seen that the value of the overall transmission of the device (which appears at the denominator of 7 in Eq. 1) is approximately identical in structures differing for the values of the interbarrier distances, due to the high number of propagating modes and to their averaging effect on transmission. Therefore it is reasonable, for this device, to separately average the numerator and the denominator of the Fano factor (over different sets of interbarrier distances), instead of averaging the values of the Fano factor for the different length sets. Therefore the mean Fano factor can

425

be approximately expressed as 7 = ( I„7; ( l - T„))/{1„T„) = 1,„{{T„) - {T^))/'Ln{Tn) (where the average is over sets of interbarrier distances). But, if the results are averaged over several sets of random interbarrier distances and the transparency of the barriers is identical for all the propagating modes, there is no dependence of the results on the longitudinal wave vector of the mode and therefore each mode gives the same contribution to the calculation. Therefore, dividing out the number of modes in the ratio, we have that 7 = {{T) — {T^))/{T), where each average is performed only on a single generical mode and depends only on the transparency T and on the number of the considered barriers. In particular, for each number of barriers, we have found, using the scattering matrix method, the expression of T and T^ as a function of the phase contributions resulting from the traversal of each interbarrier region. Then we have averaged these two expressions over all possible values of the interbarrier distances, integrating each phase contribution between 0 and In. Following this procedure, we have obtained closed-form expressions for two and three cascaded barriers. If T is the barrier transparency, for a series of two barriers we have

r , r(2-2r+r2) , , , (T^) 2(1-r) , , T^) = ^ . 7, ' and thus 7 = 1 - V T ^ = .) ' , (2) \ / 2 - r ' ^ ' ( 2 - r ) 3 ' (T) ( 2 - r ) 2 '

and, for a series of three barriers,

T) = , , r2 = ^ v/r(4-3r)' v^r(4-3r)(i6-24r+9r2)

, (r2) 3(4-8r+5r2-r3) and thus 7 = 1 - V T ^ = ,^ 7,^ ^^, ' . (3)

' (T) 1 6 - 2 4 r + 9r2 This is coherent with our numerical results, as can be seen in Fig. 1(b), where the analytical results are shown with solid symbols. The first value is also equal to the one predicted semiclassically [2, 3]. Actually, for only 2 barriers a uniform occupancy in the middle region can be defined, with a value equal to the average between the occupancies in the input and output leads. In this special case, the phase averaging procedure is equivalent to the effect of a completely dephasing mechanism [6] and therefore the semiclassical results are recovered. However, already in the case of 3 barriers, results differ from the semiclassical ones [2, 3], as a consequence of locahzation. Our results could be useful for the further investigation of experimental data obtained for superlattices [7], which do significantly differ from the expected 1/3 limit.

REFERENCES

1, C, W, J, Beenakker and M, Bilttiker, Phys. Rev. B 46, 1889 (1992), 2, M, J, M, de Jong and C, W, J, Beenakker, Phys. Rev. B 51, 16867 (1995), 3, M, J, M, de Jong and C, W, J, Beenakker, PhysicaAlM, 219 (1996), 4, G, B, Lesovik, Pis'ma Zh. Eksp. Tear. Fiz. 49, 513 (1989) [JETP Lett. 49, 592 (1989)], 5, M, Bilttiker, Phys. Rev Lett. 65, 2901 (1990), 6, H, Forster, P, Samuelsson, S, Pilgram, and M, Bilttiker, Phys. Rev B 75, 035340 (2007), 7, W, Song, A, K, M, Newaz, J, K, Son, and E, E, Mendez, Phys. Rev Lett. 96, 126803 (2006),

426

Current fluctuations in a dissipative environment Alessandro Braggio*, Christian Flindt ** andTomas Novotny**

*LAMIA-INFM-CNR, Dipartimento di Fisica, Universita di Genova,Via Dodecaneso 33,16146, Italy

^ Department of Physics, Harvard University, 17 Oxford Street, Cambridge, MA 02138, USA ** Department of Condensed Matter Physics, Faculty of Mathematics and Physics, Charles

University, Ke Karlovu 5, 12116 Prague, Czech Republic

Abstract. We have recently developed methods for calculating the statistics of a single counted quantity in a stochastic transport process described by a non-Markovian generaUzed master equation. Here, we expand on these ideas and extend our formalism to several counted quantities, allowing us to study cross-correlations between different variables. As an illustrative example, we consider charge transport through a double quantum dot capacitively coupled to a quantum point contact. We calcidate the cross-correlations between the current through the double quantum dot and the current through the quantum point contact, and demonstrate the sensitivity of this cross-correlator to measurement quantum back-action.

Keywords: GeneraUzed master equations, cross-correlations, quantum back-action PACS: 02.50.Ey, 72.70.4-m, 73.23.Hk

The dynamics of open systems can often be described via the evolution of the reduced density matrix of the system pit). For non-Markovian systems such a generalized master equation (GME) typically has the form [1, 2, 3]

p(n,t) = y dt'W(n-n',t-t')p(n',t') + f(n,t), (1) d_ '-t

dt

where the memory kernel W describes the influence of a dissipative environment on the dynamics of the system, while the inhomogeneity y accounts for initial correlations between system and environment. Here, the reduced density matrix of the system has been resolved with respect to the number of counted quantities «; (such as transferred electrons, emitted photons, or dissipated heat) contained in the vector « = («!,...,«;), and the corresponding w-resolved density matrix has been denoted by p(«,?). For a given system, the derivation of an equation like Eq. (1) may be a difficult task, but several examples can already be found in the hterature [4, 5, 6, 7].

In the following, we are interested in the statistics of the currents I = n and their cross-correlations. To this end, we define the cumulant generating function (CGF) S{x,t) corresponding to the probability distribution P{n,t) as e' ( '*) = Ln-P(«jO^'"'*' = I^„Tr{p(M,?)}e'"^, where % is the so-called counting field vector. The second equahty defines P(n,t) as the trace of the w-resolved reduced density matrix. For long times the CGF becomes linear in time, and the multivariate current cumulants can be obtained as the limit ((/('i' •'0))) = \imt-,oodtd'^^...dll.S{x,t)\:^_^Q. In Ref. [3], we derived a general expression for the CGF of a system described by a GME of the form given in Eq. (1) for a single counted quantity. As we shall now demonstrate, this derivation easily



427

carries over to the case of several counted quantities: First, we consider the probabihty distribution in ^-space, defined by the transformation P{x,t) = Y.nP{n,t)e^"^^. The corresponding GME is easily obtained from Eq. (1), and in the Laplace space it becomes an algebraic equation, which can formally be solved for p{x,z) in terms of the resolvent of the kernel ^(^ ,z) = [z — W{x,z)Y^. With the counting field vector set to zero, the system tends exponentially to a unique stationary state determined by the 1/z pole of the resolvent ^(0,z). The stationary state is given by the eigenvector corresponding to the zero-eigenvalue of Wo = W(p,z = 0), i.e., lim;^oop(0,?) = |0)), where |0)) is the normalized solution to Wo|0)) = 0. The corresponding left-eigenvector is denoted by ((0|, i.e., ((0|Wb = 0. Following Ref. [3], one can show for finite x that the imphcit solution zo{x) to the self-consistency equation zo — Ao(^,zo) = 0, where Ao(^,z) is the eigenvalue of W(^,z) with largest real-part, determines the long-time limit of the CGF, i.e., S{Xit) -^ zo{x)t- The multivariate current cumulants can thus be obtained as ((/(»i.-,0))) = dll^ ...d'^.zoix) \x^o- In the Markovian limit of the kernel W{x,z^ 0), we obtain zo(^) = Ao(^,0), consistent with previous results for the Markovian case [5, 8].

Applying the scheme developed in Ref. [3], we can express the cross-correlators in terms of the multivariate Taylor coefficients of W'(^,z) = W(^,z) — Wo =

I.Ti,...,ij,i=o ^'^'i^l'.'.tf '' f|w('i--'^-') with w(°'-'°'°) = 0 and the pseudoinverse of the kernel R defined as R = QW^^Q with g = 1 — |0))((0|. For simphcity, we only show the formula for the case of a two-dimensional counting vector X = {^-,X) corresponding to the variables n and m and the currents I = n and J = m. Generalizations to higher-dimensional counting vectors are straightforward. For the zero-frequency cross-correlator we find ((//)) = {{IJ))M + {{IJ))NM, where the Markovian term is

{{IJ))M = m (w(i'i'O) - W(i'0'0)w(0'i'0) - W(0'i'0)w(i'0'0)) |0)), (2)

and the non-Markovian contribution is

{{IJ))NM = Y. ((• '' "' ))((0| (w(i-''''i) - w(i-''''°)/?w(°'°'i)) |0)). (3) j=0, l ^ ^

Here, the stationary currents are ((/(i'°))) = (/) = ((0|w(i'°'°)|0)) and ((/(°'i))) = (/) = ((0|w(0'i'0)|0)).

We illustrate our results by considering a simple model of charge transport through a double quantum dot (DQD) with a nearby quantum point contact (QPC) used as a charge detector [10, 11, 12]. A thorough discussion of the model, including the notation used in the following, can be found in Ref. [13]. The DQD is operated in the Coulomb blockade regime close to a charge degeneracy point, where maximally a single additional electron is allowed to enter and leave the DQD. The DQD states are |0) (0 additional electrons on DQD) and \L/R) (electron in left/right quantum dot). The tunnel coupling between the left and right quantum dot is denoted by Tc, while e is the energy detuning of the two levels. The pseudo-spin system formed by \L) and \R) is tunnel-coupled to left (L) and right (R) leads via a tunnel-Hamiltonian with both leads described as reservoirs of non-interacting fermions. We will be using the symbol / to denote the current from

428

the left to right lead through the DQD. The QPC is described as a single tunneling barrier with a real energy-independent tunneling rate. Depending on the charge state of the DQD, \j) with j = 0,L,R, the barrier opacity of the QPC is modified and different values of the QPC current are obtained Ij, j = 0, L,R. In the following, we calculate the currents through the QPC and the DQD at zero temperature and in the limit of infinite source-drain biases. For IR ^ II the QPC introduces a decoherence mechanism on the DQD, since it effectively measures the electronic state of the DQD. In the following, we consider only the weakly responding limit of the QPC, such that \IR — IL\ <C (/) [14].

In order to describe charge transport through the DQD and the QPC we follow the scheme outlined in Ref. [10]. The counting variable n (m) is associated with the number of electrons counted in the right lead for the QPC (DQD) with the corresponding counting field denoted as (j> {%). The kernel W{x,^,z) in the Laplace space for the GME corresponding to the diagonal elements of the reduced density matrix of the DQD P = (PO,PL,PK)^ reads

0 YRC'^

-r{z,(P)+DLh{(p) r{z,(P) 1 (4) r{z,<j>) -r{z,<j>)-rR+DRh{<j>)

with h{(j>) = (e"^ — 1) and rate Ti (TR) for electrons to enter (leave) the DQD via the left (right) barrier between the DQD and the leads [13]. The number of electrons n that have tunneled to the right lead is increased by tunnel processes from the right dot with rate TR, and the counting factor e'^ consequently enters the corresponding off-diagonal element of the kernel. The hopping rates between the two quantum dots are r(z,(|)) = 7;2{|[z+((|))]+|[z-((|))]} with|[z] = l/zandz±((|)) =zT/e-rrf(( |)) /2, taking into account the coherent tunnel coupling Tc [13]. We note that the QPC counting field (j> enters the kernel at two different places: in the diagonal terms Djh{(j>) that describe the counting of tunneling events in the QPC charge detector with the tunneling rate Dj = Ij/e, and, most notably, in the non-Markovian hopping rates as a generalized dephasing rate rd{<j>) = [FR + {^/Dl-^/RRf - 2^/Dj^{e'^ - 1)]. For ^ ^ 0 the dephasing rate Y^ = T^{0) is due to the influence of the right lead and the nearby QPC charge detector on the coherent dynamics of the DQD [10]. We note, that if II = IR, the QPC does not detect the position of an electron on the double quantum dot and the DQD is not loosing coherence due to the presence of the QPC. The pecuhar ^-dependence of the dephasing rate Td{(j>) is related to the quantum correlations in the transport statistics of the combined DQD and QPC system, including quantum back-action due to the QPC measurement.

For the system described by the kernel in Eq. (4), we can apply our formulas for ((//)). We will consider only two cases: i) The averaged case, where decoherence is only due to the decoherence rate F^. ii) The full case, where the quantum QPC back-action is fully included. The averaged case corresponds to a calculation of the cross-correlations including only an averaged effect of the QPC on the dephasing. The stationary currents for the DQD and the QPC do not distinguish between these two cases. For the QPC current we find (/) = [4T,2rd{DRrL +DorR +DLFL) + DLrLrR{r]+4e2)]/r\ and the DQD (/) = 4Ti?TdTLTR/t^, where the denominators are f* = 4Ti?Td{2rL + TR) +

429

rLrK(r^ + 4e^). The cross-correlator ((//)) Avg for case i) is

(J) ((//))Avg = M^ IriRT.^r] {l6T,^rd{2Do-DL-DR)+4rl+rR{Do-DL))

+rLrR {Td (r4 + ([DR - DL) TL + {Do - DL) TR) (5)

+87;2r2 _ {TR {DL - 2Do) - 2DRTL)) - 2DLTI_TI^TLTR)

where Fg^ = F^ ± e^ and TLR = 2YL — Fg. In case ii) we find

{{IJ))Fun = {{tJ))AYg + p , (6)

and in particular, we see that we have in this case an additional contribution. This additional term contains information about the quantum back-action of the QPC on the DQD dynamics. Indeed, a measurement of the QPC current contains information about the state of the DQD and such a measurement consequently modifies the DQD dynamics. The main conclusion based on this result is that the back-action in this system is beyond the simple averaged approximation and that the cross-correlator is sufficiently sensitive to distinguish between the two cases. For the model considered here, the calculation could also be equivalently done by considering the density matrix of the DQD including off-diagonal elements (see Eq. (15) in Ref. [13]) with a resulting Markovian GME. The main advantage of the non-Markovian approach presented here is the possibihty to describe cases where the dynamics is intrinsically with memory, such as in the case of phonon induced dissipation. The only modification to include the dissipative dynamics is the use of the appropriate function g[z] [13]. A detailed discussion on the quantum back-action and current cumulants for the model considered here will be discussed elsewhere.

This work was supported by the INFM-CNR Seed Project, the Villum Kann Ras-mussen Foundation, the Czech Science Foundation via grant 202/07/J051, and research plan MSN 0021620834 financed by the Ministry of Education of the Czech Repubhc.

REFERENCES

1, R. Zwanzig, Nonequilibrium Statistical Mechanics, Oxford University Press, New York, 2001. 2, G, Schon, Yu, MakhUn, and A, Shnirman, Rev. Mod. Phys. 73, 357 (2001), 3, C, Hindt, T, Novotny, A, Braggio, M, Sassetti and A,-P, Jauho, Phys. Rev. Lett 100, 150601 (2008), 4, M, B, Plenio and P L, Knight, Rev Mod. Phys. 70, 101 (1998), 5, D, A, Bagrets and Yu, V, Nazarov, Phys. Rev B 67, 085316 (2003), 6, A, Braggio, J, Konig, and R, Fazio, Phys. Rev Lett 96, 026805 (2006), 7, C, Hindt, A, Braggio, and T, Novotny, A/P Conf. Proc. 922, 531 (2007), 8, C, Hindt, T, Novotny, and A,-P Jauho, Europhys. Lett. 69, 475 (2005), 9, C, Hindt, T, Novotny, and A,-P Jauho, Phys. Rev B 70, 205334 (2004), 10, S, A, Gurvitz, Phys. Rev B 56, 15215 (1997), 11, A, N, Korotkov, Phys. Rev B 63, 115403 (2001), 12, D, V, Averin and E, V, Sukhorukov, Phys. Rev Lett. 95, 126803 (2005), 13, A, Braggio, C, FUndt, T, Novotny, /, Stat. Mech. P01048 (2009), 14, R, Ruskov and A, N, Korotkov, Phys. Rev B 67, 075303 (2003),

430

Asymmetry of the excess finite-frequency noise I. Safi

Laboratoire de Physique des Solides, Universite Paris-Sud, 91405 Orsay, France

Abstract. We consider finite fi-equency noise in a mesoscopic system with arbitrary interactions, connected to many terminals kept at finite electrochemical potentials. We show that the excess noise, obtained by subtracting the noise at zero voltage fi-om that at finite voltage, can be asymmetric with respect to positive/negative fi-equencies if the system is non-linear. This explains a recent experimental observation in Josephson junctions as well as strong asymmetry obtained in typical non-linear and strongly correlated systems described by the Luttinger liqiud (LL): edge states in the fractional quantum Hall effect, quantum wires and carbon nanotubes. Another important problem where the LL model applies is that of a coherent conductor embedded in an ohmic environment.

Keywords: Finite frequency noise, AC differential conductance, Dynamical Coidomb Blockade, Luttinger Uquid PACS: 72.70.4-m,72.10.-d,73.23.-b

The finite frequency (FF) noise, obtained by Fourier transform the second cumulant of the current, has attracted a lot of interest in the mesoscopic community [1, 2], and has been possible to measure in the quantum regime, i. e. at frequencies higher than temperature. Beyond the average current, it offers already a powerful tool to reveal the charge and statistics of the charge carriers of a system when considered at zero-frequency. Nevertheless, it contains more rich informations at FF, such as on elementary excitations, typical energy scales, and especially on interactions, with the possibihty to check the underlying model or access the correlation strength etc...

It has been often stated in theoretical studies that one needs to symmetrize the current correlators, as they don't commute at different times, which leads to a FF symmetrized noise, symmetric with respect to positive/negative frequencies. Nevertheless, it has been often explored by the scattering approach at low frequencies compared to characteristic energy scales [2]. Few theoretical works went beyond this framework[3]. It has also been studied in systems where interactions intervene at any frequency scale in the FF noise such as chiral edge states in the Fractional Quantum Hall effect (FQHE), quantum wires and carbon nanotubes, described by the Luttinger liquid (LL) model, which leads to exotic phenomena such as charge fractionalization [5], spin-charge separation, and fractional statistics. It has been shown that LL is suited as well to describe a coherent conductor embedded into an ohmic environment[4]. The symmetrized FF noise has been studied in the FQHE in Ref. [6], and in Refs. [7] for quantum wires and carbon nanotubes connected to metalhc leads, where it was shown that, while the presence of the metalhc leads obscures it in the zero-frequency noise, the charge fractionalization is still present and can be extracted from the noise at high frequencies.

Nevertheless, recent experiments [8] have got access to the non-symmetrized noise [9], and thus to the emission and the absorption components of the noise spectrum given by the noise at positive/negative frequencies. What is usually measured is the excess



431

non-symmetrized noise, defined as the difference between the noise at finite voltage and at zero voltage, thus allowing to get rid of some undesirable effects. In the framework of the scattering approach, the non-symmetrized noise is not even with respect to frequency, but the excess non-symmetrized one is, thus yields again the excess symmetrized noise. This makes it difficult to argue which noise is measured in that case, and a criteria for choosing systems with an asymmetric excess noise is required.

Indeed even fewer theoretical works have dealt with the non-symmetrized FF noise beyond the framework of scattering approach [10]. It was first investigated in strongly correlated systems such as the FQHE edge states [11], a coherent conductor in an ohmic environment, and quantum wires and carbon nanotubes connected to charge reservoirs [12]. In particular, the non-symmetrized excess noise was found to be asymmetric. One could argue that such asymmetry needs Coulomb interactions to hold on, nevertheless the criteria was indeed discovered to be related to non-linearity [12, 13], as we will be explained now. This enlightens a recent experiment measuring asymmetric FF excess noise in Josephson junctions [8].

For that, we need an out-of-equilibrium generahzation of the Kubo formula. The latter might be thought to be restricted to linear transport. Nevertheless, it has been extended to the non-equilibrium case, looking at infinite systems subject to a uniform electric field, and relating the AC homogeneous conductivity to the retarded current-current correlation function computed in the presence of the finite DC bias [14] with the requirement of a stationary density matrix. This misses however the effects of the non-locahty, which are important in a mesoscopic context and where the electric field is not uniform for instance. A straightforward demonstration, extended in addition to any time-dependent Hamiltonian and valid generally for any finite mesoscopic non-linear system with many terminals, is presented in Ref. [13]. Here we focus on the stationary case. We consider a general mesoscopic system connected to many terminals under DC electrochemical potentials jU„ = eV„. Let's add a small AC modulation Vm(co)e'®* to the DC potential Vm in terminal m. The differential of the average current at a terminal n in the limit of vanishing v^: G„m{o)) = 5 < 4((o) >/5vm(co) |v^=o is an element of the AC differential conductance matrix Gy where V = (Vi,.., V/v) recalls its dependence on the DC potentials in the N terminals. One can show the generahzed Kubo formula [13]: (oG{(o) = l/hD^{(o) + iM, where D^„(?) = 9{t) ([/„(?),/„(0)]). As a consequence, we can show that the non-symmetrized FF noise matrix Sv(©), whose elements are given by:

/

oo

dte''^' [(/™(0)4(0) - < / « > < I„ >], (1) -oo

has its asymmetric part given by the dissipative AC differential conductance matrix:

S v ( - « ) - S v ( « ) = /jco9tGv((o). (2)

In particular, this is obeyed by the noise at zero voltage, i. e. at equilibrium, where the FDT yields: Sy=o(«) = 2;jcoA?(co)9tGy=o(«), with N{(o) = l /(e^®+ 1). Let's consider now the excess noise matrix, Sy" (co) = Sy(co) — Sv=o(®)- Taking the difference between Eq.(2) at finite and zero voltage vector, one gets:

S^^'{-(o)-S^^'{(o) = 2h(o[3iGv{(o)-3iGv=o{(o)]. (3)

432

WBS

SBS

<v>

T

A.

FIGURE 1. On the left: A one-channel coherent conductor with transmission coefficient 3", in series with an impedance Z((u) = S f o r (U < ffife = 1/SC, is mapped to a LL with parameter ^ = 1/(1 + /•) where r = e^Rjh. The strong (weak) backscattering limit denoted by SBS (WBS) corresponds to the tunneling (weak backscattering) regime with a dimensionless amplitude Y'j (v^). On the right: The excess non-symmetrized FF noise 5y((u) — 5y=o((y) (scaled by a renormalized reflection coefficient), as a function of frequency, in units of o^ = eV/h, for S = 0 (dashed line), R = Ihje^ (fidl line) and R = hje^ (dotted line). Here ksT/haio = 0.03.

This shows a necessary condition to get asymmetry in the excess noise between positive and negative frequencies: non-linearity, such that 9tGy depends on voltage!

Let us now illustrate this in a problem relevant for interactions and environment role in mesoscopic physics, referring to [11] for details and other typical non-linear systems.

A mesoscopic conductor embedded in an electrical circuit forms a quantum system violating Ohm's laws. The transmission/reflection processes of electrons through the conductor excite the electromagnetic modes of the environment, rendering the scattering inelastic, and reducing the current at low voltage, an effect called dynamic Coulomb blockade (DCB)[15]. This picture, vahd in the limit of a weak conductance, changes in the opposite limit of a good conductance. The description of tunneling via discrete charge states becomes then ill defined, raising the question of whether DCB survives or is completely washed out by quantum fluctuations. A challenging relationship between the DCB reduction of the current in a one-channel conductor in series with a weak impedance and the noise without impedance was proposed in [16]. We were able to explore the case of an arbitrary resistance R in series with a coherent one channel conductor with good transmission. At energies below (OR = l/RC (C the effective capacitance of the circuit) a one-channel conductor embedded in its ohmic environment behaves exactly like a point scatterer[18] in a LL liquid, with an effective LL parameter K = 1/(1 + r) (Fig.(l)). This allowed us to determine in a non-perturbative way the effect of the environment on / — V curves, and to find an exact relationship between DCB and shot noise which applies for aU energy ranges below COR and for any resistance, as well as the full counting statistics. Furthermore, such mapping was later confirmed using completely different approach for many channel system, but restricted to the high voltage regime [17]. It seems that our mapping could be generahzed to this case, provided the resistance of the channels is incorporated in the resistance of the environment R, thus the LL tums out to have more universal interest. One can also apply the first investigation

433

of non-symmetrized FF noise achieved in [11] in perturbation with respect to 1 — 5^. Fig.(l) shows strong asymmetry in the FF noise, especially when R is higher than h/e^ where the DCB is known to be strongest in the tunneling regime [19]. Measuring the FF excess noise would give a crucial test for our mapping. Notice that this offers a potential and interesting alternative to differentiate the role of intrinsic interactions in quantum wires and carbon nanotubes [12] from the role of the environment.

ACKNOWLEDGMENTS

The author thanks C. Bena and A. Crepieux for collaborations on this subject, and E. Shukhurokov, R. Deblock and C. D. Glattli for stimulating discussions.

REFERENCES

1. R.J. Schoelkopf, P.J. Burke, A.A. Kozhevnikov, D.E. Prober, and M.J. Rooks, Phys. Rev. Letters 78, 3370 (1997); R. Aguado and L.P Kouwenhoven, Phys. Rev. Letters 84, 1986 (2000); R. Deblock, E. Onac, L. Gurevich, and L.P. Kouwenhoven, Science 301, 203 (2003)

2. V. A. Khlus, Sov Phys. JETP 66, 1243 (1987); G.B. Lesovik, JETP Letters 49, 592 (1989); M. Bilttiker, Phys. Rev. Letters 65, 2901 (1990). For a review, see Y. M. Blanter and M. Bilttiker, Phys. Rep. 336,1 (2000).

3. E.V. Sukhorukov, G. Burkard, and D. Loss, Phys. Rev B 63, 125315 (2001). G. Johansson, A. Kifjack, and G. Wendin, Phys. Rev. Lett. 88, 046802 (2002); A. Kifjack, G. Wendin, and G. Johansson, Phys. Rev. B 67, 035301 (2003); A.V. Galaktionov, D.S. Golubev, and A.D. Zaikin, Phys. Rev B 68, 085317 (2003).

4. I. Safi and H. Saleur, Phys. Rev Lett. 93, 126602 (2004). 5. I. Safi, Ann. Phys. (France) 22, 463 (1997); K.-V. Pham, M. Gabay, and P Lederer, Phys. Rev B 61,

16 397 (2000). 6. C.de C. Chamon, D. E. Freed, and X. G. Wen, Phys. Rev B 53, 4033 (1996). 7. B. Trauzettel, I. Safi, F Dolcini, and H. Grabert, Phys. Rev Letters 92, 226405 (2004); H.-A. Engel

and D. Loss, Phys. Rev Letters 93, 136602 (2004); A.V. Lebedev, A. Crepieux, and T. Martin, Phys. Rev B 71, 075416 (2005). F Dolcini, B. Trauzettel, I. Safi and H. Grabert, Phys. Rev B 75, 045332 (2007).

8. P-M. BUlangeon, F Pierre, H. Bouchiat, and R. Deblock, Phys. Rev Lett. 96, 136804 (2006). E. Zakka-Bajjani, J. Segala, F Portier, P. Roche, D.C. Glattli, A. Cavanna, and Y. Jin, Phys. Rev. Lett. 99, 236803 (2007).

9. G.B. Lesovik and R. Loosen, Z. Phys. B 91, 531 (1993). M. Creux, A. Crepieux, and T. Martin, Phys. Rev B 74, 115323(2006).

10. F W. J. Hekking and J.P Pekola, Phys. Rev Letters 96, 056603 (2006). 11. C. Bena and I. Safi, Phys. Rev B 76, 125317 (2007). 12. I. Safi, C. Bena, and A. Crepieux, Phys. Rev B 78, 205422 (2008). 13. I. Safi, in preparation. 14. U. Gavish, Y Levinson, and Y Imry, Phys. Rev B 62, R10637 (2000). 15. M. H. Devoret et al, Phys. Rev Lett. 64, 1824 (1990); G.-L. Ingold and Y V. Nazarov, in Charge

Tunneling Rates in Ultrasmall Junctions, B294, H. Grabert and M. H. Devoret ed. (Plenum Press, New York, 1992), Chap. 2, p. 21.

16. D. S. Golubev and A. D. Zaikin, Phys. Rev. Lett. 86,4887 (2001). A. Levy Yeyati, A. Martin-Rodero, D. Esteve, and C. Urbina, Phys. Rev. Lett. 87, 046802 (2001). M. Kindermann and Y V. Nazarov, Phys. Rev Lett. 91,136802 (2003).

17. D. S. Golubev, A. V. Galaktionov and A. D. Zaikin, Phys. Rev B. 72 205417 (2005). 18. whose strength is related to the transmission coefficient ,S^. 19. The symmetrized noise is similar to that obtained in [17] for many channels.

434

Magnetoasymmetric noise in an Aharanov-Bohm interferometer

David Sanchez, Jong Soo Lim and Rosa Lopez

Departament de Fisica, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain

Abstract. We investigate the shot noise of an Aharanov-Bohm interferometer with a quantum dot inserted in one of its arms. We consider the case of an external magnetic flux piercing the ring. When a dc bias is apphed between the two terminals attached to the ring, a magnetoasymmetry arises in the nonhnear regime of transport. In particular, we show that the noise susceptibility is an uneven function of the magnetic field, closely fulfilling a nonlinear fluctuation-dissipation relation that connects the noise and the nonlinear conductance magnetoasymmetries. We discuss the differences arising in the high and low temperature regimes.

Keywords: Quantum noise, magnetoasymmetry, nonlinear mesoscopic transport PACS: 73.23.-b, 73.50.Fq, 73.63.Kv

INTRODUCTION

In the nonlinear regime of electron transport, screening effects can lead to a breakdown of microreversibility in mesoscopic systems [1,2]. The nonequilibrium response of the potential landscape along the system is, quite generally, an uneven function of the magnetic field, implying a magnetoasymmetry of the differential conductance. Such effect has been observed in the current-voltage characteristic of Aharanov-Bohm (AB) rings [3] and, more recently, in mesoscopic interferometers with a quantum dot embedded in one of its arms [4].

Here, we are interested in the magnetoasymmetries arising in the current fluctuations of a quantum-dot AB ring. The study of the shot noise in mesoscopic systems is an active area of current research [5]. The interest for magnetoasymmetric current fluctuations stems, in part, from recent theoretical developments [6, 7] which relate the asymmetry of the noise to the asymmetry of the differential conductance in the leading order of a voltage expansion. As a consequence, a higher-order fluctuation dissipation theorem is established between the two magnetoasymmetries. Nevertheless, explicit calculations to check the relation are based on models which treat interactions in a mean-field way. Here, we take a step further and investigate how strong Coulomb correlations affect the shot noise magnetoasymmetry.

THE SYSTEM

We consider a two-terminal AB ring with a dot inserted in one of its arms. The Hamil-tonian reads,



435

where ^ = I.a=LlR,k,a£-kaCâko'âka + lLk,K,a\We''!'c\y^Cika + h.C^, Mh =

Yâ^ddada + Urid^ridi and Jfr = la=L/R,k,a yaCî^^da + h.c^. Mc describes the energy states e ^ in the leads a = L/R, which are coupled via a direct transmission channel with amplitude W. The wavevector is denoted with k while the spin index is C7 =1,1 and the AB phase is cp. .^D models the single-level dot with energy e^ and charging energy I]. Finally, ^ accounts for the coupling "Va between the dot and the leads.

To break microreversibility in a quantum-dot AB ring, one needs a finite dc bias and f/ 7 0 [8, 9]. This can be understood from the dot retarded Green function, which in the mean-field approximation reads Gj^((«) = [« — e^ — l](^da) + ^ v ^ r c o s ^ + f r ] ^ ' , where f = r / ( l +<g) and .% = AB,/{\ + B,f, with r = Anp\'f'\^ (p is the leads' density of states and, for simplicity, we take 'fL = '^R = '^) and t, = n^p^W^. We note that the background channel induced energy shift proportional to cos is symmetric under field reversal but, in general, the dot occupation {nd^) is not an even function of ^. We can expand the occupation in powers of the voltage V, {rida) = {nda)'^^^ + {nda)'^^^V + ff{VY, where {rida)'^^^ is related to the equilibrium charge in the dot and hence it is (p-symmetric. The leading-order nonequilibrium response of the system is described by the charge susceptibility {nda)'^^\ which plays a role similar to the characteristic potentials of Ref. [1]. In general, {nda)^-^^ can be shown to be ^-asymmetric [10]. As a result, the scattering amplitude Sa a', which is a function of Gj^ [11], breaks microreversibility:

RESULTS

The average current / in the stationary limit can be found from / = (/«(?)) = —ie{[Jf,nii])/Ti using Eq. (1), where —e is the electron charge and HR = 'E.ka'^Rka'^Rko-Hence, / is expressed in terms of nonequilibrium Green functions which are determined from an equation-of-motion approach. The U term in J^D generates higher-order Green functions which are decoupled following a truncation method [12].

On the other hand, the noise power at zero frequency is given by

S=2l^J[AIii{t),AIii{t')]+)dt, (2)

where AlR^t) = iR^t) — (IR). The definition of .S" already contains two-body Green functions. We make use of a cluster expansion to write S in terms of one-body Green functions. Details of the calculation will be given elsewhere [10]. Here, we limit ourselves to the discussion of an illustrative result.

High temperature regime.—The equation-of-motion approach yields reliable results for temperatures T much higher than the Kondo temperature TK. We focus on the case of a dc bias much lower than TK and expand the current and noise in powers of V:

I=GiV + G2V^ + ff{Vf, (3a)

S = So + SiV+ff{Vf. (3b)

436

Gi is the linear conductance and fulfills the Onsager symmetry, namely, Gi(^) = Gi (—^). The leading-order rectification coefficient G2 is generally ^-asymmetric [1, 2]. Moreover, .S'o is the thermal noise and is related to Gi via the Jonhson-Nyquist relation .S'o = AKBTG\ . Thus, .S'o is ^-symmetric.

We now define the magnetoasymmetries of conductance and noise,

G_^^^^m^I^ = GM) + nv)\ (4a)

S-{<p) = '-^^^^^ = S, + rAV)\ (4b)

where we have used the even-odd behavior of the transport coefficients [3, 10]. References [6, 7] find

S4(P)=AKBTG4(P), (5)

which is a higher-order fluctuation-dissipation relation valid in the presence of a magnetic field. We numerically checked this relation for a quantum-dot AB interferometer in the case of strong correlations (IJ -^ 0°). Our results are summarized in Fig. 1. The relation is closely fulfilled in all cases. In Fig. 1 (a) we take ff = 0.1, in which case the partial waves are less likely to be transmitted along the reference arm. The electron correlations in the dot are then stronger and the deviations from Eq. (5) become more significant. Furthermore, the difference is higher for lower temperatures, as our model starts to break down due to Kondo correlations. Therefore, we attribute the deviations from Eq. (5) to the systematic errors induced by the truncation method. Despite this small inaccuracy (always smaller than 2%), the numerical results are consistent with Eq. (5). This can be more clearly observed in Fig. 1 (b), where a higher background transmission is selected (W = 0.6). In this case, dot correlations are less important and Eq. (5) is satisfied almost exactly for high temperatures (T = 40TK), for which the equation-of-motion approach is a rather good approximation.

Low temperature regime.—For very low temperatures such that T <C Tk the Fermi liquid properties of the system can be invoked to write down an exact Green function for the quantum-dot AB system even in the presence of Coulomb interaction [13, 14]. At r = 0 and in equilibrium, the retarded Green function becomes ''^^^{Ep) = (1 + t,)/2iT, where we have used the fact that the imaginary part of the interacting self-energy vanishes. As a consequence, the full transmission probability can be simplified as =5 (e) = 1 — 5i,cos^(^). In the limit F ^ 0, we find the noise,

S=^^bC05^{^) [1 - 56Cos2(^)] V+ff{yf. (6)

Therefore, the noise susceptibility .S'l is an even function of the magnetic field. From this, we conclude that the magnetoasymmetry S- vanishes at very low temperatures. Clearly, deep in the Kondo regime the charge cannot fluctuate and one obtains {n^^) = 1/2. Hence, microreversibility of the scattering matrix is recovered at finite V and U. This result holds only in the case of voltages much lower than the width of the Kondo resonance, which is of the order of KBTK.

A31

-0.012 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75

FIGURE 1. (Color online) Magnetoasymmetriesof noise 5'_((p) and current/_((p) as a function of the magnetic flux (p. We take the bandwidth D = 1, dot level e^ = —0.05, broadening 2 r = 0.031 and voltage bias V = 0.001 TK for (a) ^ = 0.1 and (b) W = 0.6.

CONCLUSIONS

We have studied the noise magnetoasymmetries in a quantum-dot Aharanov-Bohm interferometer. We find for very large charging energies a strong resemblance between the noise and the conductance magnetoasymmetries, which is consistent with a nonlinear fluctuation-dissipation relation between the magnetoasymmetries. Future theoretical investigations should shed further light on this finding.

ACKNOWLEDGMENTS

This work was supported by the Spanish MICINN Grant No. FIS2008-00781.

REFERENCES

1. D. Sanchez and M. Buttiker, Phys. Rev. Lett. 93, 106802 (2004). 2. B. Spivak and A. Zyuzin, Phys. Rev. Lett 93, 226801 (2004). 3. R. Leturcq et al.,Phys. Rev Lett 96, 126801 (2006). 4. V. Puller, Y. Meir, M. Sigrist, K. Ensslin, and T. Ihn, arxiv:0902.2706 (unpubhshed, 2009). 5. Ya.M. Blanter and M. Buttiker, Phys. Rep. 336, 1 (2000). 6. H. Forster and M. Buttiker, Phys. Rev Lett. 101, 136805 (2008). 7. D. Sanchez, Phys. Rev B 79, 045305 (2009). 8. C. Bruder, R. Fazio, and H. Schoeller, Phys. Rev Lett. 76, 114 (1996). 9. J. Konig and Y. Gefen, Phys. Rev S 65, 045316 (2002). 10. J.S. Lim, D. Sanchez, and R. Lopsz,preprint (2009). 11. T.K. Ng and P.A. Lee, Phys. Rev Lett 61, 1768 (1988). 12. C. Lacroix, J. Phys. F11,2389 (1981). 13. W. Hofstetter, J. Konig, and H. Schoeller, Phys. Rev Lett. 87, 156803 (2001). 14. R. Lopez, D. Sanchez and LI. Serra, Phys. Rev B 76, 035307 (2007).

438

Negative excess noise in gated quantum wires F. Dolcini*, B. Trauzettel^ I. Safi** andH. Grabert*

*Scuola Normale Superiore & NEST-INFM-CNR, 56126 Pisa, Italy Înstitutfiir Tfieoretiscfie Pfiysili undAstropfiysili, Univ. Wiirzburg, 97074 Wiirzburg, Germany

**Lahoratoire de Ptiysique des Solides, tJniversite Paris-Sud, 91405 Orsay France ^Ptiysilcaliscties Institut, tJniversitdt Freiburg and FRIAS, 79104 Freiburg, Germany

Abstract. The electrical current noise of a quantum wire is expected to increase with increasing applied voltage. We show that this intuition can be wrong. Specifically, we consider a single channel quantum wire with impurities and with a capacitive coupling to a metallic gate, and find that its excess noise, defined as the change in the noise caused by the finite voltage, can be negative at zero temperature. This feature is present both for large (c':^ Cq) and small (c<s:.Cq) capacitive coupling, where c is the geometrical and Cq the quantum capacitance of the wire. In particular, for c~S> Cq, negativity of the excess noise can occur at finite frequency when the transmission coefficients are energy dependent, i.e. in the presence of Fabry-Perot resonances or band curvature. In the opposite regime c < Cq, a non trivial voltage dependence of the noise arises even for energy independent transmission coefficients: at zero frequency the noise decreases with voltage as a power law when c < CqI'i, while, at finite frequency, regions of negative excess noise are present due to Andreev-type resonances.

Keywords: Excess noise, Luttinger liquid, quantum wire, quantum capacitance PACS: 72.70.+m,72.10.-d,73.23.-b

In equilibrium the fluctuations of an observable are simply related to its response function through the fluctuation-dissipation theorem. Non-equilibrium noise, in contrast, provides essentially more information than the average current. In a ballistic conductor with negligible electron correlations, for instance, the shot noise at zero temperature reads [1]

S{oi = {},V) = êVYÛ\-Tn), (1)

where r„ are the transmission coefficients of the eigenchannels of the conductor, and V is the applied bias. In recent years, formidable efforts have been made to predict [2] and measure [3] the finite frequency dependence of the noise. In contrast, not so much interest has been devoted to the voltage dependence of the noise. Intuitively, one would expect that the noise should increase with the voltage V, as also suggested by Eq. (1).

This intuition is, however, wrong in general. For a single-channel wire, for instance, Lesovik and Loosen [4] have shown that in the particular case where T is non-vanishing only in an energy window 5E, the Fermi energy lies within this energy window, and the temperature is sufficiently high (feT > 5E), the shot noise in the regime eV ~> ksT is smaller than the equilibrium noise. In the opposite limit of a multichannel clean wire it has been demonstrated [5] that, although the current noise is always increasing with V, the voltage noise may decrease with bias. These results, however, are concerned with either the high temperature regime or the semiclassical limit (number of channels tending to infinity) of transport. Since most of the experimental interest in mesoscopic



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conductors lies instead in quantum effects, the open question is whether similar behavior can occur in the deep quantum regime, i.e. for a finite number of channels at low temperatures, where Hco > feT. This paper aims at investigating this problem. We consider here a single channel quantum wire at zero temperature, and analyze the conditions under which the noise, both at zero and at finite frequency, can decrease with bias. In particular we investigate whether the excess noise

SEx{x,a),V) = S{x,a),V)-S{x,a),0), (2)

characterizing the change of the current noise due to the finite voltage with respect to the equilibrium case, can be negative. The total noise spectrum is directly related to the current spectral density 7(x,(«) through

S{x,(o,V) = —J d(o'{AJ{x,(o)AJ{x,(o')) , (3)

and is always positive, as follows from the Wiener-Khintchine theorem. A negative excess noise means that driving the system out of equilibrium by applying a voltage V reduces the noise in certain frequency regions with respect to its equilibrium value. Importantly, the determination of finite frequency noise requires that interaction effects must be accounted for. Indeed the continuity equation implies that the net flux of current flowing into the conductor equals the time rate of change of the charge Q in the conductor

L L [h j{i^;oi)-j{--;a)) = j Jxdwj{x;a)) = ia)Q{a)) , (4)

where x = ±L/2 are the locations of the edges of the conductor of length L. The wire is capacitively coupled, through a geometrical capacitance C, to its electromagnetic environment consisting of other metallic conductors nearby. The right hand side of Eq. (4) can then be interpreted as the displacement current Jd{co) = icoQ{co) through the capacitance C. Furthermore, the capacitive coupling induces a fluctuating shift Af/((«) ^ Q{(o)/C of the band-bottom of the wire which modifies the energy of the electrons, affecting in turn their scattering processes inside the conductor and therefore the current j{x, co) itself It is this feed-back process that makes the problem of determining finite frequency properties an essentially interacting problem [6]. In a DC-biased set-up, the average current / = {j\x, t)) is independent of the position x and the time t, so that the average current spectral density is simply {J{x,co)) = 27tl5{co), and the displacement current has vanishing expectation value. For the noise, however, the full frequency-dependence of the fluctuations of the current spectral density plays a role (see Eq.(3)), and the band-bottom shift induced by the displacement current fluctuations has to be taken into account.

Here we analyze the finite frequency noise for the case of a ballistic single-channel quantum wire, capacitively coupled to a gate. For simplicity, we shall treat the capacitance C as uniformly distributed along the wire, C = cL, where c is the capacitance per unit length, and L the length of the wire. The Hamiltonian of the system thus reads

440

g eV / Ml

FIGURE 1. Excess noise for a gated wire with capacitance c = Cq/10, with an impurity shifted by L/4 off the center of the wire. Regions of negative excess noise are visible at frequencies co -^ 7va>i/g, where a>i = vp/gL is the ballistic frequency and g is the parameter defined in Eq. (6).

H = Ho + Hi^p+Hc, where HQ describes the band kinetic energy of the electrons, /4np the scattering with impurities, and

Hr 1 /•i/2

2c J-L/2 5p^ (x) dx (5)

the capacitive coupling to the gate, where 5p (x) = p (x) — pp is the deviation of the electron charge density p (x) from the density pp of an electroneutral wire. The capacitance c is a crucial quantity to determine the finite frequency noise, for it relates the displacement currents to the band-bottom shift AU in the wire. In the regime c » Cq, where c^ = e^v is the quantum capacitance and v is the density of states per unit length of the electron band described by HQ, it is evident that a finite displacement current yields a very small shift AU. The term (5) is then negligibly small with respect to HQ, and therefore the problem of determining the particle current decouples from the evaluation of the displacement current. For a finite capacitance, however, the two problems have to be solved simultaneously, and the band-bottom shift cannot be neglected. In this paper we address this problem by exploiting a mapping to a Luttinger liquid model. It is, in fact, well known [7] that the problem of a quantum wire with a geometric capacitance c with respect to a gate can be mapped onto a Luttinger liquid with interaction strength

c . (6)

where Cq = e^/nhvp is the quantum capacitance. The case of large geometrical capacitance O Cq corresponds to g = 1 (no charge screening), whereas the opposite limit of c <C Cq, i.e. g ^ 0, corresponds to a completely electroneutral wire (full charge screening). In any intermediate case, only a fraction 1 —g^ of the charge is screened [7].

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In a ID wire with an impurity a capacitive coupling (5) to a gate has dramatic effects on the transport properties. The interplay of Friedel oscillations with coupling to the gate leads to a strong renormalization of the impurity strength. Furthermore, the difference in capacitive coupling strength between the gated region and the metallic lead region can yield partial reflections of charge excitations (Andreev-like reflections) [8]. Thus, the interference between Andreev-type reflections and backscattering at the impurity gives rise to notable structure in the voltage and frequency dependence of the excess noise [9], as illustrated in Fig. 1, where SEX is plotted as a function of V and ca for the case of a small capacitance c = c^/10 quantum wire. Our result shows that regions of negative excess noise are indeed present at frequencies of order coi/g. Similar conclusions can be obtained from the analysis of the ac differential conductance and non-symmetrized noise [10]. With state-of-the-art techniques using on-chip noise detection schemes, we expect that these predictions can be verified in current experimental realizations of ballistic ID conductors, such as quantum wires based on GaAs/AlGaAs heterostructures, or single-wall carbon nanotubes [11].

ACKNOWLEDGMENTS

Stimulating discussions with M. Biittiker, R. Deblock, and L.P. Kouwenhoven, and financial support from the Italian MIUR Rientro del Cervelli Program are acknowledged.

REFERENCES

1. V. A. Khlus, Sov. Phys. JETP 66, 1243-1249 (1987); G.B. Lesovik, JETP Letters 49, 592-594 (1989); M. Buttiker, Phys. Rev. Letters 65, 2901-2904 (1990).

2. B. Trauzettel, I. Safi, F. Dolcini, and H. Grabert, Phys. Rev Letters 92, 226405 (2004); H.-A. Engel and D. Loss, Phys. Rev. Letters 93, 136602 (2004); A.V. Lebedev, A. Crepieux, and T. Martin, Phys. Rev. B 71, 075416 (2005); F.W.J. Hekking and J.P Pekola, Phys. Rev. Letters 96, 056603 (2006).

3. R. J. Schoelkopf, P. J. Burke, A. A. Kozhevnikov, D. E. Prober, and M. J. Rooks, Phys. Rev. Letters 78, 3370-3373 (1997); R. Aguado andL. P Kouwenhoven, Phys. Rev. Letters 84,1986-1989 (2000); R. Deblock, E. Onac, L. Gurevich, and L. P Kouwenhoven, Science 301, 203-205 (2003); L.-H. Reydellet, P Roche, D.C. Glattli, B. Etienne, and Y. Jin, Phys. Rev. Letters 90, 176803 (2003); P-M. Billangeon, F Pierre, H. Bouchiat, andR. Deblock, Phys. Rev Letters 96, 136804 (2006).

4. G. B. Lesovik and R. Loosen, Z. Phys. B 91, 531-536 (1993). 5. O. M. Bulashenko and J. M. Rubi, Physica E 17, 638-639 (2003). 6. T. Christen and M. Buttiker, Europhys. Letters 35, 523-528 (1996); M. Buttiker, H. Thomas, and

A. Pretre, Phys. Letters 180A, 364-369 (1993); M. Buttiker, A. Pretre, and H. Thomas, Phys. Rev Letters 70, 4114-4117 (1993); M. H. Pedersen, S. A. van Langen, and M. Buttiker, Phys. Rev. B 57, 1838-1846 (1998); Ya. M. Blanter, F W. J. Hekking, and M. Buttiker, Phys. Rev Letters 81, 1925-1928 (1998); Ya. M. Blanter and M. Buttiker, Europhys. Letters 42, 535-540 (1998); I. Safi, Eur. Phys. J. B 12,451-455 (1999).

7. L.I. Glazman, I.M. Ruzin, and B.I. Shklovskii, Phys. Rev B 45, 8454-8463 (1992); R. Egger and H. Grabert, Phys. Rev Letters 79, 3463-3466 (1997).

8. I. Safi and H. J. Schulz, Phys. Rev. B 52, R17040-R17043 (1995). 9. F. Dolcini, B. Trauzettel, I. Safi, and H. Grabert, Phys. Rev B 75, 045332 (2007). 10. I. Safi, C. Bena, and A. Crepieux, Phys. Rev. B 78, 205422 (2008). 11. R. de Picciotto, H. L. Stormer, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, Nature 411, 5 1 -

54 (2001); M. Bockrath, D. H. Cobden, J. Lu, A. G. Rinzler, R. E. Smalley, L. Balents, and P L. McEuen, Nature 397, 598-601 (1999).

442

Fluctuation relations without micro-reversibility in nonlinear transport

Heidi Forster* and Markus Biittiker^

*Iivititute for Environment and Human Security, United Nations University, UN-Campus, Hermann-Ehlers-Str 10, D-53113 Bonn, Germany

^Department of Theoretical Physics, University of Geneva, CH-1211 Geneva 4, Switzerland

Abstract. In linear transport, the fluctuation-dissipation theorem relates eqiuUbrium current correlations to the Unear conductance coeflicient. Theory and experiment have shown that in small electrical conductors the non-linear I-V-characteristic of two-terminal conductor exhibits terms which are asymmetric in magnetic field and thus micro-reversibiUty is manifestly broken. We discuss a non-eqiulibrium fluctuation dissipation theorem which is not based on micro-reversibility. It connects the antisymmetric nonUnear conductance with the third cumulant of eqiulibrium current fluctuations and a noise term that is proportional to temperature, magnetic field and voltage.

Keywords: non-eqiulibrium fluctuation-dissipation theorem, magnetic field asymmetry, absence of micro-reversibiUty PACS: 73.23.-b, 05.40.-a, 72.70.4-m

Linear and nonlinear transport coefficients

The linear transport regime is governed by microscopic reversibility. Based on this principle, Onsager derived the symmetry of transport coefficients of irreversible processes [1]. For electrical conductance, this means that the linear conductance Gi of a two-terminal conductor is an even function of magnetic field. Another consequence of micro-reversibihty is the fluctuation dissipation theorem [2] which states that the equilibrium fluctuations Sgq are proportional to temperature and to the linear conductance [3] (fefi is the Boltzmann constant),

Gi(5) = Gi ( -5 ) , Seq = kBTGi. (1)

Eqs. (1) are cornerstones of linear transport theory [4]. The question whether there exist fluctuation relations which apply beyond the lin

ear transport regime has long been of interest [5]. Recently the question was raised specifically for mesoscopic conductors in the context of theoretical works that characterizes transport not only by conductance and noise [6] but in terms of the full counting statistics [7]. An early discussion is provided by Tobiska and Nazarov [8] and followed by different discussions [9, 10, 11, 12]. In particular Saito and Utsumi [12] proposed a fluctuation relation in the presence of a magnetic field. Their derivation assumes that micro-reversibility also holds beyond the linear transport regime. However it has been known both from theoretical works [13, 14, 15, 16] and from experiments [17, 18, 19, 20, 21, 22] that the current proportional to the square of the voltage (the rectification coefficient) is in general not an even function of magnetic field and thus



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o/o V

FIGURE 1. Left: The root mean squared symmetric Gs and antisymmetric GA rectification coefficients of a chaotic quantum dot with two contacts (inset) as a function of flux through the dot. <i>c = h/e is the crossover flux for the transition from 'low' to 'high' magnetic field (after Ref [15]). Right: Sketch of the noise as a function of voltage; for finite magnetic field it can drop below the Nyqiust-Johnson noise 5^^.

manifestly breaks micro-reversibility. Therefore, it is interesting to ask whether non-equilibrium fluctuation relations exist even in the case when there is a manifest departure from micro-reversibihty. The surprising answer is, yes, even for samples with magnetic field asymmetry, there exist fluctuation relations [23].

We now emphasize the leading order behavior in temperature ksT, magnetic field B, and voltage V. Expanding the current / through a mesoscopic system up to the first nonlinear contribution defines the nonlinear conductance coefficient G2 = [d^I/dV^]eq.,

I=GiV+{l/2)G2V^ GA^B. (2)

The nonlinear transport coefficient G2 can be decomposed in a part Gs that is an even function of magnetic field and exists also for non-interacting conductors and a part GA that is asymmetric in magnetic field and that exists only for interacting conductors, Gs/A = ^2(5) ± G2{—B). For a weak field the antisymmetric part is GA '~ B. In chaotic cavities G2 is a consequence of quantum interference and the coefficient fluctuates from sample to sample. It is zero on average. It's mean squared value for a magnetic flux larger than a flux quantum is given by Sanchez and one of the authors. The entire crossover from low to high magnetic field is discussed by Polianski and one of the authors [15] and is illustrated in Fig. 1. The crossover flux Oc occurs at magnetic fields generating a flux through the sample much smaller than the flux quantum Oo = h/e. Experiments on chaotic cavities were performed by Zumbiihl et al. [17]. Experiments on rings [20, 21] and carbon nanotubes [18] demonstrate that magnetic field asymmetry is generic. It has been investigated also for multi-terminal Hall bars [22]. While the work on chaotic cavities emphasizes the quantum nature of the effect, at high magnetic fields orbital effects can lead to a classical magnetic field asymmetry [16].

The magnetic field asymmetry of the nonlinear coefficient implies the absence of microscopic reversibility out of equilibrium. Large external voltages modify the electron density within the conductor, which is subject to Coulomb interaction. In other words, the local internal potential U = U{r,V) of the conductor responds to the shifts of the

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apphed voltage V and has to be determined self-consistently. Crucially, the internal potential is not necessarily an even function of the magnetic field B, but has both symmetric and anti-symmetric components. Therefore, scattering from left to right at +B does not occur with the same probability as the corresponding process from right to left at —B: This is nothing else but a lack of micro-reversibility out of equilibrium.

Fluctuation relation for nonlinear transport

We could expect that with the absence of micro-reversibihty, fluctuation relations similar to (1) would also not exist. However, this is not the case. Just like the current, also the noise S can be expanded around equilibrium for eV <C fesT, defining the linear and quadratic coefficients Si and S2,

S = Seq + SiV +[1/2) S2V^ + ... (3)

The noise susceptibihty Si = [dS/dV]eq, is the deviation from equilibrium noise linear in voltage. This contribution contains emerging shot noise. As for the nonlinear conductance, we define the (anti-)symmetric noise susceptibihty SS/A = Si(5) ±Si{—B), as well as the (anti-)symmetric third cumulant at equilibrium C^ ^ = C^^'i^B) ±€3* (—5). Surprisingly, one can derive nonlinear fluctuation relations without making use of microscopic reversibility [23]. These relations imply on the one hand that the symmetric third cumulant vanishes at equilibrium which means that it is odd in magnetic field, and that the symmetric linear coefficient of noise is proportional to temperature and to the nonlinear conductance coefficient. On the other hand it states that the antisymmetric third cumulant is composed of the corresponding linear noise coefficient and nonlinear conductance,

Ss = kBTGs, C f = 0, (f^- = 3kBT{SA-kBTGA). (4)

The third cumulant at equilibrium vanishes if the transmission through the conductor does not depend on energy. Taking into account the first order correction linear in energy, the third cumulant is proportional to [ksTf-. The magnetic field symmetry of the third cumulant at equilibrium is a consequence of micro-reversibility at equilibrium: One can show [27], that this implies even (odd) cumulants at equilibrium to be even (odd) in magnetic field, in particular C * r^ B for weak fields. Altogether, this has the following consequences: First, the symmetric noise susceptibility Ss has the same 5-field dependence as the nonlinear conductance Gs- Second, the antisymmetric contribution to noise is proportional to magnetic field, temperature and voltage for weak fields [23, 24],

SA ^ kBTBV. (5)

Thus in the presence of electron-electron interaction the noise is not an even function of magnetic field and voltage (see Fig. 1). Note that there exists a range of voltages for which the non-equilibrium noise is below the Johnson-Nyquist noise [25].

We emphasize that fluctuation relations have been derived before for systems without magnetic field [8, 9] and also in the presence of a magnetic field [12]. The symmetric

445

fluctuation relations in Eqs. (4) are identical to those from Ref. [12] which means that they remain valid even in the absence of microscopic reversibihty. The antisymmetric relations however differ: A derivation [12] based on micro-reversibihty leads to the conclusion that GA and SA in a two-terminal system both are proportional to the asymmetric third cumulant, C^' '-^ GA'~-' SA, which means that they are even functions of magnetic field for systems with a vanishing third cumulant. In contrast our derivation of the fluctuation relation in the absence of micro-reversibility permits terms asymmetric in magnetic field for the nonlinear conductance and the noise susceptibihty even if C * = 0.

Derivation

The importance of the fluctuation relation and also its generality require a careful discussion of the derivation. It is possible to derive the fluctuation relations from the full counting statistics without specifying any model of interaction. The full counting statistics of a two-terminal conductor is the probability distribution P{Q) that a number of Q charges are transmitted into the reservoirs during the measurement time t. The distribution function P{Q) is expressed by the generating function F(/A) = lnI^QP(g)e''^2^ where the counting field A is the conjugate variable to Q. In the long time limit, all zero-frequency cumulants of the current can be expressed using derivatives of the generating function with respect to the counting field, evaluated at A = 0. The mean current and the noise are given by derivatives of F with respect to the counting field, / = {e/it)[dF/d?^]i^o, S = {e^/flt) [d^F/d?^^]^^^, thus the rectification coefficient, the noise susceptibility and the third cumulant at equilibrium are exphcitly

e d^F Lr2

it dXdV^ Si

t- d^F hdl^dV

_ e^d^F 3 Pt dV (6)

with the index 0 meaning X = V = 0. We assume the temperature in the left and right reservoir to be equal, and call the ratio A = eV/ksT the affinity of the system, with V = VL — VR the apphed voltage. A magnetic field B perpendicular to the conductor is externally controlled.

It is convenient to use the notation F = F(/A, A) in order to emphasize the dependence of the generating function on the affinity. A derivation of the nonlinear fluctuation relation is based on the following properties of the generating function:

F(0,A) = 0, F{-A,A) = 0, F ( a , 0 , + 5 ) = F ( - a , 0 , - 5 ) . (7)

The first equation represents conservation of particles, which in terms of the distribution function is expressed by Y.QP{Q) = 1- More subtle is the second identity. It defines a special symmetry point of the generating function, which translates for the distribution function into a global detailed balance relation Y.QP{Q)^^ = {^^) = 1- Importantly, this can be derived without making use of micro-reversibility, and for arbitrary electron-electron interaction. The only assumptions made are that the system consisting of conductor and leads is at all times described by the number of particles in the different components, and that both the total energy and the total number of particles is conserved. For a derivation see Ref. [27]. Only the third equation in (7) makes a statement

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on the magnetic field symmetry of the generating function, it corresponds to the micro-reversibility at equilibrium. From this follows, that even (odd) cumulants at equilibrium are even (odd) in magnetic field.

The fluctuation relation (4) can be obtained from the above identities. To this end, consider the generating function with variables iX and —iX —A, expanded into Taylor series around iX =A = 0:

z./ -0 . .^ ^A-A-iXfA^ ^~ {iX)PAi

kl '^•'- pq t'-^-

The Taylor coefficients are given by /« = [d^^^F{iX,A)/d{iX)^dA^]o and fpq = [dP+iF{-iX - A,A)/d{iX)PdAi]o. As indicated by Eq. (6), the coefficients /« are directly proportional to response coefficients. From Eq. (9), a relation between the coefficients fpq and fki is defined:

Now the symmetries defined by Eqs. (7) are crucial, they determine that the coefficients foq and foq all vanish. Setting p = Om Eq. (10) and separating the first and the last term in the sum one obtains

/?o = - I ( !)(-!)" /«.?-«• (11) n=l V"/

This actually represents a multitude of fluctuation relations, relating different response coefficients of current cumulants of different order. More details on this, and a graphical representation of the fluctuation relations are given in Ref. [23]. Here, we concentrate on the first nonlinear fluctuation relation, Eq. (4), which follows from the above identity by setting q = 3:

/30 = 3(/21-/ l2) . (12)

Only from the third equation in Eqs. (7), the magnetic field symmetry of the Taylor coefficients fqo is determined. It says in particular that the symmetric part / |Q vanishes, but makes no statement on the asymmetric contribution. Identifying the Taylor coefficients /i2 and /21 as proportional to the nonlinear conductance and the noise susceptibility respectively and (anti-)symmetrizing, the fluctuation relations Eqs. (4) are directly obtained. It is important to notice that the nonlinear fluctuation relations take on the form (4) only for a two-terminal conductor. The case of a conductor with multiple terminals is discussed in Ref. [23].

