fluctuations of the inverse compressibility in disordered electron systems

PHYSICAL REVIEW B 15 JANUARY 1999-IIVOLUME 59, NUMBER 4

Fluctuations of the inverse compressibility in disordered electron systems

Gun Sang Jeon, Seongho Wu, and M. Y. ChoiDepartment of Physics and Center for Theoretical Physics, Seoul National University, Seoul 151-742, Korea

Hyun-Woo LeeCenter for Theoretical Physics, Seoul National University, Seoul 151-742, Korea

~Received 4 August 1998!

We consider interacting electrons in one- and two-dimensional disordered systems and observe within theHartree-Fock approximation that relative fluctuations of the inverse compressibility in general decrease withthe hopping strength, signaling metal-insulator transitions in the systems. Particularly, in one dimension theperturbative approach reveals two regimes where the behaviors of the relative fluctuations with the system sizeare different: In the localized regime the relative fluctuations are independent of the size while in the oppositelimit corresponding to the diffusive regime the relative fluctuations scale linearly with the system size. Tocheck the validity of the Hartree-Fock approximation, we also obtain exact results for small systems via theLanczos method, which suggests that for intermediate values of the hopping strength the relative fluctuationsare overestimated in the Hartree-Fock approximation.@S0163-1829~99!02804-0#

a

cas-diw

nu

efluThdolech

ththn

re

rahibtaths-

ostehe

ro

ng

ibu-ith

eak.

onsingdy

on,ds

ua-rma-eri-ntal

ity,he-

uc-ingoss-ereity,nger,hashasity

hein-ichThe

ofthe

I. INTRODUCTION

Fluctuation properties in mesoscopic physics havetracted much attention during the past decades,1 one well-known example of which is the universal conductanfluctuations.2 Recently, experiments on two-dimension~2D! quantum dots3–6 have shown other interesting mesocopic fluctuations. In those experiments a small quantumis weakly coupled to two current leads with a gate, whichused to vary the electrostatic potential of the dot. At lotemperatures there appears a series of sharp peaks iconductance of the dot when the gate voltage is tuned in sa way that theN-electron and the (N11)-electron states ardegenerate, and observed are two kinds of mesoscopictuations: peak-height and peak-spacing fluctuations.confining potential due to the irregular shape of theyields chaos in the corresponding classical motion of etrons in it. Accordingly, the random-matrix theory, whicdescribes statistical properties of chaotic systems,7 is ex-pected to be applicable and to give a good description ofstriking mesoscopic fluctuations in the height and inspacing of the conductance peaks. Recent experime3

showed that the peak-height distributions are in good agment with predictions of the random-matrix theory8 for boththe Gaussian orthogonal and unitary ensembles. In contthe spacing between two adjacent conductance peaks, wgives an effective measurement of the inverse compressity of the dot, turns out not to obey the Wigner-Dyson stistics predicted by the random-matrix theory applied toconstant-interaction model.4–6 Instead, the inverse compresibility exhibits Gaussian-like symmetric distributions4–6 withnon-Gaussian tails.6 Further, in some experiments4,5 the rela-tive fluctuations were measured to be far larger than thpredicted by the random-matrix theory in the constant inaction model or by the diagrammatic calculations in trandom-phase approximation.4,9

A clue to the resolution of the discrepancy has been p

PRB 590163-1829/99/59~4!/2841~7!/$15.00

t-

el

ots

thech

c-et-

eetse-

st,ichil--e

er-

-

vided in the exact diagonalization study of the interactielectrons on a 2D small disordered systems.4 As the interac-tion strength is increased, it has been found that the distrtion of the inverse compressibility becomes consistent wthe experimental observation, displaying a Gaussian pMoreover, the root-mean-square~rms! fluctuations increasewith the interaction strength, suggesting that the fluctuatiin the charging energy play an important role in determinthe inverse-compressibility fluctuations. The analytical stubased on the Hartree-Fock~HF! approximation10 alsoreached the similar conclusion that the Coulomb interacticombined with the fluctuations of the eigenfunctions, leato the enhancement of the inverse compressibility flucttions. Recently it has been proposed that the shape defotion of the dot due to the gate-voltage sweeping in expment can also contribute to the deviation of the experimeresults from the Wigner-Dyson statistics.11

Regarding the fluctuations in the inverse compressibilthere is another interesting point of view that relates the pnomena to the metal-insulator transition:12,13 In two dimen-sions it has been numerically observed that the relative fltuations decrease monotonically with the increasing hoppstrength. This behavior has been conjectured to be a crover phenomenon from the localized regime, in which tharise the universal relative fluctuations of the order of unto the metallic regime with the relative fluctuations beiinversely proportional to the linear-system size. Howevthe exact diagonalization method employed in Ref. 12restricted the numerical study to very small systems, andnot yielded the full behaviors of the inverse-compressibilfluctuations.

