PHYSICAL REVIEW 8 VOLUME 44, NUMBER 4 15 JULY 1991-II

Evolution of the quantized ballistic conductance with increasing disorder in narrow-wire arrays

Arvind Kumar and Philip F. BagwellDepartment of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02I39

(Received 4 February 1991)

We study the two-terminal Landauer conductance averaged over a parallel array of disordered narrowwires as the Fermi energy and length of the disordered region are varied. As disorder in the wires is in-creased, so that quantum diffusion becomes the dominant electron-transport mechanism, we find numeri-cally that the quantized conductance steps characteristic of ballistic transport evolve into conductancedrops after each new subband is populated. Consistent with this result, the electron localization lengthdecreases above each new subband. Adding attractive scatterers to the wires strongly modifies these re-sults due to "quasidonor levels" forming in the impurities.

I. INTRODUCTION

Discovery of the quantized ballistic conductancethrough a point contact' has greatly stimulated theoret-ical studies on the effect of impurity scattering in nearlyballistic quantum wires. ' When only a few impuritiesare present in a wire, these studies have predicted thatthe average conductance should rise after the opening ofeach new subband channel, although structure in the con-ductance of a single wire may be obscured by wave-interference fluctuations. ' ' If some of the scatterersare attractive, pronounced conductance drops before theopening of each new subband were also found tooccur, due to the formation of "quasidonor levels" inthe impurities. Thus repulsive and attractive scatter-ers result in a very different subband structure for theconductance versus Fermi energy in a narrow wire.

If disorder in the wire is increased, so that the trans-port becomes diffusive rather than nearly ballistic, itmight be expected that a fundamentally different subbandstructure should be observed experimentally' in nar-row quantum wires. Indeed, based on a modified Drudemodel and assuming that electrons scatter in the Born ap-proximation, Refs. 23—31 conclude that electron scatter-ing increases whenever a new subband becomes occupied,leading to a drop in conductance versus Fermi energyafter each new subband opening. In contrast, Refs. 3—9have argued that pronounced drops in conductanceshould occur before each new subband opening. Somestructure in the conductance versus electron density haspossibly been observed in arrays of narrow wires, '

but it is unclear what physics this structure mightrepresent or where it occurs in relation to subband mini-ma in the wires.

In this paper we calculate the two-terminal Landauerconductance of a parallel array of disordered quan-turn wires to obtain the "ensemble-averaged" conduc-tance. A "point-scatterer" model' is used to describe thedisorder in each wire. We show that there is a clear tran-sition in this model between a quantum ballistic regime,marked by increasing conductance after each new sub-band channel opens, and a quantum diffusive regime,

marked by a sharp drop in conductance when a new sub-band is populated. We associate this conductance dropwith a decrease in the electron localization length im-mediately above a subband, so that the drop in conduc-tance after each subband opening in the diffusive regimedepends on quantum diffusion, rather than the classicaldiffusion of the Drude model. Additionally, we showthat the standard "golden-rule" or "Born-approximation" scattering theory is invalid near a sub-band minimum, so that the scattering at each individualimpurity must be properly calculated to obtain thecorrect dependence of the conductance versus Fermi en-ergy. When this is done, we do not find conductancedrops after the opening of a new subband channel if theelectron diffusion is classical. Finally, both for quantumand classical electron diffusion, we find conductancedrops before the opening of a new subband channel onlywhen attractive scatterers are present in the wires.

II. MODEL FOR A DISORDERED QUANTUM WIRE

We choose a model Hamiltonian describing electronsfree to move along the x direction and confined along they direction:

+ + V, (y)+ Vd(x, y) g(x,y)2%i

=Eg(x,y ) . (I)

The confinement potential V, (y) gives rise to confinementsubbands E„such that

d , +V, (y) X.(y)=E.X.(y) .2m

Vd (x,y) =gy;5(x —x, )5(y —y, ), (3)

where the ith scatterer is located at position (x;,y, ) andhas strength y;. The conductance is obtained from the

We choose the impurity potential to be a sequence ofpoint scatterers

1747 1991 The American Physical Society

1748 ARVIND KUMAR AND PHILIP F. BAGWELL

two-probe "Landauer formula"

I eG= —= gT„,V

(4)

where T „denotes the transmission coefficient frommode n to mode m. Many individuals have contributedto our understanding of Eq. (4), as discussed in Refs.32—39 and citations therein.