Conclusion

The linear transport regime is governed by Onsager-Casimir relations and the fluctuation-dissipation theorem, both derived from the principle of microscopic re-

447

versibihty at equilibrium. In the nonlinear transport regime, interactions can not be neglected anymore. They can lead to a lack of micro-reversibility out of equilibrium and in the presence of a magnetic field. A consequence is a contribution to the mean current proportional to magnetic field and quadratic in voltage as well as a contribution to the current noise linear in magnetic field and in voltage. We have shown that even in this case fluctuation relations exist: interestingly a noise contribution proportional to temperature, magnetic field and voltage is linked to the asymmetric nonlinear conductance and the third cumulant of equilibrium current fluctuations. To our knowledge, this relation has not yet been experimentaUy verified. Mesoscopic physics with its highly controlled and accurate experiments would seem especiaUy suited to demonstrate a non-equilibrium fluctuation relation.

Acknowledgments

This work is supported by the Swiss NSF, MaNEP and the European STREP project SUBTLE. We thank D. Astumian and Y. Utsumi for correspondence and discussions.

REFERENCES

1, L, Onsager, Phys, Rev, 37, 405 (1931); H, B, G, Casimir, Rev, Mod, Phys, 17, 343 (1945), 2, A, Einstein, Ann, Phys, Lpz, 17, 549 (1905), 3, H, Nyquist, Phys, Rev, 32, 110 (1928); J, B, Johnson, Phys, Rev 32, 97 (1928), 4, R, Kubo, J, Phys, Soc, Japan 12, 570 (1957), 5, G, N, Bochkov and Yu, E, Kuzovlev, Physica A 106, 443 (1981), 6, M, Bilttiker, Phys, Rev, B 46,12485 (1992), 7, L, S, Levitov and G, Lesovik, JETP Lett, 58, 230 (1993), 8, J, Tobiska and Yu, V, Nazarov, Phys, Rev B 72, 235328 (2005); see especiaUy Sec, V, 9, D, Andrieux, and P Gaspard, J, Stat, Mech, POlOll (2006); J, Stat, Mech, P02006 (2007); D,

Andrieux, P Gaspard, T, Monnai, S, Tasaki, arXiv:0811,3687 10, M, Esposito, U, Harbola, and S, Mukamel, Phys, Rev B 75, 155316 (2007), 11, R, D, Astumian, Phys, Rev Lett, 101, 046802 (2008); Phys, Rev E 79, 021119 (2009), 12, K, Saito and Y Utsumi, arXiv:0709,4128 and Phys, Rev B 78, 115429 (2008); Y Utsumi and K,

Saito,arXiv:0810,1113 13, D, Sanchez and M, Bilttiker, Phys, Rev Lett, 93, 106802 (2004), 14, B, Spivak and A, Zyuzin, Phys, Rev, Lett, 93, 226801 (2004), 15, M, L, Polianski and M, Bilttiker, Phys, Rev Lett, 96, 156804 (2006), 16, A, V, Andreev and L, I, Glazman, Phys, Rev Lett, 97, 266806 (2006), 17, D, M, Zumbilhl et al,, Phys, Rev Lett, 96, 206802 (2006), 18, J, Wei et al,, Phys, Rev Lett, 95, 256601 (2005), 19, C, A, Marlow et al,, Phys, Rev Lett, 96, 116801 (2006), 20, R, Leturcq et al,, Phys, Rev Lett, 96, 126801 (2006), 21, L, Angers et al,, Phys, Rev B 75, 115309 (2007), 22, A, D, Chepelianskii and H, Bouchiat, Phys, Rev Lett, 102, 086810 (2009), 23, H, Forster and M, Bilttiker, Phys, Rev Lett, 101, 136805 (2008), 24, D, Sanchez, Phys, Rev B 79, 045305 (2009), 25, G, B, Lesovik and R, Loosen, Z, Phys, B 91, 531 (1993), 26, T, Christen and M, Bilttiker, Europhys, Lett, 35 (7), 523 (1996), 27, See appendix of H, Forster and M, Bilttiker, cond-mat:0805,0362

448

Excited States in an InAs Nanowire Double Quantum Dot measured by Time-Resolved

Charge Detection

Theodore Choi*, Ivan Shorubalko*, Simon Gustavsson*, Silke Schon''' and Klaus Ensslin*

'Solid State Physics Laboratory, ETHZurich, 8093 Zurich, Switzerland ^FIRST lab, ETH Zurich, 8093 Zurich, Switzerland

Abstract. We present real-time detection of single electrons in an InAs nanowire double quantum dot. Two self-aligned quantum point contacts in an underlying two-dimensional electron gas material serve as highly sensitive charge detectors for the double quantum dot. We examine the excitated states of the double quantum dot by finite bias spectroscopy. The excited states are characterized by measuring the tunneling-in and tunneling-out rates of the quantum dots.

Keywords: Electronic transport, Nanowires, Quantum Dots, Single-Electron Tunneling, Counting PACS: 73.23.Hk, 73.63.Kv, 73.63.Nm

INTRODUCTION

Probing the electronic state of quantum dots (QDs) by charge detection with quantum point contacts (QPCs) represents an elaborate method for investigating tunnel processes of single charges [1]. The high sensitivity of the conductance of a QPC to its electrostatic environment allows to measure transitions in QDs in a regime where conventional transport measurements are impeded by the limited resolution of standard current meters. However, charge detection measurements are usually based on monitoring the average conductance of the QPC and thus measuring only the change of the average population of the QDs. In this respect, time-resolved charge detection marks a significant improvement since it offers the ability to count tunnel processes of individual single charges in real-time [2, 3].

Semiconductor nanowires (NWs) are promising candidates for electronic nanoscale devices and for studying fundamental physics of low-dimensional systems [4, 5]. InAs is a particularly interesting material since it exhibits large confinement energies due to the small effective mass of the electrons. Furthermore, a large effective g* factor enhances the ability for magnetic control of spins. At the same time, strong spin-orbit interaction known for InAs could be exploited for the manipulation of single spins by electric fields [6, 7, 8]. These specific properties make InAs a unique material for realizing quantum bits in solid-state-based quantum computers by spins in coupled quantum dots [9].

Here, we present time-resolved charge detection measurements on a tunable etched InAs NW double quantum dot (DQD) using the signal of two self-aligned QPCs serving as highly sensitive charge detectors.



449

SAMPLE AND EXPERIMENTAL SETUP

An electron microscope image of the etched DQD structure and self-aligned QPCs together with a circuit scheme is shown in Fig.l. The NWs are deposited on an Al-GaAs/GaAs heterostructure with a 2DEG 37 nm below the blank surface. The 2DEG has a density of A^ = 4x10^^ cm~^ and mobility /i = 3 x 10^ cm^/Vs at T = 2 K. Electron beam lithography patterned PMMA is used as an etching mask to define the DQD and the QPCs. The structure is designed in such a way that the trenches in the 2DEG forming the QPCs by depletion of the 2DEG underneath and the constrictions in the NW forming tunnel barriers of the DQD are defined in a single step wet etching process [10, 11]. The fact that both trenches of the 2DEG and constrictions of the NW are defined simultaneously by the same etching areas ensures perfect alignment of the QPCs and the QDs. The QPCs operate as local gates to change the electron population in each QD and as sensitive charge detectors for transitions in the DQD. Compensation voltages f ggL/R were applied to the side gates in order to keep both QPCs at a constant operation point. For the presented measurements, the QPCs are operated at a slope of the conductance close to pinch-off, where we get a desirable sensitivity to transitions in the DQD. All measurements presented here are dc measurements and were performed at T = 1.8 K.

^DQD'

FIGURE 1. Electron microscope image of the etched DQD structure with QPCs and lateral side gates. A circuit scheme with the voltages and currents used in the measurement is shown in addition. A/B are the source/drain contacts of the NW. The source/drain contacts of the QPCs are the contacts 2/5 and 3/6 for the left and right QPC respectively. The electronic width of the QPCs can be changed by applying voltages to the side gates on contacts 1/4. The current is measured through both QPCs and the NW. The bias voltage on the DQD is applied symmetrically to source and drain. Scale bar: 200 nm.

FINITE BIAS SPECTROSCOPY BY COUNTING

Time-resolved measurements of the tunneling processes in the DQD become possible as soon as the tunnel rates of the barriers defining the DQD are below the bandwidth of 20 kHz of the measurement setup [12]. We tune the tunnel rates of the barriers in

450

the NW by moving to different gate voltage regions until the tunnel processes occur on a sufficiently slow time scale. Single electrons tunneling through the three barriers defining the DQD can then be counted one by one in real-time. The tunnel rate of a barrier can be determined from time traces of the QPC signals detecting transitions of single electrons across this barrier. Fig.2(a) shows a time trace of the right QPC current taken at gate voltage values where the source lead is aligned with the left QD. The transitions between the two levels reflect the equilibrium fluctuations of an electron jumping between the source and the left QD. The time during which the current stays in the lower/upper level (Tgut/in in Fig-2(a)) represents the time it takes for an electron to tunnel out or into the QD respectively. The expectation values of the tunneling times are directly related to the tunnel rates by TJJJ/Q^^ = 1/(TJJJ/O^^) given an exponential distribution of the tunneling times TQ^ /JJ^ [12].

(a) (N„N2)

200 300 Time (ms)

198 200 202 204 206 208 210 212 Vj. |(mV)

FIGURE 2. (a) Time trace for the current of the right QPC for the situation where the Fermi level of the source lead is aligned with the electrochemical potential of the left QD. The levels correspond to an electron tunneling into and out of the QD. Tunneling times are denoted by TJJJ^Q^J. (b) Finite bias triangles measured with the left QPC by time-resolved charge detection. The plotted tunneling-in rates Fjjj are extracted from time traces taken at each point of the figure. The length of the time traces is 1 s. (c) Tunneling-in rates Fjjj for a cut along the dashed line in (b). The trace exhibits a pronounced step where an excited state of the left QD enters the energy window. Level schemes illustrating the tunneling processes are given at the corresponding positions, (d) Same for the tunneling-out rates Tout along the dashed line in (b). No steps of Font can be seen.

Applying a finite bias to the DQD causes the triple points of the stability diagram to develop into triangular shaped regions [13]. Finite bias spectroscopy can be used to probe the excitation spectrum of the DQD as excited states can enter the energy window opened by the applied bias across the DQD. Fig.2(b) shows a measurement of a pair of finite bias triangles using the left QPC. The measurement was done at a gate voltage region where the tunnel rate of the right barrier was above the measurement bandwidth

451

whereas the tunnel processes of the central and the left barrier could be detected by the QPCs. The bias across the DQD is J^DQD = —3 mV. We plot the tunneling-in rate T^^ extracted from time traces taken at each point of the figure. A sharp line can be seen inside the triangles corresponding to a stepwise increase of Fjjj. We attribute this line to an excited state of the left QD entering the energy window. This is illustrated in Fig.2(c) where a cut along the dashed line in Fig.2(b) and level schemes representing the tunnel processes are shown. Distinct steps of the tunneling-in rate T^^ can clearly be seen. Using the lever arms for the conversion of voltage to energy gives a level spacing of AE « 1.4 meV for the left QD in agreement with an estimation of the spacing assuming spherical QDs with hard walls. For the trace of the tunneling-out rates Fgut plotted in Fig.2(d), no step can be seen indicating that the electrons always tunnel out of the left QD through the ground state after relaxation from the excited state. The relaxation rate of the excited state is thus much higher than the tunnel rate of the left barrier. The tunnel rate FQ of the central barrier can be estimated by taking the tunneling-in rate at the position of f 2degL where the ground states of the left and the right QD get aligned (f 2degL ~ 202.5 mV in Fig.2(c)), which yields T^^ = r(^ « 2.8 kHz. The same analysis was done for reversed bias J^DQD = 3 mV where an excited state of the right QD was detected by a step in Fgut inside the finite bias triangles with a level spacing of A£' « 1.3 meV.

CONCLUSION

In conclusion, we have demonstrated the possibility to fabricate a tunable DQD in an InAs NW with highly sensitive and perfectly aligned charge readout QPCs by a single step wet etching process. We have presented time-resolved charge detection measurements where we investigate the excitated states of the DQD by finite bias spectroscopy and by measuring the tunneling-in and tunneling-out rates of the QDs.

REFERENCES

M.Fie\detal.,Phys. Rev. Lett. 70, 1311 (1993). R. Schleser, E. Ruh, T. Ihn, and K. Ensslin, Appl. Phys. Lett. 85, 2005 (2004). L.M.K. Vandersypen et al.,Appl. Phys. Lett 85, 4394 (2004). M.T. B]6rketal.,NanoLett. 4, 1621 (2004). M.T. Bi6rketal.,Appl. Phys. Lett. 80, 1058 (2002). V.N. Golovach, M. Borhani, and D. Loss, Phys. Rev. B 67, 165319 (2006). D.V. Bulaev and D. Loss, Phys. Rev Lett. 98, 097202 (2007). C. Flindt, A.S. Sorensen, and K. Flensberg, Phys. Rev Lett. 97, 240501 (2006). D. Loss and D.P. DiVincenzo Phys. Rev A 57, 120 (1998).

10. L Shombalko et al.,Nano Lett. 8, 382 (2008). 11. T. Choi et al, New J. Phys. 11, 013005 (2009). 12. S. Gustavsson et al, Appl. Phys. Lett. 92, 152101 (2008). 13. W.G. van der Wiel et al, Rev Mod. Phys. 75, 1 (2003)

452

Universal oscillations of high-order cumulants Christian Flindt*- , Christian Fricke**, Frank Hohls**, Tomas Novotny^,

Karel Netocny*, Tobias Brandes^ and Rolf J. Haug**

'Department of Physics, Harvard University, 17 Oxford Street, Cambridge, MA 02138, USA ^Department of Condensed Matter Physics, Faculty of Mathematics and Physics, Charles

University, Ke Karlovu 5, 12116 Prague, Czech Republic **Institut ftir Festkorperphysik, Leibniz Universitdt Hannover, D 30167 Hannover, Germany

^Institute of Physics AS CR, Na Slovance 2, 18221 Prague, Czech Republic. ^ Institut fur Theoretische Physik, Technische Universitdt Berlin, D 10623 Berlin, Germany.

Abstract. We discuss our recent measurement of high-order cumidants of charge transport through a quantum dot. The cumulants were found to oscillate as functions of measurement time before reaching their linear-in-time asymptotics. A theoretical analysis revealed that such oscillations in fact constitute a universal phenomenon: for a large class of stochastic processes the high-order cumulants are predicted to oscillate as functions of basically any parameter. Here, we give an overview of these recent results, provide an outlook on future applications of our findings, and formidate a number of open questions.

Keywords: Counting statistics, cumulants, universal oscillations PACS: 02.50.Ey, 72.70.4-m, 73.23.Hk

Introduction. Fluctuations of the electrical current through nano-scale conductors have become a popular tool for characterizing and understanding stochastic charge transport processes [1, 2]. In particular, the cumulants of the distribution of transferred charges are beheved to contain detailed information about the microscopic mechanisms underlying the stochastic process. While the field, commonly known as counting statistics, so far has been dominated by theoretical studies, a number of recent experiments have brought this picture into a more balanced shape [3,4,5, 6, 7]. Experimental studies of the cumulants require high-quality statistics, which only now is becoming available via refined measurement techniques.

Very recently we studied experimentally and theoretically the cumulants of the number of charges tunneling through a quantum dot [8]. Experimentally, we found that the high-order cumulants oscillated as functions of time before reaching their linear-in-time asymptotics. From a theoretical analysis we were able to show that such oscillations in fact constitute a universal phenomenon: for a large class of stochastic processes the high-order cumulants become oscillatory functions of basically any parameter. This explained the measured oscillations of the high-order cumulants as functions of time as well as previous theoretical studies, where high-order cumulants in very different systems had been found to oscillate as functions of certain parameters [9, 10, 11, 12].

In this contribution we give an overview of our experimental findings and the theory underlying the universal oscillations of high-order cumulants. We discuss the general apphcabihty of our theory, show how cumulant oscillations can be used to extract information about the cumulant generating function, and formulate a number of open questions. For further details of our work we refer the interested reader to Ref. [8].



453

'QPC

"=" "' 10,000-

-10,000-

FIGURE 1. Experimental setup and results, a) Atomic force microscope topography of quantum dot (QD, dashed ring) and quantum point contact (QPC, dashed lines). Single electrons enter the QD from the source (5QD) at rate F^ = 1.46 kHz and leave via the drain at rate TD = 2.97 kHz. Only a single electron at a time is allowed to occupy the QD. The current running through the QPC switches back and forth between a high and a low level as an electron enters and leaves the QD. This allows us to count the number of electrons that have passed through the QD in real-time. A typical time-trace of the QPC current is shown, b) Results for the high-order cumulants ((n™)} as functions of time. The experimental results are indicated with dots, while the full lines show rate equation calculations. The error bars were obtained from numerical simulations of the experiment. Figure adopted from Ref. [8]

Experiment. Our experimental setup is shown in Fig. la. Here, electrons are driven through a quantum dot (dashed ring) operated in the Coulomb-blockade regime, where only a single additional electron at a time is allowed to occupy the dot. A large bias across the dot ensures that the charge transport is unidirectional. Electrons enter the dot with rate Ys from the source electrode (SQD) and leave with rate To via the drain (£>QD)- A nearby quantum point contact (dashed lines) is used to count the number of electrons tunneling through the dot: a separate current running through the quantum point contact is suppressed every time an electron enters the dot. The suppression is lifted as the electron leaves the dot via the drain electrode. From the switches of the current through the quantum point contact it is thus possible to count the number of electrons passing through the dot. By counting a large number of tunneling events we were able to determine the distribution of the number of transferred charges P„(?) during the time span [0, t] and in particular the cumulants of the distribution ((M™)) ,OT= 1,2,3,.... The first cumulant is the mean ((«)) = («), the second is the variance ((M^)) = (M^) — (M)^,

and the third is the skewness ((M^ )) = ((«—(«)) ). The general definition is given below. Our experimental results for the high-order cumulants ((M™)) are shown in Fig. lb.

Most notably, the cumulants oscillate as functions of time before reaching their linear-

454

in-time asymptotics (not shown). Moreover, the magnitudes of the high-order cumulants grow significantly compared to the first few cumulants which are typically on the order of unity when normalized with respect to the first cumulant. Below, we argue that oscillations and factorial growth of cumulants in fact constitute a universal phenomenon.

Theory. The following analysis apphes to a large class of stochastic processes and it is not essential whether the underlying physics is classical or quantum mechanical. Given a distribution function P„(?), the cumulants are defined as derivatives of the cumulant generating function (CGF) S(z, A) with respect to the counting field z at zero, ((M™)) = dfS{z,'k)\z-^o- Here, X denotes the set of all parameters of the stochastic transport process, including measurement time, and the CGF corresponding to Pn(t) is

S(z,A)^ln^P„(Oe"^ (1) n

In order to understand the experimentally observed oscillations of the high-order cumulants with time we study the singularities of the CGF in the complex z-plane. As recently discussed by M. V. Berry, the high-order derivatives of a function are intimately linked with the singularities of the function [13]. We thus consider the generic situation where the CGF has a number of singularities Zj, j = 1,2,3,... in the complex plane. These can be finite-order poles or branch-points. Exceptions to this scenario do exist, e.g. the Poission process whose CGF is an entire function.

Close to a singularity z :^ Zj we can approximate the CGF as S(z, A) :i; Aj/{z — Zj)^^ for some Aj and jij. For a finite-order pole, jij would be the order of the pole. Taking derivatives with respect to the counting field z, we find 5™S(z, A) -2^ { — l)'"AjBm,^j/{z — Zj)'"+'^j with Bm^nj = [jij+m— \){jij+m — 2)...jij. At first, this approximation of the derivatives close to the singularity Zj may not seem very useful, since the cumulants are defined as the derivatives of the CGF at z = 0. However, according to Darboux's theorem [13], the approximation of the derivative becomes better away from z •:^ Zj with increasing order m. This allows us to express the high-order cumulants as a sum of contributions from all singularities, i.e.

i

having introduced the polar coordinates Zj = \zj\e''^^^J. The singularities come in complex-conjugate pairs, ensuring that the cumulants evaluate to real numbers.

We are now ready to discuss a number of ubiquitous features of high-order cumulants. First, we note that the high-order cumulants generally grow factorially in magnitude with the order m due to the factors 5m,My Secondly, we notice that the singularities closest to z = 0 dominate at large orders and contributions from other singularities can thus be neglected for large enough m. Finally, we see that the high-order cumulants become oscillatory functions of any parameter among A that changes the position of the dominating singularities due to the factor gK +My) argzy xhe situation becomes particularly simple if only a single complex-conjugate pair of singularities is closest to z = 0. Denoting this pair, the corresponding expansion coefficients, and the factorial factor by z, z*, /i. A, and 5™, respectively, the high-order cumulants become

((K™)) ^ (-l)'*2A5„cos[(m + A)argz]/|zr+'* (3)

455

and we clearly see the oscillatory behavior. Returning to our experiment in Fig. 1, we can now understand the nature of the observed oscillations of the high-order cumulants with time. A pair of complex-conjugate singularities moves in the complex plane as functions of time, thereby changing argz, and thus making the high-order cumulants oscillate. The large magnitude of the oscillations is due to the factor 5^. Here, we have not given the explicit expression for the CGF relevant to our experimental setup or the analysis of the corresponding singularities and their parameter dependencies. For such details, we refer the interested reader to Ref. [8]. We note that previous theoretical studies of different systems showing oscillations of high-order cumulants as functions of certain parameters [9, 10, 11, 12] may equally well be explained within the same theoretical framework.

Experimentally, oscillations of the high-order cumulants can be used to extract information about the CGF describing the charge transfer process. For a fixed set of parameters A, the cumulants are predicted to oscillate as functions of the cumulant order m with the frequency given by the angle argz of the singularity z with the real-axis, see Eq. (3). This can be used to locate the dominating singularities of the CGF: by measuring the cumulants as functions of the cumulant order m, the angle argz can be extracted from the frequency of the oscillations, while the argument \z\ enters the magnitude of the oscillations. Additionally, the nature of the singularity, e.g. the order of a pole /i, determines the phase of the oscillations and should thus be experimentally detectable.

Conclusions. We have presented an experimental and theoretical study of high-order cumulants of charge transport through a quantum dot and shown that oscillations of high-order cumulants constitute a universal phenomenon for a large class of stochastic processes. Some open questions remain and deserve further investigation: What is the physical origin of the oscillations? What information is generally available in high-order cumulants? What do the oscillations tell us about a particular system?

Acknowledgements. The work was supported by the Villum Kann Rasmussen Foundation, the Czech Science Foundation via grant 202/07/J051, BMBF viananoQuit, DFG via QUEST, research plan MSN 0021620834 financed by the Ministry of Education of the Czech Republic, and DFG project BR 1528/5-1.

REFERENCES

1, Ya, M, Blanter and M, Bilttiker, Phys. Rep. 336, 1 (2000) 2, Quantum Noise in Mesoscopic Physics, edited by Yu. V. Nazarov (Kluwer, Dordrecht, 2003) 3, B, Reulet, J, Senzier, and D, E, Prober, Phys. Rev. Lett. 91, 196601 (2003) 4, Yu, Bomze et at, Phys. Rev. Lett. 95, 176601 (2005) 5, S. Gustavsson et at, Phys. Rev Lett. 96, 076605 (2006) 6, T. Fujisawa, T. Hayashi, R. Tomita, and Y Hirayama, Science 312, 1634 (2006) 7, A, V, Timofeev et at, Phys. Rev Lett. 98, 207001 (2007) 8, C, Hindt et al, arXiv:0901,0832 (2009) 9, S, Pilgram and M, Bilttiker, Phys. Rev B 67, 235308 (2003) 10, H, Forster, S, Pilgram, and M, Bilttiker, Phys. Rev B 72, 075301 (2005) 11, H, Forster, P Samuelsson, S, Pilgram, and M, Bilttiker, Phys. Rev B 75, 035340 (2007) 12, C, Hindi et at, Phys. Rev Lett. 100, 150601 (2008) 13, M, V, Berry, Proc. R. Soc. A 461, 1735 (2005)

456

Formulation of Time-Resolved Counting Statistics Based on a Positive-Operator-Valued

Measure Wolfgang Belzig* and Adam Bednorz'''

*Fachhereich Physik, Universitdt Konstanz, D-78457Konstanz, Germany ^Fachbereich Physik, Universitdt Konstanz, D-78457 Konstanz, Germany

University of Warsaw, Hoza 69, PL-00681 Warsaw, Poland

Abstract. It is shown how to include the influence of detector into the description of time-resolved counting statistics for a quantum point contact. Using properly constructed positive operator-valued measure, one can predict correction to the noise and avoid the paradox of negative probability.

Keywords: counting statistics, noise, quantum point contact PACS: 73.23.-b, 72.70.+m

INTRODUCTION

The investigation of current noise and full counting statistics in mesoscopic electronic circuit is an active research field, that touches upon fundamental problems quantum mechanics and measurement[l]. Here, we propose a derivation of the full counting statistics of electronic current based on a positive operator-valued measure (P0VM)[2]. Our approach justifies the Levitov-Lesovik formula [3] in the long-time limit, but can be generalized to the detection of finite-frequency noise correlations. The combined action of the projection postulate and the quantum formula for current noise at high frequencies imply an additional white noise. Our conjecture can be experimentally tested by a simultaneous measurement of high- and low-frequency noise.

QUANTUM POINT CONTACT

The quantum point contact is a narrow constriction in two-dimensional electron gas. In the vicinity of the Fermi level, for energy-independent scattering in abcence of interactions, it is described by the Hamiltonian

n

Y, dxeV{t)\\iflf^{x)\ifLn{x) + \iflf^{x)\ifM{x)\. (1)



457

where Ho = ^ dxiflVn \jfl^{x)dx\ifLn{x) - \jfl^{x)dx\ifRn{x) (2)

the fermionic operators satisfy anticommutation relations {yfA,n{x),\jfB,th{x')} = 0, {VA,ii{x),yfl^{x')} = 5AB8iiM8{x — x') for A,B = L,R. To find the scattering amplitude, it is necessary to regularize the second term in (1), namely

'Ln.\'dxz!„{xlvn) \irlfi{x)\irRn{-x) + \iflfi{-x)\irLn{x) with z„(x) -^ q„9{x). It gives

transmission T„ = cos^ {q„/fi) and reflection R„= I — T„. The current operator reads /(x) = SnSV„(i/>^ (x)i/>fi/j(x) — \j/l^{x)\jfLn{x))• We shall

introduce the following auxiliary operators

hn{Xn) = ^ { win(^) WLA(X) + l / / ]^ ( -x ) l/>i;/j(-x)),

hn{Xn) = -J-i^Lni^WLnix) " l />]^(-x) l/>iJ/j(-x)),

hiiixn) = —î/>2^(x)i/>ff/j(-x) + h.c. , hnixn) = — i/>2^(x)i/>fi/j(-x) +h.c. . (3)

for Xn = x/Vn- Note that x„ is in time units. It will be more convenient for us than length units. The Hamiltonian (1) can be now written as

H = Ho + J^ fds2z'„{s)kii{s)/e+J^ f dsV{t)[Ioii{s)-sgn{s)Iifi{s)] (4)

and current /(x) = Xn [ioh{.—Xn) —Ion{x„) —/in(—x„) —/i/j(x„)] . One can show that the above aperators satisfy the commutation relations

[Ijn{s)jM{s')] = i(?5nm8jk8'{s-s')/AK + Y,êjkie5nfhim{s)8{s-s') (5)

with Eju equal +1 for Jkl = 123,231,312, — 1 for j'A:/ = 321,213,132 and zero otherwise. The other useful commutator is [HoJjfi{s)] = —ihPj^^s).

POVM

We construct a positive definite probability of time-dependent FCS based on a POVM. Instead of the projection operator we define the more general Kraus operator

K[I] = /"z3^ê/' *[''P(*)[A*)-- (*)l/«-'P (*)/' l . (6)

Here i{t) is the Heisenberg current operator and 5^(#) denotes (anti-)time ordering. The time scale T describes internal fluctuations of the detector and depends on its temperature in general. For T ^ 0° the measurement is accurate, but the detector noise strongly affects the system by full projection. A shorter T reduces the influence of detector but induces a larger measurement error. The positive definite probability of a given I{t) is

458

defined as p [/] = TrpK^ [I]K[I] for a initial density matrix p <x exp(—Z/o/fe^) • The generating functional takes the form

.^IX] = ln(exp(/1dtx{t)I{t)/e))p = In jD<^ ^s\xM-idm\t)+x\t)li\l^ ^ (7^

where % , 0] is defined by e^^'^l = T rp#e / i t eW+2^WK»^e / f [zW-20W]/»,/(;) = JdxI{x,t)g{x — XR) is regularized current measured near the point x j > 0. Here g{x) =

e 2A?/^2nAx is a regularized Dirac delta function, T is the measurement sensitivity and Ax <C Xij is spatial sensitivity.

We can make the splitting I{t) = 'Lfi[ln{f)+In{f)] for I^{t) = Jdx g{x± XR)[±Ioii{Xn,t)-Iu{Xn,t)] and use properties [In{t)Jnif')] = 0 for ? > t',

[In it) Jn it')] = [lHt)JHt')] = '^Kit-t') for h„{t) = e " ^ / 2 v ^ T „ . Now, we are ready to calculate averages. For future convenience we shall denote s±n = s±tn. The mean current is (/(0)p = —is8,y/5xit)\x=o = 'LnS^Vit-n)Tn/j^fi-^Q^ce, conductance is robust against detector backaction.

The current correlation function reads

{5l{a)5l{b))p = -e 2 5 ^ = (^P{a,b) (8) Sxia)8xib)

fox 51 = 1-{I)p. We have finally P(a,b) = 5{a-b)/x + P^{a-b)+Po{a-b) +Pe{a,b). The Fourier

transforms of the second and the third term can be written as

^^(") - 4.2 T T-i . » . „ . N „ 2im„

^{2i9n{co) + coRw n

with 0„ defined by (??) and

2

,en{(a)::^-hni0)-m/2 (9)

Po(«) = ^ ^ X ( ^ n +^«(1 - C0S(2«?„)). (10) n

Here w(ft)) = (acot)\{'h(a/2kBT). Again with help of (??) we obtain

Pe{a,b) = yRnTn^^^~^'Tn{a-b)cOS(p{a-n.b-n) (H) n ^

with r„(?) = exp(T(/i„(?)-/i„(0))/4), w(?) = /fif(«w((«)e"''*'727r and (p{a,b) = f,dteV{t)/n.

Evaluating the second-order current correlator P{(i>) = Jdt e"^*P{t,0) for V = const, in the limit fer < ^|(«| < fi/tn < ^ / T from (9), (10) and (11), we find

e^P{(o) = I dte'"'*{5l{0)5l{t))p = (^PQ^^+S{(O)+(^P^. (12)

459

Here ^off ^ '^l^ ^^ some white offset noise due to the internal fluctuations of the detector. »S(o)) = fdtcos{cot)Trp5l{0)5l{t)/e^ is the symmetrized quantum noise. The mixing noise PA = {T/^7t^)Ri T\q{T\ \eV\/h)/T^. Here Ti = Ax/v is related to the spatial dispersion of the current detection and the function q{z) is depicted in the left of Fig. 1. The qualitative behaviour of this expression is shown in the right of Fig. 1 and is in agreement with existing noise measurements at high frequencies [4].

In future, it will be interesting to explore physical models of detector, which realize or proposed POVM-based measurement of current fluctuations at high frequencies.

ACKNOWLEDGMENTS

Our work was financially supported by the DFG through SFB 767 and SP1285.

Ph/k B iPA + PoS+Si{u;)/e'^)h/kB

{Si{u;)/e' + Poit)h/kB

eV/hw

FIGURE 1. Left: (a) Current measurement at a quantum point contact. The Gaussian distribution approximates the spatial dispersion of the measurment apparatus, refers to the dispersion of current measurement, (b) The function q{z) governs the backaction of the detector on the current measurement. Right: Qualitative prediction of the current-noise measurement at high frequencies. For the plot we assumed T = Ips, Ti = lOps and T\ = 1/2. The frequency crossover to the usual linear bias-voltage dependence takes place at 6GHz. The small mixing noise is of the order of PA(0) = l.SmK^^/^ and, hence, might be difficult to detect on top of the large offset noise i Qff = IAYJCB/^-

REFERENCES

1. Y.M. Blanter and M. Biittiker, Phys. Rep. 336, 1 (2000); Quantum Noise in Mesoscopic Physics edited by Y.V. Nazarov (Kluwer, Dordrecht, 2003).

2. A. Bednorz and W. Belzig, Phys. Rev. Lett. 101, 206803 (2008). 3. L.S. Levitov and G.B. Lesovik, JETPLett. 58, 230 (1993). 4. J. GabelH and B. Reulet, Phys. Rev. Lett. 100, 026601 (2008).

460

Magnetization fluctuations in mesoscopic conductors out of equilibrium

Rosa Lopez, Magdalena Gelabert, David Sanchez and Lloren^ Serra

Department de Fisica, Universitat de les IllesBalears, E-07122 Palma de Mallorca, Spain

Abstract. We investigate current (/), magnetization (M) and their fluctuations in a normal conductor attached to two ferromagnetic leads. For the parallel case, M vanishes independently of the contact polarization 77 but we find that the fluctuations of M depend on 77. This effect persists even in the presence of spin-flip processes (with a rate jst)- In the antiparallel case, / decreases and M increases when 77 is enhanced, reflecting the "spin-valve" effect. This tendency is inverted as long as 7sf grows. We also present the conditions for obtaining maximal fluctuations for the magnetization as a function of jst-

Keywords: Magnetization noise, spin noise, spin polarized transport PACS: 85.75.-d, 85.35.-p, 72.70.+m

INTRODUCTION

Spin-based electronics devices have recently boosted the interest in spin-dependent transport in hybrid systems. The first observation of the giant magnetoresistance effect three decades ago has opened a new area of research based on creating, manipulating and detecting spin currents and in magnetoelectronic materials [1]. Proposals like the spin field-effect transistor and ferromagnetic single electron transistors [2] have been investigated from both theoretical and experimental sides. Their applications to present technology are realiable owing to relatively long spin dephasing times [3].

We consider a simple device—a two-terminal ferromagnetic-normal-ferromagnetic (FNF) system. It is well known that transport properties of such a system depends much on the magnetic configuration of the reservoirs. A figure of merit is the tunnel magnetoresistance (TMR) that measures the relative change in the current when the contacts are either in the parallel (P) or in the antiparallel (AP) configuration [see small insets in Fig. 1]. A striking effect observed in these systems is the spin-valve effect in which half metallic contacts in the AP case yield current JAP -^ 0 and TMR^ 00. Customarily, only charge currents and their fluctuations [4] have been investigated in these systems [5]. Nevertheless, since we are concerned with the spin degree of freedon, it is natural to examine also the magnetization of the normal conductor and its fluctuations. Fluctuations of a given physical magnitude can reveal new and interesting features not present in its mean value [6]. In mesoscopic systems, e.g., current fluctuations are an indication of the presence of correlations among the carriers and they reflect its statistics as well.



461

0 0.2 0.4 0.6 0. 1

FIGURE 1. (Color online). Left panel: Average current / and magnetization M versus contact polarization r\ = r\i. Right panel: Fluctuations of current Sj and magnetization SM- The magnetic configuration (P or AP) is indicated in the insets of all figures. Equal color for the arrows means same polarization. Large (small) arrows correspond to majority (minority) spins. The lower inset in (a) depicts the equivalent circuit. We set 7] ; = fir\L a n d r = 0.5. We plot in (a)/and in (c)M for the Pease (j3 = 0.2) and distinct "j j-. We depict in (b) / and in (d) M versus r\ for j3 = — 1 (AP case) and different j^f- Right panel: (e) and (f) show fluctuations in the parallel (j3 = 1) and antiparallel (j3 = — 1) cases, respectively, versus polarization for various values of jsf- The inset in (e) corresponds to Sj versus r\ for the same values of jsf given in the legend. Magnetization fluctuations (SM) as a function of r\ are shown in (h) for different values of jst whereas (g) corresponds to SM versus jsf for different polarizations close to the half-metallic situation, i.e., 7] ^ 1.

MODEL

The two ferromagnets are attached to the normal conductor through two resistors (a G {L,R}) Ra = l/Ga where Ga = NoTa, ^a is the barrier transparency and Na the number of propagating modes. The physical properties inside the conductor are assumed to be isotropic, due to diffusive or chaotic scattering. Thus, we can consider the conductor effectively as a zero-dimensional system. This restricts our model to systems with a spin-flip length in which spatial variations of accumulated spins inside the normal conductor are absent. With these assumptions we treat the system within a spin-dependent semiclassical approach [7, 8]. In this fashion we map the system onto an equivalent circuit [see lower inset in Fig. 1 (a), left panel] where j and | spins are connected only via spin-flip events. Spin-flip scattering in the normal conductor gives rise to a spin-flip current between the spin subsystems. The ferromagnets are modelled such that they have FN interfaces with spin dependent couplings: G^, where a = {L,R} and c = {TJ i}- This fact arises from spin-dependent density of states in the ferromagnetic leads, Vaa, close to the Fermi energy. Hence, one can write the spin polarization of lead a as 77a = (vi — Va)/(V(!( + Va). In the wide band limit, the couplings are independent of energy and we have G^ = Ga( 1 + Ha )/2, where Ga = Gli + Ga- The spin dependent

462

current densities are conserved at each node in the equivalent circuit for every energy E,

i + i\ + /|t, = 0, /i + 4 - /|t = 0, /S = GS(/„ - ^ ) , (1)

where positive (negative) currents flow into (out of) the normal conductor. In Eq. (1), fa{E,) = {1 +exp{[E ±eVa]/kT)}^^ is the Fermi-Dirac equilibrium distribution function at reservoir a. We assume that/a is spin independent since spin-flip mechanisms are highly efficient in ferromagnets [9]. f^{E) is the nonequilibrium distribution function of the mesoscopic conductor to be determined from Eq. (1). The spin-flip current [7] is: /ji = Ysf{fd — Id) with jsf being the rate of spin-flip events. The electrical current flowing out of reservoir a is given by la = la+Ia where /^ = {e/h) jdEi%{E). Finally, by inserting the solution of the circuit equations, i.e., f^, in the current expression and taking VL = -VR = V/2 (KBT = 0) we arrive at / = [4/+ Gi(l - nl) + G«(l - n|)]/[4r(Gi + G«) + ^+^_]GiG«eF//i with ^± = Gi(l ± Hi) + GR{1 ± m)-On the other hand, the spin accumulation in the normal conductor is related to the magnetization that accounts for the imbalance between the two spin species: M = vJZlfjiE) - f^{E)]dE = 2vGiG«(77i - %)eK/(47(Gi + GR) + ^+^_) , where v is the conductor density of states. Hereafter, we consider a symmetric junction, i.e., GL = GR = G.

Figure 1 (see left panel) shows the behavior of the current / and magnetization M for P and AP cases. In the P configuration and for equal contact polarizations, i.e., rjL = rjR = rj, I = (e//i)(l/Gi + I/GR^^V is just given by the combination of two resistors in series. Here, both f^ and / J are simply (/t +/ff)/2, i.e., nonmagnetic even in the presence of y^t ¥= 0- When we allow different contact polarizations TJL = jirjR (0 < /3 < 1) [Fig. 1(a)], / decreases monotonically with rji and when 7sf is turned on / recovers its nomagnetic value for sufficiently large 7sf. The magnetization reflects this physical scenario as well. For y f = 0, M enhances when rji does [see Fig. 1(c)]. As 7sf increases, M diminishes and eventually, for y^i^ °° M vanishes (and the conductor becomes nomagnetic). Spin relaxation processes tend to equilibrate both spin species in the normal conductor, resulting in a nonmagnetic distribution function for the normal conductor even if the contacts are polarized. The AP case is examined in Figs. 1 (b) and (d). We take rji = —TJR = rj. The "spin-valve" effect is observed for large 77 where 7 = 0 [Fig. 1(b)] and M becomes maximal [Fig. 1(d)]. This tendency is gradually suppressed when 7sf ^ 0. Spin-flip processes try to eliminate the imbalance between up and down conductor distribution functions and, eventually, for 7sf -^ 00 the conductor becomes nomagnetic.

We now analyze current and magnetization fluctuations. In general, fluctuations are defined in terms of a zero-frequency correlator S=2Jdt (A^(?)A^(0)) where ^ can be either charge current or magnetization. In order to determine S, we assume that the contacts and spin-flip currents emit independent fluctuations (5;^ and Si?^). As a result, the spin-dependent distribution function of the normal conductor fluctuates itself. The total fluctuating current at each contact is A;^ = Si^ — G^Sf^, and the total fluctuating spin-flip current is A/jj = 5 ; | | — 7sf(5/^ — 5p). In the equivalent circuit we impose

that current fluctuations vanish at each node, i.e., Aia- ' +Aia-' ± A/jj = 0. By solving this system of equations and taking into account the nature of the fluctuations in the

463

contacts, namely, S^ = eGZlfS + f7 - 2f^fS - r{fS - Jjf], and in the spin-flip current (we assume Poissonian statistics), S[^ = e%i\f\ + / J — 2/J / j ] , we determine ?if^. Figure 1 (right panel) shows current {Si) and magnetization {SM) fluctuations for both magnetic configurations. For the P case (771 = VR = V) and no spin flips, we find Si/{Ge^V/2n) = (2 - r ) / 8 , thus Sj is independent of 77 [inset of Fig. 1(e)], while the magnetic fluctuations grow rapidly when rj -^ 1 [Fig. 1(e)]. We recall that in this case the mean magnetization is zero. When y^t is switched on, .SM becomes quenched for relatively small 7sf > 0.1. Unlike the P case, in the AP configuration (rjx = — 77ij = rj) Sj acquires a dependence on rj at y^i = 0 [see Fig. 1(f)]. In fact, Sj decreases as rj increases due to the spin-valve effect (/ is efficiently suppressed when rj ^ 1). Spin-relaxation processes reduce the spin polarization of both the conductor distribution and its fluctuations. Thus, the limit jsf -^ °° leads to Sj constant as a function of rj (compare to the P case). .SM in the AP case has a rather peculiar behavior with rj [Fig. 1(h)]. For small 7sf, .SM decreases with rj. At moderate 7sf, .SM is almost flat with rj but it shows an increasing behavior around rj ^ I. Eventually, for large 7sf the conductor becomes nomagnetic and .SM vanishes. This suggests a nonmonotonous behavior of .SM for certain values of rj close to the half metallic situation as a function of 7^/, which is confirmed in Fig. 1(g). We note that the maximum of .SM moves to higher 7sf by increasing rj as it should be expected.

CONCLUSIONS

We have investigated the current and the magnetization as well their fluctuations in a normal conductor attached to two ferromagnets for parallel and antiparallel arrangements. Interesting features arise in the parallel case where the conductor magnetization vanishes whereas its fluctuations depends on the contact polarization. For the antiparallel configuration magnetic fluctuations show a nonmonotonous behavior with the rate of spin-flip processes when the ferromagnets are almost half-metallic.

ACKNOWLEDGMENTS

This work has been supported by the Spanish MICINN Grant. No FIS2008-00781.

REFERENCES

1. For a recent review, see J. Fabian e< a/., Acta Physica Slovaca 57, 565 (2007). 2. S. Datta and B. Das, Appl. Phys. Lett 56, 665 (1990); K. Ono, H. Shimada, S. Kobayashi and Y.

Ootuka, J. Phys. Soc. Jpn. 65, 3449 (1996); J. Bamas and A. Fert, Phys. Rev. Lett 80 1058 (1998). 3. J.M. Kikkawa, LP Smorchkova.N. Samarth, and D. D. Awschalom, Science 281, 1284 (1997). 4. W. Belzig and M. Zareyan, Phys. Rev. B 69 140407 (2004). 5. See, however, J. Foros, A. Brataas, G.E.W. Bauer, and Y. Tserkovnyak, Phys. Rev. B 75, 092405

(2007). 6. Ya. M. Blanter and M. Buttiker, Phys. Rep. 336, 1 (2000). 7. A. Brataas, Yu.V. Nazarov, and G.E.W. Bauer, Phys. Rev Lett. 84, 2481 (2000). 8. D. Sanchez, R. Lopez, P Samuelsson and M. Buttiker, Phys. Rev. B 68, 214501 (2003). 9. The case of different chemical potentials in the contacts for each spin orientation will be considered

elsewhere: M.M. Gelabert et al, preprint (2009).

464

Noise characterization of a single parameter quantized charge pump

F. Hohls*, N. Maire*, B. Kaestner^, K. Pierz^, H. W. Schumacher^ and R. J. Haug*

*Institut fur Festkorperphysik, Leibniz Universitat Hannover, Appelstr 2, 30167 Hannover, Germany

^Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany

Abstract. The shot noise of a single parameter quantized charge pump is studied. The pumped current can be varied using a control gate. Quantized current plateaus / = nefp with fp the pumping frequency and e the electron charge are observed. The shot noise is minimal for each current plateau and maximal in between. Interestingly the first expected quantized current plateau at n = 1 is missing for certain control gate voltages. We use the measured shot noise to extract the probabilities for pumping none, one or two electrons at the position of the missing step. These probabilities can be used to characterize the dynamics of the non-adiabatic pumping process.

Keywords: Single electron effects, quantized charge pump, shot noise PACS: 72.70.+m, 73.23.Hk, 73.23.-b, 73.63.Kv

A quantized charge pump is a device that delivers a well controlled number n of electrons in each cycle of the driving frequency fp. Doing this with sufficiently high frequency and reliability and thus creating a current / = nefp would allow to realize a quantum standard for the ampere [1, 2, 3]. However, most approaches to realize quantized pumping [4, 5, 6, 7, 8] either lack the necessary accuracy or do not obtain sufficiently high currents. Recently a new promising reahzation was demonstrated [9]: It consists of a semiconductor quantum wire crossed by three metal gates which can be used to create potential barriers within the wire. For pumping, two of these gates were driven by phase locked high frequency signals in this initial demonstration. Recently Kaest-ner et al. have shown experimentally that driving a single gate is sufficient to achieve a quantized pumping current at very high frequency [10, 11, 12, 13]. A theoretical analysis of this non-adiabatic charge pumping promises that an optimized device could reach metrological accuracy [10].

The working principle of this new non-adiabatic charge pump is the following: The potential barriers generated by two energized Schottky gates form a quantum dot within the wire. The additional ac-voltage apphed to one of the gates first reduces one of the barriers and at the same time the dot level. The latter is reduced to below the Fermi energy, allowing the loading of electrons from the source onto the quantum dot. Then this barrier is raised, forming an isolated quantum dot. During this process some electrons are expelled dynamically from the dot and only a number n of electrons which is determined by details of the system and the gate voltages remains within the dot [14]. Applying an even higher voltage expels the electrons into the drain and finishes the pumping cycle. The number of electrons n pumped in each cycle is determined by a fine balance of



465

the dynamic confinement potential and the Coulomb energy and can be varied by the apphed dc and ac voltages.

A perfect quantized charge pump would be completely deterministic, pumping exactly n electrons in every cycle. As a result one expects vanishing low frequency noise power. For a non-ideal pump with non-zero probabilities of differing electron numbers per cycle a non-zero low frequency current noise S/(/) = S° ( / <C fp) is expected [15, 16]. Thus we can use a shot noise measurement to characterize the charge pump.

In this paper we present such noise measurements of a single parameter quantized charge pump. The pump is formed by three 100 nm-wide metalhc finger gates sitting on top of a semiconductor wire etched in a w-type AlGaAs heterostructure (Fig. Ic). Voltages are apphed to two finger gates to form potential barriers underneath. One of the finger gates is additionally driven by a sinusoidal voltage at a frequency of several hundred MHz (Fig. lb). The measurements are performed in a home built ^He cryostat at a temperature of 0.4 K using a special low capacitance probe and a very low noise current amplifier [11, 17, 18].

When varying the dc voltage applied to the finger gates and measuring the dc current we observe plateaus with quantized current values / = nefp indicating quantized charge pumping (Fig. la). In addition to the dc current the noise power is measured for frequencies / < 15 kHz. Exemplary spectra are shown in Fig. Id, one taken on a current plateau (/ = 2e/), the other one at a non-quantized current (/ = l-lef). Indeed the shot noise is strongly suppressed for the quantized current value (/ = 2el) and is nonzero for the non-quantized current value (1. lef) in agreement with our recent study in Ref. [11].

We will now focus on the measured shot noise for non-quantized current values. We examine the dependence of the current and noise as function of the voltage V2 apphed to the static gate (in Ref. [11] Vi was varied). Fig. 2a shows the normalized current as function of V2- Plateaus appear at / = 2efp, 3efp and 4efp while the / = efp plateau is missing. The measured shot noise is shown in Fig. 2b. It vanishes within the measurement resolution of AS < 10^^° A^/Hz at the / = 2efp plateau and shows minima at 3efp and 4efp, but interestingly the noise has a maximum at / = efp, marked by the dashed circle.

In Ref. [11] it was found that the non-zero noise power for current values («— l)efp < I < nefp is well modelled taking into account only two different possible pumping processes, i.e. having « — 1 or « electrons per cycle. Using this assumption one can calculate the expected noise power from the current. The result is displayed as black line in Fig. 2b. A good agreement with the measured noise is observed for V < — 110 mV and for y > —100 mV But in between these voltages near the maximum in the shot noise at the missing / = efp step it is clearly not sufficient to consider only two processes.

When regarding three different possible numbers of pumped electrons with probabilities p„-i, p„ and p„+i with pn-i + Pn + Pn+i = 1 the current and the shot noise are given by [15, 16]

I={n + pn+i-pn-i)efp and S = 2[pn-l+Pn+l-{Pn-l-Pn+l)^]e^fp. (1)

We can use these relations to determine the probabilities po, pi and p2 relevant around the missing current plateau in between 7 = 0 and / = 2efp. Fig. 2d shows the result. Starting from nearly zero current (/ o ~ 1) initially pi rises and po drops according to

466

(a)

\^^=^^

^ = ^ ^ ^ 1

\ \ w 1 ^.

90

I 2.5

2.0

1.5

1.0

0.5

0

(b) V

(d)

X T P I — ' ' ' ' ' '—I

2 i n 1 Lo u 1

1 71 1 1 1 1 1 1 —

/ vv^HV 1/ n/'' T i l / W | ^ 10 12 14

f(kHz)

FIGURE 1. a) Normalized pumped current at a frequency of / = 400 MHz as function of the two control gate voltages, b) Schematic of the measurement setup, c) SEM micrograph of the device shows the etched quantum wire that is crossed by three metallic gates, d) Selected noise power spectra for the voltage settings marked by the blue and red solid symbols in Fig. 2a. The horizontal line shows the averaged level in the range / = 5 — 15 kHz.

< "^o 0

^ -120

k a ) -1 1 1 1

w^ 1 1 1

1

1 1 1 1 1 1 1 1 1

(b) / ON • 1

i^j^jiyt^ '»,^ \—-:<^ ^ * * WW^'* ^^^ ^

-100 -80

-f ^J^^(^^ 1 1

k d ) ' ^ ^ ^ o PO CD

L O PI >

1 1 1 1 ' i * ^ * !

^ 1 < K ^ 1

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—1 1 QC>y^ ^>%J If

Ml (mV)

-115-110-105 -100 -95

V2 (mV)

0.8

0.6

0.4

0.2

0.0

FIGURE 2. a) Normalized current as function of V2 for / = 400 MHz and Vi = -130 mV. b) Measured shot noise power (Symbols, noise power average over / = 5 — 15 kHz). The line shows the expectation for only two non-zero pumping probabilities, c-hd) Current and calculated pumping probabilities throughout the initial current step.

467

Po+Pi = i- For al = efp plateau we would expect that pi reaches one while po goes to zero. But instead p2 starts to rise already for po « 0.5 and pi reaches only a maximum value of pi^max ~ 0.5 with /IQ ~ /'2 ~ 0.25 at / « efp. This reveals that the dynamics of the system in this regime does not favour a single probability, leading to the observed "missing step" and enhanced noise. This behaviour deviates from the cascade model of the dot loading described in Ref. [14]. Thus we need a new model for this regime of the charge pump which will be tested with our data.

In conclusion we have measured the current and the shot noise of a single parameter quantized charge pump as function of its control gate voltages. We have shown that the noise measurements can be used to characterize the probabihties of different electron numbers transferred in one pumping cycle. We examined the occurrence of a "missing step" which reveals an interesting regime of the dynamics of the quantum dot formed in the pump.

The authors would like to thank G. Hein, Th. Weimann, and H. Marx for support of clean room processing and sample growth. Funding was provided by the BMBF via project nanoQUIT.

REFERENCES

1. D. V. Averin, and K. K. Likharev, /. Low Temp. Phys. 62, 345-373 (1986). 2. K. Hensberg, A. A. Odintsov, F. Liefrink, and P. Teunissen, Int. J. Mod. Phys. B 13, 2651 (1999). 3. I. M. MiUs, P. J. Mohr, T. J. Quinn, B. N. Taylor, and E. R. WiUiams, Metrologia43, 227-246 (2006). 4. L. J. Geerligs, V. F. Anderegg, P. A. M. Holweg, J. E. Mooij, H. Pothier, D. Esteve, C. Urbina, and

M. H. Devoret, Phys. Rev. Lett. 64, 2691-2694 (1990). 5. L. P. Kouwenhoven, A. T. Johnson, N. C. van der Vaart, C. J. P. M. Harmans, and C. T. Foxon, Phys.

Rev. Lett. 67, 1626-1629 (1991). 6. H. Pothier, P Lafarge, C. Urbina, D. Esteve, and M. H. Devoret, Europhys. Lett. 17, 249-254 (1992),

ISSN 0295-5075. 7. M. W. Keller, J. M. Martinis, N. M. Zimmerman, and A. H. Steinbach, Appl. Phys. Lett 69, 1804-

1806 (1996). 8. J. M. Shilton, V. I. Talyanskii, M. Pepper, D. A. Ritchie, J. E. F Frost, C. J. B. Ford, C. G. Smith,

and G. A. C. Jones, /. Phys.: Condens. Matter 8, L531-L539 (1996), ISSN 0953-8984. 9. M. D. Blumenthal, B. Kaestner, L. Li, S. GibUn, T. J. B. M. Janssen, M. Pepper, D. Anderson,

G. Jones, and D. A. Ritchie, Nat Phys 3, 343 (2007), ISSN 1745-2481. 10. B. Kaestner, V. Kashcheyevs, S. Amakawa, M. D. Blumenthal, L. Li, T. J. B. M. Janssen, G. Hein,

K. Pierz, T. Weimann, U. Siegner, and H. W. Schumacher, Phys. Rev B 11, 153301 (2008). 11. N. Maire, F Hohls, B. Kaestner, K. Pierz, H. W. Schumacher, and R. J. Haug, Appl. Phys. Lett. 92,

082112(2008). 12. B. Kaestner, V. Kashcheyevs, G. Hein, K. Pierz, U. Siegner, and H. W. Schumacher, Appi. Phys. Lett.

92,192106 (2008). 13. B. Kaestner, C. Leicht, V. Kashcheyevs, K. Pierz, U. Siegner, and H. W. Schumacher, Appl. Phys.

Lett. 94,012106(2009). 14. V. Kashcheyevs, and B. Kaestner, Universal decay cascade model for dynamical quantum dot initial

ization, arXiv:0901.4102vl (2009). 15. Y. M. Galperin, O. Entin-Wohlman, and Y. Levinson, Phys. Rev B 63, 153309 (2001). 16. A. M. Robinson, and V. I. Talyanskii, Phys. Rev Lett 95, 247202 (2005). 17. A. Nauen, I. Hapke-Wurst, F Hohls, U. Zeitler, R. J. Haug, and K. Pierz, Phys. Rev B 66, 161303R

(2002). 18. N. Maire, F Hohls, T. Luedtke, K. Pierz, and R. J. Haug, Phys. Rev B 75, 233304 (2007).

468

Poor qubits make for rich physics: noise-induced quantum Zeno effects,

and noise-induced Berry phases. Robert S. Whitney

Institut Laue-Langevin, 6 rue Jules Horowitz, B.P. 156, 38042 Grenoble, France.

Abstract. We briefly review three ways that environmental noise can slow-down (or speed-up) quantum transitions; (i) Lamb shifts, (ii) over-damping and (iii) orthogonality catastrophe. We compare them with the quantum Zeno effect induced by observing the system. These effects are relevant to poor qubits (those strongly coupled to noise). We discuss Berry phases generated by the orthogonality catastrophe, and argue that noise may make it easier to observe Berry phases.

Keywords: decoherence, spin-boson model, geometric phase, Josephson devices, qubits PACS: 03.65.Vf, 03.65.Yz, 85.25.Cp

Our presentation for ICNF 2009 is entitled "Noise suppressing non-adiabaticity: observing a Berry phase alone", the details of which are in Ref. [1]. Here, we review the literature about the ways in which environmental noise can change the rate of quantum transitions (noise-induced quantum Zeno effects), and then briefly discuss the applications of these ideas to noise-induced Berry phases, such as those in Ref. [1].

Since qubits have been created and studied in experiments, theoretical descriptions of dissipation in two-level system have a renewed relevance. A "good" qubit is a fully controllable two-level system which is sufficiently weakly coupled to environmental noise that it experiences only weak dissipation (often well-described by the Bloch-Redfield equation [2]). For quantum computing, qubits need to have coherent oscillations with a quality factor of 10** or more (ten-thousand coherent oscillations before decaying). Only then will errors be rare enough that they could be fixed by error correction codes.

In contrast a "poor" qubit is a controllable two-level system which is strongly coupled to its environment, and thus experiences strong dissipation. The theory for this was discussed by Leggett et al [3], but there are still many open questions. It is worth noting that even the best qubits are only "good" for certain values of their parameters (values where the noise couples quadratically rather than linearly to superpositions of the two-levels). The qubits rapidly become "poor" away from these special points [4]. Our aim here is to reveal a little of the rich physics of such poor qubits.

Environmental noise (of classical or quantum origin) causes dissipation in quantum systems. We start this brief review with two very general and open questions.

I) What can a dissipative quantum system do that a non-dissipative system cannot?

This question is motivated by the observation that the density matrix of a dissipative two-level system, p, is described by three independent parameters, while a non-dissipative



469

two-level system is described by only two. For example, one can write

1 / l+5z Sx-\s 2 \ s x + \ S y \ " ^ • '

where Sx, Sy and Sz are real numbers. For non-dissipative (Hamiltonian) dynamics, all states must have a purity P = tr[p^] = 1, which means that s^ +Sy + s^ = I. In contrast, for dissipative dynamics we only require that s^+Sy+s^ < 1. This suggests that dissipation makes the dynamics much richer. In this article we discuss some effects of dissipation, however it is not clear to us whether all possible effects are known.

Our second question has potentially relevant quantum information processing.

II) Could noise be used to control a quantum system in a manner that cannot be achieved using traditional Hamiltonian manipulation?

We argue that this is definitely the case, and our presentation at this conference gives one example of this (see the section below about Berry phases and Ref. [1]). A more obvious example is thermalization; which drives the system to a single well-defined state, independent of the system's initial state. For example, coupling the two-level system to a zero temperature environment will cause it to decay to its ground state, irrespective of what state (mixed or pure) it was in before [5]. No non-dissipative evolution (Hamiltonian evolution) can do this. We wonder if other examples exist?

QUANTUM ZENO EFFECTS DUE TO NOISE

In the quantum Zeno effect [6], the transitions of a quantum system are slowed down (or stopped) by the fact one is observing the system. It is the quantum mechanical equivalent of saying that "a watched pot never boils" (i.e. a watched system never makes transitions). However, there is an analogy between observing a system (projective measurements of its state) and environmental noise. Indeed one can model a measurement device as a large object with many degrees of freedom interacting with the system. Thus one can ask whether environmental-noise induces a quantum Zeno effect, and (if so) what its nature is. In fact, there are a number of noise-induced Zeno effects, i.e. noise slows-down quantum transitions in a variety of (qualitatively different) ways. Also, certain noise can slightly speed up transitions (a weak anti-Zeno effect).

Quantum Zeno effect and anti-Zeno effect due to observations : Imagine a spin-half which is It) along the z-axis, in anx-axis magnetic-field, so its Hamiltonian is

•'spin — 2"^ \ \ 0

If the spin is left to evolve, it precesses around the x-axis, so that after a time njBx the spin will be | | ) . However what if we observe the spin's z-axis polarization at regular intervals Tobs much less than the time 5^ '? The observation is modelled by projecting

470

(a) ^ ^ (b)

system environment system environment

FIGURE 1. The Lamb shift. In (a) the system-levels have a smaller gap than the environment-levels. As one tums-on a system-environment interaction, there is level-repulsion which reduces the system's gap (the purple arrow on the system's upper state). In (b) the system-levels have a bigger gap than the environment-levels, so level-repulsion increases the system's gap. In most cases the environment has gaps both bigger and smaller than the system's gap, then the Lamb shift must be calculated.

out the spin-State along the z-axis as follows:

withprob.= \u\^

(3) withprob.= |vp

A simple calculation [6] shows that for i?;cTobs <C 1, the probability to find the spin up during all observations up to a time t is exp[—i?^Tobs' /4]. Thus rapid observations greatly reduce the spin's transition-rate. The anti-Zeno effect is the reverse; transitions are speeded up by the fact one is observing the system. The circumstance under which this occurs is nicely explained in Ref [7].

Weak quantum Zeno and anti-Zeno effects due to Lamb shifts : In 1947 Lamb noted that atomic Hydrogen's levels are slightly shifted with respect to the predictions of Schrodinger (or Dirac) quantum mechanics. The same year, Bethe explained this shift in terms of the interaction between the electron and the photonic excitations of the vacuum. Similar Lamb shifts occur whenever a system is coupled to an environment, and can be thought of in terms of level-repulsion, see Fig. 1. One can interprete this as a weak Zeno or anti-Zeno effect, in which the spin's coherent oscillations are either slightly slowed-down, or slightly speeded-up by a quantum environment.