In this paper, we study numerically the fluctuations of tinverse compressibility, considering the electron-electronteractions through the use of the HF approximation, whmakes it possible to investigate relatively large systems.HF approximation may provide a reasonable descriptionthe system: The long-range Coulomb interaction drives

2841 ©1999 The American Physical Society

tuityre

dyooioytuersgerionenveththucreod

re

eertiee-

enyleth

ty

ss-g

-

e

di-

the

ree-

the

-r

u-

irdrgiesan-

ct toelf-is

thees,

ract-

of

2842 PRB 59JEON, WU, CHOI, AND LEE

system to the classical limit, where we can neglect quanfluctuations through the redefinition of the effective impurpotential strength.14 Further, disorder is expected to restothe Fermi-liquid behavior.15 Indeed, the reliability of the HFapproximation has been demonstrated in a recent stu16

where the HF approximation has been found to give a gdescription of the interplay between the Coulomb interactand disorder in the quantum regime of the 1D electron stem. We investigate both the 1D systems, where the fluction effects are expected to be more conspicuous, and thones. It is shown that the relative fluctuations of the invecompressibility in general decrease with the hopping strenboth in one and in two dimensions. Particularly, via the pturbative approach we find two regimes where the behavof the relative fluctuations with the system size are differeIn the localized regime the relative fluctuations are indepdent of the size, while in the diffusive regime the relatifluctuations scale with the system size. It is also revealedthe degeneracy of the Fermi level plays a crucial role inbehavior of the average inverse compressibility and its fltuations in the metallic regime. The HF results are compawith those obtained via the exact diagonalization methand the validity of the HF approximation is examined.

II. MODEL SYSTEMAND HARTREE-FOCK APPROXIMATION

We consider interacting spinless electrons in a disordelattice of M sites, described by the Hamiltonian

H5t(^ i , j &

~ci†cj1cj

†ci !1(i

wini

11

2 (iÞ j

~ni2K !Ui j ~nj2K !, ~1!

where ci†(ci) is the creation~annihilation! operator andni

[ci†ci the number operator of the electron at sitei. The

hopping of an electron is allowed only between nearneighbors as indicated in the first term while the second trepresents the contribution of the random on-site potenwith strengthwi assumed to be distributed uniformly in thrange@2W,W#. The Coulomb interaction is included in ththird term, where the interactionUi j between the two electrons at sitei and j is given by

Ui j [e2

eur i2r j u, ~2!

with e being the dielectric constant andK is the uniformneutralizing background charge per site~in units of 1e).Here we concentrate on the half-filling case withK51/2.

The inverse compressibilityD2 for a given disorder con-figuration is obtained from the equation

D25EM /21122EM /21EM /221 , ~3!

where En denotes then-electron ground-state energy. Thstatistics ofD2 is then provided by the average over madifferent disorder configurations. In the absence of the etron hopping, it has been demonstrated numerically thatdistribution F(x) of the normalized inverse compressibili

m

dns-a-2Deth-rst:-

ate-d,

d

stmal

c-e

x[D2 /D2 , whereD2 denotes the average inverse compreibility, is apparently universal in two dimensions, displayinlarge fluctuations of the order unity.12 Further, it was arguedthat the distributionF(x) is closely related to the singleparticle density of statesg(e):

F~x!5M D2g~D2x/2!expF22ME0

D2x/2g~e8!de8G . ~4!

Recalling thatg(e)}e near the Fermi energy and that thaverage inverse compressibilityD2 is inversely proportionalto the linear system sizeN, one finds that this distribution isindependent of the system size. In particular, the exactagonalization of the Hamiltonian in Eq.~1! has demonstratedthat the relative fluctuations, defined to be the ratio ofrms fluctuationsdD2 to the averageD2 , decrease with thehopping strength and then saturate. By analogy to the thdimensional case, the behaviordD2 /D2;aB /j for aB,j,N has been conjectured in two dimensions to describecrossover from the insulating regime (dD2 /D2&1) to themetallic one (dD2 /D2;aB /N), wherej denotes the wavefunction correlation length andaB the semiconductor Bohradius.

We here employ the HF approximation to treat the Colomb interaction in Eq.~1!, and obtain

HHF5t(^ i , j &

~ci†cj1cj

†ci !1(i

wini

1(iÞ j

~^nj&2K !Ui j ni2(iÞ j

^cj†ci&Ui j ci

†cj ~5!

apart from some constant terms. In this Hamiltonian the thand fourth terms represent the direct and exchange eneof the electron-electron interactions, respectively, and thegular brackets denote the expectation values with respethe HF ground state, which should be determined sconsistently. The validity of the HF approximation

FIG. 1. The average inverse compressibility as a function ofhopping strengtht in one dimension. The data marked by squarcircles, and diamonds correspond to the system sizeN510, 30, and50, respectively. The dashed lines denote the values in noninteing clean systems (V5W50) for N510, 30, and 50 from top tobottom. The typical size of the error bars is less than the sizesymbols.

th

ibexve

e-be-ose

as

HFen

ss-

t

on-

ssa

ym

PRB 59 2843FLUCTUATIONS OF THE INVERSE COMPRESSIBILITY . . .

checked by comparing the results with those obtained viaexact diagonalization of the Hamiltonian in Eq.~1! for smallsystems.