The transmission coefficients T „ in Eq. (4) are foundnumerically by cascading together the individual scatter-ing matrices for each point defect and each intermediateregion of free propagation between defects. ' We includethe lowest five modes in our calculations, enough to un-derstand the qualitative features of the conductance, al-though we expect the inclusion of higher modes to havequantitative inAuence on our results. This numericalmodel and our particular implementation of it are bothvery reliable, since they agree with analytical results ob-tained for electron transmission through both one ' andtwo point defects in a narrow wire. Current conserva-tion is also numerically well satisfied in these simulations,giving additional confidence in the reliability of our re-sults.

For a single wire with a disordered region of length Lalong the x direction, we randomly position the scattererswith a uniform probability density over the ranges [0,W]across the channel and [O,L] along the wire. We choosea fraction f of the scatterers to be attractive (y; (0). Allthe scatterers have equal strengths

~ y; ~

= 10 feV cm . Wechoose the mean spacing between impurities along the xdirection of the wire to be 10 nm, so that there are fiveimpurities in the wire when L=50 nm and 50 impuritieswhen L=500 nm. We model the confinement using aninfinite square-well potential of width 8'=30 nm, andtake the electron mass to be 0.067 times the free mass.This choice of parameters is consistent with experimentson GaAs/Al Ga, „As heterostructures. In a more real-istic model, where the wire consists of a potential wellhaving finite depth, it will probably be necessary to in-corporate evanescent modes from the continuum in orderto obtain proper convergence of the calculation.

Impurity potential fluctuations in good-quality GaAsheterostructures are believed to be mainly weak andsmooth. Electron transport through such a smooth po-tential can be nearly "adiabatic, " producing a nearly ex-act quantization of the ballistic conductance. ' lt hasalso been argued that such a smooth scattering potentialshould give rise to enhanced electron mobility in a nar-row wire when electron transport is restricted to thelowest subband, since any carrier deAections are pri-marily due to small-angle forward scattering, which doesnot substantially degrade the electron mobility. Rough-ness along the channel edges in GaAs heterostructures, atype of disorder neglected in this work, is also believed tovary smoothly compared to the electron wavelength.Conversely, the potential in Eq. (3) is quite rough and ir-regular, and in fact produces s-wave scattering when em-bedded in a two-dimensional plane. In most calculationsusing a tight-binding Anderson model the potential isalso rough and irregular, ' since the "on-site energies"

are random.However, fair conductance quantization is still pro-

duced for electron transmission through a point-scattering potential in a narrow channel. The point-scatterer model of disorder may therefore be a reasonableone to understand the effects of wave interference be-tween scattering events in a low-dimensional conductor,even though it is a highly nonadiabatic scattering poten-tial. The point-scatterer model also gives rise to "univer-sal conductance fluctuations"' in agreement with experi-ment. In addition, electron transmission through afinite-size rectangular barrier in a wire is qualitativelysimilar to transmission through a point scatterer, indi-cating that shrinking the scatterer to zero size may not bea serious limitation. Edge roughness has not been includ-ed explicitly, but some roughness is simulated becausesome of the scatterers lie near the channel edges. For thecase of a single scatterer, the qualitative dependence ofthe electron transmission on Fermi energy for an edge de-fect is no different from that of a defect located in themiddle of the channel. Comparing our present resultsusing the scattering potential in Eq. (3) with transmissionthrough a smooth disorder potential is left to a futurestudy. We caution that the scattering effects near a sub-band crossing studied in this paper may possibly be exag-gerated or even qualitatively different when comparedwith transmission through a smoother impurity potential.

III. CALCULATED CONDUCTANCEOF A NARROW-WIRE ARRAY

In Fig. 1(a) we show the conductance as a function ofFermi energy for a single wire in the ensemble havingL=50 nm for both f=0.0 and 0.5. The conductance isseen to rise after the opening of each new subband wheth-er all the scatterers are repulsive (f =0.0) or half thescatterers are attractive (f =0.5). However, the intro-duction of attractive scatterers gives rise to pronounceddips in conductance below each subband minimum, nearthe energies of quasidonor levels splitting off from theconfinement subbands. The spacing AE from these quasi-donor levels to the next subband is of orderbE=(m/fi )(y/I/(/), close to the binding energy of astate trapped in the point defect. If the length of thedisordered region is increased to L =500 nm, as shown inFig. 1(b) for f=0.0, the resulting electron wave-interference pattern obscures any regular structure in theconductance. When f=0.5 and L=500 nm as in Fig.1(c), the conductance exhibits similar fluctuations thatobscure any underlying subband structure. Some unusualproperties of these fluctuations have been noted in Ref.17. Interestingly, the lowest subband is still approximate-ly discernible in Fig. 1(c) due to the depressed transmis-sion associated with the quasibound states.