Quantum Zeno effect due to over-damped spin dynamics There is a very pretty model of dissipative quantum mechanics in which one averages the quantum dynamics over classical white-noise. The fact the noise is white (meaning it contains all frequencies) is equivalent to saying that it is 5-correlated in time. This means one can time-slice on a timescale much less than that associated with the system dynamics, and then average over the noise in each slice independently. If the spin evolves under the Hamiltonian ^p in above, and we turn on noise along the z-axis, one finds that [8]

(4) •IT 0 0

0 -2r B,

0 \ -B:c

0

/ Sx

Sv \ s.

All

where s^ is the spin-polarization along the z-axis. Note that there is no Lamb shift here, so the noise does not modify the off-diagonal terms. One can use the level-repulsion argument to see why the Lamb shift is zero. If the white-noise were generated by an environment of quantum modes, all gaps would be equally represented. Thus there would be equal weight of modes pushing the system gap downwards as upwards, leaving the system's gap unchanged [10]. In contrast, for coloured noise the Lamb shift would be non-zero, and we would have B^ replaced by B'^ = Bx+ ff[T].

It is easy to rewrite Eq. (4) to show that s^ is given by the equation of motion of a damped harmonic oscillator; s'z + Tsz+B-^Sz = 0. Thus it has two regimes.

Under-damped {B, > T) : 5.(0 = A+e-^^+^<^''-^'^"'^' +^_e-[r-'(^'-r') '^'l^ (5)

Over-damped (5 , < r ) : 5.(0 =^+e-[r+(r'-^')'^'l^+^-e-[r-(r'-«^)'^'l^ (6)

where A^ and A_ are given hy t = 0 boundary conditions. From this we see that the frequency of coherent oscillations is reduced in the under-damped regime. By increasing r one enters the over-damped regime. As we take F ^ oo the second term in 5. decays at a rate <x F^ ' ; thus strong noise stops transitions of the spin. Consider the spin as initially |t) along the z-axis, so Sz{t = 0) = 1 and Sz{t = 0) <x Sy{t = 0) = 0, thus for large F we have A^ = 1 and A+ = 0. Then 5.(0 — exp[—i?^/(2F)], so the transition rate is suppressed by strong noise. In reviewing Ref. [8], Michael Berry [9] pointed out that this suppression of transitions in the over-damped case was a quantum Zeno effect. Indeed, in the limit F <C B^, the result coincides with that of the usual (observation-induced) quantum Zeno effect if we identify F with 21^^'.

All the above physics (under-damping and over-damping) is contained in the Bloch-Redfield equation, which is applicable whenever the decay time, 1 /F , is much greater than the environment memory-time (the memory time is zero for white-noise, so Bloch-Redfield becomes essentially exact) [11]. However often one makes an additional secular or rotating wave-approximation (e.g. as in Ref. [25]), which is only justifiable in the strongly under-damped limit (where one keeps ^[F]-terms but neglects ff[T^/Bx]-terms). As we see above, once one is beyond this under-damped limit, level-repulsion is not the only source of shifts of the coherent oscillation frequency.

A "super" Zeno effect due to the orthogonality catastrophe : The orthogonality catastrophe is a sort of extreme Lamb shift, which can only occur in a system strongly coupled to it environment. It was called "adiabatic renormalization" by Leggett et al [3], but Anderson introduced the idea earlier in the context of a fermionic (rather than bosonic) environment [12]. Each mode of the environment tries to adiabatically follow the system (spin), which means that all environment modes must shift each time the spin flips. Then the transition rate for the spin becomes B^Yl^^i (n_|n+), where \n±) is the state of the nth environment mode shifted one way or the other (depending if the spin is up or down). The overlap (n_|n+) is slightly less than one, however the product over all overlaps will be exponentially small when the number of environment modes, TV > 1. This argument neglects non-adiabatic effects (transitions of environment modes). Leggett et al [3] used renormalization group (RG) to deal with these, and concluded that certain environments (super-Ohmic) exponentially suppressed transitions , while others (sub-Ohmic) completely suppressed transitions! We call the latter a "super Zeno effect"

472

since finite noise-magnitude would completely kill transitions (cf. the examples above, where the transition occurs on a finite timescale for all finite T or T^JJ,').

Recent work[13], based on an approximate mapping between the spin-boson model and a ID Ising chain[14], suggests that Leggett et al missed one RG flow equation. Ref. [13] presented numerical results which supported the mapping, and seem to show a complete suppression of transitions; i.e. a "super Zeno effect". However we do not know of analytic calculations in this context (particularly for B^ ^ 0).

We propose bypassing the delicacies of RG analysis, by taking to heart the message of the orthogonality catastrophe and completely re-writing the original Hamiltonian (spin-half coupled to many harmonic oscillators) in terms of the shifted-environment modes. So when the spin is up (down) we work in a basis of environment modes shifted to the left (right). This exact polaron transformation [15, 16] gives us another system+environment Hamiltonian similar to the first, but with a more ugly system-environment coupling. Remarkably, it appears that the strong-coupling limit of the original Hamiltonian corresponds to a Bloch-Redfield limit of this new Hamiltonian. Initial indications are that the Bloch-Redfield analysis captures a variety of different types of strong noise-induced suppressions of spin-transitions; although it is too early to tell if they include the "super Zeno effect".

USING NOISE TO MAKE BETTER BERRY PHASES

In Ref [1], we show that if noise induces othogonality catastrophe physics in a spin's dynamics, then the spin acquires a Berry phase when the axis that the noise couples to is rotated around a closed loop. Earlier works [23] predicted noise-induced Berry phases which come from over-damped spin-dynamics, rather than an orthogonality catastrophe. We argue that in our case (unlike those earlier works), the Berry phase is relatively weakly affected by decoherence and non-adiabatic phases are exponentially small. Thus we reach the counter-intuitive conclusion that environmental noise may make it easier to measure a Berry phase.

Berry phases occur in many quantum systems [17,18], and have potential applications in both quantum computation [ 19] and metrology [20]. However in general. Berry phase do not appear "alone". If the parameters of a system's Hamiltonian complete a closed loop in a time t^, then the total phase acquired by an eigenstate is

I'total = I'dyn + I'Berry + O J ] | + O J ^ | + • • • , (7)

where the dynamic phase Odyn ° Et^, the Berry phase OBen-y ° [Et^)^, and the non-adiabatic (NA) correction o};^ °^ {Etp)^^, with E being a system energy scale (such as the gap to excitations). The Berry phase is hard to isolate (and thus hard to use for quantum computation or metrology), because it is the second term in this ?p"'-expansion of 'I'totai- To suppress the non-adiabatic terms, one must make tp large. This means 'J'dyn > ^Bv, SO One must subtract off Ojyn (using a spin-echo trick or degenerate states [22]) with extreme accuracy. For example, to get OBP with an error of 0.1%, one must make Et^ '-^ 10^, which means that Ojyn ^^ 10^ OBP. Then since one wants OBP with an error of 0.1%, one has to subtract off Odyn with an accuracy of 0.000P/o.

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In contrast, our Berry phase appears "alone"; non-adiabatic corrections are exponentially suppressed by an orthogonality catastrophe, and the dynamic phase is close to zero [24]. This can be shown by transforming to a rotating frame [21] which follows the slowly varying noise-axis. Then we arrive at a problem similar to that discussed in the previous section of the article, non-adiabatic phases are associated with spin-flips (in the new frame) and these are exponentially suppressed by the orthogonality catastrophe. We believe this is the only situation in which Ototai = 'I'Berry We speculate that this Berry phase may thereby be easier to observe (and utilize) than conventional Berry phase.

In conclusion, we note that we are currently looking at whether such noise can also make non-Abelian geometric phases (Wilczek-Zee phases [22]). Non-Abelian means ' W then ^ " is different from " ^ then ^ " . In the above case, it appears possible that if we perform two different loops ( ^ and M) with the noise, we get two different (SU(2)) geometric phases depending on which loop is followed first.

We thank M.V. Berry for comments on Ref. [1] and for informing us of Refs. [8, 9].

REFERENCES

1. R.S. Whitney, Preprint arXiv:0806.4897 2. F. Bloch, Phys. Rev. 105 1206 (1957). A.G. Redfield, IBM J. Res. Dev. 1 19 (1957). 3. A.J. Leggett, S. Chakravarty, A.T. Dorsey, M.P.A. Fisher, A. Garg, and W. Zwerger,

Rev Mod. Phys. 59, 1 (1987). 4. D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteve, andM. H. Devoret, Science

296, 886 (2002) 5. The ground state is pure, P = 1, so environments can enhance (as well as reduce) the system's purity. 6. B. Misra and E. Sudarshan, J. Math. Phys. 18, 756 (1977). 7. A.G. Kofman and G. Kurizki, Z. Naturforsch. 56 a, 83 (2001). Online at arXiv:quant-ph/0102002. 8. Ph. Blanchard, G. Bolz, M. Cini, G.F De Angelis and M. Serva, J. Stat. Phys. 75, 749 (1994). 9. M.V. Berry, Ann. N. Y Acad. Sci. 755, 303 (1995). Online at www.phy.bris.ac.uk/people/Berry_mv/ 10. Alternatively one can take a Bloch-Redfield approach, see for example Eq. (9) of Ref. [25] (dropping

the time-dependence and setting 0 = n/2). We see immediately that FQ is real for white-noise (5'(0) =constant), thus there is no Lamb shift.

11. R.S. Whitney, J. Phys. A: Math. Theor 41, 175304 (2008) 12. P W Anderson, Phys. Rev Lett. 18, 1049 (1967). 13. R. Bulla, N.-H. Tong and Vojta, Phys.RevLett. 91,170601 (2003) 14. J.M. Kosterlitz, Phys.Rev.Lett. 37,1577 (1976) 15. G.D. Mahan, Many Particle Physics (Plenum Press, New York, 1981) 16. D. Segal andD.R. Reichman, Phys. Rev A 76, 012109 (2007). 17. M.V. Berry, Proc. R. Soc. Lond. 392,45 (1984). 18. J. Anandan, J. Christian and K. Wanelik, Am. J. Phys. 65 180 (1997). 19. J.A. Jones, V. Vedral, A. Ekert and G. Castagnoli, Nature 403, 869 (2000). A. Ekert et at,

J. Mod. Opt. 47,2501,(2000). 20. J.P Pekola, J.J. Toppari, M. Aunola, M.T Savolainen and D.V Averin, Phys. Rev. B 60, R9931

(1999). R. Leone, L.P Levy and P Lafarge, Phys. Rev Lett. 100, 117001 (2008). 21. M.V. Berry, Proc. R. Soc. Lond. A, 414, 31 (1987). 22. F. Wilczek and A. Zee, Phys. Rev. Lett. 52, 2111 (1984). 23. A. CaroUo, G.M. Palma, A. Lozinski, M. Franuca Santos and V. Vedral, Phys. Rev Lett. 96, 150403

(2006). A. CaroUo and G.M. Palma, Laser Physics 16, 1595, (2006). S. Dasgupta and D.A. Lidar, J. Phys. B 40, S127 (2007). S.V Syzranov and Y Makhlin, JETP Lett. 87, 390 (2008).

24. Odyii goes linearly in the static B-field. In most qubits this can be tuned to close to zero. 25. R.S. Whitney, Y Makhlin, A. Shnirman, Y Gefen, Phys. Rev Lett. 94, 070407 (2005); NATO Science

Series II: Mathematics, Physics and Chemistry, 230, 9 (2006)

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Characterizing electron entanglement in multi-mode mesoscopic conductors

Fabio Taddei*, Vittorio Giovannetti*, Diego Frustaglia''' and Rosario Fazio*

* NEST CNR-INFM and Scuola Normale Superiors, Piazza dei Cavalieri 7, 1-56126 Pisa, Italy ^Departamento de Fisica Aplicada II, Universidad de Sevilla, Av. Reina Mercedes 2, E-41012

Sevilla, Spain

Abstract. We show that current correlations at the exit ports of a beam sphtter can be used to detect electronic entanglement for a fairly general input state. First, we demonstrate that multi-mode entanglement of electrons in a mesoscopic conductor can be detected by a measurement of the zero-frequency current correlations in an electronic Hong-Ou-Mandel interferometer By this means, one can further establish a lower bound to the entanglement of formation of two-electron input states. Second, we include the situation where electron pairs can enter the beam splitter from the same port or be separated due to backscattering. The proposed scheme allows to discriminate between particle-number and mode entanglement.

Keywords: Electronic transport in mesoscopic systems. Entanglement and quantum nonlocality. Entanglement measures, witnesses, and other characterizations PACS: 73.23.-b, 03.65.Ud, 03.67.Mn

INTRODUCTION

Generation, manipulation and detection of entangled electrons is necessary for the realization of integrated solid-state quantum computers. Among the several possibilities, a lot of attention has been devoted to the study of entanglement in multiterminal mesoscopic conductors (see Refs. [1, 2] for a review). It was shown that entanglement between spatially separated electrons can be detected by means of a beam splitter (BS) [3] which allows the incoming (and possibly entangled) electrons to be interchanged, giving rise to two-particle interference effects. As a result, the symmetry of the incoming state influences the current-noise correlations at the exit ports. In particular, Burkard and Loss [4] derived a lower bound for the entanglement of formation [5] of arbitrary mixed spin states in terms of current-noise correlations. Here we show a generalization of their results to multi-mode input states by introducing the electronic analog of the Hong-Ou-Mandel (e-HOM) [6] optical interferometer. First, we assume that a two-electron (possibly entangled) input state is injected into a BS whose arms support many propagating modes, including both orbital and spin degrees of freedom. Second, we consider the case where there is a finite probability amplitude that two electrons enter the analyzer at the same input port. This gives rise to two distinct forms of entanglement: occupation-number and mode entanglement [7, 8]. Under this generalized initial condition the analysis of the entanglement is complicated by super-selection rules induced by particle number conservation [1, 7, 8, 9]. We shaU show that, for the whole class of two-particle input states, simple data processing of the measured current cross-correlators can be used to address separately the various entanglement components [10, 11].



475

• ^ - 4

• ^ - 3

FIGURE 1. Sketch of the electronic Hong-Ou-Mandel interferometer.

T H E H O N G - O U - M A N D E L I N T E R F E R O M E T E R

The e-HOM is sketched in Fig. 1. Pairs of electrons of a given energy E above the Fermi sea, prepared in a (possibly entangled) initial state, enter the interferometer from the input ports 1 and 2. Electrons passing through the port 2 undergo an additional, controllable, phase shift â before impinging on the BS. Zero-frequency current correlations are measured at the output ports 3 and 4. The electron states, at energy E, are labeled by the indices (7, a) where j = 1, • • • 4 labels the ports of the e-HOM interferometer, and where a is the composite index {I, s} with s referring to electron spin component along the quantization axis, and I referring to the orbital channel. Following the Landauer-Biittiker scattering formulation of quantum transport [12], the current operator of the 7-th port is defined as [13]

Ij{t)-: - E hv e-""'[b]^{E)bj,a{E + na))-a]^{E)aj,a{E + na))],

E,o),a j,c(y j,c(y (1)

where v is the density of states of the leads and bjâ {E) are outgoing fermionic operators connected to the incoming ajâ{E) operators via a scattering matrix. The zero-frequency current correlations are defined as

^jf ,. hv lim -^ dh / dt2{8lj{h)5lf{t2)), (2)

where the average time and 51 j = Ij -

T Jo Jo

is taken over the incoming electronic state, T is the measurement

O N E E L E C T R O N P E R P O R T

Consider the case in which, for a given energy E, one electron per port enters the interferometer from 1 and 2:

l^^n) =Ul'â,p al„{E)alp{E)\0), E a,/3

(3)

where |0) is the Fermi sea at zero temperature and O^^j is the two-electron amplitude which we assume to be independent of E and satisfying the normalization

476

condition Xa,/31'^a./Jp = 1- A straightforward calculation of the Fano factor Fjji = Sjji/{2e^J{Ij) (If)) leads to the expression

F^4 = -T{l-T){l-w<s,), (4)

where T is the transmissivity of the BS and

w^^J^[0„^pr0p^„e'^<P"-<P^K (5)

is a (real) quantity which depends on the controllable set of phases {(pa}- By using the Chauchy-Schwartz inequality and the normalization condition of 0 „ jj it follows that for a generic input state (3) one has —l<w^<l. However, if |Oi2) in Eq. (3) is a separable state with respect to the input ports 1 and 2, it is possible to show that w^ is non negative. Negative values of w^ are hence a direct evidence of the presence of entanglement in the input state. The quantity w^ therefore acts as an entanglement witness [14] for the class of two-electron states analyzed here. Furthermore, it can be proven [4, 10] that the Fano factor of Eq. (4) is related to the entanglement of formation Ef of the input state p of the e-HOM interferometer by the inequality

with

Ef{p)>S'{W{p)), (6)

and (f(x) = H(J + •\/X(1 —X)) for x G [1/2,1] and null otherwise [here H{x) =

—xlog2X — (1 — x) log2(l — x)]. This shows that by measuring the Fano factor one can determine a lower bound for the entanglement of formation through Eq. (6).

GENERIC INPUT STATE

Let us now consider a generic input state of the form

|¥) = sin0(cos(/) | 0 „ ) + sin(/) |O22))+cos0 |Oi2) , (8)

where 9,(p G [0, n:/2] and |Oy) describe two electrons of energy E entering the BS from the ;-th and J-th port respectively (ij' =1,2) . The analysis of the entanglement contained in the state \W), can be done naturally by bipartitioning the system with respect to the port labels 1 and 2. The case 0 = 0 has been considered in the previous section. For 0 7 0 we have both occupation-number and mode entanglement. The first one is present whenever we have a non trivial superposition among terms where each lead supports a different number of incoming electrons, i.e. jOn), IO22), and IO12). The second one, instead, originates from the component IO12) of Eq. (8): it is present when the leads 1 and 2 possess one electron each and are entangled through the electronic orbital/spin modes a. The (dimensionless) current cross-correlator at the output ports 3 and 4 is

C 3 4 - ^ H m / W . 2 ^ ^ ^ ^ i ¥ ^ . (9)

477

2e2 T^^Jo

Differently from .S'34, the C34 is linear in the input state of the system for 9^0. For any given density matrix p = S^_p |*P )(*P | with \Wi) as in Eq. (8) we obtain C34(p) = 'LtP£C34(Wi), where the correlators C34(*P ) take the form

C34('P)= [ l+wcos20 + vsin20sin(2(/))]/4. (10)

Here, w and v are real quantities satisfying |w|,|v| < 1 which depend upon the interferometer phases cpa and the input state parameters. Following the derivation of Refs. [4, 10], it is easy to show [11] that £'/(p) can be lower bounded as in Eq. (6) by the quantity

F ( p ) = l -2C34(P) . (11)

It foUows that C34(p) < 1/4 implies £'/(p) > 0. Therefore, we can conclude that also in the general case in which the two electrons can enter the same port, values of C34 smaller than 1 /4 are direct evidence of entanglement in the input state. It is also possible to discriminate between occupation-number and mode entanglement through the quantities

C^4 (*I') = [1 + w cos^ 0]/4 and C^4 (*I') = v sin^ 0sin(2(/))/4. The presence of mode

entanglement can be detected by finding values of (pa such that C34 < 1 /4. Vice-versa,

we observe that any value of C34 ^ different from zero is indicative of occupation-number entanglement in the system.

REFERENCES

1. C. W. J. Beenakker, "Electron-hole entanglement in the Fermi sea", in International School of Physics Enrico Fermi Vol 162, Quantum Computers, Algorithms and Chaos, edited by G. Casati et al., lOS Press, Amsterdam, 2006, pp. 307???347.

2. G. Burkard, "Theory of solid state quantum information processing", in Handbook of Theoretical and Computational Nanotechnology, edited by M. Rieth and W. Schommers, American Scientific Publisher, Valencia, 2006.

3. G. Burkard, D. Loss, and E.V. Sukhorukov,P/;j;s. Rev. B 61, R16303-R16306 (2000). 4. G. Burkard, and D. Loss, Phys. Rev. Lett 91, 087903 (2003). 5. C. H. Bennett, D. P DiVincenzo, J. A. Smolin, and W. K. Wootters, Phys. Rev A 54, 3824-3851

(1996); W. K. Wootters, Quantum Inf Comput. 1, 2 7 ^ 4 (2001). 6. C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev Lett. 59, 2044-2046 (1987). 7. H. M. Wiseman and J. A. Vaccaro, Phys. Rev Lett 91,097902 (2003); H. M. Wiseman, S. D. Bartlett,

and J. A. Vaccaro, quant-ph/0309046. 8. N Schuch, F Verstraete and J. I. Cirac, Phys. Rev Lett 92, 087904 (2004); Phys. Rev A 70, 042310

(2004). 9. F. Verstraete and J. I. Cirac, Phys. Rev Lett 91, 010404 (2003). 10. V. Giovannetti, D. Frustaglia, F. Taddei, and R. Fazio, Phys. Rev S 74, 115315 (2006). 11. V. Giovannetti, D. Frustaglia, F. Taddei, and R. Fazio, Phys. Rev B 75, 241305 (2007). 12. M. Buttiker, Phys. Rev B 46, 12485-12507 (1992). 13. G B. Lsaovik, Pis'ma Zh. Eksp. Teor Fiz. 49, 513-515 (1989) [JETP Lett. 49, 592-594 (1989)]. 14. M. Horodecki, P Horodecki, and R. Horodecki, Phys. Lett. A 223, 1-8 (1996); M. Lewenstein,

B. Kraus, J. I. Cirac, andP Horodecki, P/;j;s. Rev A 62, 052310 (2000).

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Resistance Noise in Graphene Based Field Effect Devices

Atindra Nath Pal and Arindam Ghosh

Department of Physics, Indian Institute of Science, Bangalore INDIA 560012.

Abstract. We present a low-frequency electrical noise measurement in graphene based field effect transistors. For single layer graphene (SLG), the resistance fluctuations is governed by the screening of the charge impurities by the mobile charges. However, in case of Bilayer graphene (BLG), the electrical noise is strongly connected to its band structure, and unlike single layer graphene, displays a minimum when the gap between the conduction and valence band is zero. Using double gated BLG devices we have tuned the zero gap and charge neutrality points independently, which offers a versatile mechanism to investigate the low-energy band structure, charge localization and screening properties of bilayer graphene.

Keywords: bilayer graphene, conductivity noise, band gap PACS: 73.20.At74.40.-Fk81.05.Uw

INTRODUCTION

Graphene, a single sheet of carbon atoms, because of its unusual electronic properties, has become a potential candidate for future electronics. Specially, interest in bilayer graphene (BLG) is fueled by the abihty to control the energy gap between its valence and conduction bands through external means[l, 2]. The Bemal stacking of graphene layers in BLG leads to four orbitals, but strong interlayer coupling between the adjacent sites results in splitting to two high energy bands, leaving two low-lying states that form a zero gap semiconductor with quadratic dispersion. However, both chemical doping[3, 4] and transverse electric field[5] has been shown to open a band gap at the Fermi energy in a controllable manner, making BLG particularly attractive for nanoelectronic applications. However, intrinsic disorder has been suggested to modify the band structure in BLG as in conventional semiconductors by smearing the bands, and localizing the states in band tails [6]. Most analysis assume disorder to be static, and the influence of any kinetics or time-dependence of disorder on various properties of BLG is still poorly understood.

The resistance fluctuation, or noise measurement in graphene based devices [7] has revealed that in case of single layer graphene (SLG), the noise magnitude decreases with increasing carrier density («) as the trap potentials are screened effectively by the mobile charges. Conversely, noise in BLG increases with increasing n, which has been explained by the diminished ability of BLG to screen the external potential fluctuations in presence of finite bandgap (A^). Here, we present the low-frequency resistance noise measurements in both SLG and BLG devices. A spatially extended double-gated BLG device is used to achieve an independent tunabihty of A^ with both n and S", where (f is the transverse electric field across the electrodes, to separate the influence of band structure and carrier density on screening. Our experiments indicate that multiple



479

http://73.20.At74.40.-Fk81.05.Uw

.Mv>il ^ f

Top Gate Voltage V, (V) Vtt(V)

FIGURE 1. (a) Schematic of the double gated BLG device used in the experiment, (b) Optical micrograph of the device fabricated on the flake. The scale bar is 10 ;Um. (c) Raman spectrum of the BLG showing the typical G and 2D peaks, (d) Resistance versus topgate voltage for various back gate voltages, ranging from 30 V to —30 V (left to right) with an interval of 5 V. Insets show schematics of corresponding band structures, (e) 2D color plot of the resistance as a function of both top and backgate voltages at T = 107 K, showing that the position of the charge neutrality peak shifts with both gate voltages according to the capacitance ratio.

processes involving the charge traps are active in producing the resistance noise which is intimately connected to the BLG band structure, being minimum at A^ = 0 even if it corresponds to a nonzero n.


Device Fabrication and Gate Voltage Characteristics

Monolayer and bilayer graphene were prepared by mechanical exfoliation of graphite on top of an oxidized silicon wafer (300 nm of Si02) and characterized by Raman spectroscopy (Fig. Ic). Then field effect devices were made using standard electron beam lithography. The double gated devices were prepared similar to those reported in Refs. [8, 10]. Schematic of the vertical cross-section and optical micrograph of a typical BLG device are shown in Fig. la and Fig. lb respectively. The net carrier density on the BLG flakes is then given by n = no + ^ox^hgl^dox + ^cp^tgl^dcp-, where no is the intrinsic doping, Eox and dox are respectively the dielectric permittivity and thickness of the Si02 layer, while Ecp and dcp are those for the cross-linked polymer layer. The voltage applied on the back (doped silicon) and top (gold) gates are denoted as V ^ and Vtg-, respectively. Typically, dox ~ 300 nm and dcp ~ 100 nm were used which made the topgate about three times more effective (Fig. le)in inducing carriers in the BLG devices than the backgate {EQX ~ 4, Ecp ~ 4.5). The charge mobility of the device was estimated to be ^ 1160 cm^/Vs, which contained an intrinsic hole doping of —no ~ 5.82 x 10^ cm~^. A finite i is established between the electrodes when V ^ and Vtg are different. The resistance(/?)—Vf characteristics of the device at T f 107 K is shown in Fig. Id for several different values of V ^ spanning between —30 to +30 V. Existence of the electric

480

Back Gate Voltage (V)

^«^«Bsd^ %££!.

Top Gate Voltage (V)

3.1 1 10 100

frequency (Hz)

FIGURE 2. (a) Electrical transport and noise characterization of a dual gated BLG device. The resistance and the normaUzed noise power spectral density (NR) as functions of top gate voltages are shown for various back gate voltages: 30 V, 0 V, —30 V respectively (top to bottom). The thick solid lines are guide to the eye. The insets show typical noise power spectra SR/R^, far from the charge neutrality point for each back gate voltage, (b) The resistance and the normalized noise power spectral density (NR) as functions of back gate voltage for a single layer device at 1OOK. (c) /SR (/) at the noise minimum (Ag = 0) for various electric fields (V/nm), all of which show existence of a peak at / a; 7 Hz. Individual traces have been shifted vertically for clarity.

field-induced band gap becomes increasingly prominent at higher Vhg with increasing R at charge neutrahty.

Noise Measurements and Results

Low-frequency noise in graphene devices were measured using ac four-probe method, as well as in a five probe technique with a dynamically balanced Wheatstone bridge (for details see Ref. [9]). The excitation was kept below 500 nA, and the quadratic excitation dependence of noise at fixed R was confirmed. Typical resistance noise power spectra for BLG device are shown in the insets of Fig. 2a. The resistance noise power spectral density is often normalized as, SR^/) = JniDR^/nAcf, where JH is the Hooge parameter, and AQ « lljivc? is area of the BLG flake between the voltage probes. We found JH ^2x 10^^ sufficiently away from charge neutrahty (at«'-^ 1.5 x 10^^ cm^^), which is typical for semiconductor devices including BLG nanoribbons [7]. Here, instead of focusing on JH or noise magnitude at a specific frequency, we compute and analyze the total variance of resistance fluctuations {NR), which is essentially the normahzed noise power spectral density integrated over the experimental bandwidth.

Fig. 2a show the variation O^NR and the corresponding average resistance as functions

481

of Vtg at three different values of Vbg. For all V^g, NR shows a minimum at a specific Vtg, denoted as V^ """, and increases monotonically on both sides of V/^'"^". A similar experiment was performed with a back gated single layer device (at 100 K) (Fig. 2b), which shows that the noise magnitude increases with gate voltage in both side of the charge neutrahty point. So the observed behavior for noise in BLG is its intrinsic characteristics, which has been observed in case of BLG nanoribbons as well [7]. Although, Vf^'""^ and v^^""" are not equal in our measurements, indicating that noise minimum has been shifted away from charge neutrality in the presence of finite S". A detail analysis shows that the minimum of NR correspond to A^ = 0. Apart from that, the experimental result provide several crucial insights over charge organization and screening properties in BLG (see Ref. [10]). We have plotted fS{f) as a function / in Fig. 2c at A^ = 0 for various S", which manifests a saturation of noise at low frequency and a peak in fS{f) a t « 7 Hz. This indicates that the noise in BLG device is originating from the slow charge exchange process between the percolation cluster and the trap states in Si02 [11]. The experimental results provide a consistent framework to understand the role of disorder in the electronic and thermodynamic properties of BLG.

CONCLUSIONS

We have measured the low-frequency resistance noise in graphene based field effect devices. Using a dual gated BLG device, resistance fluctuation have been measured as a function of charge density and inter-electrode electric field. The absolute magnitude of noise is intimately connected with the BLG band structure, and shows a minimum when the band gap of the system is zero. The experiments also reveal the charge organization in BLG-based electronic devices, and the microscopic mechanism of resistance noise.

ACKNOWLEDGEMENTS

We acknowledge the Department of Science and Technology (DST) for a funded project, and the Institute Nanoscience Initiative, Indian Institute of Science, for infrastructural support.

REFERENCES

1, E, McCann and V, I, Falko, Phys. Rev. Lett. 96, 086805-4 (2006), 2, E, McCann, Phys. Rev. B 74, 161403-4(R) (2006), 3, E, V, Castro et al, Phys. Rev Lett. 99, 216802-4 (2007), 4, T, Ohta et al. Science 313, 951-954 (2006), 5, J, B, Oostinga et al. Nature Mater. 1, 151-157 (2008), 6, V, V, Mkhitaryan and M, E, Raikh, e-print arXiv:condmat/0807,2445vl (2008), 7, Y, Lin and Phaedon Avouris, Nano Lett. 8, 2119-2125 (2008), 8, D, Goldhaber-Gordon et al, Phys. Rev Lett. 98, 236803-4 (2007), 9, A, Ghosh and A, K, Raychaudhuri, Phys. Rev Lett. 84, 4681-4684 (2000); A, Ghosh et al, e-print

arXiv:condmat/0402130 vl (2004), 10, Atindra Nath Pal and Arindam Ghosh, e-print arXiv:condmat/0812,2974vl (2009) 11, B,I, Shklovskii, Solid State Commun. 33, 273-276 (1980),

482

1/f Resistance Fluctuation In Carbon Nanotubes

Hideo Akabane and Norihito Miwa

Iharaki University, Faculty of Engineering, Department of Media and Telecommunications Engineering Nakanarusawa 4-12-1, Hitachi 316-8511, Japan

akahane(a),mx. iharaki. ac.jp

Abstract. The resistance fluctuation of carbon nanotubes(CNTs) in low temperature was measured and the results were compared to that of a simulation of lattice vibration. Further, the influence of laser irradiation except the contact parts of the electrodes was studied. As the result, it was thought that the main causes of 1/f resistance fluctuation of CNTs derived from mobility fluctuation.

Keywords: 1/f resistance fluctuation, carbon nanotube, lattice vibration, Schottky barrier. PACS: 05.40.-a

INTRODUCTION

In various materials, it has been reported that the fluctuation of electrical resistance shows the 1/f type spectrum in the low frequency range. About the generating mechanism of 1/f resistance fluctuation, there is an opinion that it is caused by the mobility fluctuation, and there is also an opinion that it is caused by the fluctuation of carrier density. Recently, the 1/f resistance fluctuation in carbon nanotubes (CNTs) attracts attention[l,2]. In matters of CNTs, addition to the above origin, fluctuation of Schottky barrier, influence of the substrate and the influence of atmosphere has been pointed out as the origin of resistance fluctuation[3-6]. In this study, the resistance fluctuation of CNTs in low temperature with laser irradiation was measured and the results were compared with a simulation. As the result, the origin of mobility fluctuation was supported.

EXPERIMENT

In this study, the resistance fluctuation of CNTs in vacuum low temperature was measured. The CNTs were put in a glass capillary of 0.1mm diameter at 10 mm in length, and set on the cooling stage of a cryostat. There is a hole in the cooling stage and the temperature of capillary can partially raise by irradiation of laser beam as shown in Fig. 1. The contact parts of CNTs and an electrode are thermally coupled with the cooling stage, and there is few influence of irradiation of the laser here. If Schottky barrier is the main causes of 1/f resistance fluctuation, it should not be influenced by laser irradiation. Figure 2(a) shows the change of the resistance and Fig.



483

http://ac.jp

2(b) shows the change of power spectral density (PSD) of resistance fluctuation observed in multi-walled CNTs.

Laser Irradiation

Collimate Lens

Cooling Sage

Capillary

FIGURE 1. Schematic of the experimental setup. The middle part of the capillary is heated by laser irradiation. The contact parts of the CNTs and electrodes are thermally connected to the cooling stage.

The resistance of the sample increased with the decrease of temperature. The resistance at 30K was about 1.8 times as large as the value at 300K. On the other hand, the 1/f resistance fluctuation significantly decreased. The intensity of 1/f resistance fluctuation at 3 OK was about two columns smaller than that of 300K. The laser was irradiated when the temperature of the cooling stage reached at 3 OK.

O laser irradiation

fin

0 50 100 150 200 250 300 350

Temperature (K)

lO'

10"

V

\ S"

\'

w ^ . SSlInu 1 ^1^9 nh

1 Nil; ^

300K

lOOK

70K

30K Laser

10"' 10" 10"

(a)

Frequency (Hz)

(b)

FIGURE 2. Temperature dependences of the resistance (a) and the power spectral density of resistance fluctuations (b).

484

The resistance became the same as the case of ~50K and the intensity of the 1/f fluctuation became the same as the case of ~70K. The equivalent temperature was different between the resistance and its fluctuation. Both of them are total values of the sample, but the heat by the laser irradiation is partial. The intensity change of 1/f fluctuation by the change of temperature is remarkably large than the change of resistance. Thus, the change of total resistance by irradiated part becomes small but the change of total fluctuation becomes rather large. It thought to be that the resistance fluctuation is mainly generated in the small heating point. The similar experiments were performed using single walled CNTs and a carbon fiber as a sample. As the results of the single walled CNTs, the resistance became the same as the case of ~45K and the fluctuation became the same as the case of ~50K, and as that of the carbon fiber, the resistance became the same as the case of ~50K and the fluctuation became the same as the case of ~60K. The fundamental aspects of them were the same as the result of multi-walled CNTs. From these results, it was thought that the main causes of 1/f resistance fluctuation of CNTs were not change of Schottky barrier but the thermal vibration of the lattice.

SIMULATION

Next, we compared results of the simulation of the two-dimensional lattice vibration with the above experimental results. The triangle-lattice shown in Fig. 3(a) was used in our simulation. The fluctuation of scattering probability by acoustic phonon is shown in Fig. 3(b) [5]. In this figure, the dependence of initial displacement D which corresponds to the system temperature is shown. The dotted line indicates the case with Z3=0.01ro and sohd line indicates the case with Z3=0.03ro.

P fin

\

I j

v

\ ^

\ ^

hk.. \

%

M.

m T^

li 1 i J 1

£)=0.03ro

£)=0.01ro

10" lO' io '° lO" l o " lo'-

Frequency (Hz)

(a) (b)

FIGURE 3. Schematic of the simulation system (a) and the dependency of the initial displacement of the scattering probability fluctuations (b).

485

The difference of this initial displacement correspond to the one-hundred times difference of the temperature and it agrees with the result of Fig. 2(a). From these results, it was thought that the origin of 1/f resistance fluctuation of CNT was not carrier fluctuation but the mobility fluctuation by lattice vibration.

CONCLUTION

The resistance fluctuation of CNTs in low temperature was measured. The CNTs except the contact parts of the electrodes were heated by irradiation of a laser beam. Tthe resistance fluctuation remarkably increased and the fluctuation of the Schottky barrier was not the main origin of 1/f resistance fluctuation. Further, from the comparison of the experiments and the simulation, it was thought that the main cause of 1/f resistance fluctuation is the mobility fluctuation by lattice vibration.

REFERENCES

1. F. Liu, K. L. Wang, D. Zhang, and C. Zhou: Appl. Phys. Letters 89, 063116 (2006). 2. M. Ishigami, J. H. Chen, E. D. Williams, D. Tobias and M. S. Fuhrer: Appl. Phys. Letters 88,

203116(2006) 3. Yu-M. Lin, J. Appenzeller, Z. Chen and P. Avouris: Physica E37, 72 (2007). 4. R. Tarkiainen, L. Roschier, M. Ahlskog, M. Paalanen and P. Hakonen: Physica E28, 57 (2005). 5. G. Bosman and S. Reza: 19 International Conference on Noise and Fluctuations ICN -2007,

edited by M. Takano et al., 2007, pp. 251-256 . 6. F. Liu, M. Bao, H. Kim, K. Wang, C, Li, X. Liu and C. Zhou: Appl. Phys. Letters 86, 163102 (2005). 7. H. Akabane and R. Kawahara: 19 International Conference on Noise and Fluctuations ICN -2007,

edited by M. Takano et al., 2007, pp. 467-470.

486

Low Frequency Noise in 2DEG Channel of AlGaN/GaN Heterostructures Scaled to

Nanosize Width

Svetlana A.Vitusevich", Mykhaylo V. Pettychuk , Viktor A. Sydoruk", Thomas Schapers'', Hilde Hardtdegen", Alexander E. Belyaev", Andreas

Offenhausser" and Norbert Klein"

"Institute of Bio- and Nanosystems, Research Center Jiilich, 52425 Jiilich, Germany Taras Shevchenko National University, 01033 Kiev, Ukraine

'Institute of Semiconductor Physics, NASU, 03028 Kiev, Ukraine

Abstract. AlGaN/GaN wire structures were fabricated with widths scaling down to 185 nm. The transport property results of gateless high electron mobility structures demonstrate the formation of a high quality two dimensional electron gas (2DEG). The systematic study of the noise spectral density dependence on the width of the wire reveals nonmonotonous behavior. The peculiarities are explained by the transformation of the 2DEG to a one dimensional system by decreasing the lateral dimension. The confinement was confirmed by the increase in effective conductivity as a result of the decrease in scattering due to a stronger localization of the conducting electrons.

Keywords: AlGaN/GaN, wires, noise spectroscopy PACS:73.21.Hb,72.70.+m,71.55.Eq,72.80.Ey

INTRODUCTION

Ill-nitride materials are particularly promising for high-sensitivity label-free biosensensor applications [1], chemical and gas detectors [2] due to their unique properties such as a low density of the surface states, a high drift velocity, a highspeed response, and an availability of a high density two dimensional gas (2DEG) even under undoped conditions that avoid the doping related noise fluctuations in nanodevices. One-dimensional (ID) semiconductor nanowires offer interesting properties different from those of the bulks [3] and additional advantages, since they have an improved surface-to-volume ratio, a predicted enhanced mobility due to the ID confinement effect, and the possibility of room temperature operation based on quantum effects. Authors of Ref.[4] show that nanowires can be used for future label-free biosensor development. At the same time the increase in average relative nanowire stability the with increasing diameters of the nanowire has been found [5]. As dimensions are scaled down the material properties and underlying physics have to



487

be explored systematically and precisely to utilize the full potential of the unique properties for nano-applications. Noise spectroscopy allows not only to determine information on the signal to noise ratio, but also to extract important data concerning the materials' properties and to monitor dynamical processes with time scale resolution [6,7]. Recently we showed that the properties of AlGaN/GaN heterostructures with sufficiently large width of the channel of 100 |im can be successfully monitored by noise spectra studies [8].

In this paper, we report on the fabrication and characterisation of high electron mobility transistor structures based on epitaxially grown AlGaN/GaN layers. Electron beam lithography was used for device patterning with channels scaled to nanosize width to study the fundamental concept of scaling dimensionality.

EXPERIMENTAL DETAILS

The high electron mobility transistor structures used in this study were grown on c-plane AbOs substrates by metal organic chemical vapour deposition (MOCVD). The layer structure includes an initial 3-|im-thick undoped GaN buffer followed by a 40-nm thick unintentionally doped Alo iGaogN layer. The investigated nanowires were processed from AlGaN/GaN structures using electron beam lithography. It is known that the fabrication of ohmic contact to nanowires is more difficult as to large area materials [9]. Therefore to prevent the large contribution of the contact resistance in the total resistance of the devices we processed the structure with the long channel length of 620 |im. The ohmic contacts were produced by standard Ti/Al/Ni/Au metalhzation followed by anneahng at 900°C for 30s. I-V characteristics were measured in a wide temperature range by cooling from T=300K to temperatures below hquid-nitrogen temperature of samples mounted onto the copper finger in the Dewar with the use of a Stirhng cooler. The low-frequency noise spectra were measured in the frequency range of l-IO^ Hz at different temperatures in the range from 70 to 290K. Ultraviolet excitation was used to restore trap states after treatment of the structures at a high applied voltage of 20V.


The sheet carrier concentration was determined to be 2.2xl0'^cm"^ with a mobility of 1200 cmWs at room temperature using Hall bar mesas with voltage probes separated by a distance of 410 |im. It is important to note that the 2DEG was obtained as a result of the unique polarization induced interface charge in AlGaN/GaN heterostructures without using any doping. This avoids the ion impurity scattering and fluctuations processed due to impurity atoms. Figure la shows a schematic cross section of a set of parallel wires demonstrating the relevant design dimensions. The scanning electron beam microscopy confirms that 160 identical wires with widths W in the range from 1109 down to 185 nm were achieved reproducibly.

The effective conducting width Weff of the wires is expected to be smaller than W due to the large surface potential of GaN and the formation of surface carrier depletion regions at the mesa edges. In order to get information on the Weff of the wires we plotted their conductance extracted from the measured I-V characteristics as a function of different wire widths (Fig. lb). The conductance of single wires was determined by dividing the total conductance by the number of wires connected in parallel.

0.7 I 1

AI„.Ga„ „N

GaN

W

160 fingers W

600

W ( n m )

1000 1200

FIGURE 1. (a) A schematic cross section of the set of parallel wires demonstrating the relevant design dimensions, (b) The conductance of the single wires as a function of the geometrical width W.

The measured width, W, determined using the electron microscope and the effective 2DEG width, Weff, are listed in Table I for wire samples.

TABLE 1. Characteristic widths of AlGaN/GaN structures W(nm) 185 280 Weff (nm) - 8 0 _

360 160

472 272

715 515

925 725

1109 909

The I-V characteristics of wires with small widths demonstrate nonmonotonic temperature dependence reflecting the charge carrier redistribution in nanoscaled samples (Fig. 2a). The measured noise spectra follow a l/f dependence with y ^ I for structures with wide and nanoscale widths. By comparing the noise spectra density it is possible to choose a sample with a specific width which shows the lowest noise as well as enhanced mobility. Samples with an effective width of 160 and 272 nm revealed increased conductivity after 20V voltage treatment, while samples with 160 nm widths show one to two orders of magnitude lower noise level in comparison with samples with smaller and larger widths (Fig. 2b). As can be seen in Fig.2b samples with 160 nm widths show one to two orders of magnitude lower noise levels in comparison with samples with smaller and larger widths. A generation-recombination component of the noise is registered at decreasing temperature. The traps have a large energy barrier and a noise level which decreases with decreasing temperature from T=300K to T=200K followed by slow noise level increase in the temperature range from 200K to 70K. The later noise spectrum shifts to lower amplitudes after excitation with ultraviolet light. The low- frequency noise behaviour can be explained using a charge fluctuation model taking into account DX-like traps.

489

• —

. , —•-

4

^ ^ • - "

\y'"\^ ^^^^

1

,,

^ T ^

,.

,,-'" , " ,^ , 4 ^ ' "

,^^ - '''

100 150 200 250 300

T (K)

10° 10' 10^ 10° 10'

( ) . ^ ) . . . FIGURE 2. Experimental results obtained after 20V treatment of AlGaN/GaN wires with different channel widths, Weff(nm): 1-909, 2-272, 3-160, 4-80; (a) Resistance of samples measured as a function of temperature, (b) Noise spectra measured at V = 50mV.

The traps resulting in a noise level increase with a temperature decrease due to the occupation of the centers and due to a noise decrease at their excitation by ultraviolet light. In addition a crossover from a two dimensional to the one dimensional channel conductivity caused nonmonotonous noise level behaviour with channel width decrease.

CONCLUSION

Nanowires were fabricated from AlGaN/GaN heterostructures using ion beam etching. Nonmonotonous noise behaviour was revealed with decreasing channel width. Samples with Weff =160 nm and 272 nm demonstrate the lowest noise level. Such noise behaviour can be explained by taking into account a decrease in the lateral dimension of the wires which strengthening by surface carrier depletion resulting in a quantization of the electron energy and a transformation of the 2D confinement to a ID confined system. In this case the exchange between the conducting channel and the defects in the depletion layers decreases, which in turn causes a reduction of scattering events followed by a decrease in noise level and an increase in channel effective conductivity. This is of special importance for materials such as Ill-nitrides which are considered to be a critical driver for future information technology.

REFERENCES

1. B.S.Kang, S.J.Pearton,, J.J.Chen et al, Appl. Phys. Letters 89, 122102-1-3 (2006). 2. H.T.Wang, B.S.Kang, F.Ren et al. Appl. Phys. Letters 87, 172105-1-3 (2005). 3. K.Doi, N.Higashimaki, Y.Kawakami et al. Phys. Stat. Sol. (h) 241, 2806-2810 (2004). 4. B.S.Simpkins, K.M.McCoy, L.J.Whitman and P.E. Pehrsson. Nanotechn.lS, 1-5 (2007). 5. D.J.Carter, J.D.Gale, B. Delley and C.Stampfl. Phys. Rev. B 77, 115349-1-12 (2008). 6. N.B.Lukyanchikova. Noise Research in Semiconductor Physics. Gordon and Breach Science

Pubhshers, 411p.l996. 7. S.A.Vitusevich, S.V.Danylyuk, A.M.Kurakin et al. Appl. Phys. Letters 87, 192110-1-3 (2005). 8. S.A.Vitusevich, A.M.Kurakin, N. Klein et al. IEEE Trans.on Dev. Mater.Reliab. 8, 543-548 (2008). 9. J.S.Hwang, D.Ahn, S.H.Hong et al. Appl. Phys. Letters 85, 1636-1638 (2004).

490

1/f Noise Inside a Faraday Cage

Peter H. Handef and Thomas F. George''

Department of Physics & Astronomy and Center for Nanoscience

University ofMissouri-St. Louis, 1 University Boulevard, St. Louis, MO 6312L USA

Office of the Chancellor and Center for Nanoscience, Department of Physics & Astronomy and

Department of Chemistry & Biochemistry, Univ. ofMO-St. Louis, St. Louis, MO 63121, USA

Abstract. We show that quantum 1/f noise does not have a lower frequency limit given by the lowest free electromagnetic field mode in a Faraday cage, even in an ideal cage. Indeed, quantum 1/f noise comes from the infrared-divergent coupling of the field with the charges, in their joint nonlinear system, where the charges cause the field that reacts back on the charges, and so on. This low-frequency limitation is thus not applicable for the nonlinear system of matter and field in interaction. Indeed, this nonlinear system is governed by Newton's laws. Maxwell's equations, in general also by the diffusion equations for particles and heat, or reaction kinetics given by quantum matrix elements. Nevertheless, all the other quantities can be eliminated in principle, resulting in highly nonlinear integro-differential equations for the electromagnetic field only, which no longer yield a fundamental frequency. Alternatively, we may describe this through the presence of an infinite system of subharmonics. We show how this was proven early in the classical and quantum domains, adding new insight.

Keywords: 1/f Noise, Quantum 1/f Noise, Turbulence, Faraday Cage, Screening. PACS: 05.40.-a, 05.45.-a, 47.27.-e

I N T R O D U C T I O N

We explain here the existence of quantum 1/f noise at frequencies below the lower frequency cutoff of the electromagnetic field in the Faraday cage. One could argue against this, based on the solution of Maxwell's equations for the free electromagnetic field with the usual homogeneous boundary conditions applicable to electromagnetic Helmholtz resonators. However, both the coherent and conventional quantum 1/f effects, which together represent the observed fundamental quantum 1/f noise, are caused by the nonlinearity of the system of particle and field, e.g., of an electron (or many electrons) and the electromagnetic field. The eigenstates of the free electromagnetic field are quite different from those of the nonlinear "system of matter and field in interaction." Like any nonlinear system, this particle-field system also has an infinite system of sub-harmonics, extending down to the frequency zero.

It is true that any free electromagnetic field in an ideal (lossless, perfectly uncoupled from the external world) cavity can be expanded in terms of the cavity modes and can have only frequencies larger than the fundamental mode, which represents the lower cutoff of the frequency spectrum of the free field. However, a



491

simple example can illustrate the magnitude of the error connected with claiming that in all cases the actual electromagnetic field, describing the system of matter in interaction with the electromagnetic field in an ideal cavity, can only have frequencies that are eigenfrequencies of the free-field (empty) cavity modes. Consider a cavity with an inner diameter of the order of Im, which would have a lowest frequency of the order of the order of 100 MHz for the free electromagnetic field. A battery connected to capacitor plates encompassing most of the space in the cavity, with very small conduction between the plates, can produce an electric (and magnetic) field that slowly dies out during a year while the battery discharges. It can be completely enclosed, with the whole circuit, in that resonant cavity. This is a field that contains Fourier frequencies down to 10" Hz, but it is not a free field. It proves that the restriction to high frequencies rigorously apphes only for the free field, with no electric charges coupled in, and with no matter included.

All this may seem difficult to understand for some critics of the quantum theory of 1/f noise, because for them (1) the direct calculation that displays the nonlinearity of the system of matter and field may not be clear, or (2) the direct connection between nonlinearity and 1/f noise may not be clear, or (3) because the generation of sub-harmonics in the nonlinear system, and their ability to extend the 1/f law to arbitrarily low frequencies may not be obvious. These subjects are considered in Sec. 2 in simple, directly accessible, classical physics language, where the system of equations of motion and diffusion for electrons and holes in a semiconductor is solved together with Maxwell's equations and with the heat diffusion equation by reducing it all to a closed, third-order nonlinear equation for the components of the magnetic field at low frequencies. A similar treatment is also referred to Sec. 2 for the case of metals. In both cases, the infinite system of coupled nonlinear integro-differential equations resulting for the correlation functions of any order, describing homogeneous isotropic turbulence, is truncated. This is done by using the method introduced by Heisenberg [1] in 1946 to construct the first dynamical theory of hydrodynamic turbulence: the fourth-order correlation function is approximated in terms of the second order correlation function with a quasi-normality hypothesis. For homogeneous isotropic turbulence, this was shown by Handel to yield a universal 1/f power spectrum [2]-[6] for the field components and for the turbulent currents. Later, after many attempts to find zero-threshold instabilities that could feed the turbulence [7], Handel quantized the turbulence theory [8]-[18], noticing that the universal 1/f noise result was in fact a part of quantum electrodynamics (QED), allowing calculation of the noise magnitude.

Indeed, all these classical considerations have a quantum mechanical (QM) equivalent, where the quantum theory of 1/f noise is in fact a new aspect of QM and QED [19]-[22], and is known as quantum 1/f noise [8]-[18]. It is in essence the quantized form of the Handel's turbulence theory of 1/f noise [2]-[6]. The coherent quantum 1/f effect is derived elsewhere [12]-[15]. A generalization of Handel's turbulence theories developed for semiconductors, metals, and traffic on highways, has been obtained [12], which derives a universal sufficient criterion [12] for 1/f spectra from arbitrary nonlinear systems. There the criterion is also successfully applied to electrodynamics [12]. This provides the epistemological and mathematical basis for expecting 1/f noise at low frequency for QED.

492

Furthermore, quantum 1/f noise is not caused or even affected by the quantum interactions of charge carriers with the thermal radiation background or with any photons causing induced emission and absorption processes, so it is not affected by their energetic width. The photon modes are strongly attenuated and acquire very large width in conducting media in a wide frequency interval, but this does not affect the quantum 1/f noise that is shown to be caused only by the electromagnetic vacuum state. This thermal radiation background independence of quantum 1/f noise was rigorously derived earlier [9]-[ll], by first proving the statistical independence of spontaneous and induced radiation processes. The vacuum state does not acquire any width. This is another way through which we can understand why the quantum 1/f noise (a) is unaffected by the presence of a thermal radiation background, that was proven only contribute a small white noise term, and (b) is unaffected by the presence of a Faraday cage, a resonator, or even copper walls of infinite thickness: all these affect just the width or lifetime of the photons, not of the vacuum.

2. NONLEVEARITY OF THE SYSTEM OF MATTER AND FIELD

Maxwell's equations are linear, as long as the sources are not functions of the field. However, in reality, the field caused by the electrons acts back on them, and modifies their motion. This is seen directly, considering the electric current density j = nev of the electrons in the Ampere-Maxwell equation, and the momentum density nmv as the time integral of the force jxB per unit volume from Newton's equation. Then,

VxH = (d/dt)D+j = (e/et)D+(e/m)l[ Vx1i-(d/dt)D] xBdt» (e/m)l[VxH]xBdt (2.1) The last form is for low-frequency processes, important for 1/f noise, in which the

displacement current can be neglected. Considering B = ^iH, this is a closed equation for the magnetic field, that reflects Newton's equation as well as the Ampere-Maxwell equation, and proves our point. For an electrically-charged particle, or even an electron, quasi-elastically bound to a position of equilibrium inside an ideal cavity, with a soft elastic constant k resulting in a very long period of oscillation (m/k)''^ of the particle and of its field inside the cavity, we obtain in the same way

VxH » (e/m)l[VxH]xBdt -(ym)ndt dt'VxH (2.2) It is thus also described by a closed integro-differential equation for B as in Eq. (2.1), but with a double time-integration in the term. Equations (2.1) and (2.2) are nonlinear, and their solutions are fundamentally different from the free electromagnetic field. The closed equations for the examples with the battery and with the capacitor are even more complicated. The best example is found in our turbulence theory [2]-[6], where a large system of plasma equations for electrons and holes in semiconductors is reduced to a single closed, third-order nonlinear, equation for each of the three magnetic field components b„. For simplicity, we consider the electron-hole plasma in an unlimited volume of a symmetrical intrinsic semiconductor with equal mobilities = e/v and concentrations nn=np=n of the holes and electrons. Let us denote by Pn,Pp the partial pressures of electrons and holes, and by P the total pressure of the current carriers. Adding and subtracting the momentum transport equations for electrons and holes, and neglecting the inertial terms at low frequencies, we obtain

vv+ = (e/2c)v-xB - (l/2n)VP; vv" = 2e[E + v+xB/c] - (l/n)V(Pp - Pn), (2.3)

493

V-v+=0 (n= const); V xE =-(l/c)aB/at, V xB = 4jtenv7c, V-B = 0. (2.4)

Here v" and v are the difference and half the sum of the hole and electron drift 2

velocities, and v is a friction coefficient defining the conductivity a = 2e n/v. Fourier integral expansion and elimination all other quantities, yields [2]-[6], with s=K/k, 3 u ( k , x ) , 2 , X ^y^ + HkVk,x) = —^u(k,x)rdK<l)(k,K)u(K,0). <l>(k,K)^k'K'[- + f(s)] (2.5)

A Jo 3

-f ( - ) ; u(k,x) = (h/k3-e) e-|x|m^ Wap(k,x) ^

dx 1-s

f(s) = l - s ^ + i ( i - s ) M n s 2 s 1+s

(L/2jt)3<ba*(k,t-x)bp(k,t)>; Wap(K>) = (l/jt)JdxcoscDxJd^kw„p(k,x) = 0

4 °° 9 °° 4 °° jjjj j ii -6aR fk dkfdxu(k,x)coscDX = -h6aR f d k ^ :—- = -—6aR, (2.6) 3 •'o •'o 3 ' •'o cD + m k 3 co

which is a universal 1/f spectrum in the e=0 hmit [2]-[6]. Ql/f theory determines h&e.

REFERENCES

I. W. Heisenberg: Z. Physik 124, 628 (1948); Proc. Roy. Soc. London A 195, 1042 (1948) . 2 P.H. Handel, Phys. Rev. A 3 , 2066 (1971);. 3 . P.H. Handel, in Proceedings of the XVIIIth International Symposium on Turbulence of Fluids and

Plasmas (Brooklyn Polytechnic Institute. Press, N e w York, 1969), pp. 381-395 . 4. P.H. Handel. Physica Stat. Sol. 2 9 , 2 9 9 (1968) . 5. P.H. Handel, Z. Naturforschung 21a, 579 (1966) . 6. P.H. Handel, "Theory o f 1/f Noise in Semiconductors," Stud. Cere. Fiz. Roum. 18, 993-1055 (1966) . 7. J.O. Hinze, Turbulence (McGraw Hill, N e w York, 1959). 8. Search in Quantum 1/f Bibliography, and Integral List of Papers in Vita at www.umsl.edu/~handel. 9. P.H. Handel, in Proceedings of the II" International Symposium on 1/f Noise, edited by C M . Van

Vhet and E.R. Chenette (University o f Florida Press, Gainesville, 1980), pp. 96-110. 10. P.H. Handel, Phys. Rev. A 38 , 3082 (1988) . I I . P.H. Handel and J. Xu, in Proc. of the 5 th van der Ziel Symposium on Quantum 1/f Noise and Other

Low Frequency Fluctuations in Electronic Devices, edited by P.H. Handel and A.L. Chung (St. Louis, 1992), A lP Conference Proceedings 282 , 14-20 (1993) ISBN 1-56396-252-7.

12. P.H. Handel, in Proc. Xllth Int. Conf. on Noise in Physical Systems and 1/f Fluctuations, P.H. Handel and A.L. Chung Eds., A lP C o n f Proceedings 285 , 162-171 (1993) ISBN 1-56386-270-5.

13. P.H. Handel, Physica Status Solidi b 194, 393 (1996) . 14. P.H. Handel, IEEE Trans. Electronic Devices 41 , 2023 (1994) . 15. P.H. Handel, in Wiley Encyclopedia of Electrical and Electronics Engineering, Vol. 14, edited by J.

G. Webster (Wiley, N e w York, 1999), pp. 428-449 . 16. P. H. Handel, Phys. Rev. Lett. 34 , 1492 (1975) . 17. P. H. Handel, Phys. Rev. A 22 , 745 (1980) . 18. P.H. Handel and A.G. Tournier, "Nanoscale Engineering for Reducing Phase Noise in Electronic

Devices ," invited paper, Proc. IEEE 93, 1784-1814 (2005) . 19. T.W.B. Kibble, Phys. Rev. 173, 1527; 174, 1882; 175 ,1624 (1968); 20. T.W.B. Kibble, J. Math. Phys. 9, 315 (1968) . 21 . D. Zwanziger, Phys. Rev. D 7, 1082 (1973); Phys. Rev. Lett. 30, 934 (1973); Phys.

Rev. D 11 , 3481 , 3504 (1975); J.D. Dollard, J. Math. Phys. 5, 729 (1965) . 22. J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons (Springer, Heidelbg, 1976).

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A novel numerical approach for the frequency-domain calculation of oscillator noise

F. L. Traversa*, F. Bonani^

*Dept. Enginyerki Electronica, Universitat Autonoma de Barcelona, Spain '' Dipartimento di Elettronica, Politecnico di Torino, Italy

Abstract. This contribution presents a novel numerical approach to the estimation of the Floquet eigenvalues and eigenvectors of the perturbed system arising from an autonomous circiut (i.e., an oscillator) once the defining equations are linearized around the limit cycle representing the periodic oscillator solution. Such eigenvectors have great importance in the estimation of the fluctuations of the limit cycle itself (i.e., phase and amplitude noise) as a response to the circuit noise sources. The novelty of the method lies in the fact that all the eigenvectors can be efficiently estimated directly in the frequency domain, making use of the matrices already exploited in the solution of the Harmonic Balance system often used for the oscillator analysis.

Keywords: Oscillator, circiut noise PACS: 07.50.Hp,85.40.Bh,84.30.Ng

INTRODUCTION

The topic of noise anahsys in free running oscillators received a wide interest in the research community, especially in the fields of electronics and telecommunications, due to its extreme importance for the design and optimization of high sensitivity wireless telecommunication systems. As a consequence, a large body of literature can be found, spread over decades (see e.g. [1] for a review).

Despite its practical importance, only recently a unifying theoretical approach [2] has been provided for the fully consistent nonlinear characterization of noise in an autonomous system, formally expressed as a differential-algebraic equation (DAE):

^q (x )+ f (x )=B(x )b (0 , (1)

where hit) represents the circuit noise sources, possibly modulated by the solution dependent matrix B. Let us denote as xs(t) the (T-periodic) noiseless solution (limit cycle) of the oscillator, which satisfies the DAE for hit) = 0 (i.e., when noise is neglected).

According to [2], at least for the case of white noise sources (the much more important issue of colored noise sources has been treated separately in [3], albeit approximately but ultimately without affecting the theoretical results of [2]), the phase noise characteristics of an oscillator is given by an analytical expression as the superposition of lorentzians centered in the harmonics of the oscillation frequency /o = l/{27tT), whose amplitude is proportional to a single scalar constant ci,i given by:

ci,i = 1^ I v[(OB(xs)B(xs)^vi(0 dt=l, f fi,i{t) At (2) 1 Jo i Jo



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where the T-periodic vector vi(?) is called perturbation projection vector (PPV) [4]. The PPV is the Floquet eigenvector associated to the 0 eigenvalue of the adjoint system of the linearization of the homogeneous system (1) (i.e., for b = 0). In [4], numerical techniques have been proposed for the time- and frequency-domain estimation of the PPV, both based on an augmented version of the Jacobian associated to the system which allows for the Newton-based determination of xs{t).

