REVISTA MEXICANA DE FíSICA -l4 SUPLEMENTO 3. 7-13 DICIEMBRE 1998

Quantum and classical ballistic transport in a chaotic 2D electron channel

G.A. Luna-Acosta, M.A. Rodríguez, and A. KrokhinInstilllto de Física. Universidad Autónoma de PueblaApartado postal J--I8, 72579 Puebla. Pue., Mexico

Kyungsun NaCefllerjor Sludies in Statl.\'tical Mechanics amI Complex System. U"iI,etsity ofTexa.<i al Austin

AUSlill, Texas, 78712, USA

R.A. MéndezInstituto de P(siea, Universidad Nacional Autónoma de Aféxico

Apartado postal20-3M, 01000 México, D.F, Mexico

Recibido el 20 de enero de 1998: aceptado el 25 de enero de 1998

We rcview recent resull:;. conccrning (he quantum and c1assical dynamical propcrtics of hallistic c1cclrons in a ripplc channel. thcir transponpropcrties and its c1assical-quantum corrcspondcncc. In the classical rcgimc 01'a tinitc channc!. a signaturc 01'chaos is faund in the bchaviorof the resistance. The classical-quantum correspondence is analY7cd in terms of "quantum Poincaré plots", energy level slatistics, and certainfeatures of lhe energy-band speclra.

Keywords: Quanrum chaos; mcsoscopic systellls; ballistic lranspon

Presenramos resultados recientes sobre las propiedades cuánlicas y clásicas de electrones balíslicos en un canal ondulado, sus propiedadesde transporte y su correspondencia ciásico-cuánlica. En el regimen clásico de un canal finito, una manifestación de caos es encontrada enel comportamiento de la resistencia. La correspondencia clásico-cuántica se analiza en terminos de los "mapas de Poincaré", estadístira deniveles, y ciertas características de el espectro de bandas de energía.

Descriptores: Caos cuántico: sistemas mesoscópicos; transporte balístico

PAes: 05.45.+b; 73.23.-b; 73.20.0x

rlGURE 1. The ripplcd channcl model. Tite lOp boundary is givenby y = í + VSill (2rrx).

tures 14) has made possible the experimental realization al'mathemarica! modcls of chaos, in particular, of chaotic bil-Iiards. A relevant observable quantitiy is the linear response,e.g., the conductancc, Here we discuss sorne dynamical nndtransport pmperties 01'a system which can serve as a model oí"a rnesoscopic device in the ballistic regimc, c1assical or quan.tum. The model is a 2D eleclron channel rormed between asinusoidal (rippledJ wall and a nat wall (see Fig. IJ. The elec-tmn motion is assumed to be ballistic and the collisions withthe walls to be spccular. In the next sectioll wc will considcrthe c1assiCilldcscriplion of the motion as wcll as its transmis-sion propcrtics ami relate the behavior oí"the rcsistance to the

1. Inlroduction

Thcrc are several important questions rcgarding chaotic dc-terministic systems. Wc know that the dcllnition of chaos,whose main ingredient is sensitive dependcnce to initial con-ditions, does not apply to quantum systcrns, mainly bccauscHcisenberg's principie forbids us to spccify simultancouslyaH phase space variables and because the Schrodinger equa-tion is linear (See, e.g., Rcfs. I and 2). Nonetheless, one as-sumes that there must be sorne corrcspondcncc bctwccn theclassÍcal system and its quantum countcrpart evcn for chaoticsystems. One then asks about the quantum manifcstations, orsignatures, of c1assical chaos. This is onc of the directions 01'what is known as quantum chaos, where thcre has becn muchprogress in the last Iwenty years but stillthere are many openand fundamental qucstions (sec, e.g., Rcf. 3). On (he otherhand, one may ask about how c1assical chaos manifests it-selr in the measurable 4uanlities 01' a given physical syslem.Morcver, the systcm undcrconsideration may necd a classicaldescription to explain sorne effects and a quantum decriptionto explain others. Mesoscopie systems are the ideal arena fmstudying the cffeet of c1assical chaos on ils quantum pmp-crties as well as on its transport propcrties. Stale of the arttcchniqucs in the fabrication of semiconductor microstruc-

y

ír1

8 QUANTUM AND CLASSICAL BALLlSTIC TRANSPORT IN A CHAOTIC lO ELECTRON CHANNEL

10 x)O ••

O.OJ

••2

R _ V 1.46

0.0 I

"[2]: .- .....;. ..

ti • • 11 •• 11.'IO-'}

0.99'0.00

12

•-~ 4

oo

1.000

T

0.9911

1.0

0.5

T0.00,00 0.02 0,0"" 0.06 0,08 0.10

O.,1.0

(b)

O.,

O.,0.0 0.2 O., O., O., 1.0

VFIGURE 2. Transmissivity as a funclion of the ripple amplitude for(a) narrow (1 = 0.1) and (b) wide (1 = 5.0) channels.