In the next section, we compute the inverse compressity and its fluctuations in one and two dimensions, andamine their behaviors with the hopping strength. For connience, both the disorder strengthW and the interaction

FIG. 2. Relative fluctuations of the inverse compressibility afunction of the hopping strengtht in one dimension. The symbolare the same as those in Fig. 1. The typical size of the error bwhich are not specified in the figure, is less than the size of sbols.

e

il---

strengthV([e2/ea with a being the lattice constant! are setequal to unity. We adopt, for the most part, the freboundary conditions but also investigate the differencetween the results in the free-boundary conditions and thin the periodic-boundary conditions.

III. INVERSE COMPRESSIBILITY FLUCTUATIONS

We first investigate the 1D system ofN lattice sites. For agiven disorder configuration the inverse compressibility hbeen calculated from Eq.~3!, with the three ground-stateenergies obtained by the iterative diagonalization of theHamiltonian in Eq.~5!. Then the disorder average has betaken over more than 2000 configurations.

In Figs. 1 and 2, we plot the average inverse compreibility D2 and its relative fluctuationsdD2 /D2 . When theelectron hopping is forbidden (t50), the average inversecompressibilityD2 is shown to be of the order of unity. Itends to increase with the hopping strengtht, and in particu-lar, for larget, displays linear increase witht, in agreementwith the free-electron result

D254t cosF Np

2~N11!G , ~6!

which is represented by the dashed lines in Fig. 1. The cstant deviation from the free-electron results for larget is due

a

rs,-

FIG. 3. Normalized distributionF(x) of the inverse compressibility for several hopping strengths:~a! t50.0, ~b! t50.5, ~c! t52.0, and~d! t55.0 in one dimension. The symbols are the same as those in Fig. 1.

ousizra

or

seth

sy

ums-

hi

ess

litlicst

en-etheestls-elyonutis

alsdif-ess-elts,

thetri-sethe

thes,

ults

s,-rtedt


to the contributions of the random potential and of the Clomb interaction, and lessens gradually with the systemN. Figure 2 shows that the relative fluctuations in genedecrease with the hopping strengtht, similarly to the case of2D systems of small sizes.12 ~We will discuss later theslightly increasing behavior of the relative fluctuations fsmall hopping strength in large systems.! It is of interest tonote that for smallN the relative fluctuations tend to increawith N, especially for small values of the hopping strengThis is in contrast with the 2D case, where fort50 therelative fluctuations have been found independent of thetem size.12

The normalized distributionF(x) of x[D2 /D2 , which isshown in Fig. 3 for several values oft, displays an interestingsignature of the metal-insulator transition in the quantdots: For t50, the distribution is rather asymmetric, ashown in Fig. 3~a!. Unlike in 2D systems, however, the distribution does not appear to be universal, exhibiting the sof the peak position toward a smaller value ofx as the systemsize is increased. Ast is increased, the distribution becommore symmetric, with the width getting narrower regardleof the size. Large values oft eventually make the distributionso narrow that the relative fluctuations of the compressibibecome negligible, which is characteristic of metalsamples. Here it is noticeable that such change of the dibution appears at rather a smaller value oft as the size be-

FIG. 4. ~a! The average inverse compressibility and~b! its rela-tive fluctuations in the free-boundary conditions (s) and in theperiodic-boundary conditions (h) for N58.

-el

.

s-

ft

s

y

ri-

comes smaller. Recalling that the localization length in geral increases witht, we can infer that the shape of thdistribution is determined by the competition betweensystem size and the localization length. It is also of interto note the small tail of negative compressibility for smalt@see Fig. 3~b!#. The possibility of such negative compresibility has been reported in 2D islands, which results solfrom the disorder potential and the electron-electrrepulsion.17 At this stage, however, it cannot be ruled othat the tail of negative compressibility obtained heremerely an artifact of the HF approximation.

