local noise analysis of a schottky contact: combined

Local noise analysis of a Schottky contact: Combined thermionic-emission–diffusion theoryG. Gomila, O. M. Bulashenko, and J. M. Rubí Citation: J. Appl. Phys. 83, 2619 (1998); doi: 10.1063/1.367024 View online: http://dx.doi.org/10.1063/1.367024 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v83/i5 Published by the American Institute of Physics. Related ArticlesFrequency multiplication in nanowires Appl. Phys. Lett. 99, 153107 (2011) Modification of electrical properties of Al/p-Si Schottky barrier device based on 2′-7′-dichlorofluorescein J. Appl. Phys. 110, 074514 (2011) Effect of N/Ga flux ratio on transport behavior of Pt/GaN Schottky diodes J. Appl. Phys. 110, 064502 (2011) Non-destructive detection of killer defects of diamond Schottky barrier diodes J. Appl. Phys. 110, 056105 (2011) Extraction of voltage-dependent series resistance from I-V characteristics of Schottky diodes Appl. Phys. Lett. 99, 093505 (2011) Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

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JOURNAL OF APPLIED PHYSICS VOLUME 83, NUMBER 5 1 MARCH 1998

Local noise analysis of a Schottky contact: Combinedthermionic-emission–diffusion theory

G. Gomila,a) O. M. Bulashenko, and J. M. RubıDepartament de Fı´sica Fonamental, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain

~Received 23 September 1997; accepted for publication 5 November 1997!

A theoretical model for the noise properties of Schottky barrier diodes in the framework of thethermionic-emission–diffusion theory is presented. The theory incorporates both the noise inducedby the diffusion of carriers through the semiconductor and the noise induced by the thermionicemission of carriers across the metal–semiconductor interface. Closed analytical formulas arederived for the junction resistance, series resistance, and contributions to the net noise localized indifferent space regions of the diode, all valid in the whole range of applied biases. An additionalcontribution to the voltage-noise spectral density is identified, whose origin may be traced back tothe cross correlation between the voltage-noise sources associated with the junction resistance andthose for the series resistance. It is argued that an inclusion of the cross-correlation term as a newelement in the existing equivalent circuit models of Schottky diodes could explain the discrepanciesbetween these models and experimental measurements or Monte Carlo simulations. ©1998American Institute of Physics.@S0021-8979~98!01705-8#

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I. INTRODUCTION

Schottky barrier diodes are being extensively useddifferent applications, such as mixers and detectors, dutheir good high-frequency behavior.1 There is a vast litera-ture devoted to the description of the transport properti.e., current–voltage~I–V! characteristics, capacitance, coductance, etc., of these devices.2–4 In recent years, howeverspecial attention has been paid to the characterization onoise properties of Schottky barrier diodes.5–8 The main ap-proach currently used to interpret the experimental nomeasurement results is the phenomenological equivanoise circuit technique.5–7 Under this approach the diodecharacterized by means of the corresponding equivalentcuit. Hence, by associating each resistive element of thecuit with different noise sources~shot noise for the junctionresistance, thermal noise for the series resistance, etc.! thetotal noise of the device is evaluated.9 Despite its wide use inelectrical engineering, it should be noted, however, thatphenomenological procedure does not offer a relationtween the equivalent circuit elements and the kinetic amaterial parameters of a device~barrier height, mobility,doping, surface recombination velocity, etc.!. Moreover, itimplicitly assumes that only a given type of noise canassigned to each equivalent-circuit element and that theferent sources of noise are uncorrelated. While the firstsumption seems to hold in most of the cases, the secassumption might not, due to the existence of spatial colation between different active regions of the device.10

To overcome this situation, and to avoid anya prioriassumption regarding the different sources of noise, mrecently microscopic Monte Carlo~MC! simulation has beenperformed.8 Although the MC technique describes the carrtransport on the microscopic level and takes into accou

a!Electronic mail: [email protected]

2610021-8979/98/83(5)/2619/12/$15.00

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large variety of kinetic processes, there are some difficulin its application to the study of Schottky barrier diodes.particular, when the barrier height is sufficiently larg(.5kBT/q) a depletion of carriers of several orders of manitude occurs near the metal–semiconductor interface atvoltage biases. In such a case, the MC simulation becovery time consuming because of poor statistics of carrierthe depletion layer,8 which, in turn, controls the current annoise properties under these conditions. To overcomeproblem the special algorithm of particle multiplication hbeen realized to calculate the I–V characteristics of the diwith a high barrier.11 Whether this algorithm is suitable fothe noise calculation remains unclear. As a consequeonly the small-barrier diodes~near the flat-band conditions!have been simulated by the MC method.8 Therefore, the de-velopment of other methods able to provide theoretianalysis of the local noise properties of the Schottky contawith arbitrary ~low and high! barriers in the wide range oapplied voltage biases seems to be highly desirable.

The purpose of the present article is to present precisa simple analytical analysis of the different contributionsthe net noise of a Schottky barrier diode. To this endpropose a combined model for the noise properties of thdevices which takes into account in a common framewthe two main sources of noise present in Schottky diodnamely, the noise due to the thermionic emission of carracross the metal–semiconductor interface and the noiseto the diffusion of carriers inside the semiconductor. Bmeans of an analytical method recently developed10 we haveobtained the explicit expressions for the local noise andpedance distributions, which allow us to identify the regimof the system parameters, where one or another noise sodominates.

Furthermore, we show that the equivalent noise circpreviously reported for these contacts5–7 can be obtained aslimit of our model in some particular limiting cases. Th

9 © 1998 American Institute of Physics

license or copyright; see http://jap.aip.org/about/rights_and_permissions

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2620 J. Appl. Phys., Vol. 83, No. 5, 1 March 1998 Gomila, Bulashenko, and Rubi

equivalent-circuit parameters, such as the junction resistathe series resistance, etc., acquire closed analytical formvalid in the whole range of voltages including the crossobetween the shot noise and thermal noise limits. In additour results suggest that in order to adequately describenoise behavior of the Schottky diode under intermediateplied biases the traditionally used equivalent noise circmodel should be corrected by adding a new circuit elemcorresponding to the cross-correlation between the nsource associated with the junction resistance, and the nsource associated with the series resistance. This suggeis based on the analytical expression for the cross-correlaterm, and supported by our calculations.