Vunderlying dynamical properlies, yielding a signalure ofchaos for ballislic transport. The quantum dcscription isskelehed in Seel. 3. In Secl. 4. we discuss the quanlum-classical correspondencc of this system and in Sect. 5 v.'e giveour conclusions. The details of lhis work appear in Refs. 5-7.

FIGURE 3. (Upper panel) Retleclivity of the narrow ehannel.(Lower panel) Transmissivity of the wide ehannel. Insert: reRee-tivity.

2, Transport quantities and dynamical signa-tures, Classical regime

Wilhin the framework of lhe classieal billiard model [8), varoious effects in narrow electron channels can be understoodif the number of propagaling modes is large enough (even 3may be suffieient), see, e.g .. ReL 9. In orderIo ealeulale lrans.port quantilies. (he incoming distribuirian of particlcs enter-ing lhe lefl side of lhe ehannel musl be speeified. Assumingthe simplest conncction lo the ¡cad and complete thermal.izatian of carrices in the rcservoirs, the ¡n¡lial distribution isgiven by p(oo) = (110/21) ros (00)' Here, "o is lhe angle ofineidence and 110 is Ihe number of eleetrons (typieally la').

The motian is nol integrable in the channcl since the .f-

component of the momentum ceases to be conserved as soonas lhe ripple is luroed on. Given an in¡tial angle and height,(00, Yo) al lhe entrance of lhe ehannel, X = O, wc follownumcrieally lhe trajeelory wilhin Ihe ehannel lill il exil5 lothe lefl or lo lhe righl, and record lhe exil angle, lhe lengthof ils trajeetory wilhin Ihe ehannel and ils number of colli.sions wilh lhe wal!. This is done for eaeh of lhe 105 particlesentering lhe ehanne!. Figure 2 shows the IransmissivilY fortwo representa tive channel geometries: a narrow and a wideehanne!. Jt is clear lhal their fealures are draSlieally difer-enl. For example, the transmission in the narrow channel de-cays monotonically as the rippled amplitude v is increased,whereas for the wide ehannel il does nol. This fealure andothers, related to the mean dwelling time and mean numberofcollisions, are analyzcd in Ref. 5. Here wc conccntratc on thebehavior of lhe transmission in the regime where lhe rippleamplitude eould be considered as a perturbation. Inspeetion

01' Fig. 3 shows lhe dependence of the resistance on Ihe rip-pie amplitude: il inereases as (vh)l.46 and as (Vh)'.24 forlhe narrow and wide ehannels, respeetively. We want lo relateIhis differenee to Ihe underlying motion 01' the elassieal bal.liSlie eleelrons in Ihe ehanne!. In order lo get a panorama 01'their dynamies we plOl lhe position x and lhe x.eomponentof Ihe momentum Pn right after eaeh eollision Wilh lhe 10Pwall for several initial eonditions (aboul 20). Sinee lhe energyis eonserved, Ihe pairs (Pn, Xn) define uniquely!he motion.The mOlion in an infinitcly long channel [X = mod (2;r))is lhen a dynamieal sySlem, where plOlS of (Pn, Xn) formil5 Poinearé map [lOJ. Although lhe transmission was eom.puted wilh a finile ehannel (IWO periods), lhe Poinearé plOlShclp us to understand the transmission properties. Clearly,for nat ehannels (v = O) lhe Poinearé map (Pn, Xn) givesstraight horizontallincs sincc Po is constanl. As soon as theripplc is turoed on lhe phase spaee ehanges qualitatively, seeFig. 4a. As v is inereased in the range (O < v < 0.002)lhe phase spaee gradually ehanges from thal of an unper.lurbed ID pendulum (Fig. 4a) to thal of a perturbed ehaolieID pendulum (Fig. 4h). The ellipses correspond to trajee 10-ries eolliding almo SI perpendieularly with lhe ehannel walls,executing Iibrational motion in the wider region of the chan-ne!. The almoSl straight line aboye and below lhe separatrixof Fig. 4a eorrespond lo lranslalional orbits, moving alwaysforward (baekward) if Pn > O (Pn < O). Henee, relleetion isduc only to the librational orbits. As v ¡ncreases in [he range0.002 < v < 0.025 lhe separatrix beeomes ehaotie and ehaosalso develops in lhe upper and lower part oflhe Poincaré plol.These lhree ehaotic zones are separated by KAM curves [lOJwhich forbid the diffusion of chaotic orbiLs from one chaoticsea into anolher. Thus, the orbils eontribuling to refleetionmusl lie on lhe ellipses or on Ihe ehaotie separalrix. However