To manifest the underlying subband structure of theconductance versus Fermi energy, we plot in Fig. 2 theconductance from Fig. 1 averaged over an array of 100independent wires in parallel. Each wire has a differentrandom arrangement of the scatterers, but the length of

EVOLUTION OF THE QUANTIZED BALLISTIC CONDUCTANCE. . . 1749

3.0

2.0

o $.0

0.0E

I I I I

E3E2Fermi Energy

I I I I I I I I

. (a)f=0.0—f=0.5

the disordered region is kept fixed. In Fig. 2(a) each wirehas I.=50 nm. Some broad resonances present in the sin-gle wire are eliminated after averaging when all thescatterers are repulsive (f =0.0). When half the scatter-ers are made attractive (f =0.5), the quasibound-stateenergies vary from wire to wire, resulting in a broadeneddip in the average conductance before the opening ofeach new channel.

If we increase the length of the disordered region toI.=500 nm, as in Fig. 2(b), a fundamentally different sub-band structure of the conductance versus Fermi energyemerges. The conductance now drops abruptly after theopening of each new subband channel when all thescatterers are repulsive (f=0.0, top curve). When halfthe scatterers are made attractive (f=0.5, bottom curve),

I I I I I I I I

(b)f=0.0

I I I

l

an ).0-

!

il J

I I I I

E E1 2Fermi Energy

3.0CD

OJ

2.0-C

CD 1.0-

CJ

8 0.0o F

(a)I I I I I

E2Fermi Energy

I I I

I I I I I I I I I

I I I I I I I I

cu ( )f=0.5

rn ).0-

JI Jg JJ

I I I I I I

1 2Fermi Energy

CDOJ

1.0-

CD

C3

Q Q

1O

~ 0.1

I I I JI I I I I

E2Fermi Energy

I I I I I I I I II I

FIG. 1. Landauer conductance vs Fermi energy for a singlequasi-one-dimensional wire having (a) L=50 nm and (b) and {c)L= 500 nm. When only a few scatterers are present (a), the con-ductance varies smoothly with Fermi energy and, if some attrac-tive scatterers are present (f =0.5), dips abruptly near the"quasidonor levels" below each subband. As more scatterersare added, wave-interference conductance Auctuations in (b) and(c) obscure the underlying regular structure due to confinementsubbands.

FICx. 2. Landauer conductance averaged over an array of 100parallel wires, each having length (a) L=50 nm and (b) L=500nm. The ballistic conductance steps are rounded for the shortquantum wires in (a). A new "diffusive" subband structureemerges for the long quantum wires [(b), top curve] havingf=0.0: The conductance falls after each new subband opens.Quasidonor states are still observed for either short [(a) bottomcurve] or long [(b) bottom curve] quantum wires when f=0.5.

1750 ARVIND KUMAR AND PHILIP F. BAGWELL

the quasidonor states still give rise to a broadened con-ductance dip before the new channel opens. The neteffect of this broadened conductance dip for f =0.5 isthat the average conductance is so suppressed before theopening of- each new channel that conductance dropsafter the new channel opens are not observed. The insetof Fig. 2(b), an expanded view of the lightly boxed region,shows clearly the drop in conductance for f=0.5 beforethe second subband channel opens.

To better understand this transition from the nearlyballistic conductance in Fig. 2(a) to the diffusive subbandstructure in Fig. 2(b), we examined the variation of theconductance with the length of the disordered region.The average conductance of 100 parallel wires containingonly repulsive scatterers (f=0.0) is plotted versus L inFig. 3(a). The Fermi level is placed at energies just below(dashed), directly on (solid), and just above (dot-dashed)the second and third subband minima. The average con-ductance decreases roughly exponentially with length ateach value of the Fermi energy, as in one-dimensionalelectron localization theory. ' For short disorderedsegments L, the average conductance is seen always to in-crease with Fermi energy. However, as the disordered re-gion is made longer, a crossover length L, is found suchthat when L & L, the average conductance falls after theFermi energy passes through a new subband minimum.Consistent with this result, the electron localizationlength g, found from (G ) =exp( L/g), is —appreciablyshorter just after the Fermi energy moves into a newquasi-one-dimensional subband. This decrease in locali-zation length is systematic and repeats around each newsubband minimum in Fig. 3(a). If the scatterers are madestronger by increasing y;, the point at which the curves"cross over" occurs for a shorter length L, of the disor-dered region.