The organization of the article is as follows: In Sec.we review the thermionic-emission–diffusion theory whireflects the basic transport physics in an idealn-typeSchottky barrier diode. In Sec. III we propose the corsponding fluctuating thermionic-emission–diffusion modwhich incorporates both the fluctuations due to the diffusof carriers inside the semiconductor and the fluctuationsduced by the thermionic emission of carriers acrossmetal–semiconductor interface. Then, the analytical soluof the model is derived, and the different contributions tototal noise and local noise distributions are identified. In SIV the steady-state spatial profiles for the electric potentelectric field and carrier density are numerically obtainfrom thermionic-emission–diffusion equations, which wbe used as input data to evaluate the noise characteristicSec. V the equilibrium expressions for the analytical formlas found in Sec. III are derived and their compatibility withe Nyquist theorem shown. In Sec. VI the correspondnonequilibrium results are presented and analyzed. Finin Sec. VII we sum up the main contributions of the artic

II. THERMIONIC-EMISSION–DIFFUSION THEORY

For ideal Schottky barrier diodes the thermionemission–diffusion theory of Crowell and Sze12 has longbeen recognized as an appropriate theory to describetransport properties of these devices. The theory assuthat the two main current-limiting processes occurring inSchottky barrier diode are the diffusion and drift of carriethrough the semiconductor part of the contact and the tmionic emission across the metal–semiconductor interfaThese two processes are effectively in series, and the cuis determined predominantly by the process which causeslarger impediment to the carrier flow or, in a general casea combined action of both of them if they are comparabThe latter was pointed out as the most realistic situation3

The first process, electron transport through the semicductor, is described by means of the drift-diffusion modthat is an equation for the current coupled self-consistentlthe Poisson equation

I ~ t !

A5qnv~E!1qD~E!

]n

]x1e

]E

]t, ~1!

]E

]x5

q

e~ND2n!, ~2!


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Here, n(x,t) and E(x,t) are the electron density and thelectric field, respectively, andA is the cross-sectional areaThe drift velocity v(E) and the diffusion coefficientD(E)depend, in a general case, on the local electric field. Inmodel the current across any section of the device is theof the conducting currentJ(x,t) ~composed of the drift anddiffusion components! and the displacement current. Thcurrent is conserved and equal to that in the external cirI (t).

The second current-limiting process, the thermionemission, is described by means of a boundary conditionthe conducting current at the top of the potential barriwhich is expressed in terms of an effective recombinatvelocity v r as12,13

J~0,t !5q@n0~ t !2n0eq#v rA. ~3!

The term withn0v r represents the electron flux from semconductor to metal, withn0 being the electron density at thtop of the barrier. Then0

eqv r represents the opposite flufrom metal to semiconductor, where the quasiequilibriuelectron densityn0

eq5Nce2qfb /kBT does not depend on th

bias,Nc is the semiconductor effective density of states, afb the contact barrier height as seen from the metal side.velocity v r is given by12 v r5A* T2/qNc , whereA* is theeffective Richardson constant andT is the lattice tempera-ture.

In this article we are interested in modeling the lowfrequency plateau of the noise spectrum~above the 1/f -noiseregion!, corresponding to the time scale much longer ththe dielectric relaxation time. Due to the frequency ranchosen, the displacement current can be neglected. Thfore, by eliminatingn, the system~1!, ~2! yields a nonlinearsecond-order differential equation for the electric-field pfile,

D~E!d2E

dx21v~E!S dE

dx2

q

eNDD52

I

eA. ~4!

Moreover, by taking into account that in the frequency ranof interestJ(0,t)5I , the boundary condition atx50 for Eq.~4! following from Eqs.~3! and ~2! becomes

dE

dx Ux50

5q

e FND2n0eq2

I

qv rAG . ~5!

On the opposite side of the semiconductor the flat-baboundary condition is assumed, that is

dE

dx Ux5L

50, ~6!

where the sample lengthL is chosen to be large enough, i.emuch larger than the Debye screening lengLD5(ekBT/q2ND)1/2 to guarantee the quasineutrality condtion n(L)5ND . ~This condition is also equivalent to puttinthe Ohmic contact atx5L.) The results for a short Schottkbarrier diode (L of the order of LD) will be publishedelsewhere.14

To distinguish different regimes in the thermioniemission–diffusion theory, it is useful to introduce somemensionless parameters as follows: Since the width of


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2621J. Appl. Phys., Vol. 83, No. 5, 1 March 1998 Gomila, Bulashenko, and Rubi

depletion layer forming the potential barrier is of the orderseveral Debye lengthsLD , the typical absolute value of thmaximum of the electric field at the top of the barrier canmeasured in units ofEth5kBT/(qLD). Accordingly, thecharacteristic diffusion velocityvD appeared in the theory12

is of the order ofmEth , with m5v8(E)uE50 being the low-field mobility. Therefore, the parameterb5mEth /v r can beused to estimate the importance of different regimes limitthe current. We will see in Sec. VI when we compute timpedance of the diode, that forb!1 the transport is diffu-sion limited while forb@1 it is thermionic-emission limited~for b;1 both processes contribute similarly!.

Notice, that the parameterb only enters into the boundary condition~5!, where the term with the current is propotional tob. Therefore, in the diffusive limit whenb→0, thisterm is small in respect to others, and the electron densityn0

remains approximately at the equilibrium valuen0eq and is not

altered by the applied voltage~current!. This is equivalent toassuming that at the interface the quasi-Fermi level insemiconductor coincides with the Fermi level in the meta2

In this case the decrease of the quasi-Fermi level octhrough the depletion region. In the opposite thermionemission limit whenb is large and the boundary conditionthe interface depends strongly on the current, the quFermi level for electrons remains flat throughout the deption region~the mobility is large! and drops down inside thmetal. Therefore, the boundary condition~5! mirrors the be-havior of the quasi-Fermi level nearby the metsemiconductor contact.

III. FLUCTUATIONS IN SCHOTTKY BARRIER DIODES

A. General formulation

According to the thermionic-emission–diffusion theodescribed in the previous section two current limiting pcesses are assumed to exist in Schottky barrier diodesdiffusion of carriers through the semiconductor and the thmionic emission of carriers across the metal–semiconduinterface. Consequently, by associating, as usual, a nsource with each current-limiting process, two differesources of noise should be present in the system: a diffunoise source and a thermionic-emission one.15 While theformer accounts for the different random scattering procesoccurring inside the semiconductor, the latter describesrandom nature of the thermionic-emission processes acthe metal–semiconductor interface. Although the existeof both noise sources seems to be out of discussion, nonthe existing carrier-transport models incorporates themunified way. Concerning the equivalent circuit model, it donot incorporate them in a proper way, as will be shownlow.

Here we present a simple model in which both sourof noise are taken into account in a common framework. Tdiffusion noise is modeled, as usual, as abulk current noisesource which accounts for the velocity fluctuations16

K~x!54q2n~x!D~x!, ~7!

where the nonlocal heating of carriers is neglected.