NUMERICAL APPROACH

We propose here a novel numerical approach for the Harmonic Balance-based (HB) determination of the PPV. The numerical technique is founded on the approach presented in [5], where a general algorithm is proposed for the estimation of the Floquet eigenvectors and eigenvalues of autonomous systems linearized around a limit cycle.

Linearizing the homogeneous version of (1) around xs{t) one finds:

^[C{t)w{t)]+A{t)w{t)=0, (3)

where the T-periodic C and A matrices are the Jacobians of the nonlinear functions in (1) evaluated in the limit cycle. The adjoint of (3) is:

C T ( 0 ^ Z ( 0 - A T ( O Z ( 0 = 0 . (4)

According to [4, 6], the solution of (4) can be cast in the form z{t) = exp(—jUj:?)vj:(?), where jj^k are the Floquet eigenvalues of the linearized system. Substituting into (4) one gets:

-^ikC'yk{t) + C'Mt)-^^Mt) = 0, (5)

where x{t)is the time derivative ofx{t). The PPV is the solution corresponding to jUi = 0, always present since we are dealing with an autonomous system [6].

Time discretization of (5) and transformation in the frequency domain by means of the discrete Fourier transform (DFT) operator allows to express (5) in terms of the harmonic amplitudes \k of the eigenvector \k{t):

((CT)fQ-(AT)f)v, = M,t(CT)m, (6)

where Q is a matrix representing time derivation, and (C^)f and (A^)f are the frequency domain representations of matrices C^ and A^, respectively. Therefore, the evaluation of \k is reduced to an eigenvalue problem, thus providing a frequency domain algorithm for the estimation of the eigenvectors \k{t). Notice that (6) can be implemented without resorting to any augmented Jacobian matrix, thus making direct use of matrices already exploited for the HB determination of xs{t).

Finally, we remark that the ci,i coefficient in (2) allows to estimate the phase noise contribution only, thus completely neglecting amplitude noise. In [7] it is shown that the amplitude noise (orbital deviation) contribution is not always neghgible, and that it can be evaluated extending the formalism in [2] by exploiting also the other eigenvectors

498

y s — I

^ L" VI' RzS-

c, > C2

,^^—HH' I OTjffiy = 1

R, R,

s = ^ J e x p l ;

X + X — ^ - x =0 ^ dr ' / ,dx, dx, ^ ^ ^ ' d r ' dr

-a—-+\ v}p + a— W^ + ^lsK'pixA-l]-p = 0 dr l AT ) '- -'

~^ V^ a C,

- ^ , A = —5 ^, / = « „ , C,=-

FIGURE 1. Circuit of the Colpitts oscillator (left) and describing equations (right), both in full and normalized form. The simulation parameters are: Vcc = 15 V, j8/7 = 100, Is = 10~^ A, R\ = 28.6 Idl, i?2 = 1-4 Ul, VT = 26 mV, Q = 20 nF, C2 = 1 nF, C, = 1 jUF, L = 1.05/(4;r) mH, Re = 50 il.

Number of harmonics, a.u.

FIGURE 2. Representation of the solution as function of time on a period (left, the simulation was performed including in the HB system 120 harmonics), and estimation of the absolute error as a function of the number of harmonics included in the HB simulation (right).

Absolute frequency, MHz

, J \ FIGURE 3. Absolute frequency dependence of the oscillator phase noise (left), and time-domain representation (over one period) of the periodic scalar function / i i (t) defined in (2) (right).

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\k{t) k^ 1 of the adjoint system. The approach represented by (6) allows for a direct estimation of all the eigenvectors, and therefore is an efficient way of evaluating the \k{t) without resorting to the exphcit time-domain calculation of the full system monodromy matrix.

EXAMPLE

As an example of apphcation, we have simulated the Colpitts oscillator shown in the left part of Fig. 1. To make the system more manageable, we have exploited for the implementation the normahzed equation system shown in the right part of Fig. 1.

The nonlinear element of the circuit, i.e. the bipolar transistor, was described by a simphfied static model valid in the forward active bias region only (the circuit limit cycle xs{t) was verified to he within the region of vahdity of the model). Concerning noise, only the white transistor shot noise generators were included in the simulation.

The circuit parameters were chosen to yield a strongly nonlinear operation (/o = 474.0105 kHz), as demonstrated by the xs{t) components shown in Fig. 2 (left). The absolute error in the estimation of x^ and of the PPV as a function of the number of harmonics is also shown in the same figure. Accordingly, we estimate c = 8.37274306147111 x 10^^* s^Hz and evaluate the phase noise spectrum reported in Fig. 3.

CONCLUSION

We have developed and implemented a novel numerical approach for the estimation of phase (and, potentially, orbital deviation) noise in free-running oscillators. The methodology is based on a rigorous approach to oscillator noise evaluation, and allows for a full frequency domain calculation of the relevant quantities. This means that the same matrices employed in the HB-based determination of the oscillator's limit cycle can be exploited also for calculation of the relevant Floquet eigenvectors.

REFERENCES

1, A. Hajimiri, T.H. Lee, The design of low noise oscillators, Dordrect: Kluwer Academic Publishers, 1999,

2, A. Demir, A. Mehrotra, J. Roychowdhury, "Phase noise in oscillators: A unifying theory and numerical methods for characterization," IEEE Trans. Circ. Syst. I: Fund. Theory andApp., vol. 47, no. 5, pp. 655-674, May 2000,

3, A, Demir, "Phase noise and timing jitter in oscillators with colored-noise sources," IEEE Trans. Circ. Syst. I: Fund Theory and App., vol, 49, no, 12, pp, 1782-1791, Dec, 2000,

4, A, Demir, J, Roychowdhury, "A reliable and eflScient procedure for oscillator PPV computation, with phase noise macromodeling applications," IEEE Trans. CAD, vol, 22, no, 2, pp, 188-197, Feb, 2003,

5, F,L, Traversa, F, Bonani, S, Donati Guerrieri, "A frequency-domain approach to the analysis of stability and bifurcations in nonlinear systems described by differential-algebraic equations", Int J. Circuit Theory &App., Vol, 36, No, 4, pp, 421^39, June 2008,

6, A, Demir, "Floquet theory and non-linear perturbation analysis for oscillators with differential-algebraic equations," Int J. Circ. Theory App., vol, 28, pp, 163-185, 2000,

7, A, Carbone, F, Palma, "Considering noise orbital deviations on the evaluation of power density spectrum of oscillators," IEEE Trans. Circ. Syst. II: Express Briefs, vol, 53, no, 6, pp, 438^42, June 2006,

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Oscillator Noise Analysis: Full Spectrum Evaluation Including Orbital Deviations

Sung-Min Hong and Christoph Jungemann

EIT4, Bundeswehr University, 85577 Neubiberg, Germany

Abstract. An expression for the full output power spectrum including the orbital deviation has been derived. Using this expression, the fall output power spectrum can be readily evaluated by the Floquet analysis. From the example of Coram's two-dimensional oscillator, it is clearly shown that the orbital deviation plays an significant role under the presence of the slowly decaying mode.

Keywords: oscillators, noise, phase noise PACS: 84.30.Bv, 85.40.Qx, 85.30.Ng

INTRODUCTION

There has been considerable effort in the research field of oscillator noise analysis, especially during the last two decades [1, 2, 3]. One important result from this effort is that noise in oscillators should be decomposed into its components along the Floquet eigenvectors of the linearized system [4]. In [3], only the output power spectrum due to the phase deviation was considered, because of the fact that the orbital deviations remain small. As pointed out by many authors [5, 6], however, the orbital deviations may strongly affect the output power spectrum.

Of course, once we solve the Fokker-Planck equation which is corresponding to the oscillator system, for example in [5], the effect of the orbital deviations on the output power spectrum is readily available. However, solving the Fokker-Planck equation of a practical circuit system is almost impossible. Therefore, an approach applicable to a general oscillator with acceptable computational burden is required. In this direction, considering the orbital deviations in the evaluation of the output power spectrum was addressed in [6]. However, the authors evaluated the output autocorrelation for limited time intervals. This corresponds to evaluate the output power spectrum only at frequencies far away from the carrier. Since the near-carrier spectrum is of practical interest, we still need a unified expression applicable to all frequencies.

In this work, we present the asymptotic statistical properties of the entire output signal, which includes both the steady-state solution shifted by the phase deviation and the orbital deviations.

THEORY

If not otherwise stated, the notation follows [3]. Since the colored noise can be modeled as a sum of modulated white noise sources [7], only white noise is considered. The



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output variable at time t can be written as

n

x(f)=x,(f + a) + ^Mi(f + a)A, (1)

where a is the phase deviation, and /3j is the coefficient for the amphtude deviation ahgned to the Uiit + a) vector.

In order to derive the results shown below, we use Ito's definition of the statistical integral [8]. Also an approximation which rehes upon matching the large-T behavior of the autocorrelation, already used implicidy in [3], is adopted. After some manipulation, the output power spectrum S{(i)), which is the Fourier transform of the asymptotic autocorrelation \\mt^c^R(t, T), is given by

Sic) = Y.^.^KH^^^2,{f^^^^^^2 (2)

,Clr lai

i=2 k j:j:x.,u*4-jkco,)--^^^^^—^^ Y.ll^'^^kU*k{-Jk(Oo cii_ icm^^cn

;* i t 4 m 2 ^ 2 ,(m, hf,-^\2

^ ^ r- 9/7

where X and Ui^k are the Fourier coefficients of Xs and M,, respectively, jii is the characteristic Floquet exponent corresponding to M,, and c,,/ is defined as

1 r^ Cii' = - dtfi{t)B{x,{t))B^{x,{t))vi,{t). (3)

1 Jo For given i and k, the quantities such as ©i, ai are given by

(Oi = k(0o + 3{fii}, ai = -^co^cii-9t{jUi}, (4)

While only the first term in the RHS of Eq. (2) is considered in [3], Eq. (2) presents the full spectrum of x including the cross-correlation between the steady-state solution shifted by the phase deviation and the orbital deviations (second and third terms) and the correlation of the orbital deviations (fourth term). Once we perform the Floquet analysis of the linearized oscillator, all quantities appearing in Eq. (2) can be easily calculated. Therefore, the full spectrum of the oscillator can be calculated using Eq. (2).

EXAMPLE

Coram's two-dimensional oscillator [4] is considered. In order to consider the effect of a slowly decaying mode, one additional parameter, which represents the magnitude of

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the restoring force to the limiting cycle, is introduced. Then, this parameter happens to be — jU2 in this system.

In the polar coordinates, the governing equations are written as

r = - i U 2 ( r - r 2 ) + < „ 9=l + r. (5)

The noise source along the radial direction ^r is considered. In the following calculation, the value of cn is set to be lO '*.

When we choose the x-coordinate as an output variable, X i and L'2,1 are given by

1 1 1 1 ^M = o' ^2,l = ; + J;7 —. (6)

Z Z Z fi2

Also, for i = 2, ©1=2^, fli =2^c i i - jU2- (7)

When the absolute value of jU2 is small (weak restoring force), it corresponds to the slowly decaying mode. In this case, MI and U2 are almost aligned, therefore, it is expected that the cross-correlation between phase and orbital deviations plays an important role. We will consider two representative cases, where jU2 is either -10 (fast decaying) or -0.1 (slowly decaying). Note that since cn is fixed to a certain value, the first term in Eq. (2) predicts the same output power spectrum for different jU2 values.

Fig. 1(a) shows the cross-correlation between the steady-state solution shifted by the phase deviation and the orbital deviations. The cross-correlation has a larger magnitude and slower increase for jU2 = -0.1, since the decaying coefficient ai is close to ^k^coQ and the magnitude of f/2,yt is large. Similar behavior can be observed for the correlation of the orbital deviation, as shown in Fig. 1(b). For jU2 = -10, the decaying coefficient is large, therefore, we can expect a broad spectrum due to the orbital deviation.

Figs. 2(a) and 2(b) show the power spectrum of x for jU2 = -10 and -0.1, respectively. As expected from Figs. 1(a) and 1(b), it is clear that contributions from the orbital deviation cannot be neglected in the presence of slowly decaying mode. Especially, for jU2 = -0.1, the cross-correlation term cancels out the contributions from both the phase deviation and the orbital deviation. As a result, the output power spectrum for some frequencies is significandy lowered. This generates an output power spectrum whose slope is steeper than well-known 1//^ dependence [3].

CONCLUSIONS

An expression for the full output power spectrum including the orbital deviation has been derived. Using this expression, Eq. (2), the full output power spectrum can be readily evaluated by the Floquet analysis. From the example of Coram's two-dimensional oscillator, it is clearly shown that the orbital deviation plays an significant role under the presence of the slowly decaying mode.

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(a) (b) FIGURE 1. (a) Cross-correlation between the steady-state solution shifted by the phase deviation and the orbital deviation and (b) correlation of the orbital deviation, cn is lO^'*.

(a) (b)

FIGURE 2. (a) 5xi((U) when H2 = -10 and (b) 5xi((U) when H2 = -0.1. cn is lO"'*.

ACKNOWLEDGMENTS

The authors gratefully acknowledge financial support by the Deutsche Forschungsge-meinschaft (DFG). S.-M. Hong's work was partially supported by Korea Research Foundation Grant by the Korean Govemment(MOEHRD). (KRF-2007-357-D00159)

REFERENCES

1. F. X. Kaertner, Int. J. Circuit Theory Appl, vol. 18, pp. 485-519, 1990. 2. A. Hajimiri et al., IEEE J. Sol. State Circuits, vol. 33, pp. 179-194, 1998. 3. A. Demir et al., IEEE Trans. Circuits and Systems -1, vol. 47, pp. 655-674, 2000. 4. G. J. Coram, IEEE Trans. Circuits and Systems -1, vol. 48, pp. 896-898, 2001. 5. J. P. Gleeson et al., "Non-Lorentzian line shapes in noisy self-sustained oscillators," 2005. 6. A. Carbone et al., IEEE Trans. Circuits and Systems - II, vol. 53, pp. 438-442, 2006. 7. A. Demir, Analysis and Simulation of Noise in Nonlinear Electronic Circuits and Systems, 1998. 8. C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences,

2004.

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Experimental Study of Noise in a Frequency Synthesizer

Gurpreet Singh Sangha, Michael H.W. Hoffmann

Institute of Microwave Techniques, University of Ulm, Germany

Abstract: This paper presents new results in the noise measurements of frequency synthesizers. Noise phenomena at the output of a synthesizer are very complex. Every component in the synthesizer contributes to the total output measured noise curve. Usually, it is not possible to separate the noise sources by just measuring the noise curve at the output of a frequency synthesizer. In this paper a technique is presented that can be used to investigate the noise contribution of different devices within the synthesizer. For this work, a synthesizer circuit is built and optimized to have a low noise floor. Then stronger noise is added at different points in the synthesizer in a way that it would represent noise of a certain device in the system, and that its provenance might be identified clearly. In advancement to the previous presentations, a very complex combination of noise sources is taken into consideration.

Keywords: Synthesizer, PLL, Noise Measurement PACS: 01.30.Cc, 05.40.Ca, 02.60.Cb , 07.50.Ek,

INTRODUCTION

When designing a PLL based frequency synthesizer circuit, one of the important aspects to consider is the effects of electronic noise. The electronic noise coming from a variety of internal and external noise sources modulates randomly to the phase of an oscillator's output.

This paper contains a detailed noise analysis of a PLL based synthesizer circuit considering effects of internal and external noise sources. Theoretical analysis methodology and results from nonlinear simulations were presented by the same authors in [3],[4],[5]

NOISE ANALYSIS METHODOLOGY

The core of the synthesizer is a second order PLL (see Fig. 1), which as usual consists of a reference oscillator (XCO), a phase-detector (PD), a first order lag-lead control filter (CF), a voltage controlled oscillator (VCO), and a loop divider (1/N).

The noise behavior due to different electronic components within the synthesizer structure is considered through the design of appropriate noise sources. The additive noise sources shown in Fig. 1 represent the noise contribution of different components in the synthesizer. At the input of the PLL, the noise contribution «, from the crystal oscillator as well as the noise «M from the prescaler are considered. Inside the loop the additive noise npD of the phase detector, ncF of the control filter, phase noise tivp of the voltage control oscillator, as well as tiu of the loop divider are taken in to account.



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In contrary to the commonly used model for the frequency divider consisting of a hard-hmiter and a bandpass filter, a new superior model for non-ideal divider is already shown in the work published by the same authors in [4]. According to that model a non-ideal divider output signal would not only contain the phase noise of the VCO but it also contains a part of the amplitude fluctuations of the VCO signal.

In order to take into account amplitude fluctuations of the VCO signal, a four quadrant multiplier is used to modulate amplitude noise nvA to the VCO output signal that is being fed back to the PD through the loop divider as shown in Fig. 1.

In the PD output signal Vd(t) the terms Ruit) and Ruit) represents the effects of the XCO and the VCO amplitude fluctuations on the signal that is being fed back by the PD to the VCO. The nonlinear differential equation of this synthesizer system can be described similar to [3]. The analysis of such a nonlinear differential equation is usually mathematically very complex. The authors have already presented a novel linearization approach in [3].

The PD output signal in Fig. 1 shows that within the PLL bandwidth a part of amplitude fluctuations will be fed as tuning voltage fluctuations to the VCO input and will be converted into phase noise at the output of the PLL. This is also evident from the measurement results shown in Fig. 3(d).

XCO

".(0 « M ( 0 WpziC) "c/'C) Non-ideal

1/M PD CF -K+VHvco v.Jt) = f(RMeAt))

i fV", p»/+)(xW-n„(0

I ( t \ , -|/N „ • f—VCO pi-amplitude variation phase noise

Non-ideal

FIGURE 1. Synthesizer noise measurement setup block diagram.

MEASUREMENT RESULTS

In order to do an experimental study of noise, a frequency synthesizer circuit running at lOMHz is built. The synthesizer is based on Texas Instruments CMOS PLL chip SN74LV4046A. This chip contains a VCO as well as three different types of PDs; an XOR PD, a 3-state PD and a PD based on RS Flip Flop. Two separate chips are used; one for VCO and the other for PD. For the control filter both active and passive control filters are implemented. Measurements are done using XOR and 3-state PD as well as both active and passive filters. The results shown here are only for the 3-state PD and the active CF filter.

One of the important aspects when studying effects of noise on synthesizer is to find suitable circuits that will add noise to the PLL without influencing the dynamic behavior of the PLL circuit. Within the PLL loop active circuits based on operational amplifiers are used to add noise at different points in the circuit.

The output of the XCO and the VCO are digital signals that are first passed through bandpass filters in order to convert them into analog signals, and then noise is added. After noise addition the XCO and the VCO signals are passed through buffers (limiters) in order to convert them again into digital signal that is needed at the input

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of the PD. As mentioned before, the multiplicative noise (AM noise) is introduced through the use of a four quadrant multiplier.

The measurements are preformed using the Rohde & Schwarz FSUP Signal Source Analyzer which facilitates achievement of accurate phase noise characterization of the system.

The power source for the PLL circuit is provided by the low noise regulator M1C5209. In addition to studying influences of noise within the PLL circuit, the effects of power supply noise on the XCO output is considered. For the XCO the DC supply coming from the regulator was passed through the buffer circuit shown in Fig. 2(a). As shown in Fig. 2(b) this lead to a noise level improvement of +16dB at IkHz offset from center frequency.

The improvement in the noise performance is due to the fact that the operational amplifier has a high PSRR (power supply rejection ratio) which leads to a better noise and DC small signal variation rejection.

1CP 1(f 105 Offset Frequency (Hz)

(b)

FIGURE 2. (a) XCO power supply with buffer stage; (b) SSB spectrum of the XCO output shows improvement in the noise performance after including the buffer.

Only some of the important results from the PLL noise measurements are shown here. Diagrams (a), (b), (c) and (d) in Fig. 3 show the single side band (SSB) phase noise power spectrum density of the synthesizer circuit with noise at different points. As it was expected from theoretical and simulation results (not shown here) measurements show clearly how the noise at different points in the circuit affects the output of the synthesizer.

A white bandpass filtered noise at the input of the PD (after XCO) causes a flatting of the output spectrum within the loop bandwidth as shown in graph (a). As expected the white baseband (low pass filtered) noise source inserted directly after the PD also causes a flatting of output spectrum within the PLL bandwidth, graph (c) shows that the PLL acts as a high pass filter for the white baseband noise, which is added directly after the CF (i.e. in front of the VCO). Graph (d) shows the measurement results, where a four quadrant multiplier was used to modulate baseband white noise (AM modulation) to the signal coming out of the VCO and which is fed back to the PD through the loop divider (see Fig. 1). The VCO amplitude fluctuations as expected do play a significant role in the output spectra forming. The amplitude fluctuations are converted by the PLL into phase noise, which appears in the output signal of the PLL as it is evident from Fig. 3(d).

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CONCLUSION

This paper contains measurement results from a more complete noise analysis of a PLL based frequency synthesizer circuit considering a complex combination of noise sources. This new setup leads to a deeper understanding of the complex noise phenomenon affecting the output of the synthesizer. It shows how each point in the synthesizer circuit is contributing to the overall noise behavior. It gives deeper insight that can help in fine tuning the synthesizer circuit to reduce the total noise which appears in the output spectrum of the PLL signal.

Offset Frequency (Hz) (a) SSB-noise spectrum with BP filtered whiite noise at XCO output

lO' ' 10^ 10° 10° Offset Frequency (Hz)

(b) SSB-noise spectrum withi LP fiitered whiite noise after thie PD

10^ 10"* 10° 10° Offset Frequency (Hz)

(c) SSB-noise spectrum withi LP fiitered whiite noise after the CF

PLL output without noise 140 L . . _ ^ . .

10^ 10'' 10° Offset Frequency (Hz)

(d) SSB-noise spectrum with muitipiicative noise after the VCO

10°

FIGURE 3. Graphs (a), (b), (c) and (d) show the single side band power spectrum of the output of the PLL with noise present at different points of the synthesizer.

REFERENCES

1. Heydari, P., Analysis of the PLL Jitter Due to Power/Ground and Substrate Noise, IEEE Transactions on Circuits and Systems-L Regular Papers, 2004 .

2. Lee, T.H., Hajimiri, A., Oscillator Phase Noise: A Tutorial, IEEE Journal of Solid-state circuits. Vol. 35, No. 3, March, 2000.

3. Sangha, G.S., Hoffmann, M.H.W., Analysis and Design of a Frequency Synthesizer with Internal and External Noise Sources, AIP Conf Pro.,Vol. 780, pp.521-524, 2005.

4. Sangha, G.S., Hoffmann, M.H.W., Investigation of the Effects of Noise on a Frequency Synthesizer, AIP Conf Pro.,Vol. 922, pp.716-718, 2007.

5. Sangha, G.S., Hoffmann, M.H.W., Analysis and simulations of a frequency synthesizer with internal and external noise sources. Adv. Radio Sci., 4, 175-178, 2006, www.adv-radio-sci.net/4/175/2006.

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A fluctuation-based probe to athermal phase transitions

U. Chandni*, Arindam Ghosh*, H. S. Vrjaya''' and S. Mohan'''

'Department of Physics, Indian Institute of Science, Bangalore INDIA 560012. ^Department of Instrumentation, Indian Institute of Science, Bangalore INDIA 560012.

Abstract. We focus on athermal phase transitions where in discrete and dissipative avalanches are observed in physical observables as the system jumps from one metastable state to another, when driven by an external field. Using higher order statistics of time dependent avalanches, or noise, in electrical resistivity during temperature-driven martensite transformation in thin nickel-titanium films, we demonstrate evidence suggesting the existence of a singular 'global instability' or divergence of the correlation length as a function of temperature at the transition. These results not only establish a mapping of non-equilibrium first order phase transition and equilibrium critical phenomena, but perhaps also call for a re-evaluation of many existing experimental claims of self-organized criticality.

Keywords: athermal phase transitions, conductivity noise, martensitic transformation PACS: 64.60.Ht,05.40.-a, 81.30.Kf

INTRODUCTION

Many natural phenomena, extending from biology to material science, involve slowly driven dissipative systems that are far from thermal equilibrium, triggered only by a slowly varying external field to which the systems respond through scale-free avalanches in physical observables [1]. These systems are typically athermal with thermal fluctuations playing effectively no role in initiating the transition. The transition thus takes place instantaneously and hence at experimental time scales is drive-rate independent as well. The commonness shared by these diverse fields are (a) self-similar dynamics that lead to power law decays of the avalanches in physical observables and (b) critical exponents that can be identified with one of the universality classes [2, 3]. Many have attributed this behavior over a wide range of the driving field to the tendency of these systems to self-organize to the critical state spontaneously [2, 4]. Alternatively, incorporating on-site randomness in the Ising model leads to the random-field Ising model, which postulates the existence of a global critical point associated with a wide critical zone [5]. This critical point is associated with the divergence of a correlation length at the transition temperature, which has never been probed experimentally. Thus, the question as to whether these systems self-organize or exhibits a conventional critical phenomena still remains open. We aim at probing this fundamental problem using a fluctuation-based technique.

We focus on equi-atomic nickel titanium (NiTi) shape memory alloys, well known for their athermal martensitic transition. These systems exhibit a two-stage structural transitions from the high temperature austenite (cubic B2:CsCl) through an intermediate R-phase (rhombohedral) to a low temperature martensite (B19/B19' monoclinic) phase



511

• •.3Kfmln * 0.5K*mln O 1 K/mIn

Coollnfl - 2e&K

200 300 T(K)

1 ^ - * - * ' * * ^ IKfmln-heatlng

# 0.5K/mln-heatlng ^r 0.3K/min-heating

* • • •O ^ j { * ^ U 1K/mlr«oollng . f c i ^ ^ K y ^ ^ 0.5K/min-Gooling

^fWw^r^^ ir 0.3K/mlrv<»ollng

250 T ( K )

300 350 200 250 300 T(K)

FIGURE 1. (a) Characterization of the NiTi thin film using resistivity measurements as well as scanning tunneling microscopy, (b) Power spectral density during cooling at 285K, showing 1/f behavior for different ramp rates, (c) The frequency exponent as a function of temperature for all the three rates. Power spectral density as a function of temperature for (d) heating and (e) cooling for the three ramp rates.

[6]. Conductivity noise is measured while the sample is temperature driven at different ramp rates, thereby facilitating a direct measurement of the avalanche dynamics.


Experimental Procedure

The NiTi thin film was prepared by dc magnetron sputtering of a mosaic target, which consists of a patterned titanium disk over a circular nickel disk. The film was deposited at an Ar pressure of 2 x lO^^mbar and annealed at 480°C. The sample was characterized using standard four probe resistivity measurement using van der Pauw geometry and shows hysteresis as is shown in Figure. 1(a), which is characteristic of the underlying martensitic structural transition. The shaded regions indicate the different phases. It is to be noted that the film shows a two stage transformation behavior for both heating as well as cooling. The inset shows scanning tunnelling microscopy studies done on the film, exhibiting grain size of « 160 nm. The low-frequency noise measurements were carried out using a dynamically balanced wheatstone's bridge arrangement as proposed by Scofield [7]. The extraneous noise levels were minimized to obtain a background noise floor with Sr{f) < 10^^°V^/Hz. The sample was ramped across the transition hysteresis at three different ramp rates, and the resistivity fluctuations were recorded for time windows of « 525 seconds, which helps us in probing the avalanche dynamics in the real time. Here we present the analysis of the avalanche dynamics and its higher

512

order statistics for the three ramp rates: 0.3K/min, 0.5K/min and IK/min.

Noise Measurements and Higher Order Statistics of Avalanches

The equilibrium resistance noise has been studied before and is proposed to arise from the coupling of electron transport to structural disorder [8]. In the present context, we discuss the noise measurements during ramping [9]. The scale free dynamics and critical behavior is exhibited in the power spectral density of the resistivity fluctuations, Sp{f) ^ 1 / / " , within a temperature window as is shown in Figure. 1. We observe that Sp (/) is nearly independent of rate over three decades of frequency in the cooling cycle, as indicated by Figure. 1(b). This could be attributed to the fact that the system passes through the same set of metastable states each time a ramp is done. This essentially occurs once the disorder levels are quenched and hence it is very important that proper thermal cycling is done before the onset of the measurements. This rate independence is an essential criteria for a phase transition to be athermal, since the thermal fluctuations do not aid in crossing the energy barriers between the various metastable states. We also find that the frequency exponent a shows a clear clustering at around 1.5 for slow ramps over a broad range of temperature (320K^240K) as shown in Figure. 1 (c), which matches fairly well with the exponent observed in avalanche dynamics in Barkhausen noise, vortex avalanches in superconductors etc [4]. The small reduction in a for faster drive rate is consistent with the previous studies and can be due to the overlap of avalanches [10]. The power spectral density over the entire temperature shows clear ramp independence for the cooling cycle (Figurel .(e)), indicating that the forward austenite to martensite transformation is athermal. However, the reverse transformation is found to be rate dependent (Figurel .(d)) indicating the role played by thermal fluctuations in assisting the transformation from martensite to austenite phase [11].

It has been proposed that the higher order statistics of noise could be a sensitive probe to the long range correlations prevalent in a system during the phase transformation [12]. We hence analyze the second spectrum, or the fourth order moment, in order to quantify the extent of the non-Gaussian behavior exhibited during the transformation.Second

1000

'N 100-

C 10.

"(0 1

0.1

200K (a) 265K 320K Calculated gausslan background

cooling ^ ^ ^

0.01 0.1

f(Hz)

FIGURE 2. (a) Normalized second spectra for three different temperatures during the cooling cycle at a ramp rate of 0.3K/min. The calculated Gaussian background is plotted as well, (b) The non-Gaussian component of noise as a function of temperature for 0.3K/min ramp rate during cooling cycle. Lower panel shows the latent heat generated during the differential scanning calorimetry studies.

513

spectrum is the Fourier transform of the four-point correlation function within each temperature window and is given by S^^\f) = J^{Ap^{t)Ap^{t + T))tC0s{2nfT)dT. The non-Gaussian behavior associated with the long range correlations is exhibited as a non-white contribution to the frequency dependence of .S'(^)(/). Figure.2(a) shows the normalized second spectrum s^-^\f) = S^-^\f)/[yr^Sp{f)dfY- {JHT/L being the upper and the lower cut-off frequencies respectively), for the cooling cycle at a ramp rate of 0.3K/min. It is observed that while the second spectrum corresponding to 320K shows a Gaussian behavior, the ones for 200K and 265K clearly deviate from the expected Gaussian value. To have a qualitative understanding of the second spectrum over the entire temperature zone, we have plotted the normalized second spectrum integrated over the entire bandwidth, cr ) = jQ^s^^\f)df, for the cooling cycle as a function of temperature in Figure.2(b). Strong peaks in (y^^\f) are observed implying a sharp increase in the non-Gaussian component as a function of temperature. A direct comparison with the differential scanning calorimtery data, which is essentially the latent heat released during this first order non-equilibrium phase transition, indicates that the peaks in the second spectrum correspond to the transition temperatures from austenite to R-phase and R-phase to martensite as indicated (Similar behavior was observed during the heating cycles as well [9]). This result establishes the divergence of the correlation length at the transition temperature, giving clear evidence for a conventional critical phenomenon even though the power spectrum and the frequency exponent show a universal behavior over the entire temperature zone.

CONCLUSIONS

The work establishes conductivity noise as a sensitive tool for probing structural phase transitions exhibiting avalanche dynamics. The higher order statistics of resistivity fluctuations and the associated non-Gaussian behavior have been used to probe into the fundamental problem of the nature of the global singularity during a first order non-equilibrium phase transition. The work thus looks at first order non-equilibrium phase transition in the framework of equilibrium critical phenomenon and provides an experimental tool which could readily be adapted to many material systems.

REFERENCES

1. J. p. Sethna, K. A. Dahmen and C. R. Myers, Nature 410, 242-250 (2001). 2. L. Laurson, M. J. Alava and S. Zapperi, J. Stat. Mech. 11, LI 1001 (2005). 3. M. C. Kuntz and J. P. Sethna, Phys. Rev. B. 62, 11699-11707 (2000). 4. P J. Cote and L. V. Meisel, Phys. Rev. Lett. 67, 1334-1337 (1991). 5. J. P Sethna et al. Phys. Rev Lett. 70, 3347-3350 (1993). 6. S. Miyazaki, A. Ishida, Mat Set and Eng. A273, 106-133 (1999). 7. J. H. Scofield, Rev Sci. Instrum. 58, 985-993 (1987). 8. U. Chandni, A. Ghosh, H. S. Vijaya and S. Mohan, Appl. Phys. Lett. 92, 112110 (2008). 9. U. Chandni, A. Ghosh, H. S. Vijaya and S. Mohan. Phys. Rev Lett 102, 025701 (2009). 10. R. A. White and K. A. Dahmen, Phys. Rev Lett. 91, 085702 (2003). 11. U. Chandni, A. Ghosh, H. S. Vijaya and S. Mohan, (to he published). 12. S. Kar et al. Phys. Rev Lett 91, 216603 (2003)

514

The Study of the Discrete Quasi-Multifractal Process

A. Saichev and V. Filimonov

Nizhny Novgorod State University Gagrina ave. 23, Nizhny Novgorod, 603950 Russia

Abstract. This work is dedicated to the study of the muhifractal diffusion process, constructed using Quasi-

Multifractal Model. In contrast to other works, a multifractal spectrum, determining properties, is calculated using statistical processing of realizations of the process, where realizations were obtained by numerical simulation of the sampled model. Presented results of numerical simulations showed presence of 3 significantly different modes of quasi-multifractal processes, which we called "monofractal" "tempered multifractal" and "strongly multifractal". The "temporal" and "spectral" differences of these modes are also discussed.

Keywords: stochastic processes, multifractality, long-memory processes PACS: 05.45.Df, 47.53.+n, 02.50.-r, 02.50.Ey, 05.40.-a

INTRODUCTION

Studies of multifractal processes, which generalize the Richardson-Kolmogorov cascade model, started relatively recently. Initially they were studied with regard to fully-developed turbulence, whose velocity increments were proved to be multifractal, but now this part of the theory of stochastic processes has become a subject of great interest, since it applies to a variety of processes in different sciences ranging from fluid dynamics and high energy particle physics to biology and finance theory.

The vital issue of the theory is developing of an adequate multifractal model without self-contradictions. First models (1990-1997), based on cascade rule [1], had a number of drawbacks such as non stationary increments and absence of an explicit dependence on time. Proposed in 2000 by J.-F. Muzy et al. [2] model of Multifractal Random Walk was free of mentioned drawbacks, but it lose its meaning for high-orders moments of increments. The most adequate to the properties of multifractal processes is the Quasi-Multifractal Model, proposed in 2006 by A. Saichev and D. Sornette [3] and extended later by authors of the work [4]. According to this model the process is constructed as an exponential of long-memory process with power law memory.

This paper is discussed the Discrete Quasi-Multifractal Process based on the above mentioned continuous model. Realizations of the process and moments of their increments are discussed. Finally it is described the multifractal spectrum for different modes.



515

THE DISCRETE QUASI-MULTIFRACTAL PROCESS

Consider discrete random process X„ of the following form:

i=0 (1)

where § is zero-mean unit-variance gaussian random variable and (Oi is independent of § gaussian process with long memory, i.e. the random variable with power-law asymptotic of the covariance function. Let us consider the cot variable in the following form:

! - l

7=0 7 ' (2)

where jit is random gaussian variable and hi is the following power function:

h = {\ + i)-'l^-'^, (3)

where ^ > 0. Without loss of generality assume that variable jii has unit variance and zero mean value. As it is seen from (l)-(3) the discrete process X„ could be described with 2 significant parameters — the power (p of the kernel hi and the multiplicative coefficient p in the expression of «,.

Before proceeding to an analysis of multifractal spectra, we examine realizations of the generated discrete process. When p is small, the process X„ is obviously an almost monofractal one (Fig. la) that becomes a Wiener process as p ^ 0. As p increases, the multifractal properties of the process become more pronounced, as manifested by numerous "jumps" (Fig. lb). With further increase in p , the process is increasingly dominated by the exponential in expression (1), and the jump amplitude increases accordingly (Fig. 1 c). When p is large, X„ can be interpreted as a generalized telegraph process.

100

0

X^-100

-200

-300

V J \ . ^Wl J l

W ^ V (a)

1000 2000 3000 4000 5000

1UUU

500

0

-500

1000

1500

A i -- I ^ \ / YH-~-

(b) 1000 2000 3000 4000 5000

20000

15000

10000

5000

0

-5000

A

— ^

r " -

(c) 1000 2000 3000 4000 5000

FIGURE 1. Realizations of the discrete process X„ with (p = 0.01 andp = 0.1 (a), 0.5 (b) and 0.9 (c).

NUMERICAL EVALUATION OF THE MULTIFRACTAL SPECTRUM

Using the realizations of the discrete process X„ one can calculate its increments 5mXn = Xn+m—Xn- Simulation of a set of M > 1 realizations allows to calculate moments of

516

increments using averaging:

(4) 1 M

MJm) = - y 1=1

where XJi' is the realization number ; in the precalculated set and q is the order of moment. For performing "good" averaging, the sum in Eq. 4 should obviously contain a sufficiently large number of terms. In this work it was calculated M = 10** realizations of length TV = 10^ inset for obtaining M^(m) for each given set of parameters cp and p .

By analogy with the continuous case, we introduce the notion of a quasi-multifractal spectrum. If there exists a certain scale range

WT < m < Wi (5)

in which the dependence of the increment moment Mq{m) on the scale mL can be approximated (e.g., using the least-squares method) by the power-law dependence

Mq{m) = K^m^i, (6)

and tl,q is nonlinear function of the moment order q, we say that the initial process X„ has a quasi-multifractal properties. By analogy with the theory of turbulence, range (5) is called inertial, and the scales WT and mi are called the viscosus scale and integral scale, respectively. In this work, when analyzing the realizations with the length A = 10^ and moment orders 0 < q < 2.5, inertial range (5) has the boundaries m^ '^ 10 and Wi-5-104 .

One should note that the analytical calculation [4] of the quasi-multifractal spectrum of the process is possible only for a discrete set of points ^ = 2,4,6,. . . The values for intermediate (noninteger) orders q can be calculated only by interpolation. At the same time, the numerical simulation of the quasi-multifractal process X„ make it possible to determine the effective exponent for an arbitrary (not only for integer) q value and, thus, to obtain an almost continuous quasi-multifractal spectrum.

Figure 2 shows the results of the interpolation of the analytically calculated spectrum (thick line) and the numerical- simulation spectrum (open circles connected by a thin line). It is seen that the analytical and experimental results are in good agreement with each other. Taking into account the difference in the procedures of determining the effective exponents (in the analytical calculation, the effective exponent was introduced as the global minimum of the local exponents [4], whereas in the numerical experiment, as the approximation of the increment moment by Eq. 6), the nonlinear spectrum is due to the quasi-multifractal nature of the process, rather than to the statistical processing of incremental moments.

It is seen that the spectra of the process confirm the conclusions based on the realization of the process X„: there are three significantly different regions of the process parameters. The first, "monofractal" region (or the "weak" multifractal region), with a small parameter value p '-- 0.1 corresponds to an almost linear spectrum (the lower line in Fig. 2) and to an almost monofractal process (see Fig.la). As p increases, the deviation of the spectrum from the linear behavior increases, and the spectrum for large p values is strongly nonlinear. In the region of the significantly nonlinear spectrum (p > 0.3),

517

FIGURE 2. Quasi-multifractal spectra of the process X„ for (p = 0.01 and p = (from bottom to top) 0.1, 0.5 and 0.9. The circles connected by the thin line are the results of the numerical simulations, and the thick lines are the approximations of the analytical calculations.

two subregions can be separated. For the case of "tempered" multifractahty (p ^ 0.5), the spectrum (e.g., the middle line in Fig. 2) is well described by the square dependence obtained in the model of multifractal random walks [2]. In the region of the "strong" multifractality (p '-.^0.9), where the realizations of the process consist primarily of jumps and insignificant oscillations around certain constant levels (see Fig. 1 c), the deviations of the spectrum from the square dependence are significant (the upper line in Fig. 2).

ACKNOWLEDGMENTS

This work was supported by the Russian Foundation for Basic Research (project no. 09-02-00231 a) and by the Council of the President of the Russian Federation for Support of Young Scientists and Leading Scientific Schools (project no. NSh-1055.2008.2).

REFERENCES

1. Novikov E.A. The effects of intermittency on statistical characteristics of turbulence and scale similarity of breakdown coefficients // Phys. Fluids, A. 2, 1990. P 814-820.

2. Bacry E., Delour J., Muzy J.-F. A multifractal random walk // Physical Review, E. 64,2001. P 026103-1-026103-5.

3. SaichevA., SometteD. Genericmultifractality in exponentials oflong memory processes//Phys. Rev. E., Vol. 74, 2006. P 011111-1-011111-11.

4. Saichev A.I., Filimonov V.A. On the Spectrum of Multifractal Diffusion Process // Journal of Experimental and Theoretical Physics, 2007, Vol. 105, No. 4. P 1085-1093.

518

Analysis of Quasar Radio Wave Flux Density Fluctuations and its

Cosmological Meanings

Noboru Tanizuka Graduate Shcool of Science, Osaka Prefecture University

1-1 Gakuen-cho, Naka-ku, Sakai 599-8531, Japan

Abstract. The time series of microwave flux density variations (2.7 and 8.f GHz) from 24 quasars was analyzed in the methods of power spectral index, Higuchi's fractal dimension. Hurst exponent, Correlation dimension and Kolmogorov entropy. The aim of study is to find cosmological meanings of analyzed results by mutually referring the indices varing with red shift of quasars. A systematic change was found among them, while yet unknown of discrimination between noise and dynamics of radio source systems. Keywords: Quasar, Radio wave. Fluctuations, Time series. Spectral index. Hurst exponent. Fractal dimension, Correlation dimension, Kolmogorov entropy PACKS: 98.54.Aj, 98.54.Cr, 95.85.Bh, 98.62.Ai, 89.75.Da, 89.75.Fb

M O T I V E OF ANALYSIS

I knew several years ago that the microwave of quasars (QSOs) had been monitored continually over ten years on daily basis[l, 2]. QSOs are extragalactic radio sources and galactic objects whose distances from us are a billion light years (l.y.s) to more than ten billion l.y.s. I was interested in analyzing the time series of microwave flux density varing with time, expecting that it might give us some information of cosmic space passed by the wave and/or of its sources [3]. The time series, probably conveying information of QSO events, did not seem to have been analyzed so far except in the method of auto/cross correlations for seeking intrinsic events of radio source and nonintrinsic events such as interstellar scintillation[2]. I tried to analyze the time series in five kinds of methods of using power law standing over single to double figures of time and other dimensional scales at microwaves of 2.7 GHz and 8.1 GHz for 24 QSOs of different red shift. The data were given by Waltman[l]. ^

ANALYSIS M E T H O D S ^The Green Bank Interferometer is a facility of a National Science Foundation operated by

the NRAO in support of NASA High Energy Astrophysics programs.



519

F i g u r e 1: Spectral index a against red shift z. left: 2.7GHz; right 8.1GHz.

D

1.8

1.6

1.4

.1 • '

- °l - M

: o o

'\ ,0,

• • b b '

0

9D

1

' '

0

. .

o -

o.

---

0 1 2 red shift z

F i g u r e 2: Higuchi's fractal dimension D against red shift z. left: 2.7GHz; right 8.1GHz.

The power law of the flux density variation stands against its frequency and time scales: 1. Spectral index a is estimated from P{f) oc /~", with P the power spectral density at frequency / of the variation;[4] 2. Higuchi's fractal dimension D is from L{k) oc k~^, L the averaged absolute variation of flux density at time interval k normalized by k = 1,[5] 3. Hurst exponent H is from RN/C^N OC {N/2)^,

RN the difference between top and bottom levels of X]\[{i) = Y^'j=i{x{j) — x), x = ^ j = i ^ 0 ) / ^ ^^^ ^N ttie variance of flux density x{j) for period A ,[6] and 4. correlation dimension D2 is taken maximum of D2{m) = limg^o {logC'm(e)/loge} as m is increased, C^(e) the correlation integral at flux density scale e in m-th dimensional space of reconstructed vector {x{j),x{j + r ) , . . . ,x{j + (m — l)r)), r the time lag chosen properly[7, 8, 9]. 5. Kolmogorov (K-S) entropy K2 is given by logCm{^) ~ D2{m)log€ — 771/ 2,[9] with which the dynamics of a system of flux density variation is expected to be estimated. The quasars analyzed here are those listed in the caption of Fig. 5 and 0133+476 {z = 0.86), 0202+319 (1.47), 0215+015 (1.715), 0235+164 (0.94), 0333+321 (1.26), 0420-014 (0.92), 0828+493 (0.548, BL), 0923+392 (0.70), 1245-197 (1.275), 1328+254 (1.06), 1328+307 (0.85), 1555+001 (1.772), 1611+343 (1.40), 1741-038 (1.054), 1749+096 (0.320, BL), 2134+004 (1.93), 2234+282 (0.80) and 2251+158 (0.86), BL identified as BL Lacertae[l, 2, 3].

RESULT

The result is given for each method: 1. The range of frequency was taken in calculating a between 1.0 x 10~^ and 1.0 x 10~* Hz because of statistical accuracy and noise problems toward lower (1.1 x 10~^Hz) and higher (5.7 x 10~^Hz) frequencies, respectively, and then a stands in the range between 10 and 100 days of time scale. The result is given in Fig. 1 against red shift z of QSO. 2. Fractal dimension D stands in the range between 1 and 100 days of time scale and 3. Hurst exponent H in the range between 10 and several hundred days, and they are given in Fig. 2 and Fig. 3 against z. 4. Correlation dimension D2 refiects the nature of system's dynamics behaving in a complex state space and forming a manifold of D2, in a case of real data, superposed by noise of known and/or unknown factors[10, 11].

520

1 ' 2 red shift z

V " 2 red shift z

F i g u r e 3: Hurst exponent H against red shift z. left: 2.7GHz; right 8.1GHz.

^ i A :,f*^

-- • 4

- ^ •k

t ^ i

• -Ai

* * A k

t ; t A •

•

•k

F i g u r e 4: Correlation dimension L>2 against red shift z at 2.7GHz and 8.1GHz microwaves.

Dependence of D2 on z is shown in Fig. 4, taking 1 2 > 6 with care because of violating Ruelle's hmit[8]. Behavioral complexity of the flux density variation increases as z increases. Though the meaning of a, D, H, D2 differs each other, their dependencies on z show similar trend. 5. Kolmogorov entropy is shown against e in Fig. 5; no critical dynamical behavior is seen, but a trace of dynamics is likely left. Figure 6 shows a relation between D and H; it may serve as a tool of estimating a mechanism of the flux density fluctuations[12]. In Fig. 7, D of all QSOs given here is plotted against galactic latitude and longitude to see if D is meaningly dependent on the galactic coordinates or not. By considering that the distribution of z (instead of D) against galactic latitude and longitude[3] changes in a similar way as in Fig. 7, D is not dependent substantially on the coordinates, but on z. It stands also in the same way for the dependence of D on the equatorial coordinates [3]. Dependence of D (and other indices) on the galactic and equatorial coordinates will require greatly more samples of quasars to be estimated in a statistical accuracy, which is impossible at the moment.

DISCUSSION

For an observer at a quasar of z, the microwave frequency (2.7GIIz, S.lGHz) and the sampling period (1 day) are to be modified by the special theory of relativity, by a =

F i g u r e 5: K-S entropy against flux density scale r ( = e) at 8.1GHz of QSOs: 1226+023 (3C273, z = 0.158), 1641+399 (3C345, 0.59), 0224+671 (4C67.05, 0.524), 0336-019 (CTA26, 0.85) 1502+106 (OR103, 1.833), 0237-233 (PKS0237-23, 2.225) Calculated at time lag r = 210 and 7V=3000 by G-P method[7, 9].

o.e

0 g 0.4

0

TOY V.4

\ ^ 1226+023

r \ --- 1641+399 ;

\ \ -^- 0224+671

A****V^,^^ -^- 0336-019 ^1 "•" •-T N ^ ^ - 1502+106 v \ - ^ ^ *\ \ - ^ 0237-233

In r

521

H 1

o.a

0.8

•

b r

^

. 1 . 1 . 1

\ •

\

H

0.9

' 1 ' 1 • • •

1 J T - T ' - r

•

v.:

L'l l.S l.i

F i g u r e 6: Relation between D and H of quasars, left: 2.7GHz; right 8.1GHz.

D

1.8

1.8

•1 -

- • 0

*

10C

• • 1

*• *

«'#

1* 200

•

--

3DD 0 60 galactic latitude galactic longitude

F i g u r e 7: Distribution of D against galactic latitude and longitude of quasars (8.1GHz).

1 + z times as large as the frequency and b= \J\ — fP- times as short as the period, /? the ratio of retreating quasar speed to light speed, for example (z, /?, a, 6) = (0.1, 0.09,1.1, 0.99), (0.5, 0.38,1.5, 0.92), (1.0, 0.60, 2.0, 0.80), (2.0, 0.80,3.0, 0.60)[3].

There may be seen from Fig.l and Fig.3 discontinuity in system's dynamics around red shift z « 1 or less, and seen from Figs. 1, 2, 3 and Fig.4 more complex dynamics as z is increased. At present the complexity which may be caused by external noise added in the radio wave path and/or the radio source dynamics itself can not be discriminated, though the dynamics may be shadowed as shown in Fig. 5. Much more observations of quasars will be required for the analysis to be made effective in a statistical accuracy and for a statement implicating its result as an evidence of cosmological evolution.

A C K N O W L E D G E M E N T S

The author gives thanks to Mr. M. Takano and Dr. M.R. Khan for their help and discussions. Figures 1, 2, 3, 6 and 7 were quoted by permission from Ref. [3].

References [I] E.B. Waltman, et a l , Astro-phys. J. Suppl. Ser. 77, 379-404 (1991).

[2] R.L. Fiedler, et al., Astrophys. J. Suppl. Ser. 65, 319-384 (1987).

[3] N. Tanizuka, M. Takano, T. lEE Japan 120-C No.8/9, 1149-1156 (2000). [m Japanese]

[4] N. Tanizuka, Jpn J. Appl. Phys. 30 No.l, 171-177 (1991).

[5] T. Higuchi, Physica D31 , 277-283 (1988).

[6] J. Feder, Fractals, Plenum Press, New York, 1988, pp.149-162.

[7] P. Grassberger, LProcaccia, Phys. Rev. Letters 50 No.5, 346-349 (1983).

[8] D. Ruelle, Proc. R. Soc. Land. A 427, 241-248 (1990).

[9] R.C. Hilborn, Chaos and Nonlinear Dynamics, Oxford Univ. Press, Oxford, 2000, pp.385-383.

[10] M.R. Khan, N. Tanizuka, lEICE Trans. Fundamentals E86-A No.9, 2209-2217 (2003).

[II] M.R. Khan, N. Tanizuka, lEICE Trans. Fundamentals E88-A No.11, 3161-3168 (2005).

[12] R.V. Donner (private communication, a preprint by T. Gneiting and M. Schlather)

522

Fluctuation Theorems in Biological Physics Alexander M. Berezhkovskii * and Sergey M. Bezrukov

'^Mathematical and Statistical Computing Laboratory, Division for Computational Bioscience, CIT, National Institutes of Health, Bethesda, MD 20892, USA

Laboratory of Physical and Structural Biology, Program in Physical Biology, NICHD, National Institutes of Health, Bethesda, MD 20892, USA

Abstract. Recently formulated fluctuation theorems are highly relevant for interpretation of measurements performed on single molecules. One of these theorems can be applied to channel-facilitated transport of solutes through a membrane separating two reservoirs. The transport is characterized by the probability P„(t) that n solute particles have been transported from one reservoir to the other in time /. The fluctuation theorem establishes a relation between P„(t) and P_^{t): The ratio P^{t)IP_^{t) is independent of time and equal to SKp(nAIkgT^, where A/kgT is the affinity measured in the thermal energy units. We show that the same fluctuation theorem is true for both single- and multi-channel transport of non-interacting particles and particles which strongly repel each other.

Keywords: Diffusion, channel-facilitated transport, single-molecules, translocation statistics. PACS: 87.15.Ya; 87.16.-b; 87.16.Uv.

INTRODUCTION

These days most informative biophysical studies are performed at the single-molecule level, wherein events and process trajectories observed in experiments are intrinsically random. This randomness calls for adequate statistical methods in data analysis, and, even more importantly, for detailed understanding of fluctuations at the molecular scale. Properties of the corresponding macroscopic systems then can be explained in terms of the most basic inter-molecular interactions.

While some other examples of fluctuation theorems (see papers [1-12] and references therein) will be discussed in the talk, here we restrict ourselves to the statistics of translocations of solute particles through a channel in a membrane that separates left (L) and right (R) reservoirs containing the particles in concentrations c^ and c^ shown in Figure la. The driving force for the transport may be the difference in the solute concentrations, c^^Cj^, or a potential drop between the reservoirs, and, of course, both factors may act simultaneously. The key quantity of our analysis is the probability P^{t) that n particles have been transported from the left reservoir to the right one in time t, which is the probability that the difference between the numbers of L^-R and R^-L transitions between the reservoirs in time t is equal to n . We will see that the probabilities P^{t) and P_„{t) are related to each other and obey the fluctuation theorem [13,14]. The theorem shows that the ratio P^{t)IP_^{t) is independent of time and establishes a relation between the ratio and the affinity which may be considered as a measure of the distance from equilibrium for a non-equilibrium system [15].

CPl 129, Noise and Fluctuations, 2(f International Conference (ICNF 2009) edited by M. Macucci and G. Basso

2009 American Institute of Pliysics 978-0-7354-0665-0/09/$25.00

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NONEVTERACTEVG P A R T I C L E S

A distinctive feature of transport of non-interacting solutes is that transitions/fluxes in the L^-R and R^-L directions do not affect each other. To characterize these fluxes we introduce probabilities g^^^ («10 and g^^^ («11) that the number of transitions in the corresponding direction in time t is equal to n, « > 0, assuming that the system is in a steady state at time t = () when the observation starts.

a) J o o ^ o

o ^ 0 0

° o o 0 °

C ° ^L o

R o

o

o C„

b)

FIGURE 1. a) Schematic representation of a channel in a membrane that separates the left (Z ) and right {R ) reservoirs containing the solute particles in concentrations C^ and C^ , respectively, b) The

fate of a solute particle entering the channel from the left (upper scheme) and right (lower scheme) reservoirs. The two reservoirs and the channel are shown as a three-state system. The state in the

middle shows the channel which separates the two reservoirs.

We will describe entrance of the particles into the channel from the two reservoirs by the rate constants kl'J, I = L,R. When the potential drop is locahzed on the membrane, kfj are given by the Hill formula for the trapping rate by an absorbing circular disk on the otherwise reflecting planar wall [16] or its generalization to non-circular absorbers [17]. The probability Q^{n \t) that n solute particles have entered the channel from reservoir / in time t, is given by the Poisson distribution

a(«U) = ^(^i''^)"exp(-4^V), (1)

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where k]lj is the monomolecular rate constant, k^l' = k^^Cj, which is the inverse of the mean time between successive entrances of new particles into the channel from the reservoir / .

The fate of the particle in the channel depends on which reservoir the particle enters from. A particle entering the channel from the left reservoir traverses the channel and escapes to the right reservoir with probability P^^^ and returns to the left reservoir with probability PL^L=^~PL^R (see Figure lb). Corresponding translocation and return probabilities for particles entering the channel from the right reservoir are P^^^

When m particles enter the channel from the left reservoir, the probability that n of them pass through the channel and exit into the right reservoir while the rest m-n particles return to the initial reservoir is given by P/l^^/^"^^/w!/[«!(/w-«)!], n<m. Keeping this in mind we can write g^^^ («10 as an infinite sum

a . . ( « 11) = t—^^PL.P"QL(m I/). (2)

Using Eq. 1 it can be shown that

QL^R(" 11) = ^ ( ^ i " ^ L ^ / ) " exp(-4^)p ,^ / ) . (3)

A similar expression for Qj^^jîn \t) can be obtained if we replace L hyR and R by L inEq. 3.

We use these expressions to find probability P^"'\t) that the difference between the number of particles passing from the left reservoir to the right one in time t and the number of particles passing in this time in the opposite direction is equal to n

GO

Pl"'\t) = YQL^R(''+"\t)QR^L(k\tl « > o (4) k=Q

and

GO

P^(t) = Y.QL^R^^ I t)QR^Lik+n \t), « > 0 , (5)

where the subscript "ni" indicates that this probability describes transport of non-interacting particles. Using Eq. 3 after some manipulations we obtain

pnt)--ft(^)p V " / r— X ,/»V

"-in ^ L^R Mp

V 'în -' R^L J ' n ^ ^ ' V " - " ^L^R'în ^R^LJ {L)p ,{R , . 'P

in L^R in R^L

(6)

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where /„(z) is the modified Bessel function of the first kind, which is a symmetric function of «, /^(z)=/_^(z) and z^"'\0 is independent of the translocation direction and given by [14]

% (m)

(t) = (^i"^z.. +^i"^..z)exp[-(^,<:'P,^, +CP.^L)t]- (7)

One can formally describe each translocation as a unit step of a random walk between nearest-neighbor sites (Figure 2). The L^-R and R^L transitions correspond to steps in the forward and backward directions, respectively. Then Z^"'\t) is the probability density for the random work lifetime on a site. The probabilities W^"'^ that the random walk makes a step in the positive and negative directions are given by [14]

W^"'^=l/[l + exp[+A{cJc^,AU)/kj]], (8)

where k^ and T are the Botlzmann constant and the absolute temperature, and the affinity A{cJCj„AU) is defined by [15]

A{c^/Cj„AU) = kJln{cJCj,) + AU. (9)

Distribution P^"'\t) in Eq. 6 is the random walk propagator, i.e., the transition probability from the origin to site n in time t.

. +10 +5 -8 ^7 -5

s: -5 +4 +3 •i-2

' 0

-1 -2

, ---

-n r ^ u time, /

FIGURE 2. The number of solute particles transported from the left reservoir to the right one as a function of time in situation when there is a net flux of the particles in the Z —>• i? direction.

Denoting the potential energies of a solute particle in the left and right reservoirs by U^ and Uj^, respectively, and using the condition of detailed balance

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4' '^x^. exp(-f/, lk,T) = k^fJP,^, exp(-f/, /kj), (10)

we can express the ratio of the probabilities, P^"'\t) / P!:^'\t), in terms of the affinity

given by Eq. 9. The result is

, ., =exp 'kj

(11)

Thus, we find that probability Pj"''(0 obeys the fluctuation theorem. This form of the fluctuation theorem has been discussed by Andrieux and Gaspard in Ref [10], where they analyze ion transport in the framework of Schnakenberg's model of the ion channel [18].

STRONGLY REPELLING PARTICLES

The simplest way to model transport of strongly repelhng particles is to require that the channel cannot be occupied by more than one particle. This implies that a particle can enter only an empty channel and, once inside, it blocks the channel for the entrance of other particles from both sides of the channel. Such an assumption allows a comprehensive analytical treatment of the problem [19-21].

The Laplace transform of function x^"'' (t), which is the analog of z^"'^ (0 in Eq. 7, can be found through the Laplace transforms of four probability density functions, 'PempiO, <P,r(t), ^^L^iit), aud g}jt^jt(t) rcprescnting the probability density for the channel lifetime in the empty state, probability densities of the lifetimes in the channel for translocating particles and for non-translocating particles. Functions (p^^^it) and 'PR^R(0 are the probability densities of the lifetimes in the channel for non-translocating particles entering the channel from the left and right reservoirs and coming back to the same reservoir from which they entered. As shown in Refs. [22-24] the probability density of the lifetime in the channel for translocating particles, (p,^ ( 0 , is independent of the translocation direction. The Laplace transform of x''"'\t) is then given by [13]

1 • [PI^^PL^L'PL^L (^) + P.H'^PR^R'PR^R (^)] ^.». (^)

This implies that that the equivalent random walk in Figure 2 is non-Markovian for strongly repelling particles because the overall translocation process is viewed as at least a two-state process [19-21]. However, the probabilities of the step directions are the same for both non-interacting and strongly repulsing particles, W^"^ = W^"'^ = W^, and are given by Eq. 8.

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After the problem of finding Pj"^' (t) has been reduced to that of finding the random walk propagator, the result can be obtained by solving a set of integral equations. The Laplace transform of P^"'\t) has been derived in Ref [13] and is given by

P^^\s) , nil

i-i^W s^\-mjv_[xAs)]'

(13)

One can see that P!,''\S) satisfies {W_)"P^f\s) = {W^)"P^_:^\s) and, hence, the probability P^'^^t) obeys the same fluctuation theorem as P„^"''(0,Eq. II.

In summary, main results of this study are the expressions in Eq. 6 and Eq. 13 providing explicit solutions for P^„"'\t) and P^„"'\s) as well as Eq. II which represents the fluctuation theorem for membrane transport of both non-interacting solute particles and particles that strongly repel each other.

ACKNOWLEDGMENTS

This study was supported by the Intramural Research Program of the NIH, Center for Information Technology and Eunice Kennedy Shriver National Institute of Child Health and Human Development.

