Large negative differential resistance in a quasionedimensional quantum wireBen YuKuang Hu and S. Das Sarma Citation: Applied Physics Letters 61, 1208 (1992); doi: 10.1063/1.107596 View online: http://dx.doi.org/10.1063/1.107596 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/61/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Interactions in a coupled row of electrons formed in a quasionedimensional quantum wire AIP Conf. Proc. 1399, 337 (2011); 10.1063/1.3666391 Effects of impurity scattering on the quantized conductance of a quasi-one-dimensional quantum wire Appl. Phys. Lett. 94, 012105 (2009); 10.1063/1.3067995 Electron conduction through quasi-one-dimensional indium wires on silicon Appl. Phys. Lett. 80, 4169 (2002); 10.1063/1.1483929 Quantum cables as transport spectroscope for quasi-one-dimensional density of states of cylindricalquantum wires Appl. Phys. Lett. 77, 2015 (2000); 10.1063/1.1313247 Farinfrared emission from hot quasionedimensional quantum wires in GaAs Appl. Phys. Lett. 67, 1564 (1995); 10.1063/1.114735

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Large negative differential resistance in a quasi-one-dimensional quantum wire

Ben Yu-Kuang Hu and S. Das Sarma Joint Program for Advanced Electronic Materials, Department of Physics, University oj’bfatyland, CoiIege Park, Maryland 20742-41 I I

(Received 17 January 1992; accepted for publication 29 June 1992)

We calculate finite temperature inelastic scattering rates and mean free paths of electrons injected into a quantum wire containing a quasi-one-dimensional electron gas. We show that there is a very sharp increase in the electron scattering rate at the one-dimensional plasmon emission threshold. Based on these results, we suggest the possibility of a one-dimensional hot-electron device which possesses an 1-V curve with a sharp onset of a large negative differential resistance.

Continued advances in ultrafine lithography have re- sulted in a rapid increase in experimental work on quasi- one-dimensional semiconductor systems. Quantum wires with carrier confinements of approximately 300 A have been achieved,’ with higher quality and narrower wires expected to appear in the near future.’ There have been many proposals for applications of the properties of quasi- one-dimensional systems, such as devices based on the quantum nature of electrons at small distances,’ on the single-particle Coulomb charging energies in ultrasmall de- vices,3 and on other properties unique to one-dimensional systems,4’5 In addition to these properties, there are also unique many-body properties that are characteristic to quasi-one-dimensional systems that can also lead to inter- esting device behavior.

In this letter, we propose a novel device principle which is based on the many-body properties of a one- dimensional quantum Fermi liquid. We show that it may be possible to obtain a device with large and sudden onset of negative differential resistance (NDR) (i.e., d1/dV < 0). This sudden onset of NDR could be exploited to

produce a transistor, while the NDR itself suggests that this device might be used as an oscillator (e.g., in analogy with the Gunn oscillator or, more recently, the resonant tunneling diode6). In the proposed device principle, the predicted NDR is associated with a sharp change in the inelastic mean free path of the injected electrons at a spe- cific energy-in the ideal system at T=O, the mean free path changes from being infinite below the threshold volt- age to being zero above it.

The device principle which we propose may be exper- imentally observed in the quasi-one-dimensional version of the tunneling hot electron transistor amplifier (THETA), shown schematically in Fig. 1, which has been fabricated successfully in three and two dimensions.7 We assume that the quasi-one-dimensional device is in the extreme quan- tum limit; i.e., that all the electrons are in the lowest en- ergy subband in the device. Electrons are injected from an emitter at energies above the Fermi energy EF into a base region which contains (either through doping or electro- static confinement) a one-dimensional electron gas, and the injected electrons that travel through the base region

enter the collector on the opposite side of the base. The fraction of electrons that reach the collector depends on the mean free path (and hence the scattering rate) of the injected electrons, and the mean free path is in general a strong function of the momentum fik (or equivalently, the energy) of the injected electrons and the electron density in the base region.

In two and three dimensions, the main scattering mechanism for these electrons in the THETA devices are the coupled plasmon-optic-phonon modes.* However, in the semiconductor quantum wires in the extreme quantum limit that are currently being fabricated, the densities of the electrons in the base are low enough so that all the energy scales associated with the electron gas and opera- tion of the device (EF, plasmon energy and electron injec- tion energy) are much smaller than the optic-phonon en- ergy, and therefore the optic phonons play a negligible rae. Acoustic phonons can also be ignored because they couple very weakly to electrons in III-V semiconductors, and the associated scattering rates are on the order of lOlo s’. We assume, for the purpose of this letter, that impurity scat- tering in the wires is negligible, which is not unreasonable given the excellent and continually improving techniques for fabricating these mesoscopic systems. This last assump- tion is equivalent to assuming that the elastic mean free paths are much longer than the inelastic mean free path to be calculated in this letter-given that our calculated in- elastic mean free paths are generally a few thousand A or less, and in good quality quantum wires, elastic mean free paths are many microns, the neglect of impurity scattering is a good approximation for our purposes. Thus the main scattering mechanism for an injected electron is the inter- action with the electron gas in the base.