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On the other hand, we model the source of noise duethe thermionic emission of carriers across the metsemiconductor interface as asurfacenoise source associatewith a stochastic boundary condition. Its spectral density, asshown in Appendix A, is given by

SIc52q@ I 12I c#, ~8!

with I c5qn0eqv rA. This approximation is based on the a

sumption that the thermionic-emission processes occurside a layer of the order of the electron mean free pathlp

which is assumed to be a small fraction of the barrier widThe two sources of noise, as well as two local noise sourof Eq. ~7! corresponding to different slicesx and x8 areassumed to be uncorrelated, since the characteristic scasuch thatux2x8u@lp .

As pointed out by Van Vliet,17,18 in the case that surfacas well as bulk noise sources are present in a given systhe most convenient method to analyze the fluctuationsnoise properties of the system is the Langevin responsemulation. Essentially, this formulation consists in addingcarrier transport equations the corresponding stochastic nsources and then, by solving these equations and computhe appropriate averages, to obtain the statistical propeof the different magnitudes of interest. For the case unconsideration, this means that the fluctuation of the elecfield dEx at slicex should satisfy the linearized version oEq. ~4! with a source noise term for the currentdI x

eA LdEx52dI x , ~9!

with the operatorL given by

L5D~E!d2

dx21v~E!

d

dx1D8~E!

d2E

dx2

1v8~E!S dE

dx2

q

eNDD . ~10!

Here, dI x represents the stochastic current induced byrandom scattering of carriers inside the semiconductorhas zero mean and d-type correlation function^dI xdI x8&5AK(x)D f d(x2x8), with K(x) defined throughEq. ~7! andD f the frequency bandwidth. Furthermore, at tmetal–semiconductor interface the fluctuations satisfylinearized version of Eq.~5! with a surface Langevin-liketerm dI s present. Explicitly,

ddEx

dx Ux50

52dI s

ev rA. ~11!

Here, dI s describes the random nature of the thermioemission of carriers across the metal-semiconductor inface. It has zero mean and the dispersion^dI s

2&5SIcD f withSIc given by Eq.~8!.

Due to the requirement on the length of the samL@LD and the flat boundary condition atx5L, for the fluc-tuations of the field at this boundary we have

dEL50, ~12a!

ddEx

dx Ux5L

50, ~12b!


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with the meaning that the contribution to the noise atcontact from distances much larger than the Debye lengtscreened out.

Equations~7!–~12! constitute a complete set of equtions to analyze the noise properties of Schottky barrierodes in the thermionic-emission–diffusion approximation

B. Analytical solution

Note thatL is a linearsecond-orderdifferential operatorwith space-dependentcoefficients. To find the solution in aanalytical form we use the method recently applied forn1n semiconductor junction.10 The general solution for theelectric-field fluctuation is given by

dEx5C1r~x!1C2u~x!1r~x!E0

x u~j!

eAD~j!W~j!dI jdj

1u~x!Ex

L r~j!

eAD~j!W~j!dI jdj, ~13!

whereW(x)5r(x)u8(x)2u(x)r8(x) is the Wronskian, andr(x) and u(x) are auxiliary functions, satisfying the equtions Lr(x)50, Lu(x)50, i.e., they are the solutions of thhomogeneous equation corresponding to Eq.~9!. The expres-sions for the auxiliary functions are as follows:10

r~x!5dE/dx, ~14a!

u~x!5r~x!EC

xW~j!

r2~j!dj, ~14b!

W~x!5W~0!exp$2*0xv[E~x8!]/D[E~x8!]dx8%. ~14c!

The value of the integration constantW(0) is not actuallyimportant, since it will be canceled after substituting E~14b! into ~13!, so we can assumeW(0)51. The constantCin Eq. ~14b! can be defined by the appropriate choice forboundary condition of the functionu(x). Since it does notinfluence on the final results, we takeu8(0)50 which cor-responds to the homogeneous boundary conditions forGreen functions of the operatorL and provides the moscompact intermediate expressions. The integration constC1 andC2 in Eq. ~13! are determined by the boundary coditions at x50 and x5L. The condition ~12! impliesr(L)50 and, hence,C250. By differentiating Eq.~13! andusing Eq.~11! we obtain the stochastic value for the constaC1 as

C152dI s

ev rAr08. ~15!

Hereafter, the prime stands for a derivative, and the subin0 refers to the value of the corresponding function atx50.

The final expression fordEx then reads,


eis

i-

e

.

e

he

nts

t

ex

dEx52r~x!

ev rAr08dI s1r~x!E

0

xu~j!

c~j!dI jdj

1u~x!Ex

L r~j!

c~j!dI jdj, ~16!

where we have denotedc(j)5eAD(j)W(j). Equation~16!, together with Eq.~14!, constitutes the analytical solution to the problem posed by Eqs.~9!–~12!. It expresses theelectric-field fluctuations at pointx in terms of the noisesourcesdI x , dI s and the stationary electric field profilwhich appears throughr(x), u(x), andc(x).

It is worth noting that Eq.~16!, apart from allowing us toderive explicit expressions for the impedance field and vage and current fluctuations~see below!, it could also beused to evaluate the spatial correlations of the electric-fifluctuations ^dExdEx8& between two different points~re-gions! under nonhomogeneous conditions. These corrtions are active over several characteristic screening lenof the system.10

Now, we are in a position to evaluate the fluctuationthe voltage at the terminal ends by integrating the fluctuatof the electric field given through Eq.~16! along the system,

dV5E0

L

dExdx. ~17!

We then obtain,

dV5ZcdI s1E0

L

¹Z~x!dI xdx, ~18!

where

Zc5E02EL

ev rAr08~19!

is the impedance of the metal–semiconductor contact aciated with the thermionic-emission processes, and¹Z(x)the bulk impedance field associated with the drift-diffusiprocesses in the semiconductor. The latter consists ofterms

¹Z~x!5¹Zdep~x!1¹Zser~x!, ~20!

¹Zdep~x!5r~x!

c~x!

E02EL

r0r08, ~21!

¹Zser~x!5r~x!

c~x!E

0

x

@EL2E~x8!#W~x8!

r2~x8!dx8. ~22!

It will be shown below, that the term¹Zdep is the contribu-tion coming from the depletion layer and it dominates at locurrents, while¹Zser is determined by the doping level inside the semiconductor and is associated with the seriessistance. The latter dominates at high currents whendepletion layer disappears~flat-band condition! and the volt-age drops mainly on the bulk of the semiconductor.