REFERENCES

1. G. N. Bochkov and Yu. E. Kuzovlev, Zh. Eksp. Tear. Fiz. 72, 238 (1977). 2. G. N. Bochkov and Yu. E. Kuzovlev, Physica 106A, 443 (1981). 3. D. J. Evans, E. G. D. Cohen, and G. Morriss, Phys. Rev. Lett. 71, 2401 (1993). 4. G. Gallavotti and E. G. D. Cohen, Phys. Rev. Lett 74, 2694 (1995). 5. C. Jarzynski, Phys. Rev. Lett 78, 2690 (1997). 6. J. Kurchan, J. Phys A 31, 3719 (1998). 7. J. L. Lebowitz and H. Spohn, J. Stat Phys 95, 333 (1999). 8. R. van Zon, S. Ciliberto, and E. G. D. Cohen, Phys Rev. Lett 92, 130601 (2004). 9. U. Seifert, Europhys Lett 70, 36 (2005). 10.D. Andrieux and P. Gaspard, J. Stat Mech. POlOl 1 (2006). l l .R. D. Astumian,P/^'i. Chem. Chem. Phys 9, 5067 (2007). 12.E. M. Sevick, R. Prabhakar, S. R. Williams, and D. J. Searles, Ann. Rev. Phys Chem. 59, 603

(2008). 13. A. M. Berezhkovskii and S. M. Bezrukov, Phys Rev. Lett 100, 038104 (2008). 14. A. M. Berezhkovskii and S. M. Bezrukov, J. Phys Chem. 112, 6228 (2008). 15. L. E. Reichl, A Modem Course in Statistical Physics; Austin, University of Texas Press, 1980. 16.T. L. Hill, Proc. Natl. Acad Set USA 72, 4918 (1975). 17.0. K. Dudko, A. M. Berezhkovskii, and G. H. Weiss, J. Chem. Phys 121, 1562 (2004). 18. J. Schnakenberg, i?ev. Mod Phys. 48, 571 (1976). 19. A. M. Berezhkovskii and S. M. Bezrukov, Biophys J. 104, L17 (2005). 20. A. M. Berezhkovskii and S. M. Bezrukov, Chem. Phys 319, 342 (2005). 21. S. M. Bezrukov, A. M. Berezhkovskii, and A. Szabo, J. Chem. Phys 127, 115101 (2007). 22. A. M. Berezhkovskii, G. Hummer, and S. M. Bezrukov, P/;>'5. Rev. Lett 97, 020601 (2006). 23. J. Alvarez and B. Hajek, Phys Rev. E 73, 046126 (2006). 24. A. M. Berezhkovskii and S. M. Bezrukov, J. Phys: Condens Matter 19, 065148 (2007).

530

Applications of dynamical inference to the analysis of noisy biological time series with

hidden dynamical variables

A. Duggento*, D. G. Luchinsky*'1', V. N. Smelyanskiyf, M. Millonas** and P. V. E. McClintock*

* Department of Physics, Lancaster University, Lancaster LAI 4YB, UK ''NASA Ames Research Center, Mail Stop 269-2, Moffett Field, CA 94035, USA

"Mission Critical Technologies Inc., 2041 Rosecrans Ave. Suite 225 El Segundo, CA 90245, USA

Abstract. We present a Bayesian framework for parameter inference in noisy, non-stationary, nonlinear, dynamical systems. The technique is implemented in two distinct ways: (i) Lightweight implementation to be used for on-line analysis, allowing multiple parameter estimation, optimal compensation for dynamical noise, and reconstruction by integration of the hidden dynamical variables, but with some limitations on how the noise appears in the dynamics ; (ii) Full scale implementation of the technique with extensive numerical simulations (MCMC), allowing for more sophisticated reconstruction of hidden dynamical trajectories and dealing better with sources of noise external to the dynamics (measurements noise).

Keywords: Bayesian inference, nonlinear time-series analysis, hidden variables, dynamical inference, stochastic methods, coupled oscillators. PACS: 02.50.Tt, 05.45.Tp, 05.10.Gg, 05.45.Xt

Developing earlier works [1,2], we consider the following M-dimensional time-series ^ = {y„ = y{tn)} (tn = nh), representing N observations of the system

x(0 = f(x|c) + V ^ ^ ( 0 , y (0=g(x |b ) + V^77W- (1)

The first of Eqs.(l) defines the L-dimensional underlying stochastic dynamics (with white uncorrected noise source) and the second one defines the observed variable ^ (with an extra observational noise source). Our task is to infer the unknown model parameters, their time variations, the noise intensities and ^-trajectory: . ^={c (0 ,b (0 ,D ,M,{x„}} .

The form of the likelihood depends on the approximations of the theory. For an Euler approximation of the dynamics x„+i = x„ + /if(x*|c) + VhD^^, with x* = (x„+i +x„)/2, the minus log-likelihood function S= — \a.t{W\J{) can be written as: (see [3, 1])

2 2 „to I ^^k

2 "''^'"' ' 2

(2)

ln|M| + ^5^[y„-g(x„|b)]^M-i[y„-g(y„,x„|b)] + (L+M)Mn(2;r/i), n=\

where x„ = ' "+ ^" and summation over k is implicit in the term dJ •



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Let us assume for a moment that .^ is given. In this case, by parameterising the vector field f(x*|c) = U(x*)c = U„c, linearly in respect of its parameters, and assuming a multivariate normal prior PDF for c, we found that the posterior is also normal, and that its mean is given by [3]:

/ Af-l \ ' / Af-l

V n=0 / V n=0

'^ n=0

. j T „ - l . 1 \ ^U/m(x)

0)

The obtained result holds uniquely in the presence of additive noise (D constant). If this was not the case, then an extra parameterisation of the noise should have been employed, and heavy approximation and assumptions made in order to make the problem algebraically treatable. When ^ is not observable, a global optimization technique should, in general, be employed. In the next two following sections two example will be discussed. In the first one, a 'lightweight' implementation will be used and the use of global optimization will be avoided. In the second example the most probable state for the dynamical space will be obtained by appendix of an MCMC techinique.

LIGHTWEIGHT IMPLEMENTATION

In our first example, we decode the parameters of a system of neurons modelled by an L-dimensional system of FitzHugh-Nagumo (FHN) oscillators [5]:

^j =

ij --yt =

= - ^ y ( ^ y - « y ) ( ^ r

= -^qj + 7jVj\ = XijVj.

-^)-<ij+^j+V^j^j, Ut)Ut')) = 8ij5{t-t'),

i,j=l,...,L.

(4)

(5) (6)

where vj models the membrane potentials and qj are slow recovery variables. Parameters rji control the potential threshold for the self-excited dynamics, controlling the firing rate, and they will be considered as time-varying parameters. We assume that neither vj nor qj are read directly (i.e. 'hidden' variables), but that the measurements are made through an unknown measurement matrix X in Eq.(6). Our tasks are to: (i) reconstruct coefficients appearing in Eq.(4-5); (ii) reconstruct the mixing matrix X; (iii) reconstruct the hidden variables qj; (iv) perform tasks (i)-(iii) taking into account that some parameters might have explicit time dependence. We assume no measurement noise in Eq.(6): indeed in such systems the measurement noise is often negligible and in this way we can avoid the need for global optimization and better estimate the performance of the Bayesian inference itself. Following [2], a convenient way to treat this problem is by integration of the slow recovery variable qt and to substitute it into the top equation in Eq.(4), and consequently in Eq.(6) we obtain the explicit form for the dynamics of the readout variable:

yt = fji + Scijyj+bikik2yhyk2 + Cikik2yhyl^

+e-P'qi - J^eP(-'-'hijyjdT + Dij^j{t), (7)

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where parameters of the transformed dynamics in Eq.(7) are function of the original parameters and the matrix X. This explicit dependence is given in [2]. Although the number of base functions TV for the mixed dynamics is much larger than the number of polynomial terms in Eq.s(4-5) the inferencial algorithm exibits good performance and high speed in inferring parameters even when few of them are explicetelly time-dependent. Some results are presentented in Fig. 1.

0.4

0.2

1

(c) 1° A

r ^

T '° 3 0.5 1

—S—B—•—•—•--

FIGURE 1. Inference of r\\ and 772 from Eq.(4) for a 2-FHN system, while smoothly varying in the presence of noise. No prior knowledge of the model parameters is assumed, (a) The inferred values of r\\ (dashed red lines) are compared with their true values (full blue lines); here the ability to follow continuous evolution of the control parameters in the adiabatic limit is demonstrated, (b) The recontructed time-trace of the hidden coordinate q\(f). (c) Typical convergence of the control parameters r\j as functions of the measurement time t; the qualitative behaviour of the biggest eigenvalue of the covariance matrix is given in the box. In [2] a detailed discussion is presented.

MCMC IN AN ECOLOGICAL SYSTEM

In ecological problems the emphases are on the off-line recovery of hidden population dynamics. We therefore consider a general MCMC approach. We consider an example of predator-prey dynamics, also considered in [6, 7, 8], where the cycling dynamics of the vole population in Finnish Lapland can be modelled by the following equations for the fluctuating densities of rodents TV and their predators P (weasels, foxes, owls, and others). Dynamical inference method cannot be applied "as is" to the model of [6, 7, 8] because: (i) the noise terms are multiplicative; (ii) the predator trajectory is hidden; and (iii) the prey dynamics is measured together with some measurement noise. The first problem can be overcome by making an ad hoc change of variables. The second and third problem are more complex, but can be solved in different ways accordingly to the approximations that one can introduce. In this respect, some ways were investigated in [4]. Here, the aim is to show how an MCMC techinique can be employed for the reconstruction of the hidden dynamical sample. In particular it is very useful to analyse what happen in a one-dimensional approximation. For a detailed discussion of how to reduce to this system, and of the approximations involved, see [4, 8, 7]. The resulting one-dimesional system considered has the form:

xi = r ( l -e\ sin(2n:?+i//o))-re*' ge^i az

g2)Cl _|_ /j2 ^ ^ d -r Ononis)

y{t)=x\{t)^aob!,n{t).

(8)

(9)

(10)

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where the only observable is y{t). For the sake of simplicity we assume the noise intensities to be fixed and introduce an abbreviated vector of the unknown parameters ^ = {c,{x^}}. The MCMC algorithm can be briefly summarized as follows:

(i) Take an initial guess for ^ ( ^ ^ = {c(^),{x[ ^}}; (ii) Sample a trajectory from />(x^|x^_i,x^+i,^,D, Gobs^yt) for^ = 0, ...,i;rusingthe Gibbs sampler with Metropolis-Hastings (M-H) steps; (iii) Sample model parameters from p{^\{xt},T>,Oobsi{yt}) using the M-H algorithm; (iv) Repeat steps (ii)-(iv) until convergence is achieved. The results are summarised in Fig.2.

(a) 1.2

1.0

0 .8

0.6

0.4

0.2

0

(b) n

k f\ m

1 j il / l'

/ I r*J

/ 1 h MM y^

jy xM \j^. \^ \M )jr \M

-

-

. - - - - - " 3 4 5 6 ' " O 1 0 2 0 3 0 4 0 0 ' ^ S2 t ( yea r )

FIGURE 2. Inference parameters from Eq.(8); (a) Typical evolution of the solution of the optimization problem starting from some initial values and descending the hyperplane of the cost function defined by the posterior minus-log-likelihood; (b) Results of the MCMC calculations: convergence of the unknown predator trajectories from an initial guess (solid black line at the bottom of the figure) to the actual trajectory (solid blue line at the top of the figure) is shown by dashed red lines. The arrow indicates the direction of convergence as a function of number of iterations.

CONCLUSIONS

We have shown how the problem of dynamical inference in the presence of noise can be solved through different approaches for a fast implementation (on-line applications), investigating the boundaries of the resolution for slowly varying parameters; and for the more computationally demanding problem of global reconstruction with heavy use of MCMC for discovering the latent state variables for the extreme case of missing data.

REFERENCES

1. D. G. Luchinsky, V. N. Smelyanskiy, A. Duggento, and P. V. E. McClintock, Phys. Rev. E 11, 061105 (2008).

2. A. Duggento, D. G. Luchinsky, V. N. Smelyanskiy, I. Khovanov and P. V. E. McClintock, Phys. Rev. £77,061106(2008).

3. V. N. Smelyanskiy, D. G. Luchinsky, D. A. Timucin, and A. Bandrivskyy, Phys. Rev. E 11, 026202 (2005).

4. D. G. Luchinsky, V. N. Smelyanskiy, M. Millonas, P. V. E. McClintock, Europ. Phys. J. B 65 pp. 369-377 (2008)

5. J. Nagumo, S. Animoto, and S. Yoshizawa, Proc. Inst. Radio Engineers 50, 2061 (1962). 6. I. Hanski, H. Henttonen, E. Korpimaaki, L. Oksanen, P. Turchin, Ecology 82(6), 1505 (2001) 7. O. Gilg, I. Hanski, B. Sittler, Science 302, 866 (2003) 8. P Turchin, S.P Ellner, Ecology 81(11), 3099 (2000)

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Charge Fluctuations and Boundary Conditions of Biological Ion Channels:

Effect on the Ionic Transition Rate R. Tindjong*, D. G. Luchinsky*'+, P.V.E. McClintock*, I. Kaufman** and

R.S. Eisenberg*

*Department of Physics, Lancaster University, Lancaster LAI 4YB, UK. ^NASA Ames Research Center, MS 269-3, Moffett Field, CA, 94035, USA. "VNIIforMetrological Service, Gosstandart, Moscow, 119361, Russia.

^Department of Molecular Biophysics and Physiology, Rush Medical college, 1750 West Harrison, Chicago, IL 60612, USA.

Abstract. A self-consistent solution is derived for the Poisson-Nernst-Planck (PNP) equation, valid both

inside a biological ion channel and in the adjacent bidk fliud. An iterative procedure is used to match the two solutions together at the channel mouth. Charge fluctuations at the mouth are modeled as shot noise flipping the height of the potential barrier at the selectivity site. The residtant estimates of the conductivity of the ion channel are in good agreement with Gramicidin experimental measurements and they reproduce the observed current saturation with increasing concentration.

Keywords: ion channels, Poisson-Nernst-Planck equation, Langevin equation, self-consistent approach, charge fluctuation PACS: 87.16.Uv

INTRODUCTION

Ion channels are natural nanotubes in cellular membranes which allow the selective passage of ions into and out of the cell, thereby controlling a vast range of biological functions. Much has been learned about their structures in recent years, leading to intensive, fundamental, and applied research in both biology and physics. How ion channels are able to conduct selectively at rates comparable to free diffusion is still not well understood, despite intensive and ever-growing research. In this paper, we discuss the influence of charge fluctuations on an ion's transition through the channel. The model is used for comparison with experimental measurements of the single-channel Gramicidin current at different concentrations. The theory allows for the dependence of the interfacial concentration on the applied potential across the lipid bilayer, and is therefore able to reproduce the ion channel's fast conduction self-consistently. By taking into account the access resistance to the channel, we are able to observe the saturation of the current at higher concentrations. We consider the problem in three steps: (i) definition of the boundary conditions in a self-consistent manner, using coupled solutions of the Poisson and Nemst-Planck (NP) equations in the channel and in the bulk; (ii) theory of the transition rate through the channel in the presence of fluctuations; and finally (iii) comparison with experiment.



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METHODOLOGY

We use self-consistent Poisson and Nemst-Planck (PNP) equations

V.[£(r)V^(r)] = Y,Zjenj{r) + epex{r) i=l

(1)

and combine them with the drift-diffusion NP equation to accommodate the flux of mobile ions

dnj{r,t)

dt -y-Jj = 0, U=l,...,N),

Ji -Dj{r) Vfii ksT

nj{r)V^ (2)

The potential and the concentrations are thus determined in the bulk (i.e. away from) and close to the mouth of the channel with boundary conditions set at infinity [1]. The PNP equation inside the channel is solved analytically. The solutions obtained in the bulk and in the pore are then combined using an iterative procedure. The current and potential gradient calculated in the pore are utilised to set boundary conditions for the solution in the bulk. The bulk solution is in turn used to set the boundary conditions of the PNP equation in the pore, until convergence is achieved.

We use a ID approximation of the Poisson equation inside the channel. Applying Brownian dynamics (BD) simulations of ions in the solutions

niiXi = -miYiXi + qiQi 9UoRl

Aneecri: •'Or~ij rjf (3)

^AUoexp (^^iL.Iii] sin (in^'^ ''^ \ ae

a ra -Fch+V^myikifiiit),

we determine their arrival distribution at the channel mouth. It has been shown that this corresponds to a Poisson distribution [2]. Having estimated the arrival rates of ions at the channel mouth, we use this information to develop a theory of the transition rate through the channel. We assume that the channel is occupied most of the time by only one ion, and that the transition rate of ions through the channel is mainly determined by escape over the potential barrier at the selectivity site. Therefore the effect of the many-body ion dynamics in the bulk on ion motion in the channel is twofold: (i) a delivery of the ions to the channel mouth and (ii) modulation of the channel potential by the charge fluctuations at the channel mouth. Under these physiologically plausible assumptions we separate the ion motion in the channel from the many-body ion dynamics in the bulk. The resultant ion dynamics in the channel may be described by the overdamped Langevin equation.

myx-dV{x,t)

dx ^JlmrkBT ^{t) (4)

where the potential V{x^t) has three main contributions: (i) the potential of Coulomb interaction with ions in the bulk solution Vc\ (ii) the electrostatic potential induced by

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interaction with the channel protein Vch', (iii) the potential of Coulomb interaction with the wall charge at the selectivity site. By an averaging procedure, the effect of multi-ion motion in the bulk solution is reduced to Coulomb interaction with ions arriving at the channel mouth. The latter process can be viewed as a stochastic Poisson process or as shot noise that modulates the potential barrier for the conducting ion at the selectivity site. We estimate analytically the effect of this potential modulation on the channel conductivity. To quantify the effect of multi-ion motion in the bulk on the conducting ion at the selectivity site, we have simulated the ion's Brownian dynamics in the bulk. The mean arrival time for Na+ and Cl^ ions at the channel mouth (approximated as a cylindrical section of radius R and length R) are T+ = 365 ps and T^ = 471ps respectively. These estimates are in agreement with the theoretical estimates in [4]

where c is the ion concentration and D is the diffusion coefficient. The time evolution of the charge in the channel mouth is a Poisson process with the three main states +le, 0, and —le Fig. l(left). As a first approximation we divided the states of the channel potential affected by the charge fluctuations into two effective states: (i) a state of high conductivity, corresponding to +le, and (ii) a state of low conductivity, corresponding to a charge of either 0 or — le at the channel mouth. In this approximation the effect of three states of the potential is taken into account by asymmetry of the transition rates between the two effective states. The corresponding transition rates can be estimated as a^ = 1/ {T±), where {T±) are mean residence time of two effective states, giving (a+)^^ = 0.22ns and (a^)^^ = 0.38ns. The occupation probabilities of these two states are 0.36 and 0.64 respectively.

The effect of charge fluctuations on the channel potential is estimated by solving the Poisson equation for various positions of the conducting ion along the channel axis in the two cases: (i) when there are no other ions at the channel entrances; (ii) when there is one positive ion at the left entrance to the channel.

RESULTS AND CONCLUSION

Our self-consistent ID solutions of the PNP equations in the bulk and inside the channel allow us to determine the potential and concentration in both the bulk and the channel. On this basis, we have reconstructed the statistical properties of the charge fluctuations from 3D Brownian dynamics simulations of multi-ion motion in the bulk solution, as shown in Fig. l(left). We find that the distributions of ion arrival times at the channel mouth are exponential. These charge fluctuations strongly modulate the potential barrier for the conducting ion at the selectivity site due to amphfication of electrostatic interactions in long narrow channels of low dielectric constant. These findings have allowed us to model the mean ion transition time through the channel as an ionic escape from the potential well at the selectivity site induced by thermal fluctuations and by modulation of the height of the barrier by stochastic Poisson processes.

We can compare our simulation results with the experimental measurements of the ion current in the Gramicidin channel performed by Andersen et al. [3] in the range

537

of concentrations 50 mM to 2M. Fits to the experimental data for two values of the apphed potential 75 mV and 100 mV are shown in Fig. l(right). It can be seen from the figure that the experimental data exhibit a significant saturation of the ion current as a function of concentration, a dependence that is well-reproduced in our simulations. A more accurate model that takes into account the 3 states of the potential is currently under development.

<

o.or 0.01 0.1 C(M)

FIGURE 1. Left: Calculated charge fluctuations at the channel mouth. Positive charge is shown by the solid line and negative charge by the dashed one. Right: Comparison of theory with experiment. The current is plotted as a function of Na+ ionic concentration for a Gramicidin channel. The data points for the potential difference 75mV and lOOmV are taken from the single-channel measurements of Andersen et al [3]; the curves are from our theory for a particular set of parameter values given in [1].

ACKNOWLEDGMENTS

The work was supported by the Engineering and Physical Sciences Research Council (UK), The Russian Foundation for Basic Research, INTAS, and ESF.

REFERENCES

D. G. Luchinsky, R. Tindjong, I. Kaufman, P. V. E. McClintock and R. S. Eisenberg, "Self-consistent solution of the Poisson and Nemst-Planck equations in the bulk and in an ion channel", submitted to Phys. Rev. E. R. Tindjong, D. G. Luchinsky, P. V. E. McCUntock, I. Kh. Kaufman and R. S. Eisenberg, "Effect of charge fluctuations on the permeation of ions through biological ion channels", in M.Tacano et al. ed. Noise and Fluctuations, ICNF 2007,AIP vol. 922, Melville, New York, 2007, pp 647-650. O. S. Andersen, R. E. Koeppe, and B. Roux, IEEE Transactions on Nanoscience 4, 10 (2005). Eisenberg R S, Klosek M M and Schuss Z 1995 /. Chem. Phys. 102 1767-1780

538

Cancer growth dynamics: stochastic models and noise induced effects

B. Spagnolo'*, A. Fiasconaro^*, N. Pizzolato^*, D. Valentin*, D. Persano Adomo*, P. Caldara^*, A. Ochab-Marcinek''' and E. Gudowska-Nowak'''

*Dipartimento cli Fisica e Tecnologie Relative and CNISM, Viale delle Scienze, Ed. 18,1-90128 Palermo, Italy

^Marian Smoluchowski Institute of Physics, Jagellonian University, Reymonta4, 30-059 Krakow, Poland

Abstract. In the framework of the Michaelis-Menten (MM) reaction kinetics, we analyze the cancer growth dynamics in the presence of the immune response. We found the coexistence of noise enhanced stability (NES) and resonant activation (RA) phenomena which act in an opposite way with respect to the extinction of the tumor The role of the stochastic resonance (SR) in the case of weak cancer therapy has been analyzed. The evolutionary dynamics of a system of cancerous cells in a model of chronic myeloid leukemia (CML) is investigated by a Monte Carlo approach. We analyzed the effects of a targeted therapy on the evolutionary dynamics of normal, first-mutant and cancerous cell populations. We show how the patient response to the therapy changes when an high value of the mutation rate from healthy to cancerous cells is present. Our results are in agreement with clinical observations.

Keywords: Cancer growth models. Noise, Monte Carlo PACS: 05.40.-a,87.17.Aa, 87.15.Aa

INTRODUCTION

Although cancer is a leading cause of death in the world, it is still little known about the mechanisms of its growth and destruction [1,2].

Fluctuations may play an important role in cancer growth. In the tumor tissue, the growth rate and cytotoxic parameters are influenced by many environmental factors, e.g. the degree of vascularization of tissues, the supply of oxygen, the supply of nutrients, the immunological state of the host, chemical agents, temperature, radiations, gene expression, protein synthesis and antigen shedding from the cell surface, etc. As a result of this complexity, it is unavoidable that in the course of time the parameters of the system undergo random variations which give them a stochastic character.

The cancer growth dynamics in the presence of immune response can be modelled by the Michaelis-Menten (MM) kinetics, in which the development of tumor tissue and its reaction to immune response can be described in terms of a predator-prey model. Tumor cells play the role of "preys", and immune cells act as "predators" [3]. The Langevin equation describing our model is

x = - ^ ^ + oW)- (1)

Group of Interdisciplinary Physics; URL: http://gip.dft.unipa.it/



539

http://87.15.Aa

http://gip.dft.unipa.it/

where

f/±(x) = - y + ^ + ( /3±A)(x- ln(x+l) ) , (2)

is the MM potential switching between two configurations because of a dichotomous noise, of amplitude A and mean correlation time T, which describes realistic fluctuations in immune response (see Fig. ILeft).

Noise Enhanced Stability and Resonant Activation

The investigation concerns the evaluation of the mean escape time from the potential, which corresponds to the mean extinction time of the cancer cells. The starting position is in this case an unstable starting position, presenting, in the static or fluctuating case, a well pronounced increase of the escape time (NES effect [4, 5, 6]). This means that the noise stabilizes a metastable state in such a way that the system remains in this state for a longer time than in the absence of white noise. A different behavior, given

b I I 1 I

^I'-ie-K)? rooooo

100000

10000

1000

100

10

0.1

FIGURE 1. Left: Michaelis-Menten potential. Stars, circles and crosses, indicate maxima and minima in three configurations. Right: Coexistence of NES and RA in a three-dimensional plot of MFPT as a fimction of noise intensity and correlation time.

by the presence of the dichotomous fluctuations gives instead the opposite behavior. The presence of the fluctuations, together with the additive noise gives rise to the so called resonant activation (RA) [7, 8], which is the presence of a minimum in a nonmonotonic dependence of the escape time as a function of the characteristic time scale of the fluctuations.

In Fig. 1 (Right) the coexistence of NES and RA phenomena is shown in a three-dimensional plot of the mean first passage time as a function of the noise intensity a and the correlation time T. The two effects are acting in an opposite way in the cancer growth dynamics. Namely, the NES effect by increasing the lifetime of the metastable state delays the escape from the tumor state, while the RA effect by decreasing this lifetime leads to the extinction of the tumor. An appropriate choice of values of the noise parameters allows either to maximise or minimise the extinction time of the population. There exists a regime of a and T parameters where noise-enhanced stability is strongly reduced by resonant activation.

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Stochastic Resonance

A different investigation has been performed by considering an almost symmetric bistable Michaelis-Menten (Fig. 2(Left)), adding an external driving representing a controllable therapy, and again an external noise term ([9]). We obtain the following dynamical equation

dx dt

= x ( l - 0 x ) -/3x

1 -A[\-@{cos{2nvt))] + ^{t) (3)

where 0-symbol is the Heaviside function reflecting the on-off switch of the cyclic treatment performed with the intensity^ and frequency v. In this case the starting point of our investigation is in the right minimum of the potential, i.e. the deterministic stable cancer point. The considered treatment is so weak that no extinction is possible in a deterministic dynamics. However, the presence of noise can give rise to extinction (Fig. 2(Right)) so giving a positive contribution in cancer kinetics. However, by increasing the noise intensity a noise induced reappearance of the tumor cells become possible. By increasing again the noise intensity the presence of a synchronized extinction and reappearing of the cancer may occur, showing the so called stochastic resonance effect. The statistics of

Trajectories

0.4 -

0.3 - 1

0.2 - \

0.1 - \

*

Michaelis-Menten Potential

Without treatment / During treatment / /

/ / / /

^^_^^^y/ P = 1.48

6 = 025

FIGURE 2. Left: Pseudo-potential U{x) with j8 = 1.48, Q = 0.25 that assure bistability with an almost symmetric character (equal depths) of both potential wells. Right: Deterministic and stochastic trajectories of cancer evolution. For D = 0, the trajectory is localized in the cancer state at the minimum of the well. Low level of noise is instead able to induce the extinction (Z) = 0.011). For increasing noise intensities, the reappearance of cancer is detected (D = 0.016). For D close to 0.035, a stochastic resonance effect is visible.

both the first extinction time (FET) and the first return time (FRT) has been evaluated. The total probability of the exit (P ) and return (P ) occurrences as a function of the noise intensity calculated as the ratio of the number of exit/return trajectories normalized over the total number of experiments in the observation time window, are shown in Fig. 3(Left). The related mean values of exit and return times are plotted in Fig. 3(Right) as a function of the noise intensity Z3. In the inset of the figure we can observe the ratio MFRT/MFET. The minimum indicates the existence of a suitable noise intensity giving the optimal compromise between the effectiveness of the therapy with respect the possible reoccurrence of the cancer. This observation provides a qualitative analysis of

541

mechanisms responsible for optimization of periodic tumor therapy in the spontaneous external noise.

presence of

0.8

0.6

0.4

0.2

0

- ) \ _ 7 TSE

1

Extincion

T S R

1 1

Return Probability

-

Pg = Exit Probability Pj = Return Probability

Pj (1 - Pj)

rajconc.

1 1 1 1

10000

8000

6000

4000

2000

Mean Passage Times

N

- \ 09

\ °'

\ \ °^ - \ \ °'

Mean Exit Time (MET) —

Mean Return Time (MRT) —

METMRT Ratio

^ ^ ^

-\ /'-''''^ - —

^ — D

--

---

---

07

0.01 O02 OOl O02 O05 Om 005 006 007

Noise P ) Noise (D)

FIGURE 3. Left: Probability of the exit and return occurrence vs. noise intensity. The 'exit without return' probability Pe{l —Pr) has a nonmonotonic behavior with an optimal value of the noise D Ki 0.008. Right: Mean first exit time and mean first return time vs. noise intensity. The ratio between both measures (displayed in the inset) exhibits a minimum representing the best stochastic compromise between effectiveness of the therapy and the risk of cancer recurrence.

Imatinib-treated Leukemic Cells Dynamics

Chronic Myeloid Leukemia (CML) is a slowly progressive cancer of white blood cells. CML cells are characterized by a specific chromosomal abnormality: the Philadelphia (Ph) chromosome [10], which activates a number of cell cycle-controlling proteins and enzymes (tyrosine kinase), speeding up cell division and inhibiting DNA repairing. CML represents the first human cancer in which a targeted therapy by the tyrosine kinase inhibitor imatinib (Gleevec) leads to a striking clinical response. Unfortunately, acquired resistance to Gleevec develops in a substantial fraction of patients [11].

In this work, by using a Monte Carlo approach, we model the evolutionary dynamics of three populations of cells (heahhy cells, first-mutated cells and double-mutated leukemic cells) by simulating the stochastic behavior of cell division and mutation in a system of initially normal blood cells which can be affected by a double genetic mutation and transformed in cancerous cells. The several observed scenarios of CML patient response to imatinib are taken into account by modeling a therapy side effect of enhancement of the mutation rates. In our model, cells reproduce asynchronously: each elementary step of the stochastic process consists of a birth and a death event. In the reproduction process, one of the N cells is randomly chosen proportionally to the fitness; in the death event one of the N cells is at random eliminated.

To describe the stochastic evolution of the three types of cell populations in the absence of therapy, we have chosen the mutation rates Moi and M12 equal to 0.001 and 0.02, respectively. These values are one order of magnitude greater than the mutation rates adopted elsewhere [12] because, in this study, we are not interested in the average waiting time before the disease make its first appearance, but in the subsequent dynamic evolution. Normal cells (green line in Fig. 4a) experience the first mutation after some

542

TABLE 1. Table of model parameters.

No therapy Ideal therapy Real therapy (Case 1)

Real therapy (Case 2)



F2 10.0 Moi 0.001 M12 0.02

0.7 0.001 0.02

0.7 0.01 0.20

0.7 0.10 0.40

0.7 0.13 0.40

0.7 0.14 0.40

time interval. If the intermediate-type cells (blue line) survive for a sufficiently long time, a second-type mutation can cause the birth of cancerous cells (red line). In this case the fitness FQ and Fi are set equal to 1, while the fitness of type 2 cells has been reasonably assumed to be 10 times that of the other two populations. For this reason, the number of cancerous cells rapidly increases to the total initial value N of normal cells.

Since a successful therapy requires a basic reproductive ratio of cancerous cells lower than 1 [1], we model the ideal therapy effect by lowering the fitness parameters F2 to 0.7. All our simulations start with the patient developing CML in the absence of therapy; when the number of leukemic cells exceeds the threshold value N/3, the therapy is activated. In fig. 4b we can see that the effect of an ideal therapy is to completely eradicate the number of mutated cells and favor the restoring of normal cells. The time scaling from cell division to days is performed by assuming a complete restoration of healthy cells in almost 100 days, as experimentally observed in clinical cases of optimal therapy response [1, 2].

To simulate real cases of therapy effects on cancer evolution, we have supposed that the cell system could react to the drug administration by activating multiple genetic changes that cause an enhancement of the mutation rates from normal to cancerous cells. This assumption is supported by the experimental evidence that certain mutations increase the rate at which subsequent mutations occur [13]. This secondary effect of the therapy is investigated by modeling an increase of Moi and M12 at four different levels, as summarized in table 1. In the case 1 of real therapy (Fig. 4c), the extinction effect on type 2 cells is almost unchanged with respect to the previous case (Fig. 4b) and the first-mutant cells disappear very quickly because of the increased value of M12. When the mutation rate M12 is doubled and MQI progressively increased (cases 2, 3 and 4 in table 1), an effect of retard is observed in the recovery of healthy cells (Fig. 4d and 4e) and, in the worst case, a failure of the therapy itself is present (Fig. 4f). This means that, even if the therapy works properly by inhibiting the reproduction of leukemic cells, the cancer proliferation can still occur because of an increase of unfavorable genetic mutations.

In order to take into account also the several cases of acquired resistance to the therapy, observed in CML patients [11], we simulate the development of resistance by modeling a time-dependent linear increase of both mutation rates. In this case, after the therapy activation, the cancerous cells initially respond to the medicine, starting a decreasing trend, but when the mutation rates start to grow, the number of cancerous cells starts to rise again, quickly reaching the total value N [14].

In conclusion, we find that the response to the therapy depends on both the inhibitory capability of the medicine and on the levels of mutation rates from normal cells to

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0 .3 '-

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f

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FIGURE 4. Evolutionary dynamics of blood cells. Green line indicates the behavior of healthy cells, blue line that of first mutated cells and red line that of cancerous cells: (a) no therapy; (b) ideal therapy; (c, d, e, f) real therapy with increasing mutation rates (see table 1).

leukemic types. Our modeled evolutionary dynamics of leukemic cells are in agreement with the different temporal response of patients treated with imatinib, observed in clinical studies and summarized in Ref. [2].

REFERENCES

1. F. Michor et al., Nature Reviews Cancer 4, 197 (2004). 2. I. Roederei a/., Nature Medicine 12, 1181 (2006). 3. A. Fiasconaro, B. Spagnolo et al, Phys. Rev. E 74, 041904(10) (2006) 4. R. N. Mantegna and B. Spagnolo, Phys. Rev Lett. 76, 563 (1996). 5. A. A. Dubkov N. V. Agudov and B. Spagnolo, Phys. Rev E 69, 0161103 (2004). 6. A. Fiasconaro, B. Spagnolo and S. Boccaletti, Phys. Rev E 72, 061110(5) (2005). 7. C. R. Doering and J. C. Gadoua, Phys. Rev Lett. 69, 2318 (1992). 8. R. N. Mantegna and B. Spagnolo, Phys. Rev Lett. 84, 3025 (2000). 9. A. Fiasconaro, B. Spagnolo et al, Eur Phys. J. B 65, 435 (2008). 10. P Nowell, D. Hungerford, Science 132, 1497 (1960). 11. S. Soverini et al, Clin Cancer Res. 12, 7374 (2006). 12. Y. Iwasa, F Michor, and M. Nowak, Genetics 166, 1571 (2004). 13. L.A. Loeb, J.M. Essigmann, F Kazazi et al, Proc. Natl. Acad. Sci. 99, 16226 (2002). 14. N. Pizzolato, D. Valenti, D. Persano Adomo, and B. Spagnolo, Cent. Eur J. Phys. (2009), in press.

544

Extreme Value Analysis of Heart Beat Fluctuations

C. Pemietta* and S. Contet

*Dipartimento diIngegneria dell'Innovazione, Universita del Salento and CNISM, ViaArnesano, 1-73100Lecce, Italy, E-mail: [email protected].

''Universita del Salento, Via Amesano, I-73100Lecce, Italy.

Abstract. We have performed an extreme value analysis of the heart beat fluctuations. We have analyzed 24-h ECG time series: by considering both, the RR intervals and the increments series ARR. Sub-series, corresponding to 5-h sleeping activity and to 5-h daily activity have been also separately studied. Strong differences are found in the mean return times for high thresholds of healthy and non-healthy patients, during both daily and sleeping activity.

Keywords: Heart rate variabiUty, Fluctuations in complex systems. Extreme events PACS: 87.18.-h, 87.18.Tt,87.19.Hh

INTRODUCTION

In recent years it has become clear that many physiological signals display fairly complex dynamics, like for example long-term correlations, multifractality, non-Gaussianity etc., which reflect the complex interplay of different biological mechanisms acting and competing in highly-organized organisms [1, 2, 3, 4, 5, 6, 7, 8, 9]. In particular, it has been realized that such signals contain much more information than that catched di-recfly "by eyes" [10]. Moreover, such hidden information cannot be extracted by using conventional statistical tools. For this reason, advanced statistical methods, conceived in the context of complex physical systems, like detrended fluctuation analysis (DFA) [2, 3, 4, 5, 6, 7, 8], multifractal detrended fluctuation analysis (MDFA) [3, 6, 7] or wavelet analysis (WA)[6, 9] have been applied to the study of biomedical time series, like for example series made by heart beats or neural records [10]. However, neither MDFA nor WA methods provide a tool to classify an individual signal and new analysis methods are looked forward. Recently, some authors [11, 12, 13] have highlighted the effectiveness of extreme value analysis (EVA) in the study of complex systems. Therefore we have performed an extreme value analysis of heart beat time series.


First, we have analyzed RR time series, i.e. time series whose records are the time intervals between two subsequent R waves, as measured in 24-h ECG Holter signals [6, 10]. Two sets of signals have been considered, corresponding to 40 healthy and 90 non-healthy patients (with reduced left ventricular systolic function) [6].



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http://87.18.-h

200

150

100

0 10 20 30 40 0 10 20 30 40 50 60 70 80 90

ID number ID number

FIGURE 1. Vertical axis: root mean square deviation of the RR time-intervals measuredby 24-h Hotter ECG (time intervals are espressed in msec). Horizontal axis: patient identification number (A): healthy patients; (B): non-healthy patients. See the text for the meaning of the horizontal lines.

Actually, RR intervals are known to fluctuate in a way that reflects autonomic neural control on the sinus node, the heart first pacemaker [14]. It is generally believed [3, 4, 7, 8, 6] that a reduction of variability and/or complexity of heart fluctuations is associated with clinical risk. Yet, the strength of this association remains questionable [7, 14]. A simple insight of the above statements can be catched by considering Fig. 1, which displays the root mean square deviation of the RR intervals, GRR, for different patients belonging to set (A), healthy people, and to set (B), non-healthy people. The thick and continuous horizontal lines show the average values of GRR over the corresponding set, < ffRR >set- The dashed horizontal lines correspond to < aRR >set icTset, where a^et is the root mean square deviation of aRR calculated over each set. We see that < CTRR >setA is greater than < aRR >setB, in agreement with the association between reduced variability and higher clinical risk reported in the literature [3, 4, 7, 8, 6]. Nevertheless, Fig. 1 clearly points out that several patients belonging to the set (B) exhibit a value of aRR comparable with (and in some case even higher than) the value of aRR of patients of the set (A). In this regard, we can say that only very small aRR values contain significant (and negative) information for the individual patient. Thus, Fig. 1 well illustrates the difficulty of statistical tools to classify individual signals.

Therefore, with the purpose of exploring the predictive ability of other statistical tools, we have performed an EVA analysis on both RR series and ARR increment series. Actually, many authors [5, 7, 15] have stressed the importance of the study of the increment ARR time series, i.e. a series where the i-th record is: ARRi = RRi+i — RRi-Together with 24-h times series, shorter sub-series [6], corresponding to 5-h sleeping activity and to 5-h daily activity have been also separately studied. In fact, given the non-stationary character of the 24-h RR series associated with cyrcadian rhythms, for many purposes it is convenient to carry out a separate study of wake and sleep signals [14]. Preliminary to the EVA analysis, all the series have been normalized to satisfy the condition of zero average and unit variance. Then, by denoting x{t) the normalized

546

20

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12

10

8

6

4

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0

q=1.0

(A)

10 20 30 40 0 ID number

10 20 30 40 50 60 70 80 90 ID number

FIGURE 2. Daily activity increment time series, ARR. Vertical axis: values of the mean return times of the threshold q = 1.0 (the threshold value is espressed in root mean square units). Horizontal axis: patient identification number A): healthy patients; (B): non-healthy patients.

500

400

300

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100

q=2.5

(C)

1 il

•WVJUAI

0 10 20 30 40 0 ID number

10 20 30 40 50 60 70 80 90 ID number

FIGURE 3. Daily activity increment time series, ARR. Vertical axis: values of the mean return times of the threshold q = 2.5 (the threshold value is espressed in root mean square units). Horizontal axis: patient identification number A): healthy patients; (B): non-healthy patients.

variable, for each series/sub-series, we have considered several threshold values, q, with

<q<x„ , and we have calculated the return times, r„, of the event associated with the overcoming of each given threshold (i.e. the time intervals between two consecutive occurences of the condition \x{t)\ > q) [11, 12, 13].

Consequently, we have studied the distribution of the return times and, in particular, its probability density, $, , which shows significant deviations from the exponential distribution characterizing uncorrelated events. Furthermore, we have calculated the mean (average) retum time, Rq, and the median return time, Mq, as a function of q and

547

for the different signals (RR and increment ARR series, sets (A) and (B) of patients, full 24-h series and 5-h sub-series). Here, we show the comparison of mean return times of the increment ARR series for healthy and non-healthy people during daily activity and for two different thresholds. Precisely, Fig. 2 displays the values of i?, for g- = 1.0 (low threshold), while Fig. 3 reports these values for q=2.5 (high threshold). We can see that while low thresholds are not able to distinguish between healthy and non-healthy people, strong differences are instead found for high thresholds (associated with the occurence of extreme events). Similar differences are also found during sleeping activity. We can conclude that EVA analysis provides a very promising tool for investigating heart rate fluctuations and for the identification of clinical risk.

ACKNOWLEDGMENTS

The authors are deeply indebted to Prof. D. Makowiec (University of Gdansk) which kindly provided the time series analyzed in this work. The authors also thank Prof. Z. R. Struzik (University of Tokyo), Prof. P. V. E. McClintock (University of Lancaster) and Dr L. Urso (Clinic "Petrucciani", Lecce) for helpful and stimulating discussions.

REFERENCES

1. C. K. Peng et al., Phys., Rev. Lett, 70, 1343-1346 (1993). 2. C. K. Peng et al., Chaos, 5, 82-87 (1995). 3. P Ch. Ivanov et al., Wature, 399, 461^65 (1999). 4. A. Bunde et al., Phys., Rev. Lett., 85, 3736-3739 (2000). 5. Y. Ashkenazy et al., Phys., Rev Lett., 86, 1900-1903 (2001). 6. D. Makowiec et al., Physica A, 369, 632-644 (2006). 7. Z. R. Struzik et al., TELE Trans, of Biomed. Engin., 53, 89-94 (2006); Z. R. Struzik et al., Eump. Phys.

Lett., 82, 28005-pl-p5 (2008). 8. S. long et al., Physica A, 380, 250-258 (2007). 9. A. Xu et al.. Mod. Phys. Lett. B, 19, 59-78 (2005). 10. PhysioNet Data Bank: /;ttp://circ.ahajoumals.org/cgi/content/full/101/23/e215. 11. E. G. Altmann. H. Kantz, Phys. Rev E, 71, 056106-1-9 (2005). 12. A. Bunde et al., Phys. Rev Lett., 94, 048701-1^ (2005). 13. C. Pennetta,£'ur Phys. J. B 50, 95-98 (2006). 14. Task Force of the European Society of Cardiology and North American Society of Pacing and

Electrophysiology,£'urHeart J., 17, 354-381 (1996). 15. L A. Khovanov et al., J. Stat. Mech. pOl 016-1-15 (2009).

548

Human Sleep EEG Stochastic Artefact Analysis

Petr Sadovsky, Robert Macku

Brno University ofTechnology, Faculty of Electrical Engineering and Communication, Department of Physics, Technicka 8, 616 00 Brno, Czech Republic, [email protected]

Abstract This article describes a part of problem, why it is not possible to use a long-time spectral analysis to diagnose sleep disturbances. This article is focused especially on muscle and movement artefacts analysis as a main reason of this effect. We found that random processes which generated muscle and movement artefact signals are possible to describe by three postulates [1] for Generation-Recombination processes. The result of these processes is a 1/f 2 power spectral density.

Keywords: EEG; EEG Artefact analysis. PACS: 87.19.le

INTRODUCTION

It is known that EEG consists of many signals from many statistically independent sources. It is possible to simply divide them into three main categories:

1. Random signals (for example random signals form neurons, etc.) 2. Deterministic signal (for example rhythm, transients, etc.) (Note: Deterministic

signals are not deterministic signals to all intents and purposes. It is possible to describe shape of these signals, but time of occurrences and time of duration are definitely random.)

3. Artefacts (for example muscle and movement artefacts, EKG, 50Hz, electromagnetic disturbance, etc.)

Power spectral density of signals from the first category is of 1//" type. Power spectral density of signals from the second category contains peaks. Power spectral density of muscle and movement artefacts (third category) is of 1//^ type.

LONG TIME EEG SPECTRUM

The result of a long time spectral analysis of human sleep EEG signals is definitely the 1//^ spectrum. This is the reason why is impossible to use the long time spectral analysis for human sleep EEG diagnostic. See FIGURE 1.

EEG signals are definitely nonstationary. The nonstationarity is observed especially in the variance cr^, which is caused by movement artefacts. Mean remains almost constant.



549


http://87.19.le

In FIGURE 1 is shown a spectrum of human sleep EEG signals. The verification of stationarity of processed interval was not concerned.

These spectres are produced by Generation-Recombination Processes (G-R processes). G-R processes are typical spectral area of frequencies greater than boundary frequency fc. Spectral power density decreases \lf ^ with increase of frequency [3].

>

^

10 10 10 / / H z

FIGURE 1. Long time human sleep EEG spectrum computed from signal with artefact

10 10" 10"

/ / H z 10 '

FIGURE 2. Long time human sleep EEG spectrum computed from signal without artefact

EEG STATIONAR INTERVALS SPECTRUM

It is possible to find a long time spectrum from EEG signal, but it is necessary to previously divide EEG signals to intervals which fulfil the condition of stationarity. The rules for stationarity were used according to the Chincin rules [4]. After this analysis we obtain many intervals of different type and length. It is necessary to describe it by several parameters. Statistic moments m\, M2 and M4 were chosen as a parameters, which enables the description of single intervals and also divides them into a number of subsets.

In FIGURE 2 is shown a stationary interval spectrum from human sleep EEG. This spectrum is computed from stationary signals which contain 76 % of all intervals.

ARTEFACT ANALYSIS

According to previously verified information we assumed, that movement artefacts cause that long-time EEG spectrum is always 1//^.

We found that it is possible to simulate muscle and movement artefacts as signals of two or more statistically independent sources, but mutually conditioned. At the top of FIGURE 3 is shown a signal, marked as Sar^, of the 'quick' random process '%ari(U,i). At the bottom of fig. 1 is shown a signal marked as Sari, of the 'slow' random process '^ariif). The creation of muscle and movement artefacts is possible to describe:

nsju,t)) = ns^,,{u,t)\s^jt)) (1)

550

500

^ |f|h^t^rt JJU- II 11 I I M

-5001 I I L. _I I I I L_ 0 20 40 60 80 100 120 140 160 180 200

t/s

1

0.8

- 0.6 11 ^= 0.4

0.2

0 1 1 , 1 1 1 1 1

_

-

-

20 40 60 80 100 120 140 160 180 200

FIGURE 3. The random signal 0,3 is a result of the first type of stochastic process which is the first part of the artefact signal. The random signal Sari is a result of the second type of a stochastic process.

Both of them generate EEG artefact.

10

If f

>-? Iff

I f f

1

2

3

B

• ft J 4 ^ _ J ^^Z^jTSrrry...^^ ^ ;:::::::::i::::::::::;::::::::::;::::::::::t^ iP -^ i * -

: ' ' ' r , •

! =::==:==:=!=======:==

J

:=:==:====

•

• 0.5 1 1.5

6 8 10 12 14 16 18 20 22 24

t/s

551

FIGURE 4. There are the empirical probability density functions for length of time between artefact for Sari (at the top) and Sari (at the bottom) signals in semi-logarithmic scale.

In FIGURE 4 are shown the empirical probability density functions, which were acquired by analysis of Sars and Sar2 signals. As we can see in figure, these functions are exponential, so we can assume that it is possible to describe random processes '%ari(U,i) and '^ariif) by postulates [1] for G-R processes. The power spectral density of these signals should be of \lf ^ type, which also corresponds with results of measurement. These hypotheses were statistically tested by the Kolmogorov-Smirnov test of hypothesis.

CONCLUSION REMARKS

According to previously accomplished spectral analysis we found, that spectrum characteristic for stationary intervals of EEG signals are \lf with evident spectral peaks of periodic signals.

A spectrum of signal which contains only movement artefacts is always 1//^. Movement artefact spectrum has a higher level of power spectral density than EEG

signals. This spectrum EEG is covered on lover frequency (10 Hz or below) if this spectrum is computed from EEG signals with movement artefact.

We have analyzed random signal composed only of these movement artefacts to verify if the spectrum of \lf^ types is caused by movement artefacts.

The analysis of movement artefacts confirmed the hypothesis, that these artefacts are results of at least two mutually conditioned processes. The result is that artifacts peaks create groups.

We found two mutual independent processes which share in movement artifact generation. (FIGURE 3)

The analysis of movement artefacts and processes confirmed the hypothesis, that spectrum of \lf^ types is a result of these processes.

ACKNOWLEDGMENTS

This research has been supported by the research intention MSM 0021630503. This support is gratefully acknowledged.

REFERENCES

1. A. T. Bharucha-Reid, Elements of the theory of Markov processes and their apphcations, Dover Publications, Inc., Mineola, NY, 1960

2. P. Sadovsky, "Human Sleep EEG Analysis", Ph.D. Thesis, Bmo University of Technology, 2007. 3. S. M. Rytov, Vvedenie v statisticeskuju radiofiziku, Cast I., Slucajnyje procesy, Nauka, Moscow,

1976 4. B.R. Levin, Teorie nahodnych procesu ajeji aphkace v radiotechnice, SNTL, Praha, 1965

552

1// Noise Through Retino-Cortical Pathways Assessed By Reaction Times

Jose M. Medina'' and Jose A. Diaz

^ Center for Physics, University ofMinho, Campus de Gualtar, Braga 4710-057, Portugal E-Mail: Jmanuel@fisica. uminho.pt

Department of Optics, University of Granada, 18071, Spain E-Mail: jadiaz@ugr. es

Abstract. Certain process in the brain exhibits intrinsic fluctuations that follow flicker noise close to 1/f Fourier spectra. Examples are found from ion-channels to visual psychophysics. In the latter, modem mental chronometry plays a central role by reaction times (RT). Here we examine the existence of flicker noise in RT taking into account the first stages of color coding. Stimuli were isolated along the cardinal directions in the color space. Although the power spectral density of the RT series scales inversely with the cell size, in all cases the exponent was close to unity. Our results suggest that 1/f long-term correlations share a common mechanism from photoreceptors to the visual cortex.

Keywords: 1/f noise, visual perception, color-opponent cells, reaction time, stochastic networks PACS: 05.40.-a, 87.19.1c, 42.66.Ne

INTRODUCTION

Current models of trichromatic color vision assert that L (long-), M (middle-), and S (short-) wavelength cone signals are combined in the eye to provide three separate precortical pathways: a luminance (L+M) axis and two chromatic-opponent-cone axes, a red-green (L-M) axis, and a blue-yellow axis (S-L+M). Figure 1(a) indicates a schematic representation of the color space along the three cardinal axes. The L+M axis provides the substrate for achromatic processing and the mechanisms responsible for photometric tasks. The luminance system is supported, in part, by the magnocellular pathway [1]. Magnocellular layers present faster and less variable visual latencies. Both red-green and blue-yellow axes conform the isoluminance plane (i.e. luminance is constant, see Fig. 1(a) shaded area). In polar coordinates, hue changes are considered from the origin at center by the angular coordinate whereas the radial coordinate will indicate changes in saturation (i.e. the reciprocal of the white content). Chromatic-opponent processing is mainly located through the parvocellular and koniocellular pathways. The parvocellular division presents smaller cell size in relation to the magnocellular pathway. Parvo cells present slower latencies and carry most of the red-green (L-M) modulation at low spatiotemporal frequencies. Koniocellular layers (the smallest cells) are more sensitive to the blue-yellow information. There is also a sluggish cortical component in the S-cone opponent signals in relation to the L-M opponent signals [1].



553

http://uminho.pt

http://05.40.-a

(b)

-S+(L+M)

Reference ^ ^

+S-(L+M)

Isoluminant plane

(c)

Profile Face-on view

Spatial position 1.5 Degrees

0.2 0.4 0.6 0.8 1.0

FIGURE 1. (a) Color space used for visual psychophysics and neurophysiology, (b) CIE-1931 (x,y) chromaticity diagram used in colorimetry. Achromatic variations are orthogonal to the x-y plane. Open and solid circles indicate the reference stimulus and the selected red-green (L-M) and blue-yellow (S-

cone variations) stimuli at isoluminance, respectively. Solid diamonds are the chromaticity coordinates of the CRT phosphors. The triangle indicates the color gamut expanded by the CRT monitor at

15cd/m^. The outermost solid curve indicates the spectrum locus, (c) Spatial configuration of the visual stimuli presented.

In the study of color processing and neural conduction-time dynamics, reaction times (RT) plays a fiindamental role [2, 3]. RT can be defined as the time elapsed from stimulus presentation until a response occurs. Specfral methods have shown that the existence of infrinsic variability from frial-to-frial causes power law spectrum or flicker noise. The discovery of flicker noise in RT is a robust effect that has been confirmed in multiple observers and in most of elementary cognitive processes [4-6]. Typical levels of RT variability have presented power law specfra close to unity or 1/f noise. However unusual high RT variability associated with visual disorders has concluded higher exponent values compatible with random-walk processes [7].

The aim of the present study is to examine a plausible physiological organization of flicker noise in color vision through the different retino-cortical pathways in Fig. 1(a). First, we test if flicker noise in RT scales in relation to the cell size. This is possible if noise come from independent local sources. That is, the expected power spectral density or power spectrum should be higher when cone responses are isolated into the koniocellular pathway in relation to the parvocellular and magnocellular layers. Second, we examine if high RT variability at isoluminance (i.e. noisier for chromatic signals), present different correlation sfructure. If flicker noise has different origins in color processing, power-law noises should present different exponents when signals are isolated thought the different cardinal directions in Fig. 1(a).

METHODS

Apparatus, Stimuli and Procedure

We isolate color stimuli in each cardinal axis using methods based on colorimetry and visual psychophysics. The display system consisted of a cathode-ray tube (CRT) color monitor connected to a 8-bit graphics card in a PC. There is a linear fransformation from Fig. la to the standard ClE-1931 (x,y,Y) color space [8]. Achromatic stimuli were increments and decrements in luminance between 3 and

554

27cd/m^ at intervals of 2 cd/m^. The reference stimulus (Figure 1(b), open circle) was x=0.332, y=0.333, Y=15cd/m^. Isoluminant stimuli were fixed to the achromatic plane of 15cd/m^ (Fig. 1(b), solid circles). The reference stimulus was the same as in the achromatic case. Isolation of the red-green and blue-yellow systems was estimated by heterochromatic flicker photometry at 15cd/m^. In all cases, stimuli were uniform circular patches presented in fovea at the distance of 70cm (Fig. 1(c)). RT series were measured using a computer-based protocol between 110-3000ms (1ms accuracy). 1/f noise in RT was verified for four normal color observers in a wide range of experimental conditions in the color space. Here only time series analysis along the cardinal axes is presented. More detailed information about the setup, CRT color calibration and estimation of the L, M, S cone excitation values are in previous studies [2].

Data Analysis

There are several methods to estimate the power spectrum. The standard periodogram and the Lomb-Scargle method for unevenly sampled data [9, 10] are not suitable in the experimental conditions tested. They are less accurate [5, 9] and noisier as in the RT series example analyzed in Figure 2a. Here frequency is not Hz but in cycles per trial. A low-variance procedure was used based on the Welch's method. This method is more reliable and reduces the variance at each data point by overlapping window segments half their length. The Bartlett window was used for data windowing. If necessary, zero padding was added at the end of the time series to complete 1024 data points [9]. An eight-point power spectrum is combined from different window size: smaller and large windows were assigned for high and low frequencies, respectively (Fig. 2(a)) [5, 9].

(a) (b)

-Modified Welch's method FFT periodogram

-Lomb-Scargle method iyi.i[iilllii1l[Lj||LiL,i4;Liliil

LuiiliiULiliLlikilLiJliiiL

500

(0 £ 400

o S 300 01

E 200

c K 100

• • A

L+M axis L-M axis S-(L+M) axis

• • a • •

• * J ^ ^

(c)

-

10 ' 10 ' 10' 0 200 400 600 800 200 400 600 800

Frequency (cycles per trial) Trial number Reaction time (ms)

FIGURE 2. (a) Full logarithmic plot of the power spectrum estimation of RT series using a modified version of the Welch's method (black hue), the standard fast Fourier transform (FFT) periodogram

(gray line) and the Lomb-Scargle method (hght-gray line), (b) An example of RT (ms) time series in the L+M axis (solid hue) and in the L-M (gray hue) and S-(L+M) (hght-gray hue) axes at isoluminance. To

enhance visuahzation, data in (a) and (b) were scaled in the vertical axis by different normahzation factors, (c) Standard deviation as a function of the mean RT measured at each stimulus condition.

RESULTS

Fig. 2(b) represents an example of the RT series against the trial number along the L+M (black line), L-M (gray line) and S-(L+M) (light-gray line) color directions.

555

Both red-green and blue-yellow signals at isoluminance present, on average, longer and more variable RT values as indicated in Fig. 2(c). Figure 3 represents the power spectrum estimation from RT series along the cardinal directions in the color space.

L+M axis: -0.88±0.04 L-M axis: -0.86±0.03 S-(L+M) axis: -0.86±0.03

• L+IV! axis:-0.91 ±0.04 • L-IM axis: -0.85±0.03

S-(L+IUI) axis: -0.86±0.03

10 10' 10' Frequency (cycles per trial)

FIGURE 3. Full logarithmic plot of the power spectrum from the RT series collected along the L+M axis and the L-M and S-(L+M) axes at isoluminance. Examples in (a) and (b) correspond to two

different normal color observers, respectively. In each color axis, values indicate the estimated slope by linear regression. Errors are the standard error of the mean.

The results corroborate the existence of long-range correlations in visual perception. The magnitude of the power spectrum scales with the cell size. It was higher for the S-cone opponent axis. However, in all cases the results indicate that the exponent of the power law noise (i.e. the slope in log-log coordinates), is similar to unity but slightly lower at isoluminance. Our results conclude the existence of a common mechanism in 1/f noise and color coding and suggest that mayor deviations in the exponent may arise from color deficit observers such us those presenting dichromacy or cerebral achromatopsia.

ACKNOWLEDGMENTS

This work was supported by the Funda9ao para a Ciencia e Tecnologia and by the Center for Physics, University of Minho, Portugal.

REFERENCES

1. S. G. Solomon, and P. Lennie, Nat. Rev. Neurosci. 8, 276-286 (2007). 2. J. A. Diaz et a l , Color Res. Appl. 26, 223-233 (2001). 3. D. J. McKeefry, N. R. A. Parry, and I. J. Murray, Invest. Ophthalmol. Vis. Sci. 44, 2267-2276

(2003). 4. D. L. Gilden, T. Thornton, and M. W. Mahon, Science 267,1837-1839 (1995). 5. T. L. Thornton, and D. L. Gilden, Psychon. Buh. Rev. 12, 409-441 (2005). 6. C T. Kello et a l , J. Exp. Psychol. Gen. 136, 551-568 (2007). 7. D. L. Gilden, and H. Hancock, Psychol. Sci. 18, 796-802 (2007). 8. G. Wyszecki, and W. S. Stiles, Color science. New York, John Wiley & Sons, 1982, p. 950. 9. W. Press et al.. Numerical recipes in C, Cambridge University Press, 1992, p. 994. 10. W. Mcllhagga, J. Vision 8, 1-14 (2008).

556

The More Rehearsal, the More Noise in Timing Patterns

Miki Goan" , Takugo Fukaya , and Katsuyoshi Tsujita"

"PRESTO, Japan Science and Technology Agency, 4-1-8 Honcho Kawaguchi, Saitama, Japan ''ATR Cognitive Information Science Laboratories, 2-2-2 Hikaridai, Seikacho Sorakugun, Kyoto, Japan

^Department of Electrical and Electronic Systems Engineering, Osaka Institute of Technology, 5-16-1 Omiya, Asahi Ward, Osaka, Japan

Abstract. We implement a field study in which real rehearsals for public performance are observed in detail and, moreover, attempt quantitative analysis of the dynamics in the creative process of a drama based on the recurrence plot method. As a result, it was found that regularity in the timing pattern of the utterances is acquired first. Then we found that this pattern becomes irregular. In order to evolve a skilled creative process into art, it is not too much to say that a critical step is to create fluctuation on purpose after achieving self-organization.

Keywords: Art, Skill-acquisition, Timing, Recurrence plot, RQA. PACS: 89.65.Ef

INTRODUCTION

The purpose of this study is to clarify the skill-acquisition process for controlling the speech timing used by professional actors rehearsing a play. The approach adopted here uses the recurrence plot method, which visualizes the dynamic structure of a system in phase space. The technique makes a correlation analysis of an embedded discrete data series in a higher-ordered phase space obtained by a time series of raw data [1]. The actors belong to a company of Oriza Hirata, who is a well-known dramatist in the modem theater world. His distinctive concem is reproducing simultaneous, complexly organized conversations with lots of dialogue as naturally as possible in order to create a cocktail party effect as perceived by the audience.