In Fig. 3(b) we plot the ensemble-averaged conduc-tance versus length when f=0.5 for the same energies asin Fig. 3(a). Quantum diffusion is still evident, since theconductance still decreases roughly exponentially with L.But in contrast to Fig. 3(a), there is no crossing over ofthe conductance versus length curves so that g is roughlythe same (or increases) for increasing values of the Fermienergy. The conductance drops versus Fermi energywhen f=0.5 in Figs. 2(a) and 2(b), therefore, do not de-pend on electron "localization" phenomena, and can beseen in the conductance of each individual ensemblemember when only a few scatterers are present. Indeed,conductance drops of this sort occur if only one attractivescatterer is present in a single wire, and thereforeclearly do not depend on multiple refiections betweendifferent scatterers. In contrast, when all of the scatterersare repulsive in Fig. 2(b), the conductance drops after asubband opens occur only if the electron motion is phasecoherent over a long enough segment of the conductor.

A drop in conductance due to enhanced scatteringafter the opening of a subband channel has been arguedpreviously, ' based on a Drude model in which thecollision time is modi6ed to account for scattering be-tween quantum channels. Calculating the electrontransmission along the conductor semiclassically, by ex-actly treating the quantum-mechanical scattering at each

impurity but neglecting the wave interference betweenscattering events, is a valid approximation in many cir-cumstances. However, Refs. 23 —31 also assume that theelectron scattering at each individual impurity can be cal-culated in the Born approximation. Although the Bornapproximation adequately describes single subband trans-port, " we feel that the Born approximation (or "golden-rule" scattering rate) probably cannot be used to describemultiple subband transport along a quantum wire.

Because wave interference between different scatteringevents is neglected in the Drude approximation, a drop in

CDOJ

10

CDC3

C3

C3

O10

0.9 E„1.0 E„1.1 E„ f=0.0

I ~ I . I ~ I

100 200 300 400 500Length L {nm)

CDOJ

OV)

C

10'-0

D

8 10-2

1.0 E„1.1 E f=0.5n

I ~ I ~ I ~ I

IOO 200 300 400 500Length L {nm)

FIG. 3. Average conductance vs length L for an array of 100parallel wires having (a) f=0.0 and (b) f=0.5. Six values of theFermi energy are shown: 0.9E2 and 0.9E3 (dashed lined), E2and E3 (solid line), and 1.1E2 and 1.1E3 (dot-dashed line). The"crossing over" of the conductance curves in (a) indicates amuch shorter localization length when the Fermi energy movesinto a new subband. Quasidonor states present in (b) depressthe conductance below each subband, so there is no crossingover of the conductance curves.

EVOLUTION OF THE QUANTIZED BALLISTIC CONDUCTANCE. . .

the Drude conductance after a subband crossing shouldoccur only if the transmission at each individual scattererdecreases as a new channel is opened. However, asshown, for example, in Refs. 3—6, the electron transmis-sion through a single impurity increases when a new sub-band channel is opened. Furthermore, the golden-rule(or first-Born-approximation) approach to calculatingscatteirng rates is an approximation depending only onthe square magnitude of the scattering potential, so thatattractive and repulsive scatterers erroneously give thesame subband structure of the conductance versus Fermienergy in the Born approximation. Finally, the Born ap-proximation explicitly breaks down at a subbandminimum, as each term in the Born series becomesinfinite. Treating the first term in this series in a "self-consistent Born approximation" does not remedy theproblem, since the change in transmission at a subbandcrossing is still in the wrong direction.