From Eq. ~18! the total impedanceZ of the Schottkycontact, as well as the spectral density of the voltage fluctions SV , may be easily computed. One obtains,


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Z5Zc1E0

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¹Zdep~x!dx1E0

L

¹Zser~x!dx

[Zc1Zdep1Zser, ~23!

SV5Zc2SIc1AE

0

L

@¹Zdep~x!1¹Zser~x!#2K~x!dx

[SVc1SVdep1SVser1SVcros, ~24!

where

SVc5Zc2SIc , ~25!

SVdep5AE0

L

@¹Zdep~x!#2K~x!dx, ~26!

SVser5AE0

L

@¹Zser~x!#2K~x!dx, ~27!

SVcros5AE0

L

2@¹Zdep~x!#@¹Zser~x!#K~x!dx, ~28!

and they represent different contributions to the total voltafluctuations. On the other hand, we shall use the funcsV(x)5@¹Z(x)#2K(x), also subdivided into the corresponing componentssVdep(x)1sVcros(x)1sVser(x), in order tocharacterize the local contribution of different space regito the net noise measured between the terminals.

The spectral density of the current fluctuationsSI can bealso computed by using Eqs.~23! and ~24! as

SI5SVc1SVdep1SVcros1SVser

~Zc1Zdep1Zser!2

. ~29!

Finally, the noise temperature 4kBTn5SV /Z takes on theform

Tn51

4kB

SVc1SVdep1SVcros1SVser

Zc1Zdep1Zser. ~30!

Equations~23!–~30!, once Eqs.~19!–~22! are used, give theexact expressions for the total diode impedance, voltagecurrent fluctuations, and the noise temperature as obtaform the thermionic-emission–diffusion theory. Furthemore, they also give information about the spatial distribtion of the impedance throughout the device by means ofimpedance field¹Z(x), and the local contributions to thvoltage fluctuations by virtue of the spatial profilesV(x).These local distributions may be very useful in interpretthe results as will be seen in what follows. It is worth notithat in order to compute Eqs.~19!–~22!, and hence Eqs~23!–~30!, we only need to know the stationary electric fieprofile.

Recalling now the equivalent-circuit-noise formula fthe current spectral density of the Schottky diode5

SI52qIRj

2

~Rs1Rj !2

14kBTRs

~Rs1Rj !2

, ~31!

with Rs and Rj the series and the junction resistances,spectively, and comparing it with our Eq.~29!, it is seen, thatwe arrived at a more general result. In particular, the junct


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nded

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-

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resistanceRj usually used as a phenomenological parameto fit the experimentalI –V or noise characteristics, now cabe evaluated fromRj5Zc1Zdep and is given analytically byEqs.~19! and~21!. The series resistanceRs is equal toZser inour model and can be obtained from Eq.~22!. The noiseassociated with the junction resistance 2qIRj

2 is given in ourmodel throughSVc1SVdep, which are followed from Eqs.~25! and ~26!. Moreover, the noise term associated with tseries resistance 4kBTRs can be evaluated fromSVser givenby Eq.~27!. Finally, an additional term represented bySVcros

absent in the equivalent circuit model has appeared intheory, which is relevant, as will be shown below, in a cetain range of voltages. It is interpreted as a cross-correlabetween the voltage-noise sources associated with the jtion resistance and those for the series resistance. Moretailed discussion of all the terms will be found in the subsquent sections.

IV. STEADY-STATE SPATIAL PROFILES AND I – VPLOTS

In order to evaluate the different expressions foundthe previous section, we need to compute the corresponstationary profiles for the electric field. To find these profilwe solve numerically Eq.~4! with the boundary conditions~5! and~6! by making use of a finite difference scheme. Fthe sake of completeness, also the electron density andtential profiles will be reported. It can be shown that tdiode behavior is governed by three dimensionless pareters@for given shapes ofv(E) and D(E)]: ~i! the coeffi-cient b5mEth /v r introduced in Sec. II characterizing thtype of transport regime;~ii ! the ratio a5n0

eq/ND whichcharacterizes the height of the contact barrier, and~iii ! thecurrent, for which we use the following unitsJ5I /I R , withI R5qmEthNDA.

In the following we shall use three different values of tparameterb for a comparative analysis of different transporegimes:b520 for the case of thermionic-emission~TE!dominated transport,b50.01 for the case of diffusion~D!dominated transport, andb51 for the intermediatethermionic-emission–diffusion~TE–D! regime.

The parametera describes basically the nature of thcontact under consideration. In this sense, for a rectifycontact~high barrier! a is very small, while for not rectifyingcontacts~low barrier! a*1. In the following for our calcu-lations we takea51027. This value provides a sufficientlyhigh potential barrier ('16kBT/q at equilibrium! and a‘‘deep’’ depletion layer with the electron concentration at tbarrier topn0;1027ND . For simplicity we consider onlythe case of constant mobilitym and the diffusion coefficientD, although explicit field dependencies characteristic forhot carrier transport may be used without additional difficties. To be concrete, we takev(E)5mE, D5mkBT/q. Thelength of the sample is taken to beL515LD to guarantee thequasineutrality conditions atx5L.

Having found the electric-field profilesE(x) for a givencurrent J, the distributions of the potentialf(x) and theelectron densityn(x) can be easily restored. Knowing thtotal voltage dropV5*0

LE(x)dx1Vbi , with the equilibrium


-

eserhar

aravfat

ed

cluosythityeates

foytioacis

e-

seeetio

ib-ca

thisent

tric

d,

ter-

re


built-in voltage Vbi5(kBT/q)ln(1/a), the I –V curves canthen be obtained. In our calculations in whicha51027, thebuilt-in voltageVbi'16kBT/q.

A comparison of theI –V characteristics for the structures simulated for different values ofb is shown in Fig. 1.In all cases the qualitative behavior is similar: for low biaswhen the current is controlled by the depletion layer, thappears an exponential region, while for high biases the cacteristics becomes linear displaying the presence of seresistance effects once the depletion layer has disappeApart from a quantitative difference, the exponential behior is seen to be more pronounced for the case TE. Thethat both the thermionic emission processes as well asdiffusion processes~or a combined action of both! may in-duce a similar behavior on theI –V characteristics has to btaken into account in analyzing the properties of theseodes. In particular, care should be taken in drawing consion from an analysis of theI –V characteristics in respect tthe dominant transport mechanisms acting on a giventem. In general, precise information on the values ofdifferent contact parameters, i.e., recombination velocmobility, doping, etc., are necessary to arrive at a corrinterpretation of the results. As we will see, the same stment holds for the analysis of the noise properties of thsystems.