Drama is creation. Recently, "creative cognition" has gained popularity in cognitive science [2]. Experimental approaches are usually adopted in this area's research, but this study does not take that approach. In the conventional type of study, experimenters who already know the answers to questions ask subjects to solve them and then calculate the percentage of correct answers. In our study, on the other hand, experimenters are not researchers but artists (a drama director and actors), with the researchers observing the field of their experiment in a rehearsal hall. Because art is a creative activity that produces something new, artists as experimenters don't know the correct answers; furthermore, they themselves do not know the questions. Consequently, they must seek questions and answers by deriving tentative solutions. This is how we break into the interaction among artists (a drama director and actors) to



559

discover something and describe it. This seems to be a sensible way to solve a problem where correct answers are not known.

M E T H O D

The material for this study is comprised of video-recorded performances of the rehearsals by a director, Oriza Hirata. We examined typical multi-party conversation scenes (40 seconds long) from The Balkan Zoo (first produced in 1997). Nine actors participated in this performance-research. The target scene was repeated 94 times during rehearsals and stage performance, and thus 94 video-recorded trials were captured as digital data through image editing software (Final Cut Pro 3, Apple Inc.). Next, we marked the points where the first sounds of the first words of actors' lines were heard and advanced through them frame-by-frame because we assumed that the pattern of turn-taking of lines indicated how to capture an actor's utterance timing. The time error scale was within the video frame rate (1/30 [sec]). An utterance-timing chart was completed by this marking process. Based on it, time series data were made as follows: 1) Consider data series F(n), {n = 0,1,2, , N) whose sampling rate dt is 1/30 [sec]

(i.e., video rate). Suffix n indicates the «th sampled auditory data in the time sequence.

2) The time when the actor initiates turn-taking is marked as 4, {k = 1,2,3, , m) , where Nis assumed to be much greater than m.

3) Derive n= «*that satisfies n^dt= tk{k= 1,2,3,..., m). 4) If «i < « < «i + 5 is satisfied, then g(n) = I is given; otherwise g(n) = r, where 5 is a

parameter whose order is nearly 5 = IQdt and r is a random number such that 0 < r <0.5.

5) Calculate time delay parameters and embedding dimension [3] in the phase space using the derived time series of data g(n), {n = 0,1,2,..., N), and then define vectors v(i) {i = 1,2,...) by embedding the time series of data g(n) in the phase space. Next, in the i-j space, draw a recurrence plot diagram, where point {i,j) is plotted on the condition that distance D between two different embedded vectors v(i) and v0) in the phase space satisfies D = \v(i)-v0)\ < e, where e is a small positive number. We performed recurrence quantification analysis (RQA), a method that quantifies

differently appearing recurrence plots [4]. We selected one measure obtained by RQA to show the quantitative information on the recurrence plots: Percent Recurrence (%RECUR), which is the percentage of the number of recurrent points in the total number of plotted points. This value reflects system stability, that is, the larger the parameter is, the more stable and the less noisy the system becomes. For each time series, the neighborhood radius was 0.8 of the maximum distance separating the points in the reconstructed space.

RESULTS & DISCUSSION

We draw recurrence plots for each of the time series of data with seven delay parameters and five embedding dimensions. A diagonal hue is drawn on the {ij) plane

560

(b) (c)

ma tan m -rft ist

1 « M

zt« OD <H edu 1H* ' "

FIGURE 1. Examples of recurrence plots of time series data as timing charts of actors' turn-taking: (a) 9th trial, (b) 25th trial, (c) 93rd trial with seven delay parameters and five embedding dimensions. Recurrence plots are illustrated using the free software "Visual Recurrence Analysis" by Eugene Kononov.

image because all points are always plotted at J=i in the recurrence plots. Some periodicities are included in the time series when lines parallel to the diagonal one appear in the image. This study obtained the following results: (i) these lines were not shown at the very beginning of a rehearsal (Figure la); however, (ii) they appeared after a period of rehearsal time (Figure lb) and (iii) disappeared again on the real stage before an audience (Figure Ic).

The results indicate that there are two stages of performance improvement in the drama rehearsals directed by Hirata Oriza. The first stage is to gain regularity in the temporal patterns of utterance timing used by actors. The second stage is to intentionally fluctuate the timing gained at the first stage. It has been reported that such a two-stage model of motion learning occurs in other fields. For example, the study of applause in a concert hall reported that twitching-like applause synchronized just after the concert, but the period of applause suddenly doubled after some time [5]. In other studies of expert systems, such as hula-hoopers [6] or ceramic artists kneading clay [7], it was found that the motions of all body parts synchronize and become a single period through an unregulated period, and then multiple periodes created by sub-structures coexist at the next stage. In a theoretical sense, we think that the learning of bifurcation [8] would become a central theme in clarifying these phenomena.

Figure 2 shows time-series variation of %RECUR during all processes of rehearsals and stage performance. To investigate the difference in system stability between the first and second halves of the rehearsed trials, we conducted Student's Mests on the measure. This Mest revealed a significant difference between the two halves of the rehearsed trials: the former (7V=47, M=26.52, ^Z3=1.82) was higher than the latter iN=47, M=25.03, ^Z3=1.57) (/'(92)=4.26, j9<0.001). As mentioned before, the larger %RECUR is, the more stable and the less noisy the system becomes. The results indicate that the timing pattern of the beginning of the utterance becomes an irregular one through the rehearsals; in other words, the more rehearsals, the more noise arises in timing patterns.

Generally speaking, previous studies on the self-organization of motor learning found that the period gained by motor learning converges at a fixed ratio. However, art

561

does not aim at converging to a fine form (harmony) as a self-organized motion. Rather, art, especially modem art, pursues the beauty of asymmetrical form. Of course, this is the stage where a proficient artist learns this for himself/herself through severe training. In this manner, drama presented in front of an audience is accomplished through many rehearsals. If there exists no harmony initially, the artist can neither destroy nor fluctuate it. If we regard drama as an entire system, the audience is assumed to be moved emotionally by interpreting the drama through two phenomena: the order achieved in the whole system and the individual fluctuations intentionally made by actors trying to achieve artistic variances.

(D 01

45

Trial number

FIGURE 2. Time-series variation in percentage of recurrence points in all trials.

ACKNOWLEDGMENTS

This research was supported by PRESTO from Japan Science and Technology Agency (JST) to the first author.

REFERENCES

1. I P . Eckmann, S. O. Kamphorst and D. Ruelle, Europhys. Lett. 4, 973-977 (1987). 2. R. A. Finke, T. B. Ward and S. M. Smith, Creative Cognition: Theory, Research, and Applications,

The MIT Press, Cambridge, MA, 1992. 3. F. Takens, Lee. Notes in Math. 898, 366-381 (1981). 4. M.A. Riley, R. Balasubramaniam and M. T. Turvey, Gatt and Posture 9, 65-78 (1999). 5. Z. Neda, E. Ravasz, Y. Brechet, T. Vicsek and A. L. Barabasi, Nature 403, 849-850 (2000). 6. R. Balasubramaniam and M. T. Turvey, Biol. Cybern. 90, 176-190 (2004). 7. T. Yamamoto and T. Fujinami, "Synchronisation and Differentiation: Two Stages of Coordinative

Structure", in Proceedings of 3rd International Workshop on Epigenetic Robotics 2004, 2004, pp. 97-104.

8. J. S. T. Kelso, Dynamic Patterns: The Self-Organization of Brain and Behavior (Complex Adaptive Systems), The MIT Press, Cambridge, MA, 1995.

562

Nonlinear stochastic differential equation as the background of financial fluctuations

V. Gontis, B. Kaulakys and J. Ruseckas

Institute of Theoretical Physics and Astronomy of Vilnius University, A. Gostauto 12, LT-01108 Vilnius, Lithuania

Abstract. We present nonlinear stochastic differential equation (SDE) which forms the background for the stochastic modeling of return in the financial markets. SDE is obtained by the analogy with earlier proposed model of trading activity in the financial markets and generalized within the nonextensive statistical mechanics framework. Proposed stochastic model generates time series of return with two, the probability distribution function and the power spectral density, power-law statistics.

Keywords: Financial markets, 1/f noise, stochastic equations, q-Gaussian distribution PACS: 89.65.Gh, 02.50.Ey,05.10.Gg

INTRODUCTION

Empirical financial data exhibit the sophisticated and universal statistical properties. A variety of the so-called stylized facts has been established [1, 2], which can be seen as statistical signatures of the financial processes. The findings regarding the probability distribution function (PDF) of return and other financial variables are successfully generalized within the non-extensive statistical framework [3]. Additive-multiplicative stochastic models of the financial mean-reverting processes provide rich spectrum of shapes for PDF depending on the model parameters [4]. These models with appropriate fitting parameters do capture the distributions of returns, volatilities, and volumes but not necessarily the empirical temporal dynamics and correlations. There is empirical evidence that trading activity, trading volume, and volatility are stochastic variables with the long-range correlation [5, 6, 7] and this key aspect is not accounted for in the widespread models.

Recently we investigated the properties of stochastic multiplicative point processes analytically and numerically [8]. We derived formula for the power spectrum and related the model with the general form of multiplicative stochastic differential equations [9, 10]. Consequently, the stochastic model of trading activity based on the Poisson-like process driven by the nonlinear stochastic differential equation (SDE) was presented in Refs. [11, 12, 13, 14].

The statistical similarity of trading activity and absolute return together with the general background of non-extensive statistics give us an opportunity to model dynamics of return by nonlinear SDE. This forms a new theoretical approach of the stochastic modeling of financial variables.



563

STOCHASTIC MODEL WITH Q-GAUSSIAN PDF AND LONG MEMORY

Starting from the multiplicative point process [8] we have derived SDE [9, 10]

dx = a^(ri--]x^'^-^dt + ax'^dW (1)

which solution has power-law probability distribution function, P(x) '-^x ^, and power spectral density, S{f) '-^ l/f^, in wide range of frequencies. Here

P - l - ^ . \<K<2^ (2)

Due to the divergence of the power-law distribution and the requirement of the station-arity of the process, the stochastic equation (1) should be analyzed together with the appropriate restrictions of the diffusion in some finite interval Xmm = x < Xmax [9].

Power spectral density is determined mainly by the power-law behavior of the coefficients of SDE (1) at big values of x > Xmm- Changing the coefficients at small x, the spectrum retains power-law behavior Therefore, we propose the following modification ofEq. (1)

dx = (72(^77-1) (xo+x2)'^-'xd? + c7(xg+x2)'i/2dff. (3)

The associated Fokker-Planck equation gives q-Gaussian PDF,

r(A/2) ' -2 X V2 P(x) =- ^ ' '

with the parameter

/^xor ( (A- l ) /2 ) \xl+x

r(A/2) _ f , x2

'0 / ^ x o r ( ( A - l ) / 2 ) ^ ' ' P 4 ^2x2

^ = l + 2 / A .

Here exp^(-) is ^-exponential defined as

exp^(x)^(l + (l-^)x)VO-<?). (4)

Introducing scaled variables

X = x/xo, t = (J^XQ-^ '

we get SDE

dx= (ri-^){^+x^)'^'^xdt + {l+f)'^/Mw (5)

564

0.00010.001 0.01 0.1 1

f

FIGURE 1. PDF, P{x), of solution of Eq. (5) with parameters A = 3 and 7] = 5/2 coinciding with the analytical Eq. (6) and spectrum, S{f), in comparison with the line representing 1 / / spectrum.

with Stationary distribution

P(x) = r(A/2)

/^r((A-l)/2) \l+x v2

X/2

(6)

In figure 1 the example solutions of equation (5) with parameters A = 3 and 77 = 5/2 are presented.

EQUATION WITH TWO POWER-LAW EXPONENTS

In order to get spectrum with two power-law exponents we propose SDE

A 2

dx= [n-'^-ixe'^f ( ( l + x 2 ) l / 2 e + l)2^^"^ ' ( l + x 2 ) l / 2 e ^ l (1+X2)1

xdf- -dW. (7)

We solve Eq. (7) numerically using the method of discretization. Introducing a variable step of integration

; ( ( l+x | )V2e+i )2 hk=K^-

( l+x | ) ' i - i

the differential equation (7) transforms to the difference equations

Xk+\

tk+\ = h+K'

Xk+K'-i 1]-- (xie'^)2')x,+ K(l+x|)V2e,,

,2((l+x|)V2e + l)2

( l+x | ) '7- i •

We demonstrate an example solutions of equation (7) in figure 2 with parameters 77 = 5/2, X = 4.0, and e = 0.01. The numerical PDF fits very weU the empirical histogram of ABT stocks traded on NYSE. Model recovers fractured behavior of absolute return power spectrum.

565

r

FIGURE 2. Model calculated from Eq. (7) with the parameters 77 = 5/2, X = 4.0, and e = 0.01 PDF, P{x), (continuum line) in comparison with empirical histogram of one minute returns of ABT stocks traded on NYSE (dots) and power spectrum, 5'(x), of returns.

CONCLUSION

We propose the nonlinear SDE reproducing the fascinating statistical properties of the financial variables with ^-Gaussian PDF and fractured behavior of the power spectrum. The proposed stochastic model with empirically defined parameters reproduces the distribution of return and the correlations evaluated through the power spectral density of absolute return. Stochastic modeling of the financial variables by nonlinear SDE is consistent with the nonextensive statistical mechanics and provides new opportunities to capture empirical statistics in detail.

ACKNOWLEDGEMENTS

We acknowledge the support of the Agency for International Science and Technology Development Programs in Lithuania and EU COST Action MP0801.

REFERENCES

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

V. Plerou, P. Gopikrishnan, L. A. N. Amaral, M. Meyer, H. E. Stanley, Phys. Rev. E 60, 6519 (1999). J. P. Bouchaud and M. Potters, Theory of Financial Risks and Derivative Pricing (Cambridge University Press, Cambridge) 2003. M. Gell-Mann and C. Tsallis, Eds., Nonextensive Entropy - Interdisciplinary Applications (Oxford University Press, NY) 2004. C. Anteneodo and R. Riera, Phys. Rev E 72, 026106 (2005). R. F. Engle and A. J. Patton, Quant. Finance 1, 237 (2001). V. Plerou, P. Gopikrishnan, X. Gabaix, L. A. N. Amaral, H. E. Stanley, Quant. Finance 1, 262 (2001). X. Gabaix, P Gopikrishnan, V. Plerou, and H. E. Stanley, Nature 423, 267 (2003). B. Kaulakys, V. Gontis, and M. Alaburda, Phys. Rev £ 71, 051105 (2005). B. Kaulakys, J. Ruseckas, V. Gontis, and M. Alaburda, PhysicaA 365, 217 (2006). B. Kaulakys and M. Alaburda, J. Stat. Mech. P02051 (2009). V. Gontis and B. Kaulakys, PhysicaA 343, 505 (2004). V. Gontis and B. Kaulakys, J. Stat Mech. P10016 (2006). V. Gontis and B. Kaulakys, PhysicaA, 382, 114 (2007). V. Gontis, B. Kaulakys, and J. Ruseckas, PhysicaA, 387, 3891 (2008).

566

A Semiconductor Device Noise Model: A Deterministic Approach to Semiconductor

Device Current Noise for Semiclassical Transport

B. A Noaman and C. E. Korman

Department of Electrical & Computer Engineering, The George Washington University, 725 23' ST Washington, DC 20052, bassamn(5)gwu.edu & korman(Sjgwu.edu

Abstract. In this paper, we present a deterministic approach to calculate terminal current noise characteristics in semiconductor devices in the framework of semiclassical transport based on the spherical harmonics of the Boltzmann Transport Equation. The model relies on the solution of the Boltzmann equation in the frequency domain with special initial and boundary conditions. The terminal current fluctuation is directly related to scattering without the additional Langevin noise term added to the calculation. Simulation results are presented for the terminal current spectral density for a 1-D n*nn* structure due to elastic-acoustic and intervally scattering.

Keywords: Boltzmann Transport Equation (BTE), Langevin noise source. Spherical Harmonics (SH), Monte Carlo (MC), Plasma resonance. PACS: 61.82.Fk, ll.15.Kc, 05.40.-a, 05.40.Ca

INTRODUCTION

Noise plays an essential role when characterizing semiconductor devices. It is important to investigate electronic noise in order to minimize or at least understand its inevitable effect as technology moves towards smaller devices. Noise spectral density in semiconductor devices can be evaluated using various methods by solving the BTE. For instance, the Monte Carlo technique is a versatile approach and has the advantage that it inherently contains fluctuations''^. However, it is evaluated in the time domain. In our approach we model terminal current noise using a deterministic method directly in the frequency domain.

In previous work, a noise model was presented to compute the current noise spectral density in bulk semiconductors '"*. Furthermore, an appropriate boundary condition for this model was derived in order to describe the full dynamics of the carriers in semiconductor devices^. A similar approach based on the Green's function'' and the Langevin-Boltzmann equation^ was proposed. However, in our work we do not use the Green's function directly in the calculation, but rather use an effective distribution function in the frequency domain that is a solution of the BTE which simplifies the numerical calculations. Furthermore, we do not rely on the relaxation time approximation but rather use the collision integral in its full complexity. Finally, the current fluctuations are accounted for from the solution of the Boltzmann equation



569

http://ll.15.Kc

without the addition of the Langevin source term or any additional assumptions regarding the nature of noise.

M O D E L

For the calculation of current noise in our model, two quantities that are solutions of BTE are required: one is the steady state distribution function, which will be used as initial condition for the second quantity, and the other one is an effective distribution function in the frequency domain. Employing the Ramo-Shockley theorem to connect the instantaneous motion of carriers inside the device to the terminal current, the current autocovariance is computed as follows:

C , ( r ) = ji{x,k) ji{x',k')f{x',k'\p{x,k,z \x',k',o)- f{x,k)\ (1)

where g{x,k,T) is an effective distribution function that is a linear combination of

solutions of the BTE and it reads as follows:

|^(^^^^0+ ^«fc(^^^^^0}= o> (2)

dk'g\x,k',T is \x,k',k]- g\x,k ,T ]\ dk' S\x,k ,k'] where: L^^^^l^{i,k,T)}=

The spectral density Sj(w) is defined as the Fourier transform of the autocovariance function Q ( T ) :

5,(w)=2Re{[^rC(ry^""}=2Rej jdxdki{x,k)G{x,k,w)[ (3)

where G(x,k,w), is the Fourier transform of g(x,k,T) and satisfies the following equation:

jwG{x,k,w)-f^{x,k)i{x,k)+L^,,{G{x,k,w)} = 0 G{Q,k>o,w) = o. (4)

This method allows us to compute and calculate the current spectral density directly in the frequency domain for a given frequency.

RESULTS

In this numerical simulation, the spherical harmonics method up to the first order is used to solve the BTE, which for the 1-D case reduces to the Legendre Polynomial method^ The effect of non-parabolic band structure, as well as acoustic and inelastic intervalley phonon scattering is considered in this simulation^, and the electric field is in the [1 1 1] direction. A 0.6 um long 1-D silicon n^nn^ structure (0.2u-0.2u-0.2u) biased at 0.5 Volt has been simulated.

570

The current spectral density is expressed in terms of the odd terms of the Legendre polynomial expansion of G . In this case, it is given in terms of the first order term by the following expression:

G^ (x, k, I -tiT fdF„[X,k,w) dF„[X,k,wj] ( Ti(x,£)/„(x,s)

yjwT+f Yjy'nifjy dx -qE

ds {jwT + r'r) (5)

where G„(x,k,w) and F„(x,k,w) are the first order and zero order Legendre polynomial expansions, respectively, in the frequency domain, «?„ is the electron mass, r is the total scattering rate, / is the energy and f„(x,s) is the steady state occupation probability. Substituting this expression in to Eq. 3, one derives an explicit expression for the current noise spectral density.

The average results of the simulation are shown below: the carrier density, electric field, average velocity and average energy respectively across the device.

FIGURE 1. Simulation results of n*nn* 0.6 um structure: (a) carrier concentration and doping profile; (b) electric field; (c) average velocity; and (d) average energy.

The terminal current spectral density is shown below, which can be compared with Figure 2 in Ref. 7. It shows the well-known behavior of the current noise spectral density. Neglecting 1/f noise, the current noise spectrum is flat below 100 GHz, and it exhibits a resonant peak at the plasma frequency, which is due to the capacitance of the lowly doped region and the inductive effect of the electron's acceleration.

571

ii i iii i ii i ii? ii i iiiiii;

i* • - - * ! • - ! • - « *-..i 'iiiii \ii i i iiiiii i i ; ii i i ii ; * i i i i '.ii i i i i i i i i

i i i i i : i ii i ilili ^ i i i i i i i i i i i ii I i ii i ilili i i "*-. i i i i i i ;

ii i ii i i ii i iiiii ii i "i'*iii ID ID ID ID 10 ID 10

Fret|iieiicy (Hz)

FIGURE 2. PSD of terminal current for the 1-D n*nn* 0.6 um (0.2u-0.2u-0.2u) structure calculated by Eq. 5 and based on the first order spherical harmonics expansion

CONCLUSION

In this work, we have presented a deterministic noise model and applied it to compute the terminal current spectral density in an n^nn^ structure. The salient feature of this model is that it allows one to directly compute the current spectral density in the frequency domain. Furthermore, it relates the terminal current noise directly to electron scattering without any assumption to the nature of noise or addition of Langevin source noise.

REFERENCES

L. Varani, L. Raggiani, T. Kuhan, T. Gonzalez, and D. Pardo, "Microscopic Simulation of electronic noise in semiconductor materials and devices," IEEE Trans. Electroni Devices, Vol. 41, no. 11, pp. 1916-1925, Nov. 1994. ^ E. Starikov, P.Shiktorov, and V. Gruzinskis, J. P. Nougier, J. C. Vaissiere, L. Varani, L. Raggiani, Appl. Phys. Lett. 66 (18) (1995) ' C. Korman and I. Mayergoyz, Phys. Rev. B 54, 17620 (1996). ' ' A . Piazza, C. Korman, and A. Jaradeh, IEEE Trans. Computer-Aided Design. 18, 1730 (1999). ^ B. A. Noaman, C. E. Korman and A. J. Piazza, "A Semiconductor Device Noise Model: Integration of Poisson Type Stochastic Ohmic Contact Conditions with Semiclassical Transport" SPIE Fluctuations and Noise Conference, May 20-24, 2007, Florence, Italy. '' Christopher J. Stanton and John W. Wilkins, Phys. Rev. B 36, 1686 (1987). 'Christoph Jungemann, IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 54, NO. 5, MAY 2007 ** H. Lin, M. Goldsman and I. D. Mayergoyz, Solid-St. Electron. 35, 769 (1992) ' C. Jacoboni and L. Raggiani, Rev. Mod. Phys. 55, 645 (1983)

572

Investigation of Noise Performance of SiGe HBTs by Deterministic Simulation of Boltzmann

Equation in Two-Dimensional Real Space Sung-Min Hong and Christoph Jungemann

EIT4, Bundeswehr University, 85577 Neubiberg, Germany

Abstract. Noise simulation of a 2D semiconductor device by employing a deterministic BE solver has been reported. The Mly coupled scheme enables small-signal analysis. As an example, the RF and noise performance of a SiGe HBT, including cutoff frequency, minimum noise figure, noise resistance, and electron noise current transit time, is investigated.

Keywords: Noise, Langevin Boltzmann equation, spherical harmonics expansion, SiGe HBT PACS: 72.10.Bg, 72.20.Ht, 85.30.De

INTRODUCTION

At the semiclassical level, transport and noise in semiconductor devices can be described by the Boltzmann equation (BE). The standard approach to solve the BE, the Monte Carlo (MC) method, has many disadvantages due to its stochastic nature. Especially for noise simulation, it takes an excessive amount of CPU time to simulate noise at technically relevant frequencies.

The spherical harmonics expansion (SHE) [1], where the electron distribution function is expanded with spherical harmonics, is an alternative nonstochastic method for solving the BE. In this approach, small-signal analysis in the frequency domain is readily available. However, up to now, its apphcation to noise simulation has been reported only for one-dimensional devices [2].

In this work, we present noise simulations of two-dimensional (2D) devices using a deterministic BE solver. To this purpose, the RF and noise performance of a SiGe HBT is investigated.

SIMULATION MODELS

A BE solver based on SHE of the electron distribution function in the momentum-space [1] has been extended to 2D devices [3]. The balance equations obtained by projection of the BE onto spherical harmonics are written in the total energy space [1]. Since the Jacobian matrix contains the dependency of the balance equations on the electric potential fluctuation and the Poisson equation is solved simultaneously with the BE, rapid convergence can be obtained by the Newton-Raphson method, in contrast to previous Gummel-type attempts [1,4].



573

http://72.10.Bg

Electron transport is based on the nonparabohc six valley band structure and phonon mechanisms developed by the Modena group [5]. Apparent bandgap narrowing due to heavy doping [6] is included.

The Langevin approach is used to derive the Green's functions and power spectral densities of the distribution function directly in the frequency domain. In contrast to the MC method, small-signal analysis is available at little computational burden.

RESULTS

In Fig. 1(a) the structure of the SiGe HBT is shown. The doping and graded Ge profiles are shown in Fig. 1(b). The metallurgical base length is 24 nm. The real space is discretized with a grid of 110 by 14 nodes with a nonuniform spacing in the range from 2 to 110 nm. The uniform energy spacing is 5 meV. If not stated otherwise, spherical harmonics up to the order l„ax = 5 are included in the simulations. For comparison with the momentum-based transport model, DD and HD calculations are performed with the device simulator Galene 111 [7].

The Gummel plot is shown in Fig. 2(a). The gain at VBE = 0.65 V is 530. The peak /T evaluated by the BE simulation is 100 GHz at a collector current of 2 mA/jUm^ (Fig. 2(b)).

The spectral intensities of the base/base (Sigig), base/collector (S/^/^), and collector/collector (S/^/c) current fluctuations at 1 GHz are shown in Fig. 3(a). In the case of Sigi(,, the absolute value is shown. The frequency dependence of the base current noise, calculated using l„ax = 1, is shown in Fig. 3(b). The noise rapidly increases in GHz range. Therefore, it is very difficult to simulate by the MC method. The spatial origin of Sigig, Sigi^,, and S/ / in the middle of the emitter window is shown in Fig. 4(a).

Since both the smaU-signal conductance and the noise power spectrum are available, it is straightforward to extract useful circuit parameters. The minimum noise figure (Fig. 4(b)) and the noise resistance (Fig. 5(a)) are calculated. The electron noise current transit time, T„, which is given by Sfj^ = Iqlc^e^^'^" — 1) [8], is shown in Fig. 5(b).

CONCLUSION

In this work, noise simulation of a 2D semiconductor device by employing a deterministic BE solver has been reported. The fully coupled scheme enables smaU-signal analysis. As an example, the RF and noise performance of a SiGe HBT, including cutoff frequency (Fig. 2(b)), minimum noise figure (Fig. 4(b)), noise resistance (Fig. 5(a)), and electron noise current transit time (Fig. 5(b)), is investigated.

ACKNOWLEDGMENTS

The authors gratefuUy acknowledge financial support by the Deutsche Forschungsge-meinschaft (DFG). S.-M. Hong's work was partially supported by Korea Research Foundation Grant by the Korean Govemment(MOEHRD). (KRF-2007-357-D00159)

574

I OXID I

1 SIGE

0.30 0.40 0.50 0.6

X [tim] 0.00 0.10 0.20 0 3 0 0 4 0 0.50

(a) (b) FIGURE 1. (a) Structure of the device with an emitter width of 0.25 ;Um and (b) net-doping and graded Ge profiles of the device along the x-direction.

,—'-- Gain

^- ' ' ' - - - Base

— Collector

0.700 0.750 0.800 0.850

Base/Emitter voltage [V]

(a)

p 600,0

- 5 0 0 , 0

- 4 0 0 , 0

- 3 0 0 , 0

- 2 0 0 , 0

- 1 0 0 , 0

- 0 , 0 0 0

0

¥

1 1 1 3

120.0 -

100.0 -

80.00 -

60.00 -

40.00 -

20.00 -

0.000 -

1

- - - HD

— DD op^ o BE ^ /

, i 0^o4 te#^ , l „ „ , , <IM«

" 10" 10"' 10°

Collector current [mA/^m ]

(b) FIGURE 2. (a) Gummel plot of the device and (b) cutoff frequency, / T , at VCE = 0.8V. The cutoff frequency is extrapolated at 1 GHz.

E

Ts

< o

10' -^ — - - - B/C

_ - - - B/B

— C/C

"! 10

10 10 10 10

Collector current [mA/^m ]

? •

< 9>

10"^' -

10

i n - -

10^^ -

10 -

10-^^ -

^ ^_.,^^'^\_

^ / \

^ / \

— / =— / (0

-^Qr^

10° 10^ 10' 10®

Frequency [GHz]

(a) (b) FIGURE 3. (a) Spectral intensity of the terminal current fluctuations at VCE = 0.8 V and 1 GHz and (b) frequency dependence of 5/^/^ at VBE = 0.7 V and VCE = 0.8 V.

575

0.20 0.30 0.40 0.50 0.60

X position [i^m]

(a)

10 10 10 10 10

Collector current [mA/|am'^

(b) FIGURE 4. (a) Spatial origin of the terminal current fluctuations at Vg£ = 0.8 V and VCE = 0.8 V and (b) minimum noise figure at VCE = 0.8 V. 1 GHz.

HD

- DD

BE

10 10 10 10 10

Collector current [mA/(xm^

(a)

10 10 10 10 10

Collector current [mA/|am'^

(b) FIGURE 5. (a) Noise resistance and (b) electron current noise transit time, T„, at VCE = 0.8 V. 1 GHz.

REFERENCES

1. A. Gnudi et al., Solid-State Electron., vol. 36, pp. 575-581, 1993. 2. C. Jungemann et al., IEEE Trans. Electron Devices, vol. 54, pp. 1185-1192, 2007. 3. S. M. Hong et al., Proc. SISPAD, pp. 293-296, 2008. 4. W. Liang et al., IEEE Trans. Electron Devices, vol. 44, pp. 257-267, 1997. 5. R. Brunetti et al.. Journal of Applied Physics, vol. 52, pp. 6713-6722, 1981. 6. D. B. M. Klaassen et al., Solid-State Electron., vol. 35, pp. 125-129, 1992. 7. C. Jungemann et al.. Hierarchical Device Simulation - The Monte-Carlo Perspective, 2003. 8. G. Niu et al., IEEE Trans. Electron Devices, vol. 48, pp. 2568-2574, 2001.

576

A High Frequency Compact Noise Model for Double-Gate MOSFET Devices

A. Lazaro", A. Cerdeira , B. Nae", M. Estrada and B. Iniguez"

'^Departament d 'Enginyeria Electrdnica, Electrica i Automatica, Universitat Rovira i Virgili, 43007-Tarragona, Spain , Tel. +34977558668 Fax. +34977559605

e-mail: antonioramon. lazaro@urv. cat Section of Solid-State Electronics, Department of Electrical Engineering,

CINVESTA V, Mexico D.F. e-mail: cerdeira(a),cinvestav.mx

Abstract. Silicon-on-Insulator (SOI) MOSFETs based on multiple gate structures are excellent candidates to become an alternative to conventional bulk technologies. These devices could be used for high-frequency applications due to the significant increase in the transition frequency/j-. We present compact expressions to model the drain and gate current noise spectrum densities and their correlation for DG MOSFETs. These expressions depend on the mobile charge densities that are obtained using analytical expressions obtained from modeling the surface potential and the difference of potentials at the surface and at the center of the Si doped layer without the need to solve any transcendental equations. Using this model, the DG MOSFET noise performances are studied. The current and noise models can be easily introduced in circuit simulators.

Keywords: Noise modeling, compact modeling, Double-Gate MOSFET PACS: 85.30.Tv

I. NOISE MODEL

The DG transistor structure under analysis is shown in Fig. 1, where Na is the uniform acceptor concentration in the silicon layer with thickness equal to tst, tox is the equivalent gate dielectric thickness and L is the channel length. We are considering several noise sources. The first source is the thermal noise due to parasitic access resistances RG, RS, and RD. The other noise sources arise from the intrinsic transistor. The intrinsic contribution is modeled using two correlated current noise sources, ig, and id (admittance representation). They include the effect of diffusion noise and flicker noise in the channel, induced gate noise and shot noise. In this work, we will focus on the modehng of the thermal channel noise, the induced gate noise and the correlation coefficient. Let us consider a non uniform channel as shown in Fig. 2. The channel is split into channel sections or shdes, and the small-signal and noise source for each channel section can be derived from semiconductor equations. The local equivalent circuit (fig.2) for each channel slide is composed by the gate-to-channel capacitance, the transconductance, and the channel resistance (or conductance). Diffusion noise and gate shot noise can be incorporated into the model by adding two current noise sources for each slide.



577

http://85.30.Tv

Na Silicon Layer -Vv\^-

Gate Length L

FIGURE 1. Double-gate stracture used ^ ^ ' ^ ^ 2 Small-signal and noise equivalent circuit for a . . channel slide between x and x+Ax. An equivalent noise

source in(x) has been introduced to model the channel noise

The analysis of the active transmission line (fig. 2) using the admittance method gives the small-signal admittance matrix and its noise correlation matrix [1]. For the compact modehng of the noise, three methods are usually applied: 1) an equivalent circuit approach, 2) the impedance field method, or 3) the Langevin or Klaasen-Prins method. Reference [2] shows that the three methods are equivalent and the same final expression for the current noise densities and correlation coefficient were obtained.

Starting from [2], and using

lgAvi(Q(V')-Q(V))dr = 0

= j i2WMsC,A<il(n[ql(V')-qlYycÂ(qn(n-q„(n)dv'

--2C,A-2WMSC,A iql(V')[ql(r)-qiy"dv'

q„(V)\ j ql(r)[ql(V')-qlY' dV --2C,J-2WMsC,A{^ + Bq„{V))

we obtain the following compact analytical expressions [3]:

Q/W^ l\lg.(v'm(v')-Q(v))dr gl(V) S,dV

-ÂkT,-{2Wnfi,J,)\2C^J,

g{V) •'

\(A + Bq„{V)fgl{V)dV Vci

ofW_

D c

D c

4kT,(2WMsCM i^WMoCÂf i^CJ,) j{A + Bq„(V)fql(V)dV

4kT,i2WMsCÂy{2CM fo

j{A + BqJV)yql(V)dV yo

where the functions A,B,C,D and E are defined in [3]. The cross noise spectral density is calculated by:

(1)

(2)

578

s . joW

fr„ fv,

D c

J \gAV'){Q{V')-Q{V))dV' \o\o

giV) ''

^-^W,-2W^,C,,(t>,i2C„,(t>,)\{A + Bq„{V))gl{V)dV

joW ^—^4kT,i2WMsC,AypC^A)l{A + Bq„(V))q:(V)dV

Finally, the gate and cross spectral densities are given by:

S, =^^kT,-{lW^,CJ,Y {1C,A)" [A'C + lABD + B'E)

s... joW

'^" luK 4kT, {2W/i,C,A)' i?-CJ, ){AC + BD)

(3)

(4)

(5)

II. SIMULATED RESULTS

Figure 3 compares the results obtained using the segmentation method, the compact analytical model SDDG and the analytical model using conventional KP approach (ref [3], eq.48) for a SDDG MOSFET withL=0.5 |im and VDS=2V. A good agreement between the three methods is obtained for low drain currents. When VGS is increased, g differs more from go. Moreover, when gate length Lc decreases, it differs from the effective length Lg.

S

n

5 10"

ComiHct Mide 1 • Segmentation method

— s — Conventional KP

L=OJ Mm

?- 0.S5

I J I' a

L=50 nm —»—L=100mn —•—L=200 mn

'•^iwL. . i / T ^ ' " ^ "

VB.=1V

L=500 nm —•—L=1000 nm

1 1.5 Gate Bias(V)

0.5 1 15 2 nramVoltage(V)

FIGURE 3. Drain current noise density as FIGURE 4. Drain excess noise factor y as function of gate voltage {L=0.5 \im, t„x=2 nm, function of drain voltage {t„x=2 nm, 4=34 nm,

4=34 nm, threshold voltage 0.3V, VDS=2V) [3]. threshold voltage 0.3V, VGS=1V) [3].

Figures 4, 5, and 6 show the drain and gate excess noise factors, and the imaginary part of the correlation coefficient as a function of drain voltage ( 5=34 nm, tox=2 nm, ^Gs=lV), respectively. Due to the velocity saturation effect, Vossat is smaller as channel length scales down at high gate voltages, which results in the slight increase of thermal noise compared to that of long channel as shown in Fig. 4. In figure 5, the y9 factor tends to 4/3 for the long channel case and increases with the reduction of L. Figure 6 shows our simulation results of the imaginary part of the correlation

579

coefficient. They are slightly smaller than the value 0.4 for the long channel case, and reduce to zero at VDS=0- This coefficient decreases with channel length reduction.

4.5

4

1.5 £ r 1"

1.5

1

VB<=1V L=50 mil -»—L=100 mil . -a—L=200 mil -«—L=500 mil . -•—L=1000 mil

Drain Vo e(V)

FIGURE 5. Gate excess noise factor^ as function of drain voltage {tox=2 nm, 4=34 nm,

threshold voltage 0.3V, VGS=1V) [3].

0.4

0.35

0.3

0.25

0.15

0.1

0.05

'\

VB.= IV

0.5

•

*"* ,|,,„

-L=50 mil —«—L=500 mil -L=100mii —»—L=1000mii -L=200 mil

1 1.5 DKiiiiVoltsge(V)

•

2

FIGURE 6. Imaginary part of correlation coefficient/m('CJ as function of drain voltage {tox=2 nm, 4=34 nm, threshold voltage 0.3V, VGS=1V) [3].

i n . CONCLUSIONS

In this paper we present a compact analytical noise model using the new compact analytical model for Short Channel Symmetric Doped Double-Gate (SDDG) MOSFETs, which considers doped sihcon layer in a wide range of doping concentrations and short channel transistors. The noise compact model calculations are compared with the values obtained using the segmentation method, where the active transmission line is analyzed using circuital nodal analysis. The good agreement obtained between both methods validates the analytical expressions presented. Therefore, the proposed noise model is very promising for being used in circuit simulators with DO MOSFETs.

ACKNOWLEDGMENTS

This work was supported by the MICINN under Project TEC2008-06758-C02-02/TEC, by the European Commission under Contract 216171 ("NANOSIL"), and the Distinction of the Catalan Government for the Promotion of University Research, and the CONACYT project 56461.

REFERENCES

1. A.Lazaro, B.Miguez, Sohd-State Electron. 50, 826, (2006) 2. A.S.Roy, C.C.Enz, J-M.Sallese, IEEE Trans. Electron Dev. 53, 348 (2006) 3. A. Lazaro, A. Cerdeira, B. Nae, M. Estrada and B. Iniguez, Journal of Applied Physics, 105, 034510

(2009)

580

Shot noise analysis in quasi one-dimensional Field Effect Transistors

Alessandro Betti, Gianluca Fiori and Giuseppe lannaccone

Dipartimento di Ingegneria dell'Informazione: Elettronica, Informatica, Telecomunicazioni, Universita di Pisa, Via Caruso 16, 56122 Pisa, Italy.

Abstract. We present a novel method for the evaluation of shot noise in quasi one-dimensional field-effect transistors, derived by means of a statistical approach within the second quantization formalism, which manages to include both the effects of Pauli exclusion and Coulomb interactions. The method has been applied to Carbon Nanotubes and Silicon Nanowire Transistors. We show that noise can significantly differ from that obtained by means of the Landauer-Biittiker's formula and that the main noise source is represented by the partition noise.

Keywords: Shot noise, FETs, nanowire transistors, carbon nanotube transistors. PACS: 73.50.Td, 73.63.Nm

INTRODUCTION

In the last few years, an increasing interest has been directed to assess potential performance of quasi one-dimensional Field Effect Transistors (FETs) based on Carbon Nanotubes [1] (CNTs) and Silicon NanoWires [2] (SNWs) versus the ITRS requirements. However, an accurate investigation of electrical noise has been often neglected. In particular, Landauer-Biittiker's noise formula includes the effects of Pauli exclusion, but does not explicitly include electron-electron interaction through Coulomb force. From a numerical point of view, a self-consistent solution of the electrostatics and transport equations is mandatory in order to properly consider such effects. Here, we present a new method to compute the shot noise power spectral density, based on self-consistent Monte Carlo simulations of randomly injected electrons from the reservoirs, including Pauli and Coulomb interaction, which has been applied to ballistic CNT and SNW-FETs.

THEORY AND METHODOLOGY

As well known, Landauer-Biittiker's noise formula completely neglects Coulomb repulsion, since it does not include fluctuations in time of the potential profile and of transmission probabilities, due to randomly injected electrons through long-range Coulomb interaction. In order to consider the effect of Coulomb interaction, we have self-consistently solved the transport and the Poisson equation.

In particular, we have modelled the stochastic injection of carriers through the randomization of the occupation factors of states propagating from the contacts: the noise power spectrum can then be derived within a statistical approach and through the second quantization formalism. If | cr) is a many-particle state, the occupation number Oam{.E)



581

in the reservoir a (a = S,D) in the channel m can be either 0 or 1, and can be expressed as OamiE) = {alim{E)aam{E))o, where the operators a\^{E,t) and asm{E,t) create and annihilate, respectively, incident electrons in the source lead with total energy E in the transverse channel m. Since we are interested in current fluctuations, we need to consider an ensemble of many electrons states {| C7i ),•••, | ON) } and to compute statistical averages (). Starting from the formula of the current operator at the source, at zero magnetic field, and taking advantage of the Milatz Theorem, the zero-frequency noise power spectral density can be expressed as

.S'(O) = SpN + STT + STR + SIN

where SPN, STT, STR and SJN read:

(1)

SpN

STT

STR

SIN

nTi dE 1 S([t]„;„ l-[t]„;„ CT„,

a=S,Dl£a

2e^ nTi

dE Z 1 (ra«;/pP]«;p/^«'^'

¥P

dE X ( k£D,p£S

t'tr kp

rh' , <yDk<ysi pk •pis

lim , var dE[l,[i]s;nn'^Sn-^ [t]D;kk'^Dk \neS k£D I

(2)

where [t] is defined as [t'l't] ^ if a = .S", [t''''t'] ^ if a = Z3, and t, t' and r are the source-to-drain transmission, drain-to-source transmission and reflection amplitude matrices, respectively. Ai? is our energy step of choice. The noise power spectrum (1) is expressed as the sum of four terms. More in detail, SpN corresponds to the partition noise contribution of uncoupled modes, i.e. when t'''t and t''''t' are diagonal matrices and their diagonal elements represent the transmission probabilities in the basis of the eigen-channels. STT and STR are instead associated to the exchange correlations between transmitted and reflected states. The negative sign is a signature of the fermionic nature of the electrons, i.e. of the antisymmetric many-electrons wave-function under particle exchange. Finally, SIN represents the injection noise contribution, related to the thermal random motions of carriers in the reservoirs. Note that eq. (1) differs from Landauer-Biittiker's formula, since the matrices t, t' and r are expressed as a function of Oa, i.e. the occupation factor of injected states from both contacts.

From a numerical point of view, we self-consistently solve the 3D Poisson and Schrodinger equations, within the NEGF formalism, and considering a large ensemble of injected states, through a Monte Carlo approach (SC-MC). In particular, a quantum fully ballistic transport model has been developed by means of a mode space approach, considering four modes [3], and extending our in-house developed open source simulator -MmoTCAT) ViDES [4].

582


The simulated devices structures are depicted in Fig. la. We consider a (13,0) CNT embedded in Si02 and a SNWT with a cross section equal to 4x4 nm^. In Fig. lb, the transfer characteristics for different drain-to-source biases V^s computed applying Landauer's formula and performing SC-MC simulation are plotted as a function of the gate overdrive VGS — Vth, where VGS and Vth are the gate and the threshold voltage, respectively. As can be seen in the inset of Fig. lb, there is a very limited number of electrons inside the channel of a CNT and SNW-FET, which can make device operation extremely sensitive to charge fluctuations.

The contributions to the Fano factor of the four terms in eq. (1) are shown in Figs. 2a-d as a function of the gate overdrive for VDS = 0.5 V for CNT-FETs and SNW-FETs. As can be seen, while in the sub-threshold regime the Poissonian noise is recovered, in the strong inversion regime, noise is greatly reduced with respect to the full shot value, for both SNW-FET and CNT-FET. It is important to remark that Landauer-Biittiker's formula does not allow to quantitatively evaluate noise, since overestimation up to 180% can be obtained for SNW-FETs {VGS- Vth= 0.4 V).

In addition, the dominant noise sources in CNT-FETs is the partition noise SpN, while the injection noise Sm is equal to the 36% of the partition noise (VGS— Vth = —0-1 V). For SNW-FETs, noise due to intrinsic thermal fluctuations of carriers gives a larger noise contribution, up to the 86%o of the partition noise (VGS— Vth = —0-2 V). For all the considered bias points, exchange correlations STT and STR are always negligible for both one-dimensional structures, since electrons flow along separate quantum channels without change their transverse wavevectors.

CONCLUSION

We have developed a novel and general approach to study shot noise in nanoscale FETs. The derived analytical formula of the noise power spectrum manages to extend the validity of the Landauer-Biittiker formula including also Coulomb repulsion among electrons. Such an expression has been then exploited and implemented in a 3D self-consistent Poisson and Schrodinger solver within the NEGF formalism, in order to compute transport in one-dimensional devices, by means of Monte Carlo simulations.

ACKNOWLEDGMENTS

The work was supported in part by the EC 7FP under the Network of Excellence NANOSIL (Contract 216171), and by the European Science Foundation EUROCORES Program Fundamentals of Nanoelectronics, through funds from CNR and the EC 6FP, under project DEWINT (ContractERAS-CT-2003-980409).

583

metal gate

IjTim

1 tox=1nm

4nm

4nm

tox=1 nmt

Inm

a)

Vp3=0.5 V '

_ - Vp3=0 05V S N W , ^ ^ SC-MCVpg=0 .5V

• O SC-MC Vpg=0.05 V ,

' -0 .2 0 0.2 0.4^

gate overdrive (V) ^ -0.2 0 0.2 0.4 0.6 0,

gate voltage (V)

b) FIGURE 1. a) 3-D structures and transversal cross sections of the simulated CNT (top) and SNW-FETs (bottom). b)Transfer characteristics computed for VDS= 0.5 V and 0.05 V, obtained by SC-MC simulations and applying Landauer-Biittiker's formula. Full dots refer to CNT, empty dots to SNW. Inset: average number of electrons in CNT-FETs and SNW-FETs channel, evaluated for VDS= 0.5 V and 0.05 V.

O

^0 .6

O0.5

£0 .4

0.3

0.2

0.1

0

r Il^uil shot nois^

- ^ \ CNT

• — • PN \ Q \ -A--AIN >k ^ • -0 - 0 SC-MC • - - T L B

A "A. A .A

-0.4 -0.2 0 0.2 gate overdrive (V)

a)

-0.4 -0.2 0 0.2 gate overdrive (V)

b)

J, i Full shot noise -

• \

: - §

: A> -^ \ • - • P N A A l N

• 0 - 0 SC-MC • • • L B

X V

bSNW

v\ -\ ^

X \ ° \ \ ^ A A -

0.4 -0.2 0 0.2 0. gate overdrive (V)

-0.4 -0.2 0 0.2 O.A gate overdrive (V)

c) d) FIGURE 2. Contributions to the Fano factor F of the four terms in eq. (1) as a fiinction of the gate overdrive for V^s = 0.5 V. a)-c) Partition (PN, solid circles), injection (IN, open triangles up) and fiill noise (open circles) computed by means of SC-MC simulations and applying Landauer-Biittiker's formula (solid triangles down). b)-d) STT and STR contributions to F.

REFERENCES

1. R. Martel, T. Schmidt, H. R. Shea, T. Hertel, and R Avouris, Appl. Phys. Lett. 73, 2447-2449 (1998). 2. Y. Cui, Z. Zhong, D. Wang, W. U. Wang, and C. M. Lieber, Nano Lett. 3, 149-152 (2003). 3. A. Betti, G. Fiori, and G. lannaccone, "Shot noise in quasi one-dimensional FETs," in lEDM Tech.

Dzge^r, 2008, pp. 185-188. 4. (2008), URL http : //www.nanohub. org/tools/vides.

584

http://www.nanohub

Influence of Dopant Profiles and Traps on the Low Frequency Noise of Four Gate Transistors

A. Luque Rodriguez, J. A. Jimenez Tejada, J. A. Lopez Villanueva, A. Godoy, P. Lara BuUejos, M. Gomez-Campos

Departamento de Electronica y Tecnologia de Computadores. Facultad de Ciencias. Universidad de Granada. Avenida Fuentenueva s/n. 18071 Granada, SPAIN.

(phone: +34-958243386; fax: +34-958243230; e-mail: tejada@ ugr.es)

Abstract. This work presents a study that correlates technological parameters of SOI four-gate field-effect-transistors (G4-FET) with their output characteristics and low frequency noise in order to optimize their performance. This structure can control the position and size of the conduction channel by the application of adequate voltages to its four gates (front and back MOS gates and two lateral JFET gates). Due to this reason, many parameters can affect its behavior. We have studied the dependence of I-V characteristics and low frequency noise with parameters such as the doping profile in the channel, drain and source regions, impurities in the volume of the semiconductor, and traps in the Si-Si02 interfaces.

Keywords: Four-gate transistors, generation recombination noise, 1/f noise. PACS: 72.70.+m, 72.20.Jv, 73.40.Qv, 73.40.Lq

INTRODUCTION

The G4-FET (Fig. la) offers exciting options both in analogue and digital apphcations. It combines the advantages of both MOS and JFET transistors (see [1] and references therein). Two main modes are typical of this device: a) bulk or JFET mode where the conducting channel is controlled by the lateral gates, and it is surrounded by depletion regions (the bulk operation is promising for rad-hard, low power, high mobility and low noise applications [2]); and b) surface or accumulation MOS mode where the G4-FET can basically be seen as an accumulation mode MOSFET but with two junction-gates, one on each side of the channel. Systematic experimental data and simulations have been carried out in the past that show the bias conditions for optimized performance in these modes [1]. However, a study that shows how the doping profile [3] or traps in the structure can affect the static and noise behavior of the device is still missing.

To perform such a study we have used a 2D simulator that solves self-consistently Poisson and drift-diffusion equations in the structure [4]. This simulator incorporates a model that calculates the low-frequency noise power spectral density [5]. This model distinguishes between bulk and surface noise mechanisms (experiments showing this behavior have recently been made [6]). Our model evaluates the fluctuations of the channel cross-section produced by carrier trapping



585

http://ugr.es

Vj,, Lateral gates Vjc, Lateral gates Vj,, Lateral gates (c) Vjc, Lateral gates (d)

*n VQ, f = 0 \r~ I "n v„,^=i.4V i

Vc;,* Bottom gate V ;, Bottom gate V^;,^ Bottom gate V^;,* Bottom gate

FIGURE 1. a) Cross-section of the N-channel G4-FET structure. The drain current flow is in the z-direction. b), c), d) Evolution of the electron concentration in the channel varying the top gate

voltage, VGI, with VG2=0V, VJG=-1.5Y, VD=50mV.

and release by bulk-traps in the depletion regions, and the 1/f surface-noise produced by traps at the Si-SiOa interfaces.


In order to distinguish the effect of the traps and the doping profile, the channel is divided in three main regions: a region close to the top interface, another one close to the bottom interface and the central bulk region. The contributions of these regions to the drain current and the drain current noise have been evaluated in three operating ranges of the transistor: leakage, subthreshold and conduction regions. Typical curves of the drain current and drain current noise as a function of the gate voltage are represented in Figs. 2-3. These figures show a comparison of experimental data [7] with our numerical results.

In order to achieve the best fit in the conduction and subthreshold regions different doping profiles associated to the lateral p-n junctions have been considered. The doping profiles are modeled by uniform concentrations in the p^- and H-regions. However, the metallurgical junction of the lateral J9^-H junctions is assumed that varies along the y-axis. A position dependent metallurgical-junction of the lateral p^-n junctions models the effect of non-uniform doping profiles in the drain, source and channel regions. The importance of this parameter is two-fold: it is considered as a fitting parameter, in order to obtain the best agreement with experimental measurements of the drain current and its noise power spectral density; and it defines the shape of the conducting channel for optimum performance. In fact, a channel that is constricted in the middle (Fig. 1) is necessary in order to obtain a good agreement with the experimental current voltage curves (Fig. 2) in the conducting and subthreshold regimes.

The doping profile has no effect on the leakage regime. In this regime, impurities in the bulk have been included and characterized in order to reproduce the level of leakage current observed in the experiments. The calculated/j-KG curve is represented with squares in Fig. 2 and agrees with the experimental data (solid line). The three components corresponding to three regions in which the channel is divided (two regions close to the top and bottom interfaces and the bulk) are also depicted in this figure. These components interpret different steps appearing in the experimental

586

10-"-

10-=-

10-=-

10-^-

10-=-

10-=-

10-'=-

in-"-

Experimental:

Calculated: - • - Total • Top component w * ^

-A- Bottom component » ; ^ T ^ . W ' ' --T--Bulk component • / ^ ^

, • /

f '• f •

•

. ? . , . , - 2 - 1 0 1 2 Top gate voltage, V^^ (V)

FIGURE 2. Current voltage characteristics (ID-VQI) with FG2=0V, F / G = - 1 . 5 V , FD=50mV. Comparison of experimental data taken from [7] (solid line) with our results (symbols): the

calculated drain current and its top, bulk and bottom components.

10-

10- N

X

W" 10-' 0 W

i 10-'

10-

ra 10-' Q

10-'

1 V3,= ov, • Vj3=-1.5V,

1 V^ = 50mV

: A •

; y

•••'' / J W ^ A' / ' 'F^

1 //

• ^'17

: 1

•

.J>^ iM^y -'' u

Measurements:

Our results: —•—Sample s1 --•--Sample s2 •*• Sample s3

--T~ Sample s4

10-'

-10-'

10-'

10-'

o 10

10'

— Total Noise Current I - Bottom surface 1/f noise ^ - Bulk gr noise

-A - Top surface 1/f no ise/C- j i

Sample s4

10-' (a)

10-= 10-" 10-' 10-°

Drain current, I

10-=

(A)

10-'

(b)

/ L

-2 -1 0

Top-gate voltaje

> Vj3=-1.5V,

V^=50mV

1 2

Vc,(V)

FIGURE 3. (a) Drain current noise of the n-channel G4-FET at f=10Hz. Comparison of experimental data taken from [7] (solid line) with our results (symbols), (b) Calculated total noise

current as a function of the top-gate voltage, V^i, (solid line), and the top, bottom and bulk contributions to the total noise current (symbols).

curve. These steps define voltage ranges where any of these regions are dominant over the rest.

Different sets of values for the oxide trap concentration at the bottom and top interfaces, and for the bulk trap concentration (Table 1) are used to illustrate the effect of these three traps in different parts of the noise curve in Fig. 3a. The response of the noise curve to the variation of these parameters points out the different origin of the noise: bottom surface noise in low current regime, bulk noise in the upper subthreshold regime, and top surface noise in the conduction regime. To justify this association between the current regime and the origin of the noise, the total drain current noise and its top, bottom and bulk components are also calculated for one of these sets of traps (Fig. 3b). Three ranges are clearly distinguished in this figure, corresponding with the dominance of the noise in each of these three regions.

587

The effect of these three regions, and the traps located in them, on the drain current and on the drain-current noise can be explained by the distribution of the electrons inside the channel (Figs. lb-Id). In the low current regime (Fig. lb), a small amount of electrons are located near the bottom surface; in the subthreshold regime the electron concentration increases in the bottom-surface and bulk regions (Fig. Ic); in the on-regime, electrons accumulate near the top surface, however, the value of the concentration of electrons in the bulk and bottom surface regions is close to the dopant concentration (Fig. Id). The different distribution of the electrons when the gate voltage, VGI, changes determines the shape of current-voltage and noise curves depicted in Figs. 2-3.

The drain-current noise depicted in Fig. 3a runs in parallel with the drain current of Fig. 2. In both of them, bulk traps are included in order to interpret the leakage current and the bulk noise. However, the parameters of these traps, used to fit the experimental data, are not the same. Different bulk traps, with different capture cross-sections, are the origin of these phenomena.

As a conclusion, a simulation tool for the characterization and optimization of four gate transistors is presented. The effect of different regions and traps located in the structure is quantified in relation to the drain current and drain current noise.

TABLE 1. Sets of values for three different traps used in our calculations (oxide traps with uniform spatial distribution near the bottom and top Si-Si02 interfaces, W^j and W ,, respectively, and bulk traps, Nf). The bulk traps are located atET=Ec-0A5 eV, with capture cross sections for electrons

and holes o'„=3.68x10" cm and o"p=3.52x10" cm , respectively. Sample JV„,(cm"^eV"') A f, (cm"^eV"') Nr(cm') si 8.0x10'^ 4x10" 5x10" s2 8.0x10" 4x10" 5x10" s3 8.0x10" 4x10'^ 5x10"

_s4 1.6x10'^ 8x10" Ix lO"

ACKNOWLEDGMENTS

This work was carried out within the framework of research project No TEC2007-66812/MIC supported by the Ministerio de Educacion y Ciencia and FEDER.

REFERENCES

1. K.Akarvardar, S. Cristoloveanu and P. Gentil, IEEE Trans, on Electron Dev., ED-53, 2569-2577 (2006).

2. K. Akarvardar, S. Cristoloveanu, B. Dufrene, P. Gentil, R. D. Schrimpf, B. J. Blalock, J. A. Chroboczek and M. Mojarradi, ESSDERC Proceedings, 89-92 (2006).

3. M. H. Juang, C. 1. Ou-Yang and S. L. Jang, Solid-State Electron. 46, 1117-1121 (2002). 4. J. A. Jimenez Tejada, P. Lara Bullejos, J. A. Lopez Villanueva, F. M. Gomez-Campos, S.

Rodriguez-Bolivar and M. J. Dssn,Appl. Phys. Lett. 89, 112107-1-3 (2006). 5. J. A. Jimenez Tejada, A. Luque Rodriguez, A. Godoy, J. A. Lopez Villanueva, F. M. Gomez-

Campos, and S. Rodriguez-Bolivar, IEEE Trans, on Electron Dev., ED-5, 896-903 (2008). 6. K. Akarvardar, B. M. Dufrene, S. Cristoloveanu, P. Gentil, B. J. Blalock and M. M. Mojarradi, IEEE

Trans, on Electron Dev., ED-53, 829-835 (2006). 7. K. Akarvadar, S. Cristoloveanu and P. Gentil, Future Trends in Microelectronics, John Wiley &

Sons, 2007, pp. 277-286.

588

Analytical Frequency-Dependent Formulas of Excess Noise in Homogeneous Semiconductors

Chan Hyeong Park* and Sung-Min Hong^

*Department of Electronics and Communications Eng., Kwangwoon Univ., Seoul 139-701, Korea ^EIT4, Bundeswehr University, 85577 Neubiberg, Germany

Abstract. An analytic formula is presented for the excess noise current spectral density in homogeneous semiconductors, which originates from uniformly-distributed diffusion noise sources inside the device. As the drift transit time becomes shorter than the dielectric relaxation time, the excess noise current power becomes larger than that of thermal noise and this excess noise behaves like fidl shot noise with larger electric field. This formula shows how the carrier transit time, the dielectric relaxation time, and the collision time play each role in determining the terminal noise current. As the excess noise approaches the fall shot noise level, the power spectral density of the excess noise shows a peak around the angidar frequency determined by the inverse of the collision time.

Keywords: thermal noise, excess noise, shot noise, homogeneous semiconductor, impedance field method PACS: 72.70.4-m, 73.50.Td, 85.40.Qx

INTRODUCTION

For the compact modeling of noise current sources in resistors, often only thermal noise current sources are considered [1]. This might suggest that only thermal noise exists in homogeneous semiconductors. To understand how the ac-wise short-circuit terminal noise current behaves as the electric field inside the semiconductor increases, a physics-based approach is required to analyze noise processes in the physical system. Previously, many efforts have been given to understand noise in homogenous semiconductor resistors [2, 3, 4, 5, 6], but their analyses have been done only to zero-frequency case. In this paper, we present the analytic formula for short-circuit terminal noise current spectra for finite frequencies employing the impedance field method [7, 8,9] and show that how the excess noise occurs as the DC current level increases.