The transition from quantum diffusion to classicaldiffusion must be made by introducing additional phase-breaking scattering in the conductor. If this is done us-ing the method of Buttiker, and if the electron suffers aphase-breaking event with probability 1 between eachelastic-scattering event, the resulting classical diffusivetransport is equivalent to adding many constriction con-ductances in series (where each constriction also containsan elastic scatterer). The conductance versus Fermi ener-

gy of such a classical diffusive wire will then resembleFig. 2(a), i.e., it makes little qualitative diff'erence in theshape of the conductance versus Fermi energy whetherthe ensemble-averaged conductance is obtained by addingthe conductors classically in series or in parallel. Otherassumptions about how the electron phase is broken in-side the conductor may lead to different subband struc-tures for the conductance versus Fermi energy when thetransport is classical and diffusive. If the electron phasecould be disrupted without obtaining any extra resis-tance, such that one would add the multichannel "R /1 '

Landauer resistances in series to approach classicaldiffusive behavior, ' the ensemble-averaged conduc-tance would then resemble the "four-probe" result shownin Fig. 8 of Ref. 4. There should be little qualitativedifference in the dependence of conductance versus Fermienergy between quantum diffusion and classical diffusionif some fraction of the scatterers in the wire are attrac-tive, since the transmission through each individual at-tractive scatterer qualitatively resembles the coherentensemble-averaged transmission through the wire arrayin this case.

Many of the issues raised in this paper have also beenconsidered in connection with the quantized Hall effect.Differences between the electrical transmission throughrepulsive versus attractive scatterers have been of someinterest in the quantized Hall effect. Similarly, statesanalogous to the "quasidonor levels" of this paper canoccur due to the formation of "quasibound states" at alocal widening of the conductor in mesoscopic Hallcrosses. Electron "localization" in a narrow wire sub-ject to a magnetic field has also been studied.

IV. CONCLUSION

%e find that the electrical conductance versus Fermienergy in an array of disordered quasi-one-dimensionalwires can indeed evolve from the ballistic conductancesteps into conductance drops after a new subband opens,but for reasons totally different from those given in Refs.23 —31. In the Drude model, the conductance decreaseslinearly with the length of the wire so that the electronsare delocalized. In our calculation, we find that the elec-trical conductance in a disordered quasi-one-dimensionalwire decreases roughly exponentially with the length ofthe disordered region. The localization length associatedwith this decay is reduced when the Fermi energy crossesa confinement subband, leading to a new quantumdiffusive subband structure in arrays of long quantumwires. Therefore, we find that the occurrence of conduc-tance drops after a subband channel opening depends onquantum diffusion, and does not occur if the diffusion isclassical (as in the Drude model). Similar results shouldfollow from tight-binding transmission calculations ' ifcare is taken to exclude the formation of quasidonor lev-els in a locally attractive impurity.

These quasibound states in the attractive impurities aremanifest as pronounced conductance dips below eachsubband minimum, as reconfirmed for the ensemble-averaged conductance ' in Fig. 2 of this work. Thesesame conductance drops below a new subband alsoarise if only one or a few attractive impurities are presentin the wire. Therefore, they are essentially due tochanges in transmission occurring at a single attractiveimpurity site and do not depend on either quantum orclassical diffusion to occur, although their exact positionin Fermi energy can be somewhat modified due to elec-tron wave interference. If the potential energy is locallyattractive, so that quasidonor levels can form in an im-purity, the new subband structure arising from localiza-tion phenomena studied in this paper is modified by thedepressed transmission near the quasibound states. Inconclusion, our study shows that there is no unique"ensemble-averaged conductance" of a disordered quasi-one-dimensional wire. Instead, the ensemble-averagedconductance of such a wire depends strongly and qualita-tively on the type of disorder present.

Note added in proof. Since submission of this paper, wehave become aware of several studies treating the con-ductance of disordered narrow wires.

ACKNOWLEDGMENTS

We thank Terry P. Orlando, Dimitri A. Antoniadis,Henry I. Smith, and Kevin Delin for useful discussions.This work was sponsored by the U.S. Air Force Ofhce ofScientific Research under Grant No. AFOSR-88-0304.A.K. gratefully acknowledges support from the Semicon-ductor Research Corporation.

1752 ARVIND KUMAR AND PHILIP F. BAGWELL

'Present address: School of Electrical Engineering, PurdueUniversity, West Lafayette, IN 47907.

B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G. Wil-liamson, L. P. Kouwenhoven, D. van der Marel, and C. T.Foxon, Phys. Rev. Lett. 60, 848 (1988).

D. A. Wharam, T. J. Thornton, R. Newbury, M. Pepper, H.Ahrned, J. E. F. Frost, D. G. Hasko, D. C. Peacock, D. A.Ritchie, and G. A. C. Jones, J. Phys. C 21, L209 (1988).