The steady-state spatial profiles for the case D anddifferent currentsJ are shown in Fig. 2. The profiles displathe usual behavior: for low currents a pronounced deplelayer is observed near the metal–semiconductor interfwhile at higher currents the depletion layer progressively dappears@Fig. 2~a!#. Along with a disappearance of the depltion layer, the potential barrier also disappears@Fig. 2~c!#.The spatial distributions for another limiting case, TE caare presented in Fig. 3. Apart from quantitative differencarisen from different typical scale of the current, it is sethat for the TE case at high enough currents the deplelayer can be substituted by an accumulation layer.

V. EQUILIBRIUM NOISE AND THE NYQUIST THEOREM

The relative significance of the different terms contruting to the impedance and noise of the Schottky diode

FIG. 1. I –V characteristics for a Schottky diode operating at differentgimes: D, TE-D, TE. Current is in units ofI R5qmEthNDA.


,er-

iesed.-cthe

i--

s-e,

cte-e

r

ne,-

,snn

n

be ascertained by considering the equilibrium caseI 50, forwhich very transparent expressions can be obtained. Forcase the equation for the balance of the currnE1(kBT/q)dn/dx50 can be integrated, giving

n~x!5n0eq exp@qf~x!/kBT#, ~32!

where we have used the boundary condition for the elecpotentialf(0)50.

Furthermore, all the expressions are simplifiegiving E(L)50; W(x)5exp@qf(x)/kBT#; E(x)W(x)5LD0

2 (dr/dx), whereLD05(ekBT/q2n0eq)1/2 is the screen-

ing length corresponding to the electron density at the inface. Substituting these relations into Eqs.~20!–~22! we ob-tain

¹Zdepeq ~x!5R~x!

r~x!

r~0!, ~33!

¹Zsereq~x!5R~x!F12

r~x!

r~0!G , ~34!

-

FIG. 2. Stationary distributions of the electron densityn ~a!, the electricfield E ~b!, and the electric potentialf ~c! of a Schottky diode in the regimeD for different currentsJ.


ca

sarerll.y

-no

um

ercauset.by

is-

-

e

en-en

s inain

tral


¹Zeq~x!5R~x![1

qmAn~x!, ~35!

showing that the bulk impedance field is equal to the lo~per unit length! resistance.

Figure 4 shows the spatial profiles for¹Zeq(x) and itsconstituents¹Zdep

eq (x) and ¹Zsereq(x) calculated by using the

equilibrium distributionn(x) shown in Figs. 2 or 3~at equi-librium they are identical for all three transport regime!.The term¹Zdep

eq (x) is seen to give the main contribution nethe interfacex,5LD , since it represents the depletion laycontribution, in which the carrier density is very smaWhereas forx.7LD the impedance is determined mainly bthe series resistance term¹Zser

eq which is constant over almost the whole structure and is determined by the dodensity.

The contact impedance given by Eq.~19! is estimated bytaking into account the relationr085E0 /LD0

2 valid at equilib-rium. One gets

FIG. 3. Stationary distributions of the electron densityn ~a!, the electricfield E ~b!, and the electric potentialf ~c! of a Schottky diode in the regimeTE for different currentsJ.


l

r

Zc5kBT

q2n0eqv rA

5kBT

qIc[Rc , ~36!

whereRc is the well-known contact resistance,4 used to char-acterize the nature of the contact. Thus, the total equilibriimpedance ~resistance! of the diode is given byZeq5Rc1*0

LR(x)dx5Rc1Rdep1Rser.After integration along the structure the depletion-lay

resistance is much greater than the series resistance, beRdep/Rser;1/a@1, as it should be for a rectifying contacThe total equilibrium diode resistance is then governedthe sumRc1Rdep. Introducing the characteristic series restance asRR5EthLD /I R5LD /(qmNDA) the contact resis-tance can be evaluated byRc5RR(b/a), whereas the depletion resistanceRdep'RR /a. Therefore,Rc /Rdep'b, and oneor another resistance dominates (Rc or Rdep) depending onthe regime considered~TE or D!.

A similar spatial analysis is performed for the voltagfluctuations. For the local noise contributions in Eqs.~26!–~28! one gets

sVdepeq ~x!54kBTF r~x!

r~0!G2

R~x!, ~37!

sVsereq ~x!54kBTF12

r~x!

r~0!G2

R~x!, ~38!

sVcroseq ~x!58kBTF12

r~x!

r~0!Gr~x!

r~0!R~x!. ~39!

Hence, for the net local noise sVeq(x)5sVdep

eq (x)1sVcros

eq (x)1sVsereq (x)54kBTR(x), which is the Nyquist

theorem, sinceR(x) is the local resistance. It should bpointed out that in recovering the Nyquist theorem all cotributions, including the cross-correlation term, have benecessary.

The local noise distributions Eqs.~37!–~39! calculatedby using the equilibrium profilen(x) are shown in Fig. 5. Asa consequence of the Nyquist theorem the thick solid lineFigs. 4 and 5 coincide. Note that near the interface the mcontribution tosV

eq(x) comes from the depletion-layer termsVdep

eq (x). The series-noise term dominates in the quasineu

FIG. 4. Equilibrium bulk impedance field¹Zeq(x) and its components¹Zdep

eq (x) and¹Zsereq(x), all normalized by (qmAND)21.


sth

ge

tay

n

matee

ld atOn

ingaklynts

m--m-

dif-the

ee

a-be


region, and the cross-correlation term is active in the croover from the depletion layer to the quasineutral part ofdiode.

For the contact contribution to the voltage noise weby means of Eqs.~8!, ~25!, and~36!,

SVceq54qIcZc

254kBTRc ~40!

which again consists of the Nyquist theorem. The tovoltage fluctuations are then given bSV54kBT(Rser1Rdep1Rc). Similarly, we obtain for the cur-rent fluctuationsSI54kBT/(Rser1Rdep1Rc). These resultsshow that the Nyquist theorem is satisfied both locally aglobally, as should be.

VI. NONEQUILIBRIUM FLUCTUATIONS IN ASCHOTTKY DIODE

In this section we will discuss the results obtained froour theory for the nonequilibrium case. Note that to evaluthe different expressions derived in Sec. III B we only neto use the stationary electric-field profiles obtained in SIV.

In Fig. 6 we have plotted the bulk impedance fie¹Z(x) and its components¹Zdep(x), ¹Zser(x) for different

FIG. 5. Spatial profiles for the contributionssVdepeq (x), sVcros

eq (x), andsVsereq (x)

to the net local equilibrium noisesVeq(x), all normalized by

4kBTD f /(mAND).