Rev. Mex. Fís. 44 S3 (1998) 7-13

G.A. LUNA-ACOSTA et al. 9

-

JI:::.

-' ".:.:..-.,",::':': ~~;~:>:::==:=c=~e""''''. "0_ • ..

1.0

0.5

0.0

-o.S

.1.00.0 0.2 0.4 0.6 0.8 1.0 0.0

0.0 0.2 0.4 0.6 0.8 1.0

1.0

P" P"0.5

0.0 (@)/

FIGURE 4. Poincaré surfaces of section for narrow channels.(a) ~ = 0.001, (b) v = 0.02.

orbils on the ehaolie separatrix transmil with roughly a 50%probability sinee they exeeute random walk-like mOlion.Thus, the dominan! eontribution to refleelion in the wholeregime O < v < 0.025 comes from lbe librational orbits.Since the 1I is small in this regime. we may use the adia-batic invariance of the action along the y direction lo obtainan analytieal estimate for the dependen ce of the refleetion,R = 1 - T, on the ripple amplitude. The result, as shown inReL 5, is R ~ vl.5, whieh agrees very well wilh the numeri-cal experiments, R '" V1.46.

For the wide ehannel, the motion is radieally different,(see Fig. 5.) Here, even for very small values of v, e.g.,v = 0.001, there is already a large porlion of ehaotie phasespace because as the channel becomes wider, initially closetrajectories gel fanher apart between collisions (it is the prod-uet v, whieh gives lhe measure of ehaos). Sinee a ehaoliesignal is undistinguishable [rom random signal. wc may ex-pect the chaotic motioo of ballistic elcctrons in our ripplcdehannel to be equivalenl lo that of ballistie eleelrons eollid-ing Wilh an truly random (rough) profile. This is indeed lhecase, as can be shown by carrying out a power spcctrum anal-

FIGURE 5. Poincaré surfacc of section for widc channcls h =5.0). (a) v = 0.001. (b) v = 0.04.

ysis of the effeelive profile. For regular (ehaolie) trajeetoriesin phasc space, the power spectra gives quasiperiodic behav-ior (broadband noise). The rough lop boundary can be ex-pressed as y = 'Y+ ,¡(x), where ry(x) is a random funetionand Iry(x)1 « 1. A Taylor series funetional expansion (losecond order) of the transmission function gives a constantterm, a linear and a quadratie funelional of ry(x). Averagingover all possible lrajectories, the linear lerm disappears, giv-ing (T) ~ 1 - eonstant x r¡', where r¡ '" J(,¡(x)') isthe rms height of lhe roughness. Using lhe effeetive anal-ogy belween ehaotie and random eollisions we identify r¡with lhe amplilude of the random profile v and eoneludelhat the transmission for ehaotie motion deeays quadralieallywilh v. It follows that lhe resistan ce should increase quadral-ieally wilh the amplitude of lbe ripple. The numerieal exper-iments show, instead, that the resistance increases as v<l.24.

The rcason for such discrepancy is that the number of colli-sions in short wide channels is nOl large enough to providea good statistical average over the ensemble of effective can-dom profiles. By inereasing the length 50%, our numeriealexperiments show that R increases as V2.19. An experimen-

Rev. Mex. Fí.<. 44 53 (1998) 7- 13

10 QUANTUM AND CLASSICAL BALLlSTlC TRANSPORT IN A C1IAOTlC 2)) ELECTRON CHANNEL

FIGURE 6. Encrgy-band structure for the wide channel h = 1.5)with v = 0.04. E1 = (rlrrh) , /2.