C. S. Chu and R. S. Sorbello, Phys. Rev. 8 40, 5941 (1989); 38,7260 (1988).

4P. F. Bagwell, Phys. Rev. 8 41, 10 354 (1990)~

5P. F. Bagwell, J. Phys. Condens. Matter 2, 6179 (1990).E. Tekman and S. Ciraci, Phys. Rev. 8 42, 9098 (1990).

7A. Kumar and P. F. Bagwell, Phys. Rev. 8 43, 9012 (1991);A.Kumar and P. F. Bagwell, Solid State Commun. 75, 949(1990).

J. Masek, P. Lipavsky, and B. Kramer, J. Phys. Condens.Matter 1, 6395 (1989).

I. Kander, Y. Imry, and U. Sivan, Phys. Rev. 8 41, 12941(1990).

toF. M. Peeters, in Science and Engineering of 1- and 0- Dimensional Semiconductors, edited by S. Beaumont and C.Sotomayor- Torres (Plenum, New York, 1990).D. van der Marel and E. G. Haanappel, Phys. Rev. 8 39, 7811(1989).

' E. Tekman and S. Ciraci, Phys. Rev. B 40, 8559 (1989).S. He and S. Das Sarma, Phys. Rev. 8 40, 3379 (1989).

4A. Szafer and A. D. Stone, Phys. Rev. Lett. 62, 300 (1989).~5E. Castano and G. Kirczenow, Solid State Commun. 70, 801

(1989).M. Cahay, M. McLennan, and S. Datta, Phys. Rev. B 37,10125 (1988); S. Datta, M. Cahay, and M. McLennan, ibid.36, 5655 (1987).M. Cahay, S. Bandyopadhyay, M. A. Osman, and H. L. Gru-bin, Surf. Sci. 228, 301 (1990); M. Cahay, P. Marzolf, and S.Bandyopadhyay, in Computational Electronics: Semiconduc-tor and Deuice Simulation, edited by K. Hess, J. P. Leburton,and U. Ravioli (Kluwer Academic, Boston, 1990), p. 263; S.Bandyopadhyay and M. Cahay, in ibid. , p. 223.P. A. Lee and A. D. Stone, Phys. Rev. Lett. 55, 1622 (1985).A. C. Warren, D. A. Antoniadis, and H. I. Smith„Phys. Rev.Lett. 56, 1858 (1986).K. Ismail, D. A. Antoniadis, and H. I. Smith, Appl. Phys.Lett. 54, 1130 (1989).J. R. Gao, C. de Graaf, J. Caro, S. Radelaar, M. Offenberg, V.Lauer, J. Singleton, T. J. B.M. Janssen, and J. A. A. J. Peren-boom, Phys. Rev. 8 41, 12 315 (1990).J. Faist, P. Gueret, and H. Rothuizen, Phys. Rev. 8 42, 3217(1990).B. E. Sernelius, K. F. Berggren, M. Tomak, and C. McFad-den, J. Phys. C 18, 225 (1985).S. Das Sarma and X. C. Xie, Phys. Rev. 8 35, 9875 (1987).

~5M. J. Kearney and P. N. Butcher, J. Phys. C 20, 47 (1987).N. Trivedi and N. W. Ashcroft, Phys. Rev. 8 38, 12298(1989)~

P. F. Bagwell and T. P. Orlando, Phys. Rev. 8 40, 3735 (1989).28P. Vasilopoulos and F. M. Peeters, Phys. Rev. 8 40, 10079

(1989).T. P. Orlando, P. F. Bagwell, R. A. Ghanbari, and K. Ismail,in Electronic Properties of Multilayers and Low Dimensional-Semiconductor Structures, edited by J. M. Chamberlain, L.Eaves, and J. C. Portal (Plenum, London, 1990); P. F.Bagwell, D. A. Antoniadis, and T. P. Orlando, QuantumMechanical and Nonstationary Transport Phenomenon in

Nanostructured Silicon Inversion Layers, in Advanced MOSDeUice Physics, edited by N. Einspruch and G. Gildenblat(Academic, San Diego, 1989), Vol. 18.

3oS. Y. Qiu and M. V. Jaric, Bull. Am. Phys. Soc. 35, 492 (1990);W. Bao and Z. Tesanovic, ibid. 35, 492 (1990).M. Suhrke, W. Wilke, and R. Keiper, J. Phys. Condens.Matter 2, 6743 (1990).Y. Imry, in Directions in Condensed Matter Physics, edited byG. Grinstein and G. Mazenko (World Scientific, Singapore,1986).