FIG. 6. Spatial profiles for the local impedance¹Z(x) @normalized by(qmAND)21] for different currentsJ for the case D.


s-e

t

l

d

edc.values of the current. The term¹Zdep(x) is seen to be mostlylocalized in the depletion layer and it plays the main rolelow currents whenever the depletion layer is appreciable.the other hand, the spatial profile for¹Zser(x) is flat overalmost the whole structure with the magnitude correspondto the donor density. It is seen that its shape depends weon the bias. This clearly explains why this term represethe series resistance of the sample.

Integrating along the structure the impedance-field coponents, the depletion impedanceZdep, and the series impedanceZser are computed. By adding them to the contact ipedanceZc , the total impedanceZ is evaluated. Figure 7shows the results as functions of the applied bias in theferent transport regimes. It is seen, that for low biasesmain contribution comes from eitherZdep ~case D! or Zc

~case TE! ~or both at TE–D regime!, while at high voltagesthe series impedanceZser always dominates. Note that thseries impedanceZser is almost constant in the whole rangof voltages considered.

A similar analysis is performed for the voltage fluctutions. As before, the meaning of the different terms can

FIG. 7. ImpedanceZ vs applied voltageV at different regimes: D~a!, TE–D~b!, and TE~c!. The relative contributions from the contactZc and the bulkZser, Zdep are compared.


. I

lntseeeisehef

alioe

hp

ai

t

-uc-v-aling

tedg

tri-

-


justified by means of the local noise analysis of the diodeFigs. 8~a! and 8~b! the local noise distributionssV(x)5sVdep(x)1sVcros(x)1sVser(x) are depicted for twodifferent transport regimes~D and TE! and for two differentvalues of the current. It is seen that the termsVdep is mainlylocalized in the depletion region, the termsVser dominates inthe quasineutral part of the sample, andsVcros is important inthe crossover between the depletion and the quasineutragions and represents their cross-correlation. At low currethe depletion contribution dominates, while at high currenwhen the depletion layer has disappeared, all the componare comparable in magnitude. The main difference betwthe D and TE cases is in the position of the local nomaxima at low currents. For the case TE it lies at the intfacex50 where the electron density is lowest, while for tcase D it is displaced towards the depletion region, sincethis case the carrier density atx50 is fixed by the boundarycondition and does not fluctuate significantly.

Integrating along the device the spatial profiles forthe components, the spectral densities of voltage fluctuatSVdep, SVcros, and SVser are found. To get the total voltagnoise they should be added to the contact noise termSVc ,which is localized at the metal-semiconductor interface. Trelative significance of different components for different aplied biases is shown in Fig. 9. For low voltages the mcontribution comes from eitherSVdep ~case D! or SVc ~caseTE! ~or SVc1SVdep for TE–D case!, while at high voltagesSVser dominates. At intermediate biasesSVcros is seen to con-tribute appreciably.

Having found the total impedanceZ and the voltage-noise spectral densitySV , the spectral density of curren

FIG. 8. Spatial profiles for the local noisesV(x) @normalized by4kBTD f /(mAND)] in regimes D~a! and TE~b!. For each type of line, thecurve on top corresponds to the lower current.


n

re-ts,

ntsn

er-

or

lns

e-n

fluctuationsSI is computed from Eq.~29!. The results fordifferent transport regimes are shown in Fig. 10. A first important point to be noted is that in all cases the current fltuations display a similar behavior: for very low current leels (J&I c /I R5a/b) SI is constant, since it gives the thermequilibrium noise with different diode resistances accordto differentb. Whereas at higher currents (a/b&J&1022)all the curves exhibit a 2qI shot-noise-like behavior.19

These two different behaviors can in fact be predicexplicitly from our analytical results. Indeed, by neglectinat low biases (V&Vbi) the series-resistance and cross conbutions, one gets

SI'SVc1SVdep

~Zc1Zdep!2

, ~41!

which in the depletion approximation, valid under this lowvoltage regime, yields~see Appendix B!

SI52q@ I 12I sat#, ~42!

FIG. 9. Voltage fluctuationsSV vs applied voltageV at different regimes: D~a!, TE–D ~b!, and TE~c!. The relative contributions from the contactSVc

and the bulkSVser, SVdep, SVcros are compared~all normalized to 4kBTRR).


n-io,ef

mt,

pasen

e

t

a

tk

tothththn.

be

c-ep-mee-

heoss-ate-,

entth

of-pontheon-ar-

by

t:


'2qI, for I @I sat, ~43!

where I sat is the saturation current of the diode which icludes both the effects of thermionic-emission and diffusprocesses, and it is given by Eq.~B13!. Note, that in generalI sat is a voltage dependent function due to the diffusionfects. It is worth noting that the validity of Eq.~42! extendsover not only the ohmic regime but the hot-electron regias well ~see Appendix B!. For the case of ohmic transporhowever, theI –V characteristics takes on a simple form4

I 5I sat@exp~qV/kBT!21#, ~44!

and Eq.~42! can be rewritten as9

SI52qI coth~qV/2kBT!, ~45!

which does not include explicitly any system dependentrameter. It can be seen that the data plotted in Fig. 10 clofollow this law independently on the transport regime cosidered.

At high biases (V.Vbi) the current fluctuationsSI devi-ate from the 2qI law and tend to saturate. In this limit wmay approximateZ'Zser and SV'SVser54kBTZser, whichare independent on the bias~see Figs. 7 and 9!. Conse-quently, SI54kBT/Zser is independent of the current, thaexplains the saturation behavior at high voltages.

The effective noise temperatureTn as a function of biasin different transport regimes is depicted in Fig. 11. One csee, that starting fromTn5T at zero bias~thermal equilib-rium noise! it drops sharply towardsT/2 for low voltagesthat is a consequence of the 2qI law and is typical for theshot noise behavior.5,9 At higher biasesTn increases againapproaching the ambient temperatureT, corresponding to thediffusion noise of the series resistance. Note thatTn mayattainT or may not, depending on the length of the SchotdiodeL ~the magnitude of the series resistance!.14 The noisetemperatureTn can be used as a figure of merit in orderevaluate the effects of the different sources of noise ondevice performance. In particular, it can be seen thatmore the transport is thermionic-emission dominated,more Tn approaches the valueT/2. In this sense, one caconclude that the TE case is less noisy than the D case

FIG. 10. Current-noise spectral densitySI vs currentI at different regimes:D, TE–D, and TE. The low-current limit is shown to be nicely describedEq. ~45! ~dashed line!. SI and I are normalized by 4qIR andI R5qmEthNDA, respectively.


n

-

e

-ly-

n

y

eee

The importance of the cross-correlation term canmade more explicit by plottingSI and Tn with and withoutincluding this term. Figure 12 clearly shows that both funtions deviate significantly starting from the voltagV'10kBT/q (J'1022) when the series effects starts to apear. A similar behavior has been observed in soexperiments,6 where the noise temperature was found to dviate from the equivalent circuit results~without the crossterm! precisely as shown in Fig. 12. Therefore, a part of texcess noise observed could be assigned to the crcorrelations between the different noise sources. This stment is further confirmed by the recent MC simulation8

which also found out the discrepancies with the equivalcircuit results~compare Fig. 12 of the present article wiFig. 3 of Ref. 8!.