In strictly one-dimensional systems with a parabolic band, the only pair electron-electron scattering (where the injected electron scatters with a single particle in the base) allowed by conservation of energy and momentum is an exchange of particles, which is not a randomizing process because the electrons are indistinguishable. Multiparticle scattering (interactions of the injected electron involving two or more other electrons) is higher order in the screened interaction, and therefore we ignore them because

1208 Appl. Phys. Lett. 61 (lo), 7 September 1992 0003-6951/92/351208-03$03.00 @ 1992 American Institute of Physics 1208 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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TUNNEL BARRIER

4 -1

ii J k>k,

k+\

;~

2

L

,*gg ,F5”““ ., x b Y‘n*yN%

$0 ?r!i-r%-C;)H 11, Wkp

c!OI,LECTOR

FIG. 1. A schematic of the band diagram of a one-dimensional tunneling hot electron transistor, where electrons are injected from the emitter into the base (which contains a Fermi sea of electrons) with some fraction of the injected electrons reaching the collector. The solid (dashed) line in- dicate injection of the electrons into the base region below (above) the plasmon emission threshold (i.e., the solid line is for k < kc and the dashed line is for k> kc). The inset shows energy vs momentum loss diagram for the injected electron. The intersections of the energy vs momentum loss curve and the plasmon dispersion curve (bold line) indicates the wave vectors at which plasmons are emitted; if there is no intersection (as with the solid line), plasmon emissions are not allowed. As the energies of the injected electrons are raised above the plasmon emission threshold, the scattering rate increases dramatically (see Fig. 2), drastically reducing the fraction of injected electrons that reach the collector.

we expect their contribution to be small. The only scatter- ing mechanism left that is responsible for preventing the injected electrons from reaching the collector is the inter- action of the injected electrons with the plasmons (i.e., collective density oscillations) of the electron gas in the base.

Not all injected electrons can emit plasmons. Because the plasmon dispersion in quasi-one-dimensional systems goes as w(q) -q]log(qa) ] 1’2, where a is the width of the wire, only injected electrons with large enough kinetic en- ergies can emit plasmons (see inset of Fig. 1) . At T= 0, for a given density n, there is therefore a threshold wavevector ii,(n) below which no plasmon emission can take place. W ithin the approximations we have used and at T=O, as k is increased through kc, the scattering rate jumps from zero to inf kite (equivalently, the mean free path falls from infinity to zero); this divergence in the scattering rate at k=k, is due to phase-space reasons that are peculiar to one-dimensional systems. This result indicates that as the bias voltage is increased so that k of the injected electrons rises above k, (or equivalently if n is decreased so that kc falls below k), the jump in the scattering rate should be spectacular, and the current passing from emitter to col- lector should fall dramatically. Thermal and impurity ef- fects will broaden the divergence in the scattering rate, but as we show below, this effect persists up to relatively high temperatures.

Using the Born approximation, we calculate the finite- temperature momentum (or transport) scattering rate. I’t,k, which is the quantity that is relevant to the decay in current.g The momentum scattering rate at temperature T is given by the integral over wave vectors of the scattering

100

k/kF

FIG. 2. Momentum scattering rate Fsa of an electron in a doped one- dimensional quantum wire, as a function electron momentum, for various temperatures. The parameters used are kF=0.9 (k, is the Fermi wave vector) and r,= (2m//rf?k& =0.7 (corresponding to a= 100 A and density of 0.56X10” cm-’ in GaAs). In the inset, we show the corre- sponding mean free path, Ik= Q/I-~,~

probabilities, weighted by the change in momentum; i.e.,”

rt,k=2

1 ’ l-exp[wk(q)/k,T] [l--feq(kS-q)l’ (1)

Here, ok(q)=E(k+q) --E(k) [where E(k) =fi2k2/2m is the electron kinetic energy], V,(q) is the Coulomb matrix element for electrons in the lowest subband of a square quantum well of width a with hard walls, f,,(k) is the Fermi-Dirac distribution at temperature T, and .s(q,a) is the dielectric function of the one-dimensional electron gas, within the random phase approximation (RPA).”