FIGURE 7. Encrgy-band structure for narrow channcls (')' = 0.1).(a) v = 0.001. (b) v = 0.02.

percent of the width). Here, perturbation theory to secondarder was necessary lO find good agreemcnt with nurncri-cal data. lt is intercsting to note that the expansion pararnc-tcr for (he quantulll perturbation theory turos out to be 1//''(2.

whereas in ilS classical description. the perturbation param-eter is vh, This feature is similar 10 the \Vell kno\Vn kiekedrotor model (sec. e.g., Ref. 12), where classically. the non-linear parameter is the strength of the kiek l\, but quantummechanically it is a comhination of I< and the time betweenkicks. As the ripplc amplitude inereases (see Fig. 7b) the re-pulsion is so strong that the lowest energy banus dissappear.ami only a discrcte sel of cigenstates are allowed lo cxisl inthc chanllcl, conducting no currcnt.

k

(b)

E/E?

o. ,k

-o . .,

E/E~

o. ,

(al

E/E~

, ,

The problem of calculating quantum-mechanically the trans-mission propcrtics ror the rippled channel involvcs thenumerical evaluation of the S matrix together with theLandauer-Buttiker formula (see. e.g., ReL 4). Here, \Vecon-sider instead the quantum dcscription of ballistic electronsin an infinitely long (periodic) rippled channel.lt is an eigen-value problem for the Schródingerequation subject to Dirich-let boundary conditions. The solution was carried out inReL 6. \Vhere the reader is referred to for details. Recallthat in the e1assical deseripJion the top profile \Vas modu-lated by a sin (2rrx) function. In the quantum description.\Ve no\V chose a cos (2,'I'X) profile for symmetry reasons.We approached the problem by first performing a eoordi-nate transformation, sueh that both profiles appear nat. Atransformation that accomplishes this is: u = x and v =y/b + vcos (2JTx)]. In these ne\V coordinates the bound-ary conditions for the new wave function w(u, v) are simply'1'(11.0) = O = '1'(11,1) but the covariant Laplacian in thenew (curvilincar) coordinate becomes quite complicated.

Because the Hamiltonian is a periodic function ofthe coordinate u fOI" the infinitely long channel, the en-ergy eigenstates satisfy B1och's thcorem. WE(U, v) =ex!, (iku)<t>d u, v), \Vhere k(E) is the Bloch \Vavevector andthe state <t>dl1, v) is the Bloch funetion \Vith the periodie-ity of the wall. The wave function is then cxpandcd in adouble Fourier series with each term satisfying the Dirich-let boundary conditions. Thc final step is the numerical sol u-tion af the eigenvalue problem in the (11, v) coordinates. Forgiven values of v and l' we can then obtain the correspond-ing energy band spectra for the electron. Figure 6 sho\Vs theenergy-band speetra for the \Vide channel ("( = 1.5) \Vithv = 0.04. C1early. the energy for the nat channel is givenby E = (r,2/2)(k2 + (JT1nh)2) and the energy band spcctraobtained from it. is very similar to Fig. 6. except with no levelrepulsion. This appears to be in c1ear contrast with the classi-cal description where the same value of v changed drasticallythe dynamics; [rom regular to globally chaotic. see Fig. 5. Inthe next section \Veshall come back to this point as \Ve lookat the c1assical-quantum correspondence of this system. An-alytical calculations using quantum degenerate perturbationtheory showed to be in excellcnt agrcement, even to first or-der. \Vith the numerieal results in the energy range of Fig. 6.

Representative speetra for the narro\V channel (-y = 0.1)is sho\Vn in Fig. 7 for v = 0.001 and 0,02, Inspection afFig. 7a shows that the splitting of levels is largc for ccrtaineigenstates even for such small ripple amplitude (only one

3. Quantum dynamics

tally measurable quanJity is the resisJivity p of thechannc!,which, according to Ihe well-known Landauer formula [11 J.p _ (1 - T) /T. relates transmissivity with resisiJivity. Ac-cording 10 Lhisanalysis we conclude that resistivity ¡ncreases(in the region of small v) proporJionally to v3/2 or v2• de-pending on whether the underlying dynamics is regular orchaotic, respcctivcly.

Re\'. Mex. Fis. 44 S3 (1998) 7-13

G.A. LUNA.ACOSTA el al. 11

FIGURE 8. Poincaré surface of section for the narrow channcl withv = 0.02.

-.....~---------1

;¡;

11'¡

!-Pr:L..JI f: ¡.,

Xno

o

-1-lf

4. Quantum-c1assical correspondence

FIGURE 9. Quantum Poincaré maps (or the narrow channel withv = 0.02 and k = O. (a), (b). (e), and (d) eorrespond lo energylevels 1, 2, 7, and 7, respectively. PE = (2E)1/2, where E is theenergy of the eigenstate.