VII. DISCUSSION

In this article we have presented an analytical analysisthe noise properties of ann-type Schottky barrier diode under forward bias conditions. Our study has been based ua noise model which incorporates in a unified way bothnoise due to the diffusion of carriers through the semicductor and the noise due to the thermionic-emission of c

FIG. 11. Noise temperatureTn vs applied voltageV at different regimes: D,TE–D, and TE.

FIG. 12. Current-noise spectral densitySI vs currentI for the regime TE–Dwith ~solid line! and without~dashed line! the cross-correlation term. InseThe same for the noise temperatureTn .


suantiadamd.

a

ta

ioearlly

eeedt

rig

bth

pe

l o

re

le

weut

uga

ubean.

ulenthonm

es

dg intua-

beseped

usisere-

hede-p-

theas

Di-lhe

ter-oml

as-

ityctor

m

lybutas

nts


riers across the metal–semiconductor interface. By meanthe analytical solution of the model, the different contribtions to the net noise of the diode have been identified,with the help of a local analysis, assigned to different sparegions of the device. Furthermore, the relevance of theferent contributions depending on the characteristic pareters and the applied voltage range has been analyzeparticular, it has been shown that at low biases (V&Vbi) thenoise properties are dominated either by the diffusion of criers in the depletion layer near the interface (b!1) or bythe thermionic emission of carriers across the mesemiconductor interface (b@1). Here,b is the coefficientbetween the diffusion velocity and the contact recombinatvelocity. At high biases (V*Vbi), the noise properties arshown to be dominated in all cases by the diffusion of criers through the quasineutral region of the device. Finastarting from the applied voltageV;Vbi at which the deple-tion layer becomes small, an additional contribution has bidentified. Due to its spatial localization in the crossover btween the depletion layer and quasineutral region of thevice, this additional term has been argued to representcross-correlation between the voltage noise sources onated in these two parts of the device.

A detailed comparison between the results predictedmeans of our model and those predicted by means ofconventional equivalent circuit noise technique has beenformed. In particular it has been shown that:~i! The so-called junction resistanceRj , whose origin is notclearly established in circuit analysis, consists in generatwo contributions:Zc and Zdep, for which we give explicitexpressions.~ii ! The series resistanceRs is given throughZser, whosevalue can also be obtained from the corresponding expsion.~iii ! The 2qIRj

2 behavior associated withRj at low currentlevels is in fact given throughSVc1SVdep, which reproducesthis type of behavior at low voltages, but holds in the whorange of biases. Note, that both the thermionic emissioncarriers through the metal–semiconductor interface asas the diffusion of carriers in the depletion region contribto this behavior.~iv! The noise due to the series resistance is given throSVser, which reproduces correctly the well known thermnoise behavior.~v! There is an additional contributionSVcros to the voltagefluctuations not present in the conventional equivalent circanalysis. This contribution is due to the cross-correlationtween the fluctuations generated in the depletion layerthose in the quasi-neutral region of the semiconductorstarts to become relevant in the transition from the 2qI be-havior towards the series-resistance noise~near the flat-bandconditions!. It has been argued that this additional term coexplain some discrepancies observed between experimmeasurements or MC simulations and the results ofequivalent circuit model precisely under near flat-band cditions. It should be pointed out that mm-wave and sub-mwave mixers operate precisely at this regime,7 so that ourtheory can be useful for the local-noise analysis of thdevices in order to improve their performance.


of-dl

if--In

r-

l-

n

-,

n-e-hei-

yer-

f

s-

ofll

e

hl

it-d

It

dtale--

e

It is worth remarking that our theory provides closeanalytical expressions to evaluate all the terms appearinthe impedance as well as in the voltage and current fluctions under the full range of applied biases. Hence, it canused to improve the existing equivalent circuit model. Theexpressions can in fact be easily inserted into the develonumerical programs which calculate the steadyI –V charac-teristics and electric field profiles for Schottky diodes, thadding also information about the local and global noproperties of these systems. In addition, the method psented in this article, besides its fruitful application for tSchottky barrier diodes, it can be incorporated into anyvice model described by means of the drift-diffusion aproach and its modifications.21 Our work then offers newperspectives on what concerns the spatial analysis ofnoise properties of space-charge-limited devices, suchn1 –n–n1 diodes, field-effect transistors, etc.

ACKNOWLEDGMENTS

Two authors~O.M.B. and G.G.! acknowledge support bythe Generalitat de Catalunya. O.M.B. is also grateful toreccion General de Ensen˜anza Superior of Spain for financiasupport. This was was supported by the DGICYT of tSpanish Government under Grant No. PB95-0881.

APPENDIX A: SPECTRAL DENSITY OF LOW-FREQUENCY CURRENT FLUCTUATIONS ATA METAL–SEMICONDUCTOR INTERFACE

The current crossing at the metal–semiconductor inface consists of electrons emitted individually and at randin both directions. IfI ms5qn0

eqv rA is the current from metato semiconductor andI sm5qn0v rA is the current from semi-conductor to metal, then the net current isI 5I sm2I ms asgiven by formula~3!. Since both random processes aresumed to be statistically independentI ms andI sm exhibit fullshot noise.9 Therefore, for the low-frequency spectral densof current fluctuations generated at the metal–semiconduinterface one gets

SIc52q~ I sm1I ms!52q~ I 12I ms!52q~ I 12I c!, ~A1!

whereI c[I ms.

APPENDIX B: UNIVERSALITY OF THE 2 qI LAW FORTHE CURRENT NOISE IN DIFFERENTTRANSPORT REGIMES

The purpose of this appendix is to show explicitly froour analytical results that the 2qI behavior in the currentspectral densitySI observed at low currents emerges not onin the thermionic-emission case, as it would be expected,also for the diffusion case and the combined TE–D casewell.