In Fig. 2, we show the results of our calculation of It+ and the corresponding mean free path, lk=Uk/IZk (where uk=fik/m is the electron velocity). In the case of T-O, the Born approximation It& is exactly zero up to the plasmon emission threshold, and it diverges as (k- kc) “’ as k-k,’ . As the temperature is increased, the divergence becomes a finite peak due to the broadening of the plasmon line due to Landau damping, and the peak shifts to higher energies. The shift of the peak is due to an upward shift in energy of the plasmon dispersion curve with increasing temperature, which is a well-known phenomenon in plasma physics.” In one dimension, the plasmon dispersion for small q is

02(q) =: qqy l+c!?), (2)

where (u2) denotes the average v2 over the distribution of the electron gas in the base. For a one-dimensional Fermi gas, to order T2,

(3)

explicitly showing the upward shift in the plasmon disper- sion with increasing temperature. The sharp drop in the mean free path persists to relatively high temperatures

1209 Appl. Phys. Lett., Vol. 61, No. 10, 7 September 1992 B. Y.-K Hu and S. Das Sarma 1209 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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(here on the order of tens of degrees for the parameters chosen), and therefore should be experimentally observ- able. We believe that this sharp drop in the inelastic mean free path will produce a large NDR in quantum wires as the injected electrons pass through the threshold energy.

Before concluding, we note that RPA exactly repro- duces the collective mode spectrum for the exactly soluble Luttinger model for one-dimensional systems. l3 Since our results are based mainly on this collective mode behavior, the RPA should be a good approximation for our purposes.

To conclude, we have shown that, due to the sudden onset of a very large rt,k due to the emission of plasmons in a doped quasi-one-dimensional quantum wire, a one- dimensional THETA device could show an I-Y curve with a sudden onset of large negative differential resistance. This characteristic could have applications in switching devices or oscillators. Note that in higher dimensional electron systems there is a plasmon threshold as well where the onset of plasmon emission occurs. The effect in higher di- mensions, however, is not dramatic because the ideal mean free path does not change from being infinite below the threshold to zero above (as it does in the one-dimensional system) since single particle scattering contributes in higher dimensions, in contrast to one dimension. Thus, our proposed NDR in quantum wires is a specific one- dimensional many-body property.

This work was supported by the U.S. ONR and U.S. ARO.

‘A. S. Plaut, H. Lage, P. Grambow, D. Heitman, K. von Klitzing, and K. Ploog, Phys. Rev. Lett. 67, 1642 (1991); A. R. Goiii, J. S. Wiener, J. M. Calleja, B. S. Dennis, L. N. Pfeiffer, and K. W. West, ibid. 67, 3298 (1991).

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3For a review, see, D. V. Averin and K. K. Likharev, in Mesoscopic Phenomena in Solids, edited by B. L. Altshuler, P. A. Lee, and R. A. Webb (Elsevier, Amsterdam, 1991), pp. 173-271.

‘See, e.g., H. Sakaki, Jpn. J. Appl. Phys. 19, L735 (1980); D. A. B. Miller, D. S. Chemla, and S. Schmitt-Rink, Appl. Phys. Lett. 52, 2154 (1988); S. Briggs, D. Javanovic, and J. P. Leburton, ibid. 54, 2012 (1989).

‘For reviews, see, Nanostracture Physics and Fabrication, edited by M. A. Reed and W. P. Kirk (Academic? New York, 1989); S. Datta and M. J. McLennan, Rep. Prog. Phys. 53, 1003 (1990).

‘E. R. Brown, T. C. L. G. Sollner, and C. D. Parker, Appl. Phys. Lett. 55, 1777 (1989).

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“R. Jalabert and S. Das Sanna, Phys. Rev. B 40, 9723 (1989); and references therein.

“The total finite-temperature Born approximation scattering rate is ac- tually infinite due to the divergence in the factor (I-exp[o,(q)/ ksq ] -’ at small q. However, these small q scattering events have little effect on the degradation of the current, which is the quantity of interest here. Therefore, we calculate the physically more meaningful quantity rr,k.

“See, e.g., D. Pines and P. Nozieres, The Theory of Quantum Liquids (W. A. Benjamin, New York, 1966; reprinted by Addison-Wesley, Red- wood City, CA, 1989), Chap. 4.

“Q. P. Li and S. Das Sarma, Phys. Rev. B 43, 11768 (1991). I2 See , f ., e g N. A. Krall and A. W. Trivelpiece, Principles of Plasma Phys-

its (McGraw-Hill, New York, 1973; reprinted by San Francisco Press, San Francisco, 1986), Chap. 8.

13Q. P. Li, S. Das Sarma, and R. J. Joynt, Phys. Rev. B 45, 13713 (1992).

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