In lhe e1assiea! analysis, we were able 10understand lhe trans-port properties of lhe ehanne! wilh lhe aid of the Poinearéplots in phase spaee. As il is well known, in quantum me-chanics, phase space trajectorics have no rncaning due to theincertainty principIe. However, this does not preclude the ex-istence o( a rneaningfull quanturn rnechanical phase spaceof probablities. Wigner's funetion densily [13J pw(q,p) =J ds >ji' (q + s) >ji (q - s) was lhe tirsl quanlity to provide uswith this eoneept. It has several properlies, one of whieh isthat its integration over the rnomentum (coordinate) is theamplitude squared of the wave function in coordinatc (mo-mentum) representation. Howcvcr, Pw can acquire ncgativcvalues. To remedy lhis problem, Husimi [14J smoothed ilwith a minimun uncertainty gaussian wave packet centeredal a phase poinl (qo,Po). This was shown lObe equivalenltothe square of the projection of the eigenstate with a coherentslate a, Pll(qo, po) =1 (a 1 >ji) 12

Our system has lwo degrees of freedom, (x,y), and in or-der lo form a "quanlum Poinearé map" (QPM) we must alsoseleel a Poinearé surfaee of seetion. In lhe elassieal plols, lhesurfaee of seelion was ehosen to be lhe lop protile, ahhoughwe eould equally have ehosen lhe lower, flal protile. For lheQPMs, it is technically more convenient to choose the lowerprofilc, which also gives syrnrnetrical plots with rcspect to x.We cannot simply set y :::::O in the wave function bccause ofthe Dirichlct boundary conditions on the wall, but we can beas close to the wall as desired by considering a Taylor seriesexpansion to first order. Thc first term is zero and the secondis S(x)y, where S(x) = 8>ji /8y Iy=o. This means lhal veryc10se tu the surface, the wave function is separable and wccan use lhe Husimi density funetion detined for S(x). In ad-dition, we musl take eare of the periodieily of lhe ehannel,lhe delails of whieh are explained in ReL 6. Figure 8 shows

lhe e1assieal Poinearé plot for the narrow ehannel wilh rippleamplilude v = 0.02. This was oblained for lhe same geomel-rieal paramelers as Fig. 4b but wilh a eosine protile (insteadof sine) and wilh the bOllOm protile as the Poinearé surfaee.This ehange is in order lOcompare with the QPMs. Figure 9shows lhe QPMs of lhe eigenslales 1, 2, 7, and 8 al k = O.The first 2 statcs are gaussian states; i,e., SHO eigensta[es, inagrcement with the classical motion associated with the libra-brational motion observed in the central region of the classi-cal map. As the energy inereases (2nd lo 4th levels not shownhere) lhe supporloflhe quanlum states exlend lo lhe larger el-lipses, characteristic of 10 pendulum states. Further ¡ncreasein the energy forces lhe stales [o have their support on the re-gion corresponding to the c1assical chaotic separatrix (see the71hand 81h states). NOle lhat the 81h slale has lhe highesl den-sily preeisely on lhe unslable tixed poinl of period one. lhesequanlum-elassieal corresponden ce should be refleeled in lhespeelra also: the equidislanl spaeing of the 4 lowesl Iyingslales corresponds lo lhe SHO level spaeing. NOle, moreover,lhal the spaeing remains almost equidistant also for lhe 51h lo81h slales. We know that lhe spaeing in the libralional regimedecreases in the neighborhood of the separtrix of an inte-grable pendulum. We believe lhis is the effeel of level repu!-sion induced by the dynamical chaos in the separatrix region.The small ehaotie struelure (Iike lhe liny higher order reso-nant islands) obscrved in the classical maps is not reflectedin these QPMs beeause the size of lhe minimum uneerlainlyarca h is mueh larger than them. Figures 10 and 11 show lheQPMs al k = 0, for levels 452, 453, and 1005, respeetively.Figure 12 is the eorresponding e1assieal plot. NOliee how lhelevels 452 and 453 have their support on lhe period 8 unslableand slable tixed poinls, respeelively. The eigenslale 1005 lies

Rev. Mex. Fir. 44 S3 (1998) 7-13

12 QUANTUM ANO CLASSICAL BALLlSTlC TRANSPORT IN A CHAOTIC 20 ELECTRON CHANNEL

5. Conclusions

5Jo

1.4

0.2

0.6

(s)

In this article we have reviewed sorne recent results concern-ing the quantum and dynamieal properties of ballistie elec-trons in a rippled channel, as well as their effect 00 its trans-port properties. It was shown how an analysis of the phasespaee dynamies reveals a transport signature of ehaos in the