Neglecting the series resistance effects at low currewe have

SI'SVc1SVdep

~Zc1Zdep!2

. ~B1!


thonnc

r.a

Eq

-

ag

ef-ote

-c

ve

e

-

.

tal-

qui-the

r

sitythe

des

es,ize

ni,


Furthermore, for these values of the applied biasdepletion-layer approximation holds for which the electrconcentration inside the depletion layer is neglected, sin(x)!ND .2 For this caser(x)5dE(x)/dx becomes

r~x!'H r0 , 0,x,w,

0, w,x,L,~B2!

wherer05qND /e andw is the length of the depletion layeUnder these conditions the electric field is approximated2

E~x!'H E0~12x/w!, 0,x,w,

0, w,x,L.~B3!

As a result for the depletion-layer impedance by using~21! we have

Zdep5E02EL

eAr08E

0

w dx

D~x!W~x![

CE

vD, ~B4!

whereCE5(E02EL)/(eAr08), and we have defined the diffusion velocityvD by22

1

vD5E

0

w dx

D~x!W~x!. ~B5!

The expressions for the contact impedance and voltspectral density follows from Eqs.~19! and ~25!,

Zc5CE

v r, ~B6!

SVc5CE2 2q@ I 12I c#

v r2

5CE2F2qI

v r2

14q2An0

eq

v rG . ~B7!

Now, the only unknown term in Eq.~B1! is the depletionvoltage-noise spectral densitySVdepwhich can be found fromEq. ~26!

SVdep54q2CE2AE

0

w n~x!dx

D~x!W2~x!. ~B8!

To proceed further we integrate Eq.~1! with the help of theintegrating factor 1/W(x) finding the following relation:

n~x!

W~x!5n~0!1

I

qAE0

x dx8

D~x8!W~x8!. ~B9!

Substituting this relation into Eq.~B8! and using the identity~can be proved by integrating by parts!

E0

w

f ~x!F E0

x

f ~x8!dx8Gdx51

2 F E0

w

f ~x!dxG2

, ~B10!

with f (x)51/@D(x)W(x)#, one arrives at

SVdep54q2CE2AS n~0!

vD1

I

2qAvD2 D . ~B11!

Now we can use the boundary condition forn(0) given byEq. ~3! and substitute all the terms into Eq.~B1!. One gets


e

e

s

.

e

SI52qF I 12qAn0

eqv rvD

v r1vDG[2q@ I 12I sat#, ~B12!

where

I sat5qAn0eq v rvD

v r1vD~B13!

is the saturation current of the diode including both thefects of thermionic-emission and diffusion processes. Nthe universality of Eq.~B12! which holds for different trans-port regimes~D, TE, and TE–D! and conduction mechanisms ~ohmic, hot-electrons!. For the D case under ohmiconditions a similar derivation of the formula~B12! has beenperformed by van der Ziel.23

1T. W. Crowe, R. J. Mattauch, H. P. Ro¨ser, W. L. Bishop, W. C. B.Peatmean, and X. Liu, Proc. IEEE80, 1827~1992!.

2E. H. Rhoderick and R. H. Williams,Metal-Semiconductor Contacts, 2nded. ~Clarendon Press, Oxford, 1988!.

3H. K. Henisch,Semiconductor Contacts~Clarendon Press, Oxford, 1984!.4S. M. Sze,Physics of Semiconductor Devices, 2nd ed.~Wiley, New York,1981!.

5M. Trippe, G. Bosman, and A. van der Ziel, IEEE Trans. MicrowaTheory Tech.34, 1183~1986!.

6A. Jelenski, E. L. Kollberg, and H. H. G. Zirath, IEEE Trans. MicrowavTheory Tech.34, 1193~1986!.

7A. Jelenski, A. Gru¨b, V. Krozer, and H. L. Hartnagel, IEEE Trans. Microwave Theory Tech.41, 549 ~1993!.

8T. Gonzalez, D. Pardo, L. Varani, and L. Reggiani, Appl. Phys. Lett.63,3040 ~1993!; J. Appl. Phys.82, 2349~1997!.

9A. van der Ziel,Noise in Solid State Devices and Circuits~Wiley, NewYork, 1986!.

10O. M. Bulashenko, G. Gomila, J. M. Rubı´, and V. A. Kochelap, Appl.Phys. Lett.70, 3248~1997!; J. Appl. Phys.83, 2610~1998!.

11M. J. Martın, T. Gonza´lez, D. Pardo, and J. E. Vela´zquez, Semicond. SciTechnol.11, 380 ~1996!.

12C. R. Crowell and S. M. Sze, Solid-State Electron.9, 1035 ~1966!; Re-printed in Semiconductor Devices: Pioneering Papers, edited by S. M.Sze~World Scientific, Singapore, 1991!, pp. 414–427.

13An extension of this boundary condition to the case of non-ideal mesemiconductor contacts has been worked out in G. Gomila, A. Pe´rez-Madrid, and J. M. Rubı´, Physica A233, 208 ~1996!.

14G. Gomila, O. M. Bulashenko, and J. M. Rubı´ ~unpublished!.15This reasoning may be made more rigorous by considering the none

librium thermodynamics arguments. For a general formulation oftheory, see e.g.: S. R. de Groot and P. Mazur,Non-equilibrium Thermo-dynamics~Dover, New York, 1984!. For its application to semiconductosystems, see G. Gomila and J. M. Rubı´, Physica A234, 851 ~1997!.

16C. M. van Vliet, IEEE Trans. Electron Devices41, 1902~1994!.17K. M. van Vliet, J. Math. Phys.12, 1981~1971!.18K. M. van Vliet and H. Mehta, Phys. Status Solidi B106, 11 ~1981!.19It is the custom to call each type of noise with the current spectral den

SI52qI as a shot noise. Actually, the genuine shot noise is related toPoissonian ballistic electron flow like that occurred in vacuum-tube dio~Ref. 9!, or in ballistic semiconductor devices~Ref. 20!. Here, this relationappears as a consequence of the~near-!exponentialI -V characteristicsunder combined diffusive and thermionic-emission transport regimwhich are apparently not ballistic. To use the term ‘‘shot’’ to characterthe type of noise under this situation is not completely adequate.

20T. Gonzalez, O. M. Bulashenko, J. Mateos, D. Pardo, and L. ReggiaPhys. Rev. B56, 6424~1997!.

21P. J. Price, J. Appl. Phys.63, 4718~1988!.22For ohmic conduction our definition reduces to the conventional one~Ref.

4!: vD5(mkBT/q)$*0w exp@qf(x)/kBT#dx%21 where f(0)50 has been

used.23A. van der Ziel, Physica B94, 357 ~1978!.


local noise analysis of a schottky contact: combined

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