DERIVATION OF THE ANALYTICAL FORMULA

To calculate the ac-wise short-circuit noise current in a homogenous n-type semiconductor bar, only diffusion noise sources are considered. With the drift-diffusion scheme with velocity relaxation time T« considered [10, 11], the Fourier-transformed noise current density 5/„(x, (o) of frequency O) at position x due to the diffusion noise source of frequency O) at position x' reads:

5Jn{x,(0) = 1 + j(OZu

dSij/ VD 5n Vt d8n

dx L ND ND dx (1)



589

where 5Y and 5n are the Fourier transforms of the electric potential fluctuations and the electron density fluctuations, respectively. VD is an applied bias voltage at a drain terminal, L the sample length, a the conductivity. No the donor concentration, and Vt = ksT/q the thermal voltage. The solution of the normalized electron density fluctuation, then reads:

c fgh'^Q _ gVo^j L-h{i-':'o) _ g-M^-'''o)^ forxo < X'Q

cfeî-ô-gVoj fg-^2(i--«;o)_g-Ai(i--«;o)j forxo>x^

where XQ = x/L, MQ = n/No- Here the eigenvalues Ai 2 are given by:

Ai,2(co) = ^ ± ( ^ ^ ) ^ / l + ^ ( ^ ^ ) -CO2T„T, + JCOT, (3)

where LD{= \/£sVt/qND) is the extrinsic Debye length, Zd{= es/qjinNo) is the dielectric relaxation time and Zt{= L?-/jinVo) is the carrier transit time with drift velocity v^ = finVo/L. Also, c is given by C{X'Q,(O) = q/[It{(o){?,i - A2)e(î+^2K(g-A2 _ g-A.i)] ^ /;(©) = aAVt/{l + jO)Tu)L, and A is the sample's cross-sectional area. The drain current fluctuation due to the current source out of the ground terminal and flowing into position x' with value —qe^"" is calculated to be:

jj I qx' ^/Deî-e^2 + eî(i"-'o)(e^2-l) + e^2(i"-'o)(l-eî) G„(x, ©)-—+— 7 m — u ^^'

Since the power spectral density (PSD) of the short-circuit noise current is given by S5i{(o) = jQ4NDADn{(o) \dG^{x,(o)/dx\ dx, we can show that for homogeneous n-type sihcon resistors, the PSD of the short-circuit terminal noise current due to diffusion noise sources distributed throughout the sample is given by

where D„ (co) =£)„ (0) /(1 + ©^ T^ ) is used for diffusion constant of velocity fluctuations at frequency co [4, 7], and G is the DC conductance. F^^, the excess noise factor is:

F - ( c o ) - 2(Ai + A2) |Ai|2|e^2_i| ZiJ_/g2Re(Ai)_l\ 2Re(Ai) I J

2Rea,) I J h+lt I

|Ai|2|A2|2|e^2-eî|2

2Re(A2) I i / ^Kei ^^^^ (6)

SIMULATION RESULTS

Figure 1 (a) and (b) show the calculated short-circuit noise current spectra as a function of frequency with a sample length L of 0.8 jUm and 1.0 jUm with cross-sectional area

590

v=^ V L = 0.8fim

ji__ = 1416cmWs

t^ = 0.21 ps

•X X X X X x ^

/ ^ = 1 V

i = 1 fim

. A'^=10"cm"'

H__ = 1416cm*/Vs

T = 0.21 ps

- ^ - • S , .

-+-*«„

^*:.

X > « K X S « ; ^

!

1

u 1^ I; '1 J

(a) (b)

FIGURE 1. (a) Short-circuit noise current spectra at the applied voltage Vp of 1 V and the sample length L of 0.8 fxm and (b) 1 fxm, respectively.

Length, L (urn)

FIGURE 2. Short-circuit noise current spectrum at 10 Hz at the applied voltage V^ of 1 V as a function of the sample length L.

A of 1 jum^. A peak around 1 THz occurs as the excess noise increases toward the full shot noise level. In Fig. 2, as L decreases, the excess noise crosses over the thermal noise around L = 6jUm.

Fiure 3 (a) is the magnitude of the electron density fluctuations 5no{xo;x'Q,Q)) versus position xo when the perturbation point XQ = 0.5 and / = 100 GHz with VD as a parameter. As VD increases toward 1 V, the shape of the magnitude of 5no gets asymmetric toward the drain. Figure 3 (b) shows the magnitude of the derivative of electron current Green's function of the drain terminal at L = IjUm from low to high frequencies, to show that the magnitude of the vector Green's function is large around the source region compared to the drain side at low frequency, meaning that the spatial distribution of diffusion noise source contribution to the drain terminal noise current is dominated by the source side up to tens of GHz frequencies. However, as frequencies increases towards 100 GHz, the vector Green's function shows rather different behavior and the spatial contribution

591

T

^ T

1.0

0.S

0.6

0.4

0.2

on

V^=O.0V

0.2V 0.4V 0.6V 0.8V 1.0V

/ = 100 GHz

Af^=10"cm"'

ji = 1416cm' /V8

T = 0.21 ps

/ ^^-^lU

k -<•

—x^

-

-

-

1 =0 6 _ ^ 1

10

1 0.1

-\

v..

1^=1 V

Af^=10"cm"'

H^=1416cm'/V8

T = 0.21 ps

"~~ -- -

' /=10 Hz /=1MHz /= 1 GHz /=10GHz

- - - /= 50 GHz /= 100 GHz

1

• \ > , : V\

(a) (b)

FIGURE 3. (a) Magnitude of electron density fluctuations 5no(xo;xQ = 0.5, (u) at /=100 GHz with VD as a parameter, (b) Frequency dependence of the magnitude of the derivative of the Electron Green's function (or the electron vector Green's function) for the drain terminal at L = l;Um and VD = IV.

becomes comparable throughout the sample region around 100 GHz.

ACKNOWLEDGMENTS

This work has been supported by Nano IP/SoC Promotion Group of Seoul R&BD Program in 2009, and by Hynix Semiconductor Inc.

REFERENCES

1. P R . Gray, P. J. Hurst, S. H. Lewis and R. G. Meyer, Analysis and Design of Analog Integrated Circuits 4thEd.,Wiley, 2001.

2. R. Landauer, Phys. Rev. B, 47, 16427-16432 (1993). 3. G. Gomila and L. Reggiani, Phys. Rev B, 62, 8068-8071 (2000). 4. F. Bonani and G. Ghione, Noise in Semiconductor Devices: Modeling and Simulation, Springer-Verlag,

2001. 5. S.-M. Hong, H. S. Min, C. H. Park, and Y. J. Park, in Proc. of SPIE Noise in Devices and Circuits II,

vol. 5470, 2004, pp.16-27. 6. M. Lax, W. Cai, and M. Xu, Random Processes in Physics and Finance, Oxford University Press,

2003, pp.211-226. W. Shockley, J. A. Copeland and R. P. James, in Quantum Theory of Atoms, Molecules and the Solid State, edited by P. -O. Lowdin, Academic, 1966, pp. 537-563.

8. G. Ghione and F. FUicori, IEEE Trans. Comput.-Aided Design Integr Circuits Sys., 12, 425^38 (1993).

9. F. Bonani, G. Ghione, M. R. Pinto, and R. K. Smith, IEEE Trans. Electron Devices, 45, 261-269 (1998).

10. M. Lundstrom, Fundamentals of Carrier Transport 2nd Ed., Cambridge University Press, 2000. 11. C. Jungemann and B. Meinerzhagen, "On the High-Frequency Limit of the Impedance Field Method

for Si," in Noise and Fluctuations-2005, edited by T. Gonzalezs et al., AlP Conference Proceedings 780, American Institute of Physics, New York, 2005, pp. 799-802.

7

592

Simulation of Microplasma Noise in PN Junctions

p. Koktavy, P. Sadovsky and M. Raska

Brno University of Technology, Faculty of Electrical Engineering and Communication, Department of Physics, Technicka 8, 616 00 Brno, Czech Republic

koktavy@feec. vuthr.cz, http://www.feec.vuthr. cz

Abstract. Local avalanche breakdowns take place in the neighbourhood of PN junction local defects at sufficiently high reverse voltages. These breakdowns give rise to microplasma noise. Mathematical description of the noise type was suggested on the basis of its analysis. Verification of results was carried out by means of microplasma noise simulation.

Keywords: PN Junction, microplasma noise, avalanche breakdown, simulation, statistical characteristics. PACS: 85.30.Kk, 73.50.Td

INTRODUCTION

In a semiconductor PN junction there are locahzed regions, which cause the PN junction reverse breakdown voltage to be reduced. The local avalanche breakdowns in the neighbourhood of these defects give rise to microplasma noise [1]. The behaviour of this type of noise depends on the measuring circuit parameters. We studied the statistical characteristics of this type of noise in a low-impedance circuit, the device being powered from a constant voltage source. In this case, the microplasma noise has the form of approximately rectangular impulses, the amplitude of which is constant, the impulse inception and duration being random.

THEORETICAL CHARACTERISTICS OF MICROPLASMA NOISE

The microplasma bistable behaviour may be described with a two-state stochastic process of generation-recombination type (G-R process) [2]. Among the most important microplasma noise characteristics, there are the impulse duration distribution density/(rO and the neighbouring impulse separation distribution density y(ro). These quantities versus time plots show an exponential shape frequently,

/ ( r , ) = r e - , / ( r „ ) = ge-^^», (1)

where the quantities g and r are the generation and recombination coefficients.



593

http://www.feec.vuthr

http://85.30.Kk

The noise power spectral density can be expressed in a form

S,{(o)- ^ligr {g+r)[{g + rf +co^]

(2)

where /M is the current microplasma noise amplitude.

EXPERIMENTAL CHARACTERISTICS OF MICROPLASMA NOISE

An example of the microplasma noise experimentally obtained waveform is depicted on Fig. 1. We can observe groups of short pulses separated by longer time delay here. Some changes of G-R spectrum (Lorentzian) appear in the frequency domain, too. A ripple of the noise power spectral density at the the cut-off frequency is seen on Fig. 2. The neighbouring impulse separation distribution density (Fig. 3) showed certain deviations from the exponential curve. This phenomenon can be misinterpretted as superposition of two independent G-R processes.

10 10 10 10

/"/Hz

FIGURE 1. Measured microplasma FIGURE 2. Measured power noise noise waveform, solar cell K2. spectral density, solar cell K2.

10^

10"

10"

10'

10"

' ^^^^^?'L^ ' .'•

0.05 0.10 0.15 0.20

f/ ms

FIGURE 3. Impulse separation probability distribution density, solar cell K2.

0 0.05 0.10 0.15

f/ ms

FIGURE 4. Generation coefficient time dependence, solar cell K2.

0.20

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TIME BEHAVIOUR OF GENERATION COEFFICIENT

The experimental impulse separation probability distribution density /o(ro) being determined, the generation coefficient versus time dependence during the impulse being OFF can be found by means of the formula [3]

g(t)-1

l - j / o ( 7 o ) d r o

-/o(0. (3)

The resulting course is depicted on Fig. 4. We can express the generation coefficient in the form

g(t) = ae-" + : (4)

where a = 7.26-10^ s\ b = 3.55-10^ s"' and j 3.55-10''s"\ The generation coefficient time dependence is in connection with decrease of free

charges concentration in the breakdown region neighbourhood after the discharge termination. The recombination coefficient was constant in our case, r = 2.06-10^ s"\

SIMULATION OF MICROPLASMA NOISE

Simulation of microplasma noise time waveform with the time dependent generation coefficient according to Eq. (4) was realized on the basis of the G-R process postulates, see Fig. 5. Hence the noise power spectral density was found. Fig. 6. The simulated courses are in very good accordance with the measured characteristics.

FIGURE 5. Simulated microplasma noise waveform, time variable g.

w

>

10

10

10

10

10

10

- "''/^'*«"^ \ ^ v -X^

^"' ^

10" 10- 10"

/"/Hz

10= 10°

FIGURE 6. Simulated power noise spectral density, time variable g.

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The time waveform of the simulated microplasma noise with the constant generation coefficient, g = go = S.SSTO"' s"\ is depicted for comparison on Fig. 7. It is clear, that both characteristics are quite different from measured characteristics on Fig. 1 and Fig. 2.

>

FIGURE 7. Simulated microplasma noise waveform, constant g.

FIGURE 8. Simulated power noise spectral density, constant g.

CONCLUSION

The methodology for the time dependence of generation and recombination coefficients evaluation was presented in the manuscript. It was found out that deviations of time waveform and power spectral noise density from theoretical courses can be caused by the time dependence of the generation coefficient in consequence of free charges concentration decrease in the breakdown region neighbourhood after the discharge termination.

A C K N O W L E D G M E N T S

This paper is based on the research supported by the Grant Agency of the Czech Republic, grant No. 102/07/0113 and the project VZ MSM 0021630503.

REFERENCES

1. p. Koktavy, R. Macku, P. Paracka, O. Krcal, Microplasma noise as a tool for PN junctions diagnostics. WSEAS Transactions on Electronics. 200H. 4{9). p. 186- 191. ISSN 1109-9445.

2. A. Bharucha-Reid, Elements of the theory of Markov processes and their applications. Dover Publications, Inc., Mineola, NY, 1960.

3. M. Raska, P. Koktavy, Application of Microplasma Noise Statistical Characteristics to Studying the PN Junction Heating in the Neighbourhood of Local Defects. WSEAS Transactions on Electronics. 2008. 4(1). p. 202 - 207. ISSN 1109-9445.

596

A New Circuit Topology For The Realization Of Low Noise Voltage References

C. Ciofi, G. Scandurra, G. Cannata

Dipartimento di Fisica della Materia e Ingegneria Elettronica, University of Messina Salita Sperone 31, 1-98116 Messina, Italy

Abstract. A new topology for the realization of high stability, very low noise voltage references is presented that relies on a self biasing circint configuration based on a low noise operational amplifier. The series connection of a number of forward biased low noise diodes is used as an internal reference for generating any desired output voltage. The circint includes a temperature control system for obtaining very high stabiUty coupled to very low noise. As an example, a 2.5 V voltage reference with an output power spectral density as low as 80 nV/VHz at 100 mHz and less than 10 nV//VHz for frequencies larger than 1 Hz was obtained with a temperature stability better than 2 n.V/°C (less than 1 ppm/°C).

Keywords: Low noise voltage reference, high stabiUty, low frequency noise measurements. PACS: 07.50.-e 07.50.Hp 84.30-r

INTRODUCTION

Very low noise voltage sources are often required as part of a low frequency noise measurement systems. Especially those voltage sources employed for biasing the devices under test must be characterized by a very low level of noise in order not to contribute to the background noise of the system[l,2]. Solid state voltage references are characterized by a very high level of noise, especially at very low frequencies, and therefore they cannot be used in high sensitivity noise measurement systems. Batteries are often used as very low noise voltage sources, however they need to be recharged and, moreover, the supphed voltage is not constant as it is a function of the charge state. A few possible implementations of fixed and/or programmable very low noise voltage sources have been proposed in the past[3-5]. In [5] we exploited the non linear characteristic of low noise silicon diodes for filtering out the noise generated by a solid state reference. In its simplest form, such a system can be realized as in Fig. la. Each diode is obtained starting from SSM2220 low noise PNP transistors. The number N of diodes sets the output voltage, while the resistance Rs has little effect on it. The value of Rs, however, sets the current flowing through the diodes, thus also setting the value of their small signal equivalent resistance. In the case of the SSM2220, in the current range of a few mA, the voltage drop across each diode is about 0.64 V. Therefore, with N=4, we can obtain an output voltage of about 2.56 V. The basic idea underlying the approach is that, while the voltage drop across the diodes can be a significant fraction of the voltage supplied by the solid state reference, their equivalent resistance can be made to be a small fraction of the resistance Rs, thus



599

causing a large attenuation of the voltage noise that is generated by the solid state reference at all frequencies. This approach proved to be effective, but two main limitations remained: a) it is not possible to finely adjust the output voltage to any desired value; b) the noise produced by the solid state reference, while considerably reduced, is still the limiting factor for obtaining lower noise levels. To solve these problems, we devised the new circuit configuration reported in Fig. lb.

NEW VOLTAGE REFERENCE

At tum on, the circuit in Fig. lb is unstable. The output voltage evolves until a stable operating point or saturation is obtained. It can be shown that the situation in which the diodes are forward biased corresponds to a stable operating point.

AD586

a)

NxV,

FIGURE 1. a) a simple implementation of a low noise voltage reference as proposed in [5]; b) new approach; c) small signal eqinvalent circuit for noise analysis.

The stable operating point can be calculated from the non linear system in Eq. 1, obtained by assuming the virtual short circuit condition for the operational amplifier.

v,-m D

Rr

Vr.=NVr 1 + ^ ^ 1

(1)

In Eq. 1, /o is the saturation current of each diode and Vj^KT/q (K: Boltzmann constant; T: absolute temperature; q: electron charge). Eq. 1 can be simplified when we observe that for a forward current ID in the order of a few mA, the voltage drop VD across each diode in Fig. lb is about 0.64 V. By a proper selection of N and of the ratio R2IR1, any desired output voltage can be obtained. As far as the output noise is concerned, it can be estimated with reference to the small signal equivalent circuit in Fig. Ic, where rd=VjJlD is the differential resistance of each diode. Because of the very small value of rj (in the order of a few ohms), the contribution of the noise introduced by the resistance RD can be neglected. For the same reason and if /?i<100 Q., the

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contribution of the equivalent input current noise sources of the OP27 can be neglected. In the reasonable approximation RD»Nrd, we have:

By using the second equation in Eq.l, Eq. 2 can be written in the following way:

Svo=H^\ {S,„+NSj+^\4KTR, ; - ^ > 1 (3)

From Eq. 3, and since Se„«NSed, it is apparent that it is convenient to select N in such a way as to have NVD as close as possible to Vo- Since Ri must be maintained below 100 Q., the current flowing through the series of R2 and Ri may be in the order of several mA. Therefore, it is mandatory to employ excess noise free resistors, otherwise a significant contribution of flicker noise would add to Svo in Eq. 2. With a proper selection of N, the output noise can be reduced to be in the order of the equivalent input voltage noise of the operational amplifier.

As far as the stability of the output voltage is concerned, we have to face the fact that the temperature coefficient KVD of the forward voltage drop across a diode is about -2 mV/°C. The temperature coefficient Kvo of the output voltage Vo is:

Kvo

( ^ , A f.r \ 1 + -

V

A^^VD ,. . , K,^ (4)

In order to reduce the temperature dependence, we must resort to a control system for the diode temperature. This can be accomplished quite easily and with excellent results by means of the following approach. We resort to a high stability solid state reference, such as the AD586 by Analog Devices, and we obtain by means of partition a voltage equal to the one we desire at the output of the low noise voltage reference. The number of diodes N and the ratio R^Ri are selected in such a way as to have, at the highest expected operating temperature, an output voltage slightly larger than the desired one. A proportional integral controller, such as the one used in [5], is employed in order to drive a heater consisting of one or more resistors glued on top of the cases of the SSM2220 transistors. Once the system is turned on, the proportional integral controller acts in such a way as to rise the temperature of the diodes until the output voltage Vo drops to, and is maintained at, a value very close to the reference voltage regardless of the changes in the ambient temperature.

In order to test the new approach, a 2.5 V low noise voltage reference was designed and tested. To simphfy the design of the temperature control system, we employed one single SSM2220 IC, into which two transistors are present, notwithstanding that better results would have been obtained with A =3 {VulVrn'^.^). We set /?i=100 Q. and /?2= 105 Q., while RD is 300 Q., resulting in a bias current of about 4 mA. The reference voltage for the temperature control was obtained starting from an AD586 5V solid state reference and by a resistive divider employing 0.1% metallic film resistors. With the temperature control off, Vo=2.5 V for a room temperature of about 36 °C. The output noise was measured by resorting to a JFET input ultra low noise amplifier whose background noise is reported in Fig.2. The noise at the output {Vo) and across the series of the two diodes is reported in the figure. It must be noted that the voltage

601

across the diodes is itself a reference voltage (about 1.3 V) with exceptionally low noise. The results of noise measurements do confirm that the noise introduced by the diodes is much smaller than the noise introduced by the OP27. The output spectrum is consistent with what expected as due to the equivalent input voltage of the operational amplifier only, save in the very low frequency region. For /<400 mHz, additional noise is introduced as the result of the influence of the temperature control system, as it is demonstrated by the power spectrum measured when the control is off. The power spectrum at the output of the "low noise" AD586 voltage reference is also reported for comparison. The very good noise performances are coupled to very good performances in terms of stabihty. The output voltage was monitored while the temperature of the metallic box into which the reference was hosted was made to slowly change by a few degrees centigrade. A temperature stabihty better than 2 }iV/°C was estimated, corresponding to less that 1 ppm/°C.

T 1 1—I—I—r-r-| 1 1 1 1—I—I—TT

AD586

0,1 Frequency (Hz)

FIGURE 2. Results of noise measurements on the 2.5V low noise reference. BNii the equivalent input noise of the JFET ultra low noise ampUfier. DN is the noise across the diodes {BN is not subtracted).

REFERENCES

1. C. Ciofi, G. Giusi, G. Scandurra, B. Neri, Flue. Noise Lett. 4, L385-L402, (2004). 2. C. Ciofi, B. Neri, Jour. Phys.D 33, R199-R216 (2000). 3. L. Baracchino, G. Basso, C. Ciofi, B. Neri, IEEE Trans. Instr Meas. 46, 1256-1261, (1997). 4. C. Pace, C. Ciofi, F. Crupi, IEEE Trans. Instr Meas. 52, 1251-1254, (2003). 5. C. Ciofi, G. Cannata, G. Scandurra, R. Merlino, Fluct. Noise Lett. 7, L231-L238, (2007).

602

A Very Simple Low Noise Voltage Preamplifier For High Sensitivity Noise Measurements

G. Cannata, G. Scandurra, C. Ciofi.

Dipartimento di Fisica della Materia e Ingegneria Elettronica, University of Messina Salita Sperone 31, 1-98116 Messina, Italy

Abstract. An ultra low noise voltage preampHiier is presented that, while characterized by excellent noise performances, is based on a very simple topology. Indeed, the choice has been made of maintaining the component count to a minimum and of avoiding, by design, that any trimming or caUbration step be reqinred. This was done in order to make the reaUzation of this piece of instrumentation as simple as possible for any researcher interested in noise measurements. The ampUfier is AC coupled with a low corner frequency below 100 mHz and a useful bandwidth in the order of a few kHz. The voltage gain is 80dB and the eqinvalent input voltage noise is 14, 1.4 and less than 1 nV/VHz at 100 mHz, 1 Hz and fory>10 Hz, respectively.

Keywords: Low noise voltage amplifier. Low frequency noise measurements. PACS: 07.50.-e 84.30.Le 84.30.-r

INTRODUCTION

The sensitivity of a noise measurements system is set by the Background Noise (BN) of the instrumentation employed for the realization of the measurement chain. In the case of low frequency voltage noise measurements, the most important contributions to the BN usually come from the equivalent input current and voltage noise sources of the voltage preamplifier. While at very low frequencies (f<lO Hz) the best performances in terms of Equivalent Input Noise Voltage (ElVN) are obtained by resorting to BJT input stages, the corresponding relatively high Equivalent Input Current Noise (EICN) limits their apphcations to those cases in which a DC coupling is possible to a noise source whose equivalent impedance is not larger than a few hundreds ohms. Therefore, JFET input stages are preferred for the realization of general purpose low noise amphfiers since, because of the very low bias and equivalent noise currents, AC coupling to high impedance sources down to a few tens of mHz is possible without degradation of the BN of the system. Notwithstanding that better noise performances can be obtained with dedicated designs [1,2], most researchers usually employ standard and quite dated pieces of instrumentation such as the EG&G (now Signal Recovery) 5113 for their experiments. Indeed, some degree of skill in electronic prototyping is required for designing a very low noise amplifier. Moreover, because of the wide dispersion in the characteristics of large area JFETs, a pre-selection of the specific devices to be used in order to reduce the need for trimming and calibration is required. This may be discouraging for researchers interested in noise measurements without a specific knowledge of electronic circuitry.



603

In this work we specifically address this issue and we propose a very simple topology for a JFET input voltage preamplifier that, while characterized by excellent low frequency noise performances, does not require any calibration or trimming step and can be built using quite standard components easily available from any general electronic supply company, with the only exception of the input JFET pair (Interfet IF3602) that, at present, can only be obtained from official sale representatives.

NEW LOW NOISE VOLTAGE AMPLIFIER

The complete schematic of the low noise preamplifier is reported in Fig. 1. The circuit within the gray area can be regarded as a JFET input Low Noise Operational Amphfier (LNOA). In order to simplify the discussion, let us assume that Rc=0 and that the Instrumentation Amplifier IAI be characterized by a flat response (Gain=100). As the bandwidth of the INA131 is 70 kHz, this is a sensible approximation up to a few kHz. The open loop gain AVLN of the LNOA can be approximated as follows:

l+STr IRj^Cc', Rn — Rn Rn (1)

where gm is the transconductance of the JFETs and Afl=100 is the gain of/AI.

LNOA

C..

N A A A ^

lA.

FIGURE 1. Complete schematic of the proposed low noise voltage amphfier. Component values and description are reported in Table 1. Vm, and Vgs are 12 V lead acid batteries.

TABLE 1. Component list for the circtnt in Fig. 1. Component Component type Component value h Ml, IA2 Ry R2

Rss RAI,RAI

CAI,CA2

Rc,Cc

Low noise JFET pair Instrumentation Amplifier 0.1%, l/4W,Wirewound

5 X 10 kn , 0.1%, 1/8 W, in parallel, metaUic film 2x3,3 kfJ in parallel, metallic film

Polypropylene

Interfet 1F3602 1NA131 (Gain=100)

lon. ikn 2kn

1,65 kn 3.3 Mn, 1 Mn 22nF, lOnF 68 n, 33 nF

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Typical pinch off voltages for the IF3602 fall in the range between -1.5 and -0.5 V and therefore the bias current for each JFET is in the order of 4 mA. In this condition, the value of gm is about of 70mA/V, so that Avzjvo=14xlO . The frequency of the dominant pole introduced by Cc, that is required for frequency compensation, is /c=l/(27n;c)=l-2kHz. Therefore we have a resulting Gain Bandwidth Product (PGB) of about 17 MHz. As far as the noise is concerned, we can easily calculate the power spectral density SVEQ of the EIVN source in Fig. 2 by taking into account the noise contributions coming from the JFETs, from the load resistances RD and from the EIVN of/AI. It results:

^VEQ • ^ênFET + 1

isAf ^'ênRD + '^ enAD 1 + (2)

where ênf^r and SenAD are the power spectra of the EIVN of the JFETs and of hi, respectively, and Se„RE>=4kTRo is the thermal noise of the load resistances. With the typical values for the quantities in Eq. 2, at low frequencies SVEQ reduces to ISenPET- At higher frequencies, the contribution due to SenAo can not be neglected any longer. At frequencies larger than 1 kHz, we have 5e„i7£7^0.5 UV/A/HZ, 5e„Az>= 12 UV/A/HZ, and therefore the contribution due to SenAo equals that of SenPET at/=9 kHz. It is apparent that it would be convenient to have fc as large as possible in order to reduce the contribution of SenAD- However, increasing fc does reduce the stability margin of the system. Indeed, in order to be able to set/c=1.2 kHz, a zero must be added {Re in series with Ce) in order to cancel out the dominant pole introduced by /AI. The values of the AC coupling network components (CAI, RAI) are such that the pass band extends down to a few mHz and that the contribution of the thermal noise of RAI, assuming much smaller source impedances, is negligible with respect to SVEQ for frequencies larger than 100 mHz.

Ri

R^ p A A / S

V,N^ C

Al

*—o

-VEQ

'R Al

FIGURE 2. Simplified equivalent circuit of the fist stage of the amplifier in Fig. 1

The gain of the first stage in Fig 1 is 100, with a bandwidth in excess of 100 kHz. The overall bandwidth is however limited at 70 kHz because of the pole introduced by IA2- Because of the gain of the first stage, the noise introduced by hi and by the second AC coupling stage {CAI, RAI, fi=23 mHz) is negligible. Therefore, in the bandwidth between 100 mHz up to a few kHz, the amplifier behaves as a voltage amplifier with a gain of 80dB and an equivalent input noise essentially coincident with the sum of 2SenFET and the thermal noise contribution ofRi (0.4 UV/A/HZ). The measured EIVN of

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the preamplifier and the result of noise measurements on a 1 kQ resistor are reported in Fig. 3. It can be calculated that the EIVN at 0.1, 1 and 10 Hz is 14, 1.4 and 1 UV/A/HZ. TO better appreciate these results, it is worth noting that in the case of the popular low noise amplifier EG&G 5113 the EIVN is 7500, 750 and 105 UV/A/HZ at the same frequencies.

^

10'

10'

10'

10

If > \

n k" P % -

'H, -BN (inDut shorted)"'

It

- 1 kQ rpsistnr at innut_

'*%\llL^ )UlLgMt im 'WW B i l l

0,1 10 frequency (Hz)

FIGURE 3. Equivalent input voltage noise of the new low noise amplifier. The eqinvalent input noise with a 1 kfJ resistance at the input is also reported in the figure.

As it is apparent, excellent performances are obtained notwithstanding the very simple topology that has been used. A few aspects of the design are worth of some discussion. The input offset voltage of the LNOA, which is due to JFET mismatch, can be quite large. Offset voltages as large as 50 mV can be present. The DC path from the output of Mi to the gate of the rightmost JFET in Fig. 1 allows for automatic compensation of the input offset voltage of the JFET pair. Offset voltages in excess of 100 mV can be tolerated as the DC voltage at the output of IAI may range from -10 to lOV. The AC coupling stage at the output of IAI is required in order to remove this large DC component before amplifying the AC component. The relatively large DC current flowing through the series of Ri and R2 requires that these resistances be excess noise free. Indeed, high quality wire wound resistors are used for Ri and /?2-The very same problem may arise because of the current flowing through RDI and RDI-In this case, however, the noise is reduced by the gain of the JFET differential stage, and therefore the parallel connection of five 10 kQ. metallic film resistors was sufficient to make this noise contribution negligible. The flicker noise generated in the Rss resistor is rejected by the differential configuration of the circuit.

REFERENCES

1. C. Ciofi, G. Giusi, G. Scandurra, B. Neri, Flue. Noise Lett. 4, L385-L402, (2004). 2. B. Neri, B. Pellegrini, R. Saletti, IEEE Trans. lustrum. Meas. 40 2-6 (1991).

606

About Quartz Crystal Resonator Noise: Recent Study

F. Sthaf, S. Galliou", J. Imbaud", X. Vacheref, P Salzenstein", E. Rubiola' and G. Cibiel''

"Time and Frequency Dept, FEMTO-STInstitute, UMR CNRS 6174, ENSMM, UFC, UTBM ENSMM, 26 Chemin de I 'Epitaphe, 25030 Besangon cedex, France

Microwave and Time-Frequency Department, CNES, Toulouse, France

Abstract The first step, before investigating physical origins of noise in resonators, is to investigate correlations between external measurement parameters and the resonator noise. Tests and measurements are mainly performed on an advanced phase noise measurement system, recently set up. The resonator noise is examined as a function of the sensitivity to the drive level, the temperature operating point and the tuning capacitor.

Keywords: Flicker Noise, Quartz Resonator. PACS: 77.65.Fs

INTRODUCTION

The Centre National d'Etudes Spatiales (CNES), Toulouse, France and FEMTO-ST Institute, Besancon, France, have initiated a program of investigations on the origins of noise in bulk acoustic wave resonators. Several European manufacturers of high quality resonators and oscillators are involved in this operation [1]. Tests and measurements are mainly performed on an advanced phase noise measurement system, recently set up for this program [2-3]. The instrument sensitivity, that is, the background phase noise converted into Allan deviation, is of lO"'"*. Understanding quartz crystal resonator noise is a complex problem and requires a long-term study. This will be performed in two main steps.

The first step is to investigate correlations between actual geometrical data, external parameters of influence and the measured resonator parameters. The second step will consist in building a noise model for bulk acoustic wave resonators from a more microscopic basis. The inherent noise of a quartz crystal resonator is now clearly identified as 1/f noise and it is the main limitation of ultra-stable quartz crystal oscillators [4, 5]. For this study, according to a macroscopic point of view, three research axes have been investigated: resonators sensitivity to the drive level, effect of temperature (influence of the operating point versus the turn over temperature) and the frequency shift (pull-up). First results of the analyses are presented in this paper. Several sets of measurements have been compared.



607

http://77.65.Fs

NOISE CHARACTERIZATION METHOD

Carrier suppression technique is used in the bench principle. The general idea of this passive method (Fig. 1 a) consists in reducing the noise of the source as much as possible [4-5]. Indeed, when resonators exhibit a very weak noise, the noise of the source is always higher than that of the quartz crystal resonator. Thus, the direct feeding of the driving source signal through only one resonator does not permit to extract the resonator noise from the output resulting noise. On the other hand, the source signal can be subtracted when passing through two identical arms equipped with identical resonators (the devices under tests: DUT). Then the contribution of the source is cancelled while inner noise of both resonators is preserved because one resonator noise is not correlated to the other one. When the carrier suppression is achieved (less than -75 dBc is acceptable), the resulting signal only made up noise from both resonators, is strongly amplified and mixed with the source signal to be shifted down to the low frequency domain and processed by the spectrum analyzer. In such a way, noise to be measured from both resonators can be brought up at a higher level than the driving source noise. Moreover, the noise floor of the bench can be measured with resistors substituted for crystal resonators. Fig. lb shows the measurement system. Boxes concept gives a lot of facilities to build the system.

FIGURE 1. (a) Principle of the measurement bench, (b) Resonator phase noise measurement bench.

Fig. 2 gives a typical result of the measured single side-band power spectral density of the phase fluctuations, 5S(J) = S 10)12. The floor of the short-term stability of the resonator is classically given by the Allan standard deviation. Considering the 1/f resonator frequency fluctuation of the resonator, the measured value at 1 Hz of the power spectral density of phase fluctuations, .S'/i Hz), is used to give the flicker floor due to the resonator.

.isa

A -*rfpt

1

r • ' i %

II ^ i • • i li i

VJ " Id,

\ r

^ 1 J ^ W w ^

i V i

|[

1 11 11 ][

\\%A i "

i li

Wwi** 1

i\\\

'y floor fr ^^ -'21n25^(l//z)

with FL the Leeson frequency andfies. the resonance frequency of the resonator.

10

FIGURE 2. Typical ^ff) of 5 MHz SC-cut resonator. In this case ay_f,„„j =1-10"

608


Drive Level effect

Measurements have been carried out on 10 MHz, SC-cut, quartz crystal resonator pairs from manufacturer A and 5 MHz SC-cut resonators from manufacturer B. As the resonator dissipated power in oscillator is about 100 iW, the measurement power is varied to 20, 100 and 200 iW. The tuning series capacitor Q remains constant when power is changed. The frequency shift induced by the amplitude-frequency effect is compensated by the output frequency of the synthesizer. Fig. 3 a shows the Allan standard deviation versus the crystal dissipated power Pxtai- The trend of the curves is similar in regard to the 10 or 5 MHz resonators. Fig. 3b shows ay floor according to the crystal dissipated power density. The crystal power density is computed considering the acoustic volume of the vibration [6]. Influence of this parameter on the noise level is not demonstrated.

i —-jjc =1

-f-ManLJfaBfainrA:l{)|i#tz [

T i -^ i

I r T

i 1 I

• MguitAbuBTBiSMHi

'ManufKtunfC: lOMHi

I

P„„(liW) P.i„di!™sity|mW/<:in')

FIGURE 3. (a) cry_fiooj vs. crystal dissipated power, (b) cry_fiooj vs. crystal power density. For the 10 MHz resonators, Q is equal to 60 pF and for the 5 MHz, Q is equal to 33 pF.

Effect of Temperature

The influence of the temperature is particularly studied according to the operating point (Top) versus the quartz crystal turn over temperature (Tto). Fig. 4a shows the frequency temperature behaviour of the quartz crystal resonator. Thermally controlled ovens have been used in order to control the quartz crystal temperature by step lower than 0.05°C [7]. Noise measurements have been done on classical 10 MHz, SC-cut quartz crystal resonator. Fig. 4b shows that the modification of the temperature operating point has not a real influence on the resonator noise.

T.. 1

: 1 • •

•

1 1

•

1

. ' •

|,.

' ' ] ,

J ! I 0 1 ! J J S 1 0 1 ! 3

ST-T„-T.p(''C) S T - T B - T , , ^ ^

FIGURE 4. (a) Frequency variation vs. temperature of a SC cut resonator, (b) Allan standard deviation < y_fioor of a resonator pair measured versus the temperature shift 8T = Tto - Top.

609

PuU-Up Capacitor

Fig. 5 a illustrates the frequency shift induced by the serial capacitor. In Fig. 5b Allan standard deviation of a 10 MHz, SC cut resonator is given according to Ct when capacitor values have been chosen at 10 pF, 60 pF and 220 pF. All noise values are inside the error bars except for the smallest value of Ct. This simple modelization shows that a small value of the tuning capacitor Ct should not be chosen. The best configuration is given according to the parallel static capacitor of the resonator [3].

(a)

-co

1

1

50 lim Xili ) - / \ 4IIII i j : - ] j

15(1 2(MI 2.511

FIGURE 5. (a) Frequency shift versus Ct of a 10 MHz, SC-cut resonator, (b) Allan standard deviation, ' y_fioor, according to Q.

CONCLUSION

Several sets of measurements have been compared. The resonator noise is observed according to the input power in the bench arm. About the temperature influence, a stable oven allows to reject the relative frequency fluctuations due to the oven stability at a lower level than the quartz crystal noise. Because of the measurement errors, the resonator noise seems not influenced by the resonator operating temperature. The resonator noise is also measured at a constant resonator power according to a wide range of series capacitors. The tuning capacitor modifies the overall impedance, but, a correlation between load impedance, tuning capacitor and the flicker noise of the resonator is not shown. The second step will now consist in building a noise model for bulk acoustic wave resonators from a more microscopic basis. Such a conference is a unique opportunity for us to compare our work with that of the noise community and to deal with noise physicists.

REFERENCES

1. S. Galliou et al, Proc. Joint Meeting IEEE Ann. Freq. Cont. Symp. and European Freq. and Time Forum, Genova, Switzerland, 2007, pp. 1176-1181.

2. F. Sthal et al, Proc. Joint Meeting IEEE Ann. Freq. Cont. Symp. and European Freq. and Time Forum, Genova, Switzerland, 2007, pp. 254-260.

3. F. Sthal et al, Proc. 20th European Freq. and Time Forum, Toulouse, France, 2008, FPE-0057.pdf 4. F. Sthal et al, IEEE Trans. Ultrason. Ferroelec. Freq. Contr., vol. 47, 369-373, (2000). 5. E. Rubiola et al, IEEE Trans. Ultrason. Ferroelec. Freq. Contr., vol. 47, 361-368, (2000). 6. D.S. Stevens et al, J. A. S. A., vol 79, 6, 1811-1826, (1986). 7. F. Sthal et al. Electronics Letters, vol. 43, no. 16, 900-901, (2007).

610

High Precision Noise Measurements at Microwave Frequencies

Eugene Ivanov and Michael lobar

School of Physics, University of Western Australia, 35 Stirling Hwy., Crawley, 6009, WA

Abstract . We describe microwave noise measurement system capable of detecting the phase

fluctuations of rms amplitude of 2-10 radl^Hz • Such resolution allows the study of intrinsic fluctuations in various microwave components and materials, as well as precise tests of iundamental physics. Employing this system we discovered a previously unknown phenomenon of down-conversion of pump oscillator phase noise into the low-frequency voltage fluctuations.

Keywords: microwave interferometry

PACS: 07.57.-c

The concept of interferometric measurements at microwave frequencies was first suggested in the late 50's, but it took almost 40 years before its high potential was fully uncovered [1, 2]. This was largely assisted by the arrival of high electron mobility transistors capable of amplifying the relatively strong microwave signals without adding any noticeable noise of their own. A combination of microwave circuit interferometry and low-noise signal amplification enabled the "real time" noise measurements with sensitivity approaching the standard thermal noise limit (STNL). The latter, when expressed in terms of rms phase fluctuations, is given by 4^BTol{Pinc«)' where, fe is the Boltzmann's constant, !„ is the ambient temperature, a is the insertion loss in the test sample and i^„cis the power incident on the test sample. Current work describes microwave noise measurement system with spectral resolution exceeding the STNL by at least 3 times due to the efficient use of signal power associated with power recycling.

Diaphragm

T

From pump source

ToFFT S /Analyzer

Phase-shifter

Attenuator

Pre-amplifier Carrier suppression

control

Legend DBM - double-balanced mixer LNA - low-noise amplifier T - sample/device under test AD - amplitude detector

FIGURE 1. Interferometric noise measurement system with power recycling



611

http://07.57.-c

A schematic diagram of the noise measurement system is shown in Fig. 1. It consists of a standing wave interferometer based on a Magic Tee waveguide coupler and a microwave readout system which includes a low-noise microwave amplifier (LNA), reference phase-shifter and a double-balanced mixer (DBM). A distinctive feature of the measurement system in Fig. 1 is a distributed resonator formed by an inductive diaphragm and a short-circuited section of a waveguide with a variable phase-shifter to adjust its resonant frequency. The use of the waveguide components instead of micro-strip ones improves the efficiency of power recycling (due to the reduced loss of the distributed resonator) and eliminates technical noise sources from inside the interferometer (some micro-strip components, such as directional couplers, exhibit an excess noise when exposed to relatively high level of microwave power). A test sample can be either inserted directly into the waveguide or, if it is a coaxial device, coupled to it via adapter. The smaller the loss in the test sample, the higher the sensitivity of the measurement system. This is due to the extended lifetime of the distributed resonator which enhances the random phase/amplitude modulation of microwave carrier by the non-thermal fluctuations in the test sample. The microwave signal reflected from the distributed resonator interferes destructively with a fraction of the incident signal at the interferometer's 'dark port'. This cancels the carrier of the difference signal while preserving noise modulation sidebands caused by fluctuations inside the interferometer. Those noise sidebands are amplified and demodulated to DC in a non-linear mixing stage. Amplification of the residual noise before its demodulation overcomes the relatively high technical fluctuations of the mixing stage making the effective noise temperature of the entire readout system close to its physical temperature. The phase noise floor of the measurement system at p,„^ =i w is shown in Fig. 2

(curve 1). At Fourier frequencies f > 5 kHz it approaches 2 •70"''''rafif/V^ which corresponds to the highest spectral resolution achieved both in the microwave and the optical domains [3].

-:. 10-'

SB

0) (0

a (0

10'" .

10-" .

10-'

10-

':

\ 1

- ' '<' '-*%iyykii.. .(. . . , . , , •ii?ipj

STNL

, / IM

I

!

:

1

-

10 100 1000 10000

f(Hz)

FIGURE 2. Noise floor of a 9GHz measurement system: curve 1 is the overall noise floor, curve 2 is the noise floor due to fluctuations in the microwave electronics of the readout system

612

The horizontal line in Fig.2 corresponds to the STNL calculated for a conventional interferometer assuming that: i „ =1W, T = 300K and a = 0.5dB. As follows from the

data in Fig. 2, the spectral resolution of the phase measurements is slightly more than a factor of 3 better than the STNL. Curve 2 in Fig. 2 shows the phase noise floor due to the technical fluctuations in the electronics of the microwave readout system (LNA & DBM assembly). It was measured with the input of the LNA terminated. The increasing divergence between two noise spectra at low Fourier frequencies could be attributed to intrinsic fluctuations inside the interferometer which are likely associated with the ambient temperature upsetting its balance. The rough peaks in the noise spectrum are of vibration and acoustic origin. In the above case, a piece of hollow waveguide -lOcm long was used as the test sample; the phase sensitivity was maximized by adjusting the aperture of the diaphragm and its distance from the symmetry plane of the Magic Tee.

At low Fourier frequencies the spectral density of phase fluctuations inside the interferometer was found to be almost independent on the type of components used. Fig. 3 shows the phase noise floors of different measurement systems we tested, including: (i) conventional coaxial interferometer [2]; (ii) coaxial interferometer with power recycling [4] and (iii) current waveguide based interferometer. All these noise spectra converge to the same power law: 7.7-70"*// yrad/^Hz) atf<lHz

10-'

I 10'" .-

10'

10-'

10'

-8

-9

10

11

1

lu 2

3

!

, 10 100

f(Hz)

1000

FIGURE 3. Noise floor of different 9 GHz interferometric measurement systems at low Fourier frequencies: conventional coaxial interferometer (curve 1), coaxial interferometer with power recycling (curve 2), waveguide

interferometer with signal recycling (curve 3).

The very high resolution of the above phase measurements was achieved by paying serious attention to various technical noise sources. First, a careful balancing of the interferometer (carrier suppression of the order of 80 dB) was required to avoid the excess 1/f - voltage noise due to saturation of the microwave LNA. Secondly, the signal of the microwave pump oscillator was band-pass filtered to remove the higher

613

order harmonics from its spectrum. Such harmonics are not attenuated when the interferometer is balanced and saturate the LNA. Thirdly, we have to deal with the enhanced sensitivity of the measurement system to pump oscillator frequency fluctuations due to the dispersion in the distributed resonator. This resulted in a highly non-monotonic spectrum of mixer voltage noise with a deep trough at a few kHz and a tall peak around 15 MHz. The latter was found to have been saturating the preamplifier (used in front of the FFT spectrum analyzer) producing the excess voltage noise with the "white" spectrum. Introduction of a passive low-pass filter in front of the pre-amplifier solved this problem. We've also discovered an additional type of noise inherent to narrow-band interferometers only. It turned out to be related to the down-conversion of high frequency phase fluctuations of the pump oscillator signal. The point is that the narrow bandwidth of the interferometer makes filtering of the pump oscillator phase fluctuations less efficient. As a result, the oscillator phase noise at high Fourier frequencies (above few MHz) is synchronously demodulated to DC when interacting with the replica of itself inside the mixer. We simulated these "noise-noise" interactions by considering the propagation of a phase-modulated signal through the dispersive network. We also managed to induce this effect artificially by modulating frequency of the pump oscillator with a band limited random voltage noise centered around 10 MHz. The problem of the excess noise in the recycled interferometer was solved by sending the pump oscillator signal through a high-Q cavity resonator.

Concluding, we demonstrated the possibility of "real time" noise measurements at microwave frequencies with spectral resolution 3 times below the standard thermal noise limit. This was achieved by combining the principles of microwave circuit interferometry with power recycling and minimizing the influence of technical noise sources on the measurement process. We also discovered that under certain conditions an excess "white" noise appears at the output of the "recycled" interferometer due to down-conversion of high frequency phase fluctuations in the spectrum of the incident signal.

ACKNOWLEDGMENTS

This research is supported by the Australian Research Council.

REFERENCES 1. A. L. Whitwell and N. Williams, Microwave J., pp. 27-32, 1959. 2. E. N. Ivanov, M. E. lobar, and R. A. Woode, IEEE Trans, on UFFC, 45, no. 6, pp. 1526-1536, 1997 3. P. Fritschel, G. Gonzalez, B. Lantz, P. Saha,M. Zucker, Phys. Rev. Lett, 80, no. 15, pp. 3181-3184, 1998. 4. E. N. Ivanov and M. E. lobar, IEEE Trans, on UFFC, 49, no. 8, pp. 1160-1161, 2002

614

Extraction and Analysis of Noise Parameters of On Wafer HEMTs up to 26.5 GHz

Alina Caddemi, Giovanni Crupi, and Alessio Macchiarella

Dipartimento di Fisica della Materia e Ingegneria Elettronica, University of Messina, SalitaSperone 31, 98166-Messina, Italy

Email: (caddemi, giocrupi, alessio. macchiarella}@ingegneria. unime. it;

Abstract. This paper presents a procedure for extracting the four noise parameters of on wafer scaled HEMTs based on AlGaAs/GaAs heterostructure. This procedure rehes on 50-Q noise figure measurements up to 26.5 GHz by determining the equivalent temperatures associated to the intrinsic resistances of the small signal equivalent circuit. The non-quasi-static effect associated to the gate-drain resistance Rg is accounted for by the noise model. A good agreement between measured and simulated noise figure is obtained and scaling of the noise parameters is reported.

Keywords: HEMT, Noise figure, On wafer, Small signal modeling. PACS: 52.70.Gw

INTRODUCTION

A full characterization in terms of noise (N-) and scattering (S-) parameters of microwave devices is a basic need for the computer aided design (CAD) of wideband low-noise amplifiers. The S-parameters can be measured by using a vector network analyzer, whereas the N-parameters can not be directly measured by a single instrument. Several papers in the scientific literature report theoretical and experimental analysis of the FET noise properties [1-5]. In [1], no correlation is introduced between the gate and drain noise sources. In fact, only two frequency-independent noise constants Tgs and Tds, which are the temperatures associated to gate-source resistance Rgs and to the drian-source resistance Rds, have to be determined. This model does not take into account the non-quasi static effect of Rgd (see Fig. 1) [6], and then its equivalent temperature Tgd [5]. In this paper, we present the procedure employed to obtain the four N-parameters of GaAs HEMTs based on a 50-Q measurement system up to 26.5 GHz by determining the equivalent temperatures associated to the intrinsic resistances (i.e., Rgs, Rgd, Rds) of the small signal model.

MEASUREMENT PROCEDURE

The four N-parameters of a Device Under Test (DUT) can be defined as follows:



615

http://52.70.Gw

\Y,-Y„,\ FDUT = - min +Rn (1)

where FDUT is the noise figure, Fmm is the minimum noise figure, Rn is the noise resistance, Yo=Go+jBo is the optimum source admittance, and Ys=Gs+jBs is the source admittance that in our case is equal to 20 mS.

The measurement set-up to obtain these noise parameters is illustrated in Fig. 2. The input and output stages consist of adapters, bias tees, cables, and probe tips. The DUTs are on wafer AlGaAs/GaAs HEMTs with different gate widths: 100, 200, and 300 |im biased at VGS = -0.6 V and VDS = 2.5 V ( b = 9.3, 19.9, 31.4 mA). The procedure employed to extract the four N-parameters consists of the following steps: 1) Measurement of the S-parameters of the HEMTs over the frequency range 0.5 GHz

- 50 GHz by using a E8364A PNA (Precision Network Analyzer). 2) Determination of an accurate linear model (see Fig. 1). The extrinsic circuit

elements, which are assumed to be bias independent, are obtained from S-parameters measurements under "cold" pinch-off condition, (i.e., VDS = 0 V and VGS < Vpo ). After removing the effects of the extrinsic elements, the intrinsic ones are obtained at each bias point from the intrinsic admittance (Y-) parameters.

3) Measurement of the noise figure of the whole system consisting of the DUT, the input and the output stages (Fm). This measurement is performed over the frequency range 0.5 GHz - 26.5 GHz by using an E8975A NFA (Noise Figure Analyzer). The system noise figure is given by the well known Friis formula:

where FIN and GIN are the noise figure and the available gain of the input stage, FDUT and GDUT are the noise figure and the available gain of the DUT, and FQUT is the noise figure of the output stage.

4) Calculation of the available gain of the DUT by using its S-parameters. 5) Determination of the S-parameters of the input and output stages [4]. This step

allows us to calculate GIN, FIN, and FQUT. 6) Calculation of the noise figure of the DUT by the Friis formula. Then a first order

fitting is performed to obtain a smoother FDUT as a function of the square of the frequency (see Fig. 3).

7) Determination of the equivalent temperatures Tgs, Tgd, Tds obtained for reproducing the first order fitting (see Fig. 4). Note that Tgs is selected equal to the room temperature.

8) Simulation of the four N-parameters by using the final noise model (see Fig. 5). In Fig. 5, we can notice that Fmm shghtly increases with larger devices, as reported in [7]. Moreover, by increasing the gate width, we obtained that Rn decreases while the magnitude of Yopt increases.

CONCLUSIONS

A method based on 50-Q noise figure measurements that allows to extract the N-parameters of GaAs HEMTs has been presented. The accuracy of the obtained noise

616

model has been verified by the good agreement between measured and simulated noise figure for scaled devices up to 26.5 GHz.

ACKNOWLEDGMENTS

This work was supported by "IMT-ARSEL" project prot. RB1P06R9X5 with financial support by Itahan MlUR and "CMOGAN" project through the contribution of the Italian Ministero degli Affari Esteri, Direzione Generale per la Promozione e la Cooperazione Culturale.

REFERENCES

1. M. Pospieszalski, IEEE Trans. onMicrowave Theory Tech. 37, 1340-1350 (1989). 2. A. Caddemi, A. Di Paola, M. Sannino, IEEE Trans. onMicrowave Theory Tech. 47, 6-10 (1998). 3. P. J. Tasker, W. Reinert, B. Hughes, J. Braunstein, M. Schlechtweg, "Transistor Noise Parameter

Extraction Using a 50-Q Measurement System," in IEEE MTT-S Int. Microwave Symp. Dig., 1993, Atlanta, GA, pp. 1251-1254.

4. J. Gao, C. L. Law, H. Wang, S. Aditya, G. Boeck, IEEE Trans, on Microwave Theory Tech. 51, 2079-2089 (2003).

5. T. Felgentreff, G. Olbrich, P. Russer, "Noise Parameter Modeling of HEMTs with Resistor Temperature Noise Sources," m IEEE MTT-S Int. Microwave Symp. Dig., 1994, San Diego, CA, pp. 853-856.

6. G. Crupi, D. Xiao, D. M. Schreurs, E. Limiti, A. Caddemi, W. De Raedt, M. Germain, IEEE Trans, on Microwave Theory Tech. 54, 3616-3622 (2006).

7. B.-U. H. Klepser, C. Bergamaschi, M. Schefer, W. Patrick, W. Bachtold, "LowNoise Optimization of InP HEMT's," m Indium Phosphide and Related Materials, 1995, Hokkaido, Japan, pp. 397-400.

gate g

Cpg,

Rd ^i drain V\AA/ -»-nnRP-t—°

Cpd/2 Cpd/2

FIGURE 1. Small signal equivalent circuit.

r H BIASIEE

POWER SDTPLY I3631A

NETWORK ANALYZER

Ea364A

SERIAL NOISE SOURCE N4002A

NOISE FIGUEE ANALYZER

EB574A

FIGURE 2. S-parameter and noise measurement set-up.

617

200 400 600 800 f (GHz^)

200 400 600 f (GHz^)

0 200 400 600 800 ^ (GHz^)

(a) (b) (c) FIGURE 3. Measured FDUT (symbols) and first order fitting (dashed lines) vs. square of frequency of HEMTs at VQS = -0.6 V and VDS = 2.5 V. The gate width is: 100 |xm (a), 200 |xm (b), 300 |xm (c).

200 400 600 f (GHz^)

(a)

200 400 600 800 f (GHz^)

200 400 600 800 f (GHz^)

(b) (c) FIGURE 4. Simulated FDUT (solid lines) and first order fitting (dashed lines) vs. square of frequency of HEMTs at VQS = -0.6 V and VDS = 2.5 V. The gate width is: 100 |xm (a), 200 |xm (b), 300 |xm (c).

(c) (d) FIGURE 5. Simulated F„i„ (a), R„ (b), Y„pt (c), r„pt (d) for HEMTs at VQS = -0.6 V and VDS = 2.5 V. The maximum radius for the polar plot of Y„pt is 60 mS. The gate width is: 100 |xm (squares), 200 |xm (triangles), 300 |xm (asterisk).

618

Possible Correlation between Flicker Noise and Bias Temperature Stress

Paul-Jurgen Wagner*, Thomas Aichinger^, Tibor Grasser*, Michael Nelhiebel^ and Lode K. J. Vandamme**

'Christian Doppler Laboratory for TCAD in Microlectronics, Institute for Microelectronics, Technische Universitat Wien, 1040 Wien, Austria

^Kompetenzzentrum fUr Automobile und Industrieelektronik, 9524 Villach, Austria ** Department of Electrical Engineering, Eindhoven University of Technology,

5600 MB Eindhoven, The Netherlands

Abstract. A link between Bias Temperature Stress (BTS, NBTI) and flicker noise (1//-noise) is explored by comparing flicker noise data to charge pumping data. Large-area devices are shown to initially have very low, bias independent normalized flicker noise. After BTS the normalized noise increases considerably and becomes gate bias dependent. Small-area devices are shown to exhibit bias dependent burst noise (RTS) in addition to flicker noise, regardless of BTS.

Keywords: ReUabiUty, MOSFET, BTS, NBTI, charge pumping, flicker noise PACS: 85.30.Tv, 85.40.Qx

INTRODUCTION

When subjected to strong-inversion bias and high temperatures, the drain current of MOSFETs degrades, a phenomenon known as Bias Temperature stress (BTS). The drain current degradation is often described as an increase of the threshold voltage, but other parameters, foremost the carrier mobility and the sub-threshold slope, degrade as well. The exact physical mechanism responsible for BTS are still controversial, but there is ample evidence that both interface states, possibly created by breaking the bonds of passivating hydrogen [1], and oxide traps play a role [2].

Since flicker noise has been used as a diagnostic tool in various places before [3, 4], we conducted a series of flicker noise measurements on MOSFETs that previously experienced BTS degradation. To assess the amount of degradation, the increase in interface state density was monitored using charge pumping measurements [5].

METHODOLOGY

The devices measured were pMOSFETs with W/L = 50|am/ 10|am and fox = 30nm. We studied three wafers that differed only in the back-end-of-line processing, described in [6]. One process variant resulted in a high initial interface trap density Nno, but showed a comparatively low increase AA it after BTS; this wafer is referred to as wafer A. Wafer B had both medium initial interface traps and medium increase of traps after BTS. The third wafer (C) exhibited a high AA it, resulting in the highest post-stress interface state



621

http://85.30.Tv

10-

10-

10-

10-

<K

X ^ . . " X j ; X X ^ . x X X X X X > s t X X

r, 7.3 X10-15 v2

^Vis = s.gxio-ity^

/ fresh device O

stressed device •

background noise x

X X X X X X X X „ " X

X

10" lOi 10^ 10^ lO'* 10^

/ (Hz)

FIGURE 1. Noise spectrum of a large-area MOSFET (wafer C) biased at Vgs = -1.54V before and after BTS and respective least-squares fits. The crosses show the background noise (I^ = 0).

density, despite the fact that this wafer's initial interface state density was lowest. On every wafer at least two neighbouring devices were measured using constant-baselevel charge pumping at 2 MHz [7]. Next, on every wafer one device was stressed for 10^ s at Vgs = —17.5 V and 175 °C, and charge pumping was done again immediately upon release of stress.

Then, noise measurements were performed in the linear region of the MOSFETs (|Vds| < 0.3 V) at gate voltages Vgs = -1.54V,-3.07V,-4.59V. Figure 1 depicts the spectra for a fresh and a stressed device at weak inversion. Although the bias point is approximately the same, the noise power density is tenfold for the stressed device. Prior to the noise measurements, the /d(Vgs)-characteristic of a fresh device was measured, and the SPICE level-1 model was fitted yielding the parameters Vt = —0.95 V, /3 = 1.23 X 10-4 A/V2, and 6 = 0.128V-1.

In addition to the large-area transistors, small-area transistors with W/L = 2.4|am/2.6|am were examined. Figure 2 shows that with these devices the noise is not conveniendy described by a pure l//-dependence. Because of the smaller number of free carriers the Lorentzians of distinct traps may be visible, and their superposition yields l//^-noise with /appreciably deviating from unity, as predicted by the criterion A?<l/(47ra)in[8].

Using the empirical relation [9]

/il=const d

^'•ds

yds=const ds

a Nf

(1)

an a was calculated for every device at every bias by taking foSvis {fo)/^^^ at /o = lOHz. Care was taken to verify a 1//-dependence of Sy^^ around /o, which was the case for all large-area transistors. Assuming a homogeneous channel, the number of carriers in

622

4.4x10-'^ V^/Hz (//IHz)!-'

o _6 .3x lO-"V^ I LgxlO-'^VyHz _ *'* / + l+(2?t.0.9ms./)2

wafer C,Vgs = - 3 . 0 8 V O

wafer A, Vgs = - 4 . 5 9 V •

background noise x

FIGURE 2. Noise spectra of small-area MOSFETs (unstressed) for different biases. The spectra clearly deviate from the 1 //-form: One can partly be fitted with a much higher slope, thus more resembling a 1 //^-spectrum. The other one can be fitted by a superposition of a 'true' 1 //-component and a Lorentzian. This situation is characteristic for the presence of a single dominant trap. These traps were also visible in the time domain.

the channel A was obtained via A = L /[^qr^^), where r^^ is the (measured) channel resistance at Vds ~ 0, ^ the elementary charge, and the carrier mobility ji was calculated from the SPICE model parameter /3.

RESULTS

The fresh devices showed very low a values around 10^^ that were only weakly dependent on the gate voltage. The values were quite similar for all three wafers. The stressed devices exhibited considerably higher noise power, corresponding to higher a values, that moreover turned out to be bias dependent: For weak inversion, a of the stressed devices was up to ten times the value of the fresh ones, where at strong inversion fresh and stressed devices had comparable a values, cf. Figure 3.

Since the conductivity is proportional to the product of carrier number A and carrier mobility ji, assuming that both A and ji fluctuate independently allows to split a = a^ + UN- In a first order approximation, mobility reduction and parasitic resistances are negligible, hence a^ is independent of gate bias. It seems likely that the unstressed (fresh) devices just show this kind of flicker noise, i.e. Of = % , which is a bulk noise effect. Continuing with this interpretation, for stressed devices aN = a — a^ = as — (X[ = Aa is the flicker noise component due to carrier number fluctuations. According to the McWhorter theory, UN °= V^g*' ^s nicely confirmed by Figure 3.

623

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10-

^

4

C

<i <;

•

1

1 •

wafer R O

• • n

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10-

If-. I>

J

Aa = 7|i ( ^ g / i V

water A •

wafer R *

^ wafer C * "^....

• • •

• • • .

f •

0.6 0.8 1.0 2.0 4.0 0.6 0.8 1.0 2.0

V: = \V,s-V,\ (V) K* = |Vgs-Vt| (V)

4.0

FIGURE 3. Left: Dependence of calculated a values on the effective gate voltage (empty symbols: fresh devices; solid symbols: stressed devices). Right: Increase of a due to BTS, and least squares fit to the data, indicating Aa =< 1/ g*-

CONCLUSIONS

The low-frequency noise behaviour of large-area pMOSFETs subjected to bias temperature stress was investigated. Unstressed devices showed very low and gate bias independent a values around 10^^, pointing to a bulk origin for the flicker noise in these devices. After BTS, the devices showed considerably increased, gate bias dependent a values, indicating that in competition with the bulk mobihty noise, a surface-provoked noise component of the McWhorter type emerges.