M. Biittiker, Phys. Rev. Lett. 57, 1761 (1986).R. Landauer, Z. Phys. 8 68, 217 (1987).M. Biittiker, Y. Imry, R. Landauer, and S. Pinhas, Phys. Rev.8 31, 6207 (1985).M. Biittiker, IBM J. Res. Dev. 32, 317 (1988).

37M. Biittiker, in Electronic Properties of Multilayers and LowDimensional Semiconductor Structures (Ref. 29).R. Landauer, J. Phys. Condens. Matter 1, 8099 (1989).R. Landauer, in Analogies in Optics and Micro-Electronics,edited by W. van Haeringen and D. Lenstra (KluwerAcademic, Dordrecht, 1990).J. G. Williamson, C. E. Timmering, C.J.P.M. Harmans, J. J.Harris, and C. T. Foxon, Phys. Rev. B 42, 7675 (1990).L. I. Glazman, G. B. Lesovik, D. E. Khmel'nitskii, and R. I.Shekhter, Pis'ma Zh. Eksp. Teor. Fiz. 48, 218 (1988) [JETPLett. 48, 238 (1988)].

4~A. Yacoby and Y. Imry, Phys. Rev. B 41, 5341 (1990).M. Buttiker, Phys. Rev. 8 41, 7906 (1990).

44H. Sakaki, Jpn. J. Appl. Phys. 19, L305 (1980).45A. Kumar, S. E. Laux, and F. Stern, Appl. Phys. Lett. 54,

1270 (1989).R. Landauer, Philos. Mag. 21, 863 (1970); P. W. Anderson, D.J. Thouless, E. Abrahams, and D. S. Fisher, Phys. Rev. 8 22,3519 (1980).

The exponential decrease of conductance with L at E=E2and E=E3 we find surprising, since Ref. 7 shows that waveinterference between propagating modes does not determinethe electron transmission properties at a subband minimum.The "localization" must therefore arise from a differentmechanism.

48We note that more ensemble members are needed to averageover fluctuations if attractive scatterers are present, so thatthe curves in Fig. 3(b) are not as smooth as the ones in Fig.3(a). We have also explored making all the scatterers attrac-tive (f =1.0), and find behavior qualitatively similar to thef=0.5 case. However, the average transmission is generallylowered as a greater fraction of the scatterers are made at-tractive.R. Landauer, Philos. Mag. 21, 863 (1970).

5OM. Biittiker, Phys. Rev. B 33, 3020 (1986).P. F. Bagwell and T. P. Orlando, Phys. Rev. B 40, 1456 (1989).P. F. Bagwell, Ph.D. thesis, Massachusetts Institute of Tech-nology, 1991.R. J. Haug, R. R. Gerhardts, K. v. Klitzing, and K. Ploog,Phys. Rev. Lett. 59, 1349 (1987).

54R. L. Schult, H. W. Wyld, and D. G. Ravenhall, Phys. Rev. 841, 12 760 (1990).J.M. Kinaret and P. A. Lee, Phys. Rev. 8 43, 3847 (1991).

56Y. Joe and S. E. Ulloa, in Nanostructures: Fabrications inPhysics, edited by T. P. Smith III, S. D. Burger, D. Kern, andH. Craighead, MRS Extended Abstract No. 26 (MaterialsResearch Society, Pittsburgh, 1990),p. 47, and unpublished.H. U. Baranger, Phys. Rev. 8 42, 11 479 (1990).

58G. Y. Hu and R. F. O' Connell, Phys. Rev. 8 43, 12 341 (1991).

EVOLUTION OF THE QUANTIZED BALLISTIC CONDUCTANCE. . . 1753

J. A. Nixon, J. H. Davies, and H. U. Baranger, Phys. Rev. B43, 12 638 (1991).L. Wang, S. Feng, and Y. Zhu, Bull. Am. Phys. Soc. 36, 357(1991).

~V. Mitin, Bull. Am. Phys. Soc. 36, 412 (1991).~S. Choudhuri, S. Bandyopadhyay, and M. Cahay, in Proceed-

ings of the Second International Symposium on Ãanostructures

and Mesoscopic Systems, edited by W. P. Kirk (Academic,New York, in press).

6 H. Kasai, K. Mitsutake, and A. Okiji, J. Phys. Soc. Jpn. (to bepublished).H. Tamura and T. Ando (unpublished).M. K. Schwalm and W. A. Schwalm (unpublished).