FIGURE 13. Level spacing distribution P(s) Cor the wide chan-nel (¡ = 1.5) with /.1 = 0.04. The total statistics is 2(x)() levels(200 < N < 2200) at k = 0.01. The dotled (,olid) line corre-sponds lo the Poisson (Wigner-Dyson) distribulion.

raodom matrix theory conjecture [17], which states thal thelevel spaeing distribution (LSD) should be Poissonian mWigner-Dyson, if the underlying c1assical dynamics is reg-ular or ehaolie. respeetively. We have performed the levelspacing statistics to check if OUT system supports the coo-jeeture [7]. Fm laek of spaee, we willlimit ourselves here toexamine a few aspeets of this analysis. Consider the energy-band speetra (Fig. 6) of the wide ehannel with v = 0.04.Reeall that its classieal dynamies is globally ehaotic, (seeFig. Sb). The LSD is earried out for a given fixed value ofthe Bloch wave vector. Figure 13 eorrcsponds to the LSD fork = 0.01 ealculated fm 2000 levels. We see thal the data fitsthe Wigner-Dyson distribution for Gaussian orthogonal en-sembles (GOE). in agreement with the eonjeeture. Extensivenumerical experiments showed that such a distribution is thesame for any value of k in the Brillouin lOne exeepl for k = Oand k = 1/2. The LSD fm k = O (not shown) approaehes.instead, the Poissonian distribution, characteristic of regulardynamies. This apparent diserepaney is resolved by notieingthal: (1) the energy eigenstate is also a parity eigenstale fork = O and k = 1/2 but is not fm any other k and (2) theconjecture assumes that a1l discrete syrnmetries have becoremoved. By ehosing definite parity states. the GOE distribu-tion is reeovered. We also found, that the LSD eorrespondingto the speetra of Fig. 7b gives a distribution which is interrne-diate betwecn Wigner-Dyson and Poissonian. This is consis-tent with the classieal dynamies having a mixed phase spaee.(see Fig. 4b).

x

..

P,"

px

entirely on the ehaotie sea of the eorresponding elassieal map.The appearanee of small high density struetures of Fig. 11is probably the phase space manifestation of the so-ealled"sears" predieted by Heller [15], whieh are the sites of un-stable periodic orbits. There is ao infinite numbcr oC unstableperiodic points in the chaotic sea bUI it seems that the quan-tum states will find their higher support on the lowest periodunstable points.

A standard proeedure for studying classieal-quantum eor-respondenee of ehaotie systems is to analyze its level statis-tics (see. e.g .• Ref. 16). This eorrespondenee is based on the

FIGURE 12. Poincaré surface oC section Cor the narrow channeJwith (v = 0.04).

xFIGURE 11. Quantum Poincaré map Cor the narrow channel withv = 0.04 and k = O for the level IDOS.

FIGURE 10. Quantum Poincaré maps for the narrow channel withv = 0.04 and k = O. (a) level452. (b) level453.

Rev. Mex. Pis. 44 83 (1998) 7-13

G.A. LUNA-ArOSTA et al. 13

behavior of!he resistance. Similarly, a c10se correspondencewas found between the c1assical phase space, the quantumphase space of probablilies, and fealures of lhe cncrgy-bandspcctra. Moreover, the quantum signature of chaos was cvi-dent in !he level spacing statistics of the band-energy speclra.

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8. C.W. Beenaker and H, Houten. in Solid Srate Physics: Ad-vanees in Research and ApplicariollS, editcd by H. Ehrcnreichand OS. Tumbull (Acadcmic, New York. 1991).Vol.44. p. 1.

Acknowledgments

We acknowlcdge partial sup0r! from NSF-CONACyT pro-gram, No. E120-3341.

9. K.L. Shepard. rvt.L. Roukes, and B.P. Van der Gaag. Phys. Hel'.Le//. 64 (1990) 1154.

10. A.l Lichtenberg and M.A. Lieberman. Regular and SrochasticMorion, (Springer-Yerlag, New York. 1982).

11. R. Landauer. IHM J. Res. Dev. 1 (1957) 223; Phi/os. Mag. 21(1970) 863.

12. F.M.lzrai1ev.Phys. Rep. 196 (1990) 299.

13. E.P.Wigner.Phys. Rev. 40 (1932) 749.

14. K. Husimi. Prac. Phys. Math Soco Jpn. 22 (1940) 264.

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