REFERENCES

1. K. Jeppson, and C. Svensson, Journal of Applied Physics 48, 2004-2014 (1977). 2. V. Huard, M. Denais, and C. Parthasarathy, Microelectronics Reliability 46, 1-23 (2006). 3. J. Scofield, T. Doerr, and D. Fleetwood, IEEE Transactions on Nuclear Science 36,1946-1953 (1989). 4. X. Li, and L. K. J. Vandamme, Solid-State Electronics 35, 1477-1481 (1992). 5. G. Groeseneken, H. Maes, N. Beltran, and R. de Keersmaecker, IEEE Transactions on Electron

Devices ED-31, 42-53 (1984). 6. M. Nelhiebel, J. Wissenwasser, T. Detzel, A. Timmerer, and E. BertagnoUi, Microelectronics Reliabil

ity 45, 1355 - 1359 (2005). 7. T. Aichinger, and M. Nelhiebel, IEEE Integrated Reliability Workshop 2007 Final Report pp. 63-69

(2007). 8. L. Vandamme, and F. Hooge, IEEE Transactions on Electron Devices 55, 3070-3085 (2008). 9. F N. Hooge, T. G. M. Kleinpenning, and L. K. J. Vandamme, Reports on Progress in Physics 44,

479-532(1981).

624

Low Frequency Noise Evolution of AlGaN/GaN HEMT after 2000 hours

of HTRB and HTO life tests

Charlotte Sury, Amaud Curutchet, Nathalie Malbert and Nathalie Labat

Lahoratoire IMS, Universite Bordeaux 1, 351 cours de la Liberation, 33405 Talence Cedex, France

Abstract The low frequency drain noise of High Electron Mobility Transistor based on AlGaN/GaN heterostructure on SiC substrate is analysed before and after 2000 hours of DC life tests. The life tests have weakly affected the drain current noise of this technology. The experimental results show that the degradation should not be located in the channel.

Keywords: AlGaN/GaN HEMT, Low frequency noise, HTRB and HTO life tests, reliability. PACS: 72.70.+m

INTRODUCTION

This study deals with a low frequency drain noise analysis of High Electron Mobility Transistor (HEMT) based on AlGaN/GaN heterostructure. Low frequency noise gives important data to assess the quality and reliability of a technology [1, 2, 3, 4]. It reflects the sheet carrier density and their transport properties. In particular, the 1/f noise in ohmic regime reflects conductivity fluctuations of carriers. In this paper, the 1/f drain current noise in AlGaN/GaN HEMTs is analysed before the life tests and after 1000 hours and 2000 hours of life tests.

EXPERIMENTAL BACKGROUND

The 0.25nm HEMTs have been fabricated by MOCVD on SiC substrate. The epitaxial structure is composed of a buffer layer, a non doped 1.7 im GaN layer, and a non doped 22 nm Alo.24Gao.76N barrier layer. The channel sheet carrier density is 1.05xl0"cm"^. The devices are passivated with SiN. The ohmic contacts are Ti/Al/Ni/Au stacks set by evaporation. The HEMTs present a double-fmger gate, which length is 0.25|^m and width is ISO im. The drain-source distance is 3.25|^m.

All components are issued from the same wafer. Eight components have been submitted to HTRB life tests (High Temperature Reverse Bias) [5], with a channel temperature of 175°C. Sixteen components have been submitted to HTO life tests (High Temperature Operating) [5], respectively eight ones with the channel temperature Teh fixed at 175°C and eight ones with Teh = 150°C.



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http://Alo.24Gao.76N

STATIC CHARACTERIZATION

The static parameters measured for all components of each set before and after the life tests have shown a weak dispersion. A drain current density in the range of 0.9A/mm at VGS = 0 and a peak transconductance of 305 mS/mm at VDS = 8 V were observed in virgin devices. After 2000 hours of all life tests, a decrease of the drain current at VGS = 0 of about 20% has been observed [6]. The threshold voltage has shifted of 0.1 to 0.2 V and the transconductance peak has decreased of about 5%. A very weak evolution of the static parameters has been observed between 1000 hours and 2000 hours of life tests as shown in figures lA and IB.

VDS

(A)

(V)

FIGURE 1. IDS-VDS characteristics for VGS=0 to -4V at to, after 1000 hours and 2000 hours of HTO life test (A) and HTRB life test (B) performed with T,h = 175°C

L O W FREQUENCY NOISE CHARACTERIZATION

Drain Current Noise Comparison at V G S = 0

The drain current noise spectra measured at to, after 1000 hours and 2000 hours of life test have been compared for a gate bias VGS of 0 and a drain current of 10 and 80mA. Data plotted in figures 2A, B and C are respectively related to the HTO life tests performed at 150°C and 175°C and the HTRB one.

At low values of the drain current such as IDS= 10mA, the drain current noise presents no dispersion before the life tests and a very weak increase after 1000 hours and 2000 hours of life test. At higher values of the drain current such as IDS= 80mA, the LF noise has increased after the HTO test at 150°C and the HTRB life test.

For all components, the noise spectra shape has not changed after the life tests. The drain current noise spectra have been fitted with a home made software to identify the 1/f noise component and the generation-recombination (g-r) noise components. One g-r contribution has been detected at to with a cut-off frequency around 40 Hz and it has not evolved after the life tests. From the evolution of the drain current noise with the temperature, a trap with activation energy of 0.8eV has been identified in the 300K to 360K temperature range at to. As no g-r noise component has appeared after the life tests, no new trap has been created.

626

(A)HTO-150

1E-13

1E-U

1E-16

lE-16

lE-17

1E-13

1E-19

1E-20

1E-21

'*'***fi"

H T O T 1 7 5 X

_.lDs=80rrA

V6S=0

- to - lOOCh - 2CCKIh

* t^^ feMKj ' * ' ' *^ f>v

flHzl

(B)HTO-175

S„(A'/Hz)

13

14

13

17 •

18

19

20

los=10mA J^

HTRB175°C

— . liis=eOmA

V<JE=0

- t o -1000(1 - JOOBi

^Sl*'Hrt ^ ^

1E+O0 1E-KH lE-KH ie+03 1E+D4 1E-K)5

(C) HTRB

1E-11

• ;HTRE175'C • • • » • :HTOTi50'C — i w o n i ;HTOT175*C —20CIOh

(D)

FIGURE 2. Comparison of the drain current noise spectra at VGS=OV for lDs=10mA and 80mA before and after the life tests: (A) HTO 150°C , (B) HTO 175°C, (C) HTRB.

(D) Drain current noise spectral density normalized to the drain current square for devices submitted to the three life tests at to, and after 1000 and 2000 hours of ageing test.

The 1/f noise normalized to the drain current square allows the quantification of the observed degradation (figure 2D). This evolution is very weak at low drain current but increases with the drain current. Indeed, as observed on the evolution of the I-V characteristics, the devices begin to operate in the saturation regime at lower VDS after the life tests than before (figure 1). The increase of the 1/f noise is more pronounced after the HTRB test than after the HTO tests.

Drain Current Noise Comparison for different V Q S

The drain current noise evolution with the gate polarization at a fixed drain current in ohmic regime brings information related to the location of the noise source responsible for the degradation. As expected, the drain current noise increases as the gate bias becomes closer to the threshold voltage of the device. This usual behaviour is consistent with the onset of the saturation regime as well at to than after the life tests. This is stated clearly through the evolution of RDS with VGS plotted in figure 3.

627

lE-08

lE-12 -

lE-13

RDs(Olims)

lE+02

• . HTRB MS'C • : HTOT 150'C A : HTOT 175'C

to 1000h 2000h

lE+01

lE+00

\- lE-01

lE-02

lE-03

-2.5 -1.5 -1 -0.5

VosOO FIGURE 3. Drain-source resistance and drain current noise spectral density normalized to the drain

current square for devices submitted to the three life tests at to, and after 1000 and 2000 hours of ageing test.

As described previously, the evolution of the 1/f noise after the life tests is weak (figure 3). The more the HEMT is biased close to the pinch off, the more predominant the noise source located in the 2-DEG channel. This means that the degradation is not located in the channel.

Moreover, our previous studies on Transmission Line Model patterns have shown that the noise sources located in the 2-DEG channel and at the ohmic contact interface induce equivalent contribution to the LF drain current noise for this technology [6].

CONCLUSION

This study presents the comparison of the drain current noise of AlGaN/GaN HEMTs issued from the same wafer which have sustained 2000 hours of life tests. The life tests have weakly affected the drain current noise of this technology. The experimental results show that the degradation should not be located in the channel.

ACKNOWLEDGMENTS

This work is supported by ANR CARDYNAL research project. The authors would like to thank all the partners at lEMN-Lille, Alcatel Thales 3-5 Lab and INL-Lyon.

REFERENCES

1. N. Labat et al., Microelectronics Reliability 44, 1361-1368 (2004). 2. L.K.J. Vandamme et al., IEEE Trans, on Electron Devices, Vol. 41, 2176-2187 (1994). 3. P. Vahzadeh et al., IEEE Trans, on Device and Materials Reliability, Vol. 5, No. 3, 555-563 (2005). 4. A. Sozza et al.. Microelectronics Reliability 46, 1725-1730 (2006). 5. A. Sozza St aX., Microelectronics Reliability 4S, 1617-1621 (2005). 6. N. Malbert et al.. Internal report, CARDYNAL ANR Research report (2008).

628

Low Frequency Noise in High Speed SiGe:C HBTs after Forward and Mixed-Mode Stress

Malick DIOP''^ Cedric LEYRIS", Nathalie REVIL", Mathieu MARIN", and Gerard Ghibaudo''

"STMicroelectronics, 850 rue J. MonnetBP16, 38926

Crolles cedex, France ''IMEP-LAHC, 3 Parvis Neel,

38016 Grenoble Cedex 1, France,

Abstract Low Frequency Noise performances were investigated before and after stress under high forward current and mixed-mode stress. Improvement of i/f noise after stress has been obtained in high injection after forward stress for the first time. This phenomenon was attributed to a defects passivation mechanism during stress. However, higher level of excess noise is usually observed after mixed-mode stress which establishes the most critical one.

Keywords: BiCMOS, Bipolar transistor, SiGe:C, high speed, reliability, noise, self heating. PACS: Replace this text with PACS numbers; choose from this list: http://www.aip.org/pacs/index.html

INTRODUCTION

One of the figure of merite of modern heterojunction bipolar transitors is their high speed and low noise performance giving them a no negligible advantage on CMOS devices for sensitive applications and analog circuits requiring high frequency performances. SiGe technology has been demonstrated to be exceptionally promising for mm-wave application such as 77 GHz automotive radar, 60GHz WLAN [1]. Nevertheless, in order to achieve the maximum of frequency, HBTs are generally subject to high current density and/or high voltage around or beyond it breakdown voltage (BVcEo) able to induce serious damage on device reliability and performance. Low Frequency Noise performances before and after stress under high forward current and mixed-mode will be discussed in this study, in these particular conditions.

TECHNOLOGY DESCRIPTION

The architecture of this HBT (fig.l) is a fuUy self-aligned (FSA) double-polysilicon structure using Selective Epitaxial Growth (SEG) of the SiGe:C base (B). The process uses a standard collector module (n+ buried layer / collector sinker), a self-aligned selective implanted collector (SIC), a boron doped SEG SiGe:C base, and an arsenic



629

http://www.aip.org/pacs/index.html

in-situ doped monoemitter (E). An inside spacer module is employed to obtain emitter widths WE < 0.13jum. The inside spacer module used is a D-shaped spacer formed by a thin oxide and a thick nitride. The HBT front-end fabrication is finalized with a spike activation annealing, Cobalt silicidation and contact formation. The technology is based on a 0.13-jum CMOS core process with a copper BEOL [2].

NITRIDE SPACER

SiGe:C INTRINSIC BASE

Pl^+Sinker;: NEURIED '•: '• iN Sijifcel iP'-x-x-x-iMPLANT::::;:::::-:':'^^ FIGURE 1. Schematic cross section of FSA

HBT

The tested devices, exhibit peak dc current gain around 2000 and breakdown voltage BVcEo of 1.6V. The cut-off frequency fT, which represents the high-speed performance of the device, is around 250GHz at the Nominal Current (NC) density around 10 mA/pi.m^.

LOW FREQUENCY NOISE MEASUREMENTS

We focus on stress impact on i//"noise performances at room temperature. It is well know that the most relevant source of noise in bipolar transistors is the base current [3]. The noise is typically proportional to h" and could be generated by the fluctuations of carriers mobility (fl^i) and/or carriers number {o^2). In this last case a can varies from 1.8 to 2.2 depending of the SiGe profile and collector doping one [4]. Fluctuations on IB are represented by the noise power spectral density (PSD) SIB:

F J.

Where Kp correspond to model parameter used in SPICE. For all the studied devices, noise

PSD has been characterized in the same conditions before any stress. Fig.2 shows the measured SIB at lOHz as function of IB. A typical IB" dependence is obtained, with a ranging from 2.11 on first device and 2.19 on second device. The extracted a value is in line with a noise generated mechanism due to carriers number fluctuations. In what foUows, our study will be oriented on two stress mode conditions, forward and mixed-mode.

(1)

l.E-08 l.E-07 l.E-06 IB(A)

l.E-05 l.E-04

FIGURE 2. Measured SiB at 10 Hz as a function of IB

Noise performances evaluation will be focused on what happened in middle and high injection which are more representative of typical operation mode of HBTs.

630

Forward stress

Forward bias stress was applied on the studied devices with different stress current (/£•). Noise PSD before and after stress is shown in fig.3. For device 1, stressed at emitter current of lE=21mA and VCE=1-5V Aaxvag 6000s, higher level of excess noise has been systematically observed whatever the injection level.

Moreover, for lB=10|iA (high injection), generation recombination (G-R) centers contributes to an Lorentzian spectrum (1//) noise signature as depicted in fig 4. linked to a trapping/detrapping process inside the emitter - base oxide space area [5] is characterized.

10 100 1000 10000 100000 Frequency (Hz)

FIGUR E 3. SIB spectra for fresh (bold line) and stressed (line) device in forward bias stress

l.E-19

l.E-20

latter stress AE=0.3x3.7nm' Ibstress.=21mA@VCE=1.5V

• Lorentzian 1

Lorentzian 2

l.E-23 ,

l.E-24

100 1000 Frequency (Hz)

10000 100000

FIGURE 4. SIB spectra for fresh and stressed device in forward bias stress: Lorentzian fit

Figure 5 shows PSD before and after stress for device 2, stressed at lE=28mA and VcE=l-5V during 6000s. Unlike what was seen in previous case, we observe a slight shift of the noise PSD at O.ljiA (middle injection), no 1/f noise evolution for h measurement of 10 jiA (high injection) and for the first time an improvement of 1/f excess noise for IB=50JIA (high injection) after stress. This new phenomenon was attributed to a passivation of preexisting defects which do not participate to the transport before stress. Simulation of junction temperature where defects was created, reveals strong self heating procreated at 28mA higher than those at 21mA able to anneal some creating and/or preexisting defects [6].

^ Unusual behaviour . = 50uA AE=0.3x3.7|im^

Istress=28mA;VcE=1.5V

High injection

1000 10000 Frequency (Hz)

100000

FIGURE 5. SIB spectra for fresh (bold line) and stressed (line) device in forward bias stress

Mixed mode stress:

Investigations were also done under mixed-mode stress conditions. It is called mixed - mode stress because it presents not only the conventional E/B damage

631

response but also B/C degradation on DC characteristics. These conditions are generally used in RF and mixed-signal applications to achieve maximum circuit performance. Device 3 have been stressed at IE=100mA and VCB=2VbeyoaA BVCEO in order to create avalanche current.

Fig.6 plots the measured SIB at lOHz as function of IB before and after stress. The parameter a close to 1.92 for fresh device was fond around 0.8 after mixed-mode stress. This change in noise mechanisms was attributed to trap assisted tunneling conduction induced under stress.

For IB=0.5JIA we plot in fig.7 PSD which exhibit a strong increase of excess noise level confirmed by Lorentzian illustrating a serious damage induced by hot energetic carriers due to strong polarization.

lE-19

lE-20

-^lE-21

^ lE-22

^ lE-23

lE-24

lE-25

l.E-08

; y =

; after stress

, before stress

V

= 4E-16)( 0J804

AE=0.69xl4.58|im-f=10HzVcb=0V

* - T , , , 1.9257

y = 2E-llx

lE-21 after stress AE=0.69xl4.58nm'

Jstress=10mA/nm';VCB=2V T=300K

• Lorentzian

lE-05 l.E-07 lE-06

IB(A)

F I G U R E 6. Measured SiB at 10 Hz as a function of IB before and after mixed - mode stress

lE-26

10 100 1000 10000 100000 Frequency (Hz)

FIGURE 7. SIB spectra for fresh (bold line) and

stressed (line) device after mixed - mode stress

CONCLUSION

This study has clearly demonstrated the impact of high forward and mixed - mode reliability stress on i/ifnoise in Fully Self-Aligned bipolar transistors.

For the first time, improvement of i//"noise after stress has been obtained in high injection after forward stress. This phenomenon was attributed to a defects passivation mechanism during stress. Nevertheless, several experiments need to be done for validate this unusual behaviour explanation.

After mixed - mode conditions, combination of strong current and voltage has lead to a highest level of excess noise, more than two decades which never been observed in forward.

REFERENCES

1. G.Avenier et al, In Proceedings of the Bipolar/BiCMOS Circuits and Techn. Meeting 2008. 2. P. ChevaUer et al, IEEE JSSC, pp.2025-2034, October 2005 3. G.Niu et al, IEEE JSSC, vol 36, n°9, 2001 4. J.D. Cressler, In Proceedings of SPIE, vol 5844, 2005. 5. M.Diop et al, IEEE Journal of Microelectronic Reliability, vol 48, p. 1198 - 1201, 2008 6. M.Diop et al accepted to International Reliability Physic Symposium2009

632

Low-frequency Noise Characterizations of GaN-based LEDs With Different Growth Parameters

W.K. Fong, K.K. Leung, and C. Surya Department of Electronic and Information Engineering and

Photonics Research Centre The Hong Kong Polytechnic University

Hong Kong, China

Abstract . We investigated the effects of the material growth parameters on the properties of 1// noise in GaN-based LEDs under high current stress. Two sets of growth parameters were used in which the fluxes for trimethyl indium (TMl) and triethyl gallium (TEG) have been varied. TEM results show that type A devices (TMI/[TMI+TEG] = 64.7%) produces much sharper GaN/InGaN interfaces than type B devices (TMI/[TMI+TEG] = 79.9%). Detailed characterizations of the optoelectronic and low-frequency noise properties of the devices were conducted. It is noted that both type A and B devices exhibited degradations in both optoelectronic and low-frequency noise when subjected to current stress. We performed systematic investigations on the degradations of the properties of the devices as a function of the stress time. Experimental results indicate type B devices exhibit much higher rate of degradations than type A devices. The experimental results show that the interface quality of the devices has strong impact on the optoelectronic and low-frequency noise properties of the devices.

Keywords: GaN, flicker noise, light emitting diodes, degradation, multiple quantum wells PA OS: 72.70.4-m, 72.80.Ey, 73.21.Eg

INTRODUCTION

Ill-nitrides are the materials of choice for the development of high-power LEDs [1]. Device reliability is an important issue for such high-power applications. Lhe hfetimes and reliability of the devices are strongly affected by the crystal quahty of the films. Lo date sapphire is the most commonly used substrates for the growth of III-nitride heterostructures. Due to the large lattice mismatch between GaN and the sapphire substrate high dislocation density is typically found in the GaN films. Lhis may significandy affect the reliabihty and hfetimes of the devices. It is therefore important to optimize the growth conditions to enhance the electrical, optical and reliability performance. In this paper we present experimental data on the properties of low-frequency noise in GaN LEDs fabricated using different growth conditions for the active regions of the devices. Lhe devices were subject to high current stressing and the degradations of the low-frequency noise properties were investigated as a function of the stress time. Lhe degradation mechanism will be investigated based on the experimental results on the optoelectronic and low-frequency noise properties of the devices.



633

EXPERIMENT

Galhum nitride epitaixial layers were grown on (0001) sapphire substrates using a Thomas Swan close-coupled showerhead MOCVD reactor. A 30 nm thick GaN nucleation layer was deposited at 520°C. This was followed by a 100 nm thick of undoped GaN epilayer deposited at 1035°C and a pressure of 200 Torr during. A ditertiarybutyl silane source was used for the growth of n-doped GaN [2,3]. The carrier concentration of the n-layer is 3xlO^*cm" . The active layers consist of five GaN/InGaN quantum wells (QWs). The InGaN MQWs were grown at 700°C and the GaN barriers were grown at 850°C. The doping concentration for the top p-layer is lO^ cm" . Finally, a 150nm thick p-GaN layer of doping concentration -4x10^^ cm" was grown at 1010°C on top of the MQWs. High transmittance Ni/Au (5nm/5nm) and Ti/Al (lOnm/lOnm) layer were used for the p-type and n-type ohmic contacts respectively.

To investigate the mechanism of device and low-frequency noise degradations we have used two different growth processes as indicated in table 1 below. TABLE 1: The growth conditions for type A and type B devices

InGaN well (3nm)

GaN barrier (8nm)

Type A devices TEG= 10.56 nmol/min TMl = 19.39 nmol/min TMl / (TMl + TEG) = 64.7% 27.54 nmol/min

Type B devices TEG = 3.52 nmol/min TMl = 13.96 nmol/min TMl / (TMl + TEG) = 79.9% 27.54 nmol/min

A lower TMI/(TMIH-TEG) ratio was used for the growth of type A devices while the total flux had been increased substantially. The increase in the flux for type A devices results in three-fold increase in the growth rate for the MQWs. The InGaN quantum well growth time was varied such that both types of devices have the same well thickness of 3nm. Type A MQW gives similar EL emission peaks as the type B devices although a reduced TMI/(TMIH-TEG) ratio. This indicates higher In incorporation efficiency can be obtained when growing InGaN well using condition A. Furthermore, it is found that MQWs grown using condition A exhibit much sharper GaN/InGaN interface as shown in the TEM image in Fig. 1 as the out-diffusion of In is substantially reduced due to the lower TMI/(TMI + TEG) ratio.

The electrical, optical and low-frequency noise properties of the devices were characterized as a function of the stress time, fj, which is the time the device was subjected to a high stressing current. The I-V characteristics for devices A and B are demonstrated in Fig. 2 below, in which lines Al and Bl represent the data for t = 0, A2 and B2 are the data for t = 12 hr, and A3 and B 3 are the data for t, = 24 hr. It is observed that both devices show similar forward characteristics. The type A device exhibit higher leakage current in general. Increase in the leakage current is observed in both devices due to the application of the stressing current. The EL spectra of the devices were measured using an integrating sphere and the results are shown in Fig. 3 below. Significant degradations in the EL spectra are also observed for both devices A and B with fj. The degradations of the optoelectronic properties of the devices suggest the generation of locahzed states as a result of the application of the stressing current. The results corroborate with the experimental data on the low-frequency noise properties for the devices as a function of fj as shown in Fig. 4. In general, it is found that type B devices exhibit much higher rates of degradation.

634

FIGURE la: TEM picture of a typical type A F J G U R E lb: TEM picture of a typical type B device device.

0.01

1E-3

O) 1E-4

^ 1 E - 5

h 3 1E-6

g> 1E-7

1E-8

•

f o i ^ ' ^ I ^ . 1 . 1

^^X^m

. / | ! ^ #

<r 1

1

1

n

1

o B1 • A1 A B2

Voltage, V

FIGURE 2: The I-V characteristics for devices A and B at different stress times.

400 410 420 430 440 450 460 470 480 490 500

Wavelength, nm

FIGURE 3: EL spectra for devices A and B at different stress times.

Our results show that high InGaN growth rate enhances the In incorporation. It is because low growth rate results in significant In desorption and out-diffusion during growth. In addition, preferential etching of In from the surface by hydrogen, generated from the cracking of NHs. Therefore, all the MQWs in this study were grown using N2 carrier gas in order to minimize the etching of In. All these effects result in crystal faults and

FIGURE 4: The current noise power spectra for ^^even Strain will build up in the MQW devices A and B after subjected to different stress Structure. Furthermore, significant spiral times. growth occurs at low InGaN growth rate which increases the surface roughness significantly[4]. F. Scholtz et al [5] had also shown that the strain in the MQW has

635

strong effects on In incorporation. The local high In-content InGaN clusters are seen in type B sample as shown in the TEM micrograph clearly indicates excessive strain in the MQWs among type B devices. In contrast, type A MQW shows abrupt interface and no obvious In composition fluctuation as shown in the TEM micrograph. This accounts for the substantially higher degradations in the low-frequency noise properties for type B devices because large quantity of defect states may be generated in highly strain MQWs such as those found in type B devices when the structures are subjected to a large stressing current. Type A devices, on the other hand, show better hot electron hardness as indicated in the slower increase in the low-frequency noise levels for this group of devices.

CONCLUSION

In conclusion, we have conducted systematic investigations on the degradations of the optoelectronic and low-frequency properties of GaN LEDs. The experimental results indicate that significant degradations in the low-frquency noise and the optoelectronic properties of the devices when subjected to a high stressing current. It is observed that the group of devices with higher strain in the MQWs also exhibit higher rates of degradation of device properties. This strongly suggests strains in the MQWs play an important role in the reliabihty of the devices.

ACKNOWLEDGMENTS

This work was supported in part by a grant from the Innovative Technology commission under the Guangdong/Hong Kong Scheme (Project no. GHP/031/07GD). Further support is provided by a University Research Grant of the Hong Kong Polytechnic University.

REFERENCES

1. S. Nakamura, T. Mukai, and M. Senoh, Appl. Phys. Letters 64, 1687-1689 (1994).

2. S. Leu, H. Protzmann, F. Hohnsdorf, W. Stolz, J. Steinkirchner, and E. Hufgard, J. Crystal Growth 195, 91-97 (1998).

3. C.J. Deatcher, C. Liu, M.G. Cheong, L.M. Smith, S. Rushworth, A. Widdowson, and I.M. Watson, Chem. Vap. Deposition 10 187-190 (2004).

4. S. Keller, S.F. Chichibu, M.S. Minsky, E. Hu, U.K. Mishra, and S.P. DenBaars, J. Cryst. Growth 195, 258-262 (1998).

5. F. Scholtz, J. Off, A. Kniest, and O. Ambacher, Mater. Sci. Eng. B 59, 268-273 (1999).

636

Multi-parameters Characterization of Electromigration Noise in Metal Interconnection

Liang He'', Lei Du'', Yi-Qi Zhuang'' and Jun-Lin Bao''

" School of technical physics, Xldlan University, Xl'an 710071, China.

' Institute of microelectronics, Xldlan University, Xl'an 710071, China

Abstract. In order to overcome the shortcomings of power spectrum density (PSD) method In noise

analysis, two non-hnear methods were adpoted to analyse electromigration noise In metal Interconnection.

Correlation Integral result showed the dominant component of noise altered from random component to

determinate component during electromigration process, which meant that the noise was changed from

random signal to chaotic dynamic signal. Multlscale entropy result showed the complexity of nosle

decreased dulrng electromlgralton process, which reflected the disorder degree of electromigration system

decreased continually.

Key words: noise, electromigration, correlalton Integral, multlscale entropy

PACS: 05.45.Tp, 05.40.-a, 05.45.-a

INTRODUCTION

With the continueal miniaturization of integrated circuit, electromigration in metal interconnetction becomes a considerable problem [1]. Electromigration noise contains a lot of information fi-om system interior. The traditional characterization of noise using PSD has some shortcomings. First, PSD is a global average result in arithmetic, and so can not reflect local information in time domain. Secondly, The failure of device is usually a complicated dynamic process, it possibly induces non-stationary or/and non-linear noise. PSD method is more suitable for analyzing stationary and linear noise but the noise of multiple structures. Thirdly, noise signal involves much abundant information such as singularity, complexity, chaos and non-Gaussian etc., which can not be get only by PSD analysis.

In this paper, correlation integral and muitiscale entropy are adpoted to analyze the nondeterminacy and complexicity of electromigration noise respectively. The results show that the two methods can reflect the change of noise characteristic during electromigration process, and through combining with electromigration physical space, they can characterize the evolvement of electromigration interior physics process.

THE RESULT OF CORRELATION INTEGRAL

Correlation integral was presented by P. Grassberger and I. Procadccia, it is a basic tool for non-linear djmamics research[2]. The arithmetic shows the relation between

correlation integral C(r) and specified value ris C(r) x r"'-'"\ and power exponent



637

v(m) which is called correlation dimension is the slope of log C(r) versus logr.

correlation integral is used to distinguish random signal and chaotic signal[3]. Fig.l is the correlation integral of Al interconnection electromigration noise, (a)

shows at the earlier stage, the slope of linear area increases as the embedded vector increases, which indicates the dominant component of noise is stochastic. During this stage, vacancy appears as defect, and its concentration increases linearly with time. It causes electron scattering probability increases as well.The relationship between electron scattering probability and diffusivity can be presented as:

qz _ q M--

m Pm (1)

where r is electron mean fi-ee transition time, P is electron scattering probability, m

is electronic effective mass, q is electric quantity of a charge. Based on Hooge equation,

the relationship between current noise and diffusivity can be written as:

S,{f)=^{J-S,f=^{nq^\E[SS (2)

where S, is the power spectral density of current noise, a^ is Hooge coefficient, N is

the total number of carriers, 5 is the corss-sectional area of interconnection, n is

electron concentration, E is electric-field intensity. Current noise during early electromigration generates from electrons' random scattering by a mass of vacancies.

vy i[>'

(a) rb)

Fig.l Correlation integral of electromigration noise, (a) earlier stage; (b) intermediate stage

Fig. 1(b) is the correlation integral of electromigration intermediate stage. v(m)

present convergent trend when m increases, it indicates that deterministic component in electromigration noise increases during this period. At electromigration intermediate stage, vacancy concentration has been saturated, and vacancies accumulate at grain boundary, so vacancy concentration begins to decrease, the number of microstate and disorder extent of scattered particle decrease as well, stochastic component in

638

electromigration noise decreases correspondingly. Fig.2 is the correlation integral of electromigration advanced stage. Once m

reaches to a certain value, V(OT) converges to an extremum. This reflects the main

component of electromigration noise has changed form stochastic to deterministic. With the accumulation of vacancies, void nucleation will happen in this stage. Comparing void with mesoscopic conductor open chaotic cavity, we find they are both micro size, and they have similar irregular shape. Electronic transport in chaotic cavity has typical chaotic character, it induces chaotic noise with deterministic dynamic[4], so we can deduce that void scattering will induce chaotic noise too, so deterministic component becomes the dominant component in noise .Fig.3 is a chaotic signal phase diagrams after void nucleation, it presents infinite nested self-similar structures, which is an important characteration of chaotic signal, proves the noise is definitely a chaotic signal with deterministic dynamic mechanism.

Fig.2 correlation integral of

electromigration noise (advanced stage)

Fig.3 phase diagrams of electromigration chaotic signals

THE RESULT OF MULTISCALE ENTROPY

Multiscale entropy was first presented by Costa M. et al[5], it was used to character the complexity of time series[6]. Fig.4 is multiscale entropy of electromigration noise, every curve corresponds to a time t during electromigration porcess. The result shows, at the earlier stage, the value of multiscale entropy is greater and tends to convergent, it means noise is more irregular and complicated; at the advanced stage, after t=20h, the value decreases obviously and not convergent any more, that means noise becomes more regular, the complexity of noise decreases, which reflects the change of noise generation

r

Fig.4 Multiscale entropy of electromigration noise Fig.5 resistance change during

electromigration process

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Fig.5 is a plot of resistance change during electromigration process, at t=20h, resistance change abruptly. So at the moment of void nucleation, multiscale entropy decreases obviously and not convergent any more.

Multiscale entropy describes the complexity of electromigration noise, however, thermodjmamics entropy can reflect the number of microstate, which is called disorder degree. Because there is a relationship between signal entropy and thermodynamics entropy, it is possible to characterize the disorder of physics system with multiscale entropy of noise. Accroding to reference, based on the hypothesis that every microstate is independent and equiprobable, signal entropy and thermodynamics entropy have a relationship as:

— = k\D.2 (3) H

where, S is the thermodynamics entropy of system, H is signal entropy, in this paper, it is the multiscale entropy, k is the Boltzmann constant. During electromigration process, the multiscale entropy of noise decreas, so the thermodynamics entropy of system decreas, that indicates the disorder degree and microstate of electromigration system decrease as well.

CONCLUSION

Correlation integral and multiscale entropy are applied to analyze electromigration noise. Though nondeterminacy and complexicity analysis, the change of electromigration microstate can be obtained. The two methods combine with PSD, provide a more comprehensive multi-parameters characterization for electromigration.

REFERENCES

[1] K. N. Tu, "Recent advances on electromigration in very large scale integration of interconnects",

Journal of Applied Physics, Vol. 94, No. 9, 2003.

[2] P. Grassberger, I.Procaccia, "Characterization of Strange Attractors", Physical Review Letters, Vol.

31,No.l,1983.

[3] James Theiler, " Spurious dimension from correlation algorithms apphed to hmited time-series data" ,

Physical Review A, 1986,34: 2427-2432.

[4] Mikhail L P 2005 Ph.D Dissertation (Ithaca:Comell University) 23

[5] Costa M, Goldberger A L, Peng C K 2002 Phys. Rev Lett. 89 068192

[6] Costa M, Goldberger A L, Peng C K 2005 Phys. Rev E 71 021906

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Micro-plasma Luminescence And Signal Noise Used To Solar Cells Defects Diagnostic

Vanek Jiri", Koktavy Pavel'', Dolensky Jan", Vesely Ales", Chobola Zdenek", Paracka Petr

"Department ofElectrotechnology, Faculty of Electrical Engineering and Communication, University of Technology Brno, Udolni 53, 602 00 Brno

Department ofPhyzics, Faculty of Electrical Engineering and Communication, University of Technology Brno, Technicka 8, 602 00 Brno

'Department of Phyzics, Faculty of Civil Engineering, University of Technology Brno, Veveri28, 602 00 Brno

Abstract. This work deals with the usage of signal noise and micro-plasmas luminescence for solar cells diagnostic. When high electric field is applied to PN junction of solar cell with some technological imperfections it produces in tiny areas of enhanced impact ionization called micro-plasmas which could lead to deterioration in quality or destruction of PN junction. On this account it is possible to use methods which indicate presence of micro-plasma in junction and enable quality and quantitative description of tested cells.

Keywords: Micro-plasma, luminescence, noise, solar cell. PACS: 05.40.Ca, 06.20.-f 84.60.Jt

INTRODUCTION

The need for increasing the solar cell efficiency implies the application of selective processes within the framework of their structure preparation technology. In most cases, the selective processes require more sophisticated technology and diagnostic method. The one of the future diagnostic method used for examination of quality can be the detection of micro-plasma and the noise diagnostic.

Micro-plasma noise is commonly observed in the reverse biased p-n junctions at the onset of the avalanche breakdown. This type of noise was been first described in details 1952[I]. McKay[2] found out that in unstable state the current PN junction looks like a sequence of square pulses of fixed amplitude. Meanwhile the lengths of the pulses and pauses between them vary in a random manner.

Local areas with increased concentration of donor or acceptors or with another defect which can develop lower breakdown voltage in reverse biased PN junction exist. It is possible to indicate it by these methods:



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By detecting of emitted radiation from the point of defect during formatting micro-plasma discharges. (Fig.2) This method is possible to use in optoelectronic devices and solar cells. By scanning of time response of reverse biased current (Fig. 3) By measurement of VA characteristics using current power source supply. By measurement of effective value of narrowband current noise reverse biased current or voltage (Fig. 4)

FIGURE 1. The picture of the electroluminescence measurement (I=3A, U=1.3V, t=30s) of solar cell Z08051. In the picture can be seen the marks of transport equipment used during manufacturing. On the right can be be detected marks of fingerprint and on the left side damaged passivation.

FIGURE 2. The picture of detected luminescence of microplasma (U=5V, 1=1.06A, t=30s) of solar cell Z08051. The damaged surface by the fingerprint on the right side is producing strong source of radiation. Also the damaged passivation is the

source of microplasma light.

FIGURE 3. Example of two-state impulse noise in tested solar cell Z08051.

FIGURE 4. Example of r.m.s. value of noise current IN versus the increasing ramp reverse

voltage UR, solar cell Z08051.

Reverse biased current through PN junction in the area of homogenous breakdown is formed basically only by the current of these local defects. In places of increased concentration of free charge carrier with regard to small area cross-section (from lO"'"* m^ to 10"' m ) the high current density exists (from 10^ A.m" ) which can lead to

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strong local heating-up and sequentially to local diffusion or thermal breakdown of PN junction.

Research of breakdown voltage on temperature dependence confirms hypothesis that it is avalanche breakdown in consequence of impact ionization in strong electrical field. The important characteristic of avalanche breakdown is coefficient of impact ionization which is strongly dependent on the value of electrical field applied on the PN junction

10° 10 10^ 10^ 10^ 10'

Up/V f /Hz

FIGURE 5. The noise spectral density as a function of forward voltage for solar cells

NO.Z08051.

FIGURE 6. The noise spectral density versus frequency for solar cells No.Z08051

RL = 1 k Q , at Up = 0 V, Up = 0.55 V and Up = 0.73 V

We are tested also excess noise of forward-biased sample. Fig. 5 shows a noise power spectral density versus DC forward voltage plots as picked up across a load resistance of RL = 100 £2, RL = 1 kQ., RL = 10 kQ. and RL = 100 kQ. for specimen No Z08051, at a mean frequency of 1 kHz and a bandwidth of 20 Hz.

A curve showing two peaks is observed. The first peak occurs at a voltage of Umaxi = 0.5 V, whereas the second, at Umax2 = 0.65 V.

The maximum noise voltage is observed in the case of the PN junction dynamic resistance matching the load resistance value. This makes us to suppose two different defects in the PN junction regions being connected in parallel to each other.

The behaviour of the characteristic over 0.9 V provides us with indications concerning the contact quality. A fast growth of excess noise in this region leads us to a conclusion that the contact duality is poor.

Fig. 6 shows the noise voltage spectral density versus frequency plot for specimen No. Z08051. The noise voltage was picked up across the load resistance RL = 1 kQ.. The different curves have been measured at three DC forward voltages, namely, UFI = 0 V, corresponding to the measuring setup background noise, UF2 = 0.55 V,

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corresponding to the first peak, and, finally, UF2 = 0.73 V, corresponding to the second peak. The noise spectral density versus frequency plot is off" type, where n =0.85.

CONCLUSION

The first suggested method for diodes screening is based on tracing the microplasma noise current vs. time behaviour. The reverse-biased bi-stable current PN junction characteristics depend on the reverse voltage, the diode load impedance and the temperature. In the bi-stable behaviour region, the oscilloscope displays random rectangular impulses pertaining to either one or several regions of reverse voltages UR.

The second method is based on tracing the noise current (voltage) r.m.s value versus reverse current plots. Reverse biased diodes having no microplasma are expected to generate homogeneous-breakdown avalanche process induced noise only. Microplasma diodes will show at lower voltages an additional noise component, corresponding to the bi-stable conductivity due to the individual microplasma regions.

A large number of micro-plasma regions, which partially overlap each other, occur in large-area PN junction. Repeated measurements coincide with each other. The above described method indicates the PN junction technological condition very well.

The comparison of pictures taken during microplasma luminescence and during electroluminescence confirms the presumption of origin of microplasma in defect areas in PN junction.

ACKNOWLEDGEMENTS

This paper is based on the research supported by the Grant Agency of the Czech Republic, grant No. 102/09/0859 and the project VZ MSM0021630516.

REFERENCES

1. PEARSON, GL. and SAWYER, B. Proc. Inst. Radio Engrs., 1952, vol. 40, p. 1348. 2. McKAY, KG. Avalanche Breakdown in Silicon. Phys. Rev., May 1954, vol.94, no.4, p. 877-884. 3. CHYNOWETH, AG. and PEARSON, GL. Effect of Dislocations on Silicon p-n Junctions. J. Appl.

Phys., 1958, vol. 29, no. 7, p. 1103-1110. 4. BATDORF, RL. et al. J. Appl. Phys., 1960, vol. 31, p. 1153. 5. McINTYRE, RJ. Theory of Microplasma Instability in Silicon. J. Appl. Phys., June 1961, vol. 32,

no. 4, p. 983-995. 6. MOLL, JL. and OVERSTRAETEN, R. VAN. Charge Multiplication in Silicon p-n Junctions. Sol.

St. EL, March-April 1963, vol. 6, no. 2, p. 147-157. 7. HAITZ, RH. et al. Avalanche Effects in Silicon p-n Junctions. 1. Localized Photomultiplication

Studies on Microplasmas. J. Appl. Phys., June 1963, vol. 34, p. 1581-1590. 8. CHAMPLIN, KS. Microplasma Fluctuations in Silicon. J. Appl. Phys., July 1959, vol. 30, no. 7, p.

1039-1050. 9. RUGE, 1. and KEIL, G. Mutual Interactions between Microplasmas in Silicon p-n Junctions. J.

Appl. Phys., Nov. 1963, vol. 34, no. 11, p. 3306-3308. 10. KOKTAVY, P., SIKULA, J. Reverse Biased P-N Junction Noise in GaAsP Diodes with Avalanche

Breakdown Induced Microplasmas. Fluctuation and Noise Letters, ISSN 0219-4775, 2002, roc. 2, c. 2, s. L65-L70.

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Low Frequency Noise Measurement of Reverse Polarized Silicon Carbide Schottky Diodes

Arkadiusz Szewczyk, Barbara Stawarz-Graczyk

Gdansk University of Technology, Faculty of Electronics, Telecommunications and Informatics Department of Optoelectronics and Electronic Systems

11/12 Gahriela Narutowicza Street, 80-952 Gdansk-Wrzeszcz, Poland

Abstract. The results of 1/f noise measurements results of reverse polarized silicon carbide Schottky diodes are presented. Devices show significant dispersion in both, static and noise data. Some of the devices show also RTS noise.

Keywords: Low Frequency Noise, Diode, Silicon Carbide. PACS: 07.50.Hp, 85.30.-z

INTRODUCTION

Sihcon carbide, SiC, is a novel material for electronics. It offers higher band gap, higher breakdown electric field and higher thermal conductivity in comparison to other materials as silicon or gallium arsenide. Therefore, it is used in power and HF electronics as a material for switching elements as Schottky Diodes and MESFETs, that can work with higher switching frequencies and higher junction temperature, up to 175°C [1]. Due to its high thermal conductivity, SiC is widely used in power LEDs and other power elements as a material for substrate.

Although the material is well known its technology for electronic use is been continuously developed and improved.

Nowadays several producers, as CREE and Infineon offer wide range of SiC Schottky diodes. They cover the forward current from 1 to 10 A for single diode and have reverse voltage usually equal 600 V or 1200 V.

DEVICE STUDIED

The aim of the study was to examine the quality, reliability and behavior of the silicon carbide Schottky diodes. The measurements included static and noise measurement in both, forward and reverse polarization.

The devices used in this study are commercially available SiC Schottky diodes of CREE (CSD02060, CSD04060) [2] and Infineon (SDT04S60) [3] with forward current ID = 2 A / 4 A and reverse voltage UR = 600 V. For each type of diodes we measured 20 devices.



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M E A S U R E M E N T RESULTS

During the studies on diodes both, the static, I = f(U) and noise characteristics of the devices was measured.

Static Characteristic Measurement

The static, I(V) characteristic was measured in forward and reverse polarization. In forward polarization all studied devices showed excellent behavior. The I = f(U)

characteristics have perfect reproducibility and are fully comparable with producers data.

However, in reverse polarization characteristics of diodes show significant dispersion, as it is shown in Fig. 1. It must be pointed here, that all devices meet producers data: the maximum measured reverse current is lower than typical values given in technical data [2, 3].

FIGURE 1. Reverse I = f(U) characteristics of studied devices: (a) diodes CSD02060, (b) diodes CSD04060, (c) diodes SDT04S60.

Low Frequency Noise Measurement

Regarding the reverse characteristics, we supposed that the observed dispersion will be visible also in noise data, as the 1/f noise is proportional to current (1) [4]:

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S=^^, (1) /

where .S" is the power spectral density in [A^/Hz], Kp is proportionality coefficient, / is a current and/is frequency.

The low frequency noise measurement setup for reverse polarized diodes is shown in Fig 2. The setup comprises of (i) high voltage supply for diode polarization, (ii) current to voltage converter equipped with additional amplifier, (iii) analog to digital converter with antialiasing filter and (iiii) computer equipped with Lab View software for measurement control and data acquisition and analysis.

A/D Converter

PC LabView

FIGURE 2. Measurement setup for diode noise measurement in reverse polarization.

The noise was measured in reverse polarization range UR = 0 to 700 V in frequency range from IHz to 1000 Hz. Typical group of power spectral density calculated from measurement data is shown in Fig. 3. As one can see, the spectra show distinct 1/f character and the noise level increase with reverse current increasing.

In Fig. 4. we show the power spectral density of current of 5 selected diodes of each group measured at reverse voltage UR = 400 V. As is clearly seen, the dispersion in noise data corresponds to dispersion in reverse current at this polarization.

FIGURE 3. Typical group of power spectral density of the diode current for different values of reverse voltage UR (here the data for one of CSD04060 device).

CONCLUSIONS

Studied devices show excellent behavior in forward polarization, however in reverse polarization significant dispersion in reverse current is observed. The phenomena seem not to be dependent on producer or device parameters. Noise measurements in reverse polarization show dispersion in power spectral density and

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the character of the dispersion in noise is similar to the one observed in reverse current.

(a) CSD04060 @ U„ = 400V

FIGURE 4. Dispersion in 1/f spectra for 5 selected devices of each group: (a) CSD02040, (b) CSD04060, (c) SDT04S60.

During the noise measurements we observed in several devices the RTS phenomena. It was observed in higher polarization voltage and the intensity of RTS (amplitude and pulse duration) increased in time.

More work is therefore required on noise of the devices in order to established methods to evaluate device properties including detection of weak RTS in power spectrum, avoiding high voltage polarization.

ACKNOWLEDGMENTS

This work was supported by Polish Ministry of Science and Higher Education -project No. PBZ-MEiN-6/2/2006.

the

REFERENCES

1. A. Agarwal, R. Singh, S-H. Ryu S-H. et al., "600 V, 1- 40 A, Schottky Diodes in SiC and Their Applications", http://www.cree.com/products/pdf/PWRTechnicalPaperl.pdf.

2. Cree Technical Data, http://www.cree.com. 3. Infineon Technical Data, http://www.infineon.com. 4. CD. Motchenbacher, J.A. Connelly, „Low Noise Electronic System Design", John Wiley & Sons,

1993.

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http://www.cree.com/products/pdf/PWRTechnicalPaperl.pdf

http://www.cree.com

http://www.infineon.com

Noise Harvesting

L. Gammaitoni*, F. Cottone^, I. Neri* and H. Vocca*

*N.i.P.S Laboratory, Dipartimento di Fisica, Universita di Perugia, 1-06123 Perugia, Italy, and Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, 1-06123 Perugia, Italy

^Stokes Institute, University of Limerick ER02-009, Limerick Co., Ireland

Abstract. Kinetic energy harvesting has been the subject of a significant research effort in the last twenty years. Unfortunately most of the energy available at the microscales comes in the form of random vibrations with a wide spectrum of frequencies while standard harvesting methods are based on linear oscillators that are resonantly tuned in narrow frequency ranges. In this paper we present a novel approach based on the exploitation of nonlinear stochastic dynamics and show that, under proper conditions nonUnear oscillators can beat the standard linear approaches with significant increase in the harvesting eflScency. For the sake of demonstration we present experimental residts from a toy-model bistable oscillator made by a piezoelectric inverted pendidum.

Keywords: noise, nonlinearity, energy harvesting, piezoelectric PACS: 05.10.Gg, 89.20.Ff, 85.40.Qx

Powerful computers as small as shirt buttons, micro-sensors invisibly dispersed in the natural environment, human and animal health control devices that can be ingested or implanted; all these scenarios have something in common: the need for a reliable, cheap, durable, efficient powering. For this reason ambient energy harvesting has been the object of a large research efforts aimed at providing an autonomous solution to the powering of small electronic mobile devices.

Vibration harvesting is an interesting option due to the almost universal presence of mechanical vibrations. Present working solutions for vibration-to-electricity conversion[l, 2] are based on oscillating mechanical elements that convert kinetic energy via capacitive, inductive or piezoelectric methods. Mechanical oscillators are usually designed to be resonantly tuned to the ambient dominant frequency. However, in the vast majority of cases the ambient vibrations have their energy distributed over a wide spectrum of frequencies and tuning is not always possible due to geometrical/dynamical constraints[3]. To overcome these difficulties it has been recently proposed[4] a different approach based on the exploitation of the properties of non-resonant oscillators. Specifically we demonstrated that a bistable oscillator, under proper operating conditions [4], can provide better performances compared to a linear oscillator in terms of the energy extracted from a generic wide spectrum vibration.

THE EXPERIMENT

In order to discuss the advantages of nonlinear oscillators in terms of energy harvesting, we considered an inverted pendulum, whose dynamics has been modified in order to accommodate both linear and nonlinear behaviors in the same physical system. A schematic of the experimental apparatus is available in [4]. The inverted pendulum is



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realized with a four-layer piezoelectric beam (mod. T434-A4-302 4-Layer Bender by Piezo system inc.) made by Lead Zirconate Titanate (PSI-5A4E) 60 mm of free length, clamped at one end. The piezoelectric beam has a width of 5 mm and a thickness of 0.86 mm. The pendulum mass is a steel cylinder 140.0 mm long and with diameter of 4.0 mm, with attached three magnets (each magnetic dipole moment is 0.051 Am^ ). The tip magnet is faced by a similar magnet with inverse polarities placed at a distance A and held in place by a massive structure. The distance A can be adjusted via a micrometric displacement control system. The voltage signal from the piezo is measured through a load resistor RL placed in parallel with piezoelectric output voltage terminals and sampled.

The effect of ground vibration force is mimicked by applying an arbitrary magnetic excitation on two small magnets attached near the base of the pendulum. Under the action of the excitation the pendulum oscillates, alternatively bending the piezoelectric beam and thus generating a measurable voltage signal. The dynamics of the inverted pendulum tip can be controlled with the introduction of an external magnet conveniently placed at a certain distance A and with polarities opposed to those of the tip magnet. The external magnet introduces a force dependent from the distance A between the two magnets that opposes the elastic restoring force of the bended beam and, as a result, the inverted pendulum dynamics can show two stable equilibrium positions. In fact, when the external magnet is far away, the inverted pendulum behaves like a linear oscillator whose dynamics is resonant with a resonance frequency determined by the system parameters. This situation accounts well for the usual operating condition of traditional piezoelectric vibration-to-electric energy converters. On the other hand, when A is small enough the random vibration makes the pendulum swing in a more complex way with small oscillations around each of the two equilibrium positions and large excursions from one to the other^

In Fig. 1 we computed the power by measuring the voltage drop V over a resistive load, under the influence of a random vibration with Gaussian distribution and exponential autocorrelation function. The average electrical power is plotted as a function of the magnet distance A, for three different values of the noise standard deviation. In all the cases the power increases rapidly from the linear case (large magnet distance) up to a maximum value and then decreases when the magnets become closer and closer Specifically three different regimes can be identified:

1) Large magnet distance. The pendulum dynamics is characterized by quasi-linear oscillations around the single minimum located at zero displacement, in correspondence with the vertical position of the pendulum. This condition accounts for the usual performances of a linear piezoelectric generator

2) Small magnet distance. The potential energy is bistable with a very pronounced barrier between the two wells. In this condition and for a given amount of noise, the pendulm swing is almost exclusively confined within one well and the dynamics is characterized once again by quasi-linear oscillations around the minimum of the confining well.

3) In between of the two previous cases, there is a range of distances where the power

Seearepresentative video of the pendulum dynamics at h t t p : / /www. n i p s l a b . o r g / n o d e / 1 6 7 6

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' 1 —

a= 1,2 (mN) a = 0,6(mN) o = 0,3(mN)

10 A„ 15

A (mm) 20 25

FIGURE 1. Piezoelectric oscillator mean electric power as a function of A for three different values of the noise standard deviation cr. When A » A^ the potential U{x) shows a single minimum. The continuous curves have been obtained from the numerical solution of the stochastic differential equation (1). Both in the experiment and in the numerical solution, the stochastic force has the same statistical properties with correlation time T = 0.1 s. The rms is computed after zero averaging the time series (expected relative error in the numerical solution is within 10%).

reaches a maximum value. In this condition the pendulum dynamics is highly nonlinear and the swing reaches its largest amplitude with noise assisted jumps between the two wells. As it is well evident in Fig. 1, the maximum values of the output power exceed by a factor that ranges between 4 and 6 the value obtainable when the magnet is far away. This indicates a potential gain for power harvesting between 400% and 600% compared to the standard linear oscillators.

T H E M O D E L

In order to quantitatively account for the experiments we developed a dynamical description of the inverted pendulum based on the following equation of motion:

mx dU{x)

dx •rx-KyV{t) + a^{t) (1)

The first term on the right hand side accounts for the conservative force, where U{x) is the potential energy of the pendulum:

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U{x)=K^ + {a^ + bA^)-^l^ + cA^ (2)

with K, a, b and c representing constants related to the physical parameters of the pendulum [4].

The second term on the right hand side of (1) — yx, accounts for the energy dissipation due to the bending and —KyV{t) accounts for energy transferred to the electric load RL with coupling equation:

V(t) V{t)=KcX--^ (3)

C and Kc are respectively the capacitance and the coupling constant of the piezoelectric sample. Finally, CJ< (?) accounts for the vibration force that drives the pendulum. ^ (?) represents a stochastic process with the same statistical properties of the magnetic excitation. In Fig. 1 we plot with continuous line the computed power {V^)/RL as obtained by the numerical solution of the equation of motion. All the parameters have been measured from the experimental apparatus and introduced into the equation. As it is apparent the agreement between the experimental data and the model is very good indeed.

In conclusion we have shown that under proper conditions nonlinear oscillators can beat the standard linear oscillators with significant increase in the harvesting efficency. It is worth noticing that the improvement obtained in moving from the linear to the nonlinear dynamics is qualitatively independent from the details of the dynamics, here represented by the analytic form of the bistable potential under consideration [4]. Moreover the dynamical features discussed here are not limited to the sole piezoelectric energy conversion but can be applied also to other principles, e.g. capacitive and inductive. They can be apphed also to micro and nanomechanical resonators [5] where noise driven dynamics are considered as a promising option [6].

ACKNOWLEDGMENTS

The authors gratefully acknowledge financial support from Ministero Itahano della Ricerca Scientifica (PRIN 2007) and European Commission (FPVI, STREP Contract N. 034236 SUBTLE: Sub KT Low Energy Transistors and Sensors).

REFERENCES

1, S, Roundy, P.K, Wright, Smart Mater, Struct, 13 (2004) 1131, 2, Y,C, Shu, I,C, Lien, Smart Mater, Struct, 15 (2006) 1499, 3, S,R, Anton, H,A, Sodano, Smart Materials & Structures 16 (3): R1-R21 JUN 2007, 4, F, Cottone, H, Vocca, L, Gammaitoni, Phys, Rev, Lett, 102, 080601 2009, 5, Z,L, Wang, J, Song, Science, 2006 Apr 14;312(5771):242, 6, L, Gammaitoni, A,R, Bulsara, Phys, Rev, Lett, 88, 230601 2002

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Spectral analysis of electromagnetic and acoustic emission stochastic signals

Tomas Trcka' , Pavel Koktavy' , Bohumil Koktavy^

'Brno University of Technology, Faculty of Electrical Engineering and Communication, Technicka 8, 616 00 Brno, Czech Republic.

^Brno University of Technology, Faculty of Civil Engineering, Zizkova 17, 602 00 Brno, Czech Republic.

Abstract. Application of mechanical stress leads to micro-cracks formation in stressed solid dielectric materials. Generation of these cracks is accompanied by generation of the noise signals of electromagnetic (EME) and acoustic (AE) emission, which can be measured by appropriate sensors. Spectral analysis of these noise signals can be used for better diagnostics of mechanically stressed solid dielectric materials. The group of realizations with damp kvaziharmonic waveforms of EME signals is the most significant for analysis of the crack generation. In this special case, the crack walls make this characteristic type of moving, after the crack opening. This specific group of signals allows spectral analysis of measured EME and AE signals. The spectral analysis provides the better view of the crack generation and it allows study relations between EME and AE signals during this process (in the time and the frequency domain).

Keywords: Micro-cracks, electromagnetic emission, acoustic emission, spectral analysis. PACS: 46.50.+a, 81.05.Qk, 81.70.Ex.

INTRODUCTION

Noise signals of electromagnetic (EME) and acoustic (AE) emission can be observed when the solid dielectric materials are mechanically stressed. The generation of cracks in the solids is accompanied by the redistribution of the electric charge. The crack walls are electrically charged and their vibrations produce time variable dipole moments [1]. Hence the individual cracks become an electromagnetic field sources, which can be measured by appropriate sensors. This effect is known as electromagnetic emission (EME).

The signal of the acoustic emission (AE) is generated simultaneously with the EME signal. The time lag of the AE signal after the EME signal provides information about the distance of the crack from the AE sensor.

EXPERIMENTAL

Our fully automated set-up was used for simultaneous measurement of the EME and AE noise signals. The measured specimens are prepared from EXTREN 500



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composite material based structural profiles. The composite material under study consists of a combination of a fibre glass reinforcement and a resin binder. The binder protects the reinforcement from mechanical damage, maintains the structural profile shape and transfers the tension into the reinforcement. The measured specimens are blocks of dimensions (50-52) mm x (58-61) mm x 10 mm. The applied mechanical stress was perpendicular to the reinforcing glass fibre direction. The block diagram of our experiment set-up is in detail described in [2].

SPECTRAL ANALYSIS

EME and AE signals have the random nature. The significant tool for random signals description is the power spectral density (PSD). The basic method for estimation of the power spectral density of an TV-samples record is the periodogram. The periodogram can be calculated from a discrete Fourier transform (DFT) of the input signal. It can be proved that the periodogram is an unbiased but also not a consistent estimate. The classical non-parametric (Bartlett and Welch) methods for PSD estimation use the periodogram. These methods are based on averaging periodograms. The averaging periodograms is always trade-off between the variance decrease of periodogram and the frequency resolution.

In the case of Bartlett method, the input signal of TV samples length is divided into K non-overlapping segments length of M. The Bartlett periodogram is then obtained as the average of K periodograms from the individual segments. The Bartlett periodogram is again asymptotically unbiased, but now the variance is l/K of the classic periodogram variance.

The Welch method allows overlapping of the individual segments. The total number of segments then increases and the variance of Welch PSD estimation decreases. The second modification is application of a window function to individual segments prior to computing the periodogram. The window function alleviates the discontinuities near the boundaries and reduces the spread of the spectral energy into the side-lobes of the spectrum. The Welch PSD estimation is then obtained by averaging of the individual modified periodograms.

The realizations with damped kvaziharmonic waveforms of EME signals (Fig. 1 and Fig. 4) were selected from the set of measured waveforms. These realizations were analyzed by a code created in the MATLAB environment. It allows the interactive changes of many individual parameters (for example the length of segments, the window function, the reciprocal overlap size, etc). The individual parameters influence on the final power spectral density of analyzed signals may be easily studied thanks to this code.

Due to the reproducibility and results comparison, it was necessary to choose just one power spectral density estimation method and exactly specified used parameters. In our case, the Welch method with the Manning window function, segments length of 256 samples and reciprocal overlap of 75 % has been chosen. This solution is a compromise between the frequency resolution, the amplitude variance and the computational difficulty.

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Analyzed signals may be divided into two specific groups. The first one consists of the realizations with identical dominant frequencies of both signals. Fig. 1 shows the analyzed EME and AE signals time domain, which belong to this group. The dominant frequency of the signals from Fig. 1 is the same (312.5 kHz) (Fig. 3 and Fig. 4). Fig. 2 illustrates the time behaviour of the both signals dominant frequencies. It is obvious that these dominant frequencies remain almost the same in time.

Illti'^^*mmmm<^,^,^^.,-12,0 EME AE

150 t /us

150 200 250

g. 1 Time behaviour of EME and AE signals

o "

Fig. 2 Dominant frequencies in time

Fig. 3 PSD of EME signal Fig. 4 PSD of AE signal

100 150 200 250

f/HS

_l ^ _ r ^ L _

.........

1 EME AE

150 t /us

Fig. 5 Time behaviour of EME and AE signals Fig. 6 Dominant frequencies in time

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In the second group, the dominant frequencies are different. The example signals in the time domain from this category can be found on Fig. 5. The dominant EME signal frequency is 449.2 kHz (Fig. 7), while the dominant frequency of AE signal equals to 332.1 kHz (Fig. 8). Fig. 6 shows the second group of signals dominant frequencies time behaviour. Observed frequencies do not change dramatically in time again.

Fig. 7 PSD of EME signal Fig. 8 PSD of AE signal

CONCLUSION

The realizations with damped kvaziharmonic waveforms of EME signals were selected from the set of measured waveforms. These realizations were analyzed by a code created in the MATLAB environment. The dominant frequencies of EME and AE signals were the initial results of the created program. We discovered that the analyzed signals may be divided into two specific groups. The first one consists of the realizations with identical dominant frequencies of both signals. In the second group the frequencies are different. The second purpose of the code was to track time behaviour of the dominant frequencies observed in the first step. The dominant frequencies almost do not vary in time.

The spectral analysis and the code, described in this paper, can be used for improvement of noise diagnostics of mechanically stressed solid dielectric materials.

ACKNOWLEDGMENTS

This paper was supported by the Grant Agency of the Czech Republic under Grant 102/09/H074 and the project VZ MSM 0021630503.

REFERENCES

1. KOKTAVY, B.; KOKTAVY, P. Noise spectroscopy of mechanically stressed insulating materials. ID Noise and Fluctuations. Melville, New York, AIP. 2007. p. 298 - 301. ISBN 978-0-7354-0432-8.

2. KOKTAVY, P. Experimental study of electromagnetic emission signals generated by crack generation in composite materials. Meas. Sci. Technol. 20 (2009) 015704. ISSN 0957-0233.

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