The perturbative approach in the metallic regime revean interesting crossover from the localized regime to thefusive regime. In the mesoscopic scale the inverse compribility has three kinds of contributions: one from the levspacing in the kinetic energy due to the finite-size effecanother from the level fluctuations due to impurities, andthird from the electron-electron interaction. Since the conbution from the interaction is of a higher order in the inversystem size than that from impurities, we may neglectthird contribution in large systems (N@1). Thus, only twoenergy scales, the level spacing in the kinetic energy andcorrection to the single-particle energy due to impuritie

FIG. 5. Comparison of the exact and the Hartree-Fock resfor ~a! the inverse compressibility and~b! its relative fluctuations asa function of the hopping strengtht. The data marked by squarecircles, and diamonds correspond toN56, 8, and 10 in the HartreeFock approximation, and the data marked by triangles, invetriangles, and pentagons correspond toN56, 8, and 10 in the exacdiagonalization method.

heog

asel tty

doalhe

theion

n-are

thethens

on-o

theom--to ae

ermiary

ul

le

cor-less


compete with each other. In the half-filling case with tfree-boundary conditions, which has been considered innumerical calculations, the competition of the two enerscales leads to two regimes: In the regimeN@(t/W)2 thelevel spacing is given by the ordert/N, thus far smaller thanthe level correction due to impurities, which is estimated(1/N)( iwi;W/AN. In this case both the average invercompressibility and the rms fluctuations are proportionaW/AN, leading to the relative fluctuations of the order uniregardless ofN, t, andW. In the opposite limitN!(t/W)2,on the other hand, the average inverse compressibility istermined by the kinetic energy-level spacing, which is stillthe order oft/N, while the rms fluctuations are proportionto W2/t. The corresponding relative fluctuations take tscaling form

dD2

D2

;NS W

t D 2

. ~7!

The fact that the localization lengthj takes the form18

FIG. 6. Comparison of the exact and the Hartree-Fock resfor ~a! the inverse compressibility and~b! its relative fluctuations asa function of the inverse system size 1/N. The data marked bysquares, circles, and diamonds correspond tot50.0, 0.5, and 5.0 inthe Hartree-Fock approximation, and the data marked by trianginverted triangles and pentagons correspond tot50.0, 0.5, and 5.0in the exact diagonalization method.

ury

s

o,

e-f

j}S t

WD 2

~8!

for W!t in the long wire suggests that such change inbehavior of the relative fluctuations reflects the transitfrom the localized regime withj,N to the diffusive regimewith j.N. Our numerical results, which also reflect the iteraction effects as well as the above two contributions,in good agreement with these predictions.

We now discuss how the boundary conditions affectbehavior of the inverse compressibility. Figure 4 showsaverage inverse compressibility and its relative fluctuatiofor N58 as functions of the hopping strengtht, both in thefree-boundary conditions and in the periodic boundary cditions. The striking difference in the behaviors from the twtypes of boundary conditions can be observed. Unlikefree-boundary conditions, where the average inverse cpressibility grows linearly witht, the periodic-boundary conditions have the average inverse compressibility saturatefinite value of the order of unity. Accordingly, the relativfluctuations remain finite in the limitt→`. The origin ofthis behavior can be attributed to the degeneracy at the Flevel in a free-electron system under the periodic-bound

ts

s,

FIG. 7. ~a! The average inverse compressibility and~b! its rela-tive fluctuations as functions of the hopping strengtht in two di-mensions. The data marked by squares, circles, and diamondsrespond to the system sizeN54, 6, and 8, respectively. The typicasize of the error bars which are not specified in the figure is lthan the size of symbols.


FIG. 8. Normalized distribution of the inverse compressibility for various hopping strengths:~a! t50.0, ~b! t50.2, ~c! t50.5, and~d!t53.0 in two dimensions. The symbols are the same as those in Fig. 7.

inin

n

eseonfrthtso

Fildlita

itasn

Toiz

on

earval-fonoth

sti-

Ac-ua-d

atitha-

heichbe-are

2Das

-enatility

ely

conditions. In such a system of sizeN54M , the (2M )th andthe (2M11)th single-electron levels are degenerate, leadto vanishing inverse compressibility. In consequence, theverse compressibility for larget has only the contributionsfrom disorder and from the electron-electron interaction adoes not grow very large even in the limitt→`, yieldingnonzero relative fluctuations. For the system of sizeN54M12, on the other hand, the average inverse compribility and its relative fluctuations show qualitatively thsame behaviors as those in the free-boundary conditisince there does not exist such degeneracy. Since theboundary conditions are presumably more realistic thanperiodic-boundary conditions in view of small quantum dowe have adopted the free boundary conditions in most ofnumerical calculations.

Finally, in order to check the validity of the employed Happroximation, we have diagonalized exactly the Hamtonian in Eq.~1! through the use of the Lanczos methoFigure 5 shows that the HF approximation produces quatively the same results in the whole range oft as those ob-tained from the exact diagonalization method. Quanttively, however, it appears that the HF approach hagrowing tendency to overestimate the relative fluctuatiofor intermediatet('0.5) as the system size increases.confirm this tendency, we have performed exact diagonaltion up to the system sizeN520, for three values of thehopping strength (t50.0, 0.5, and 5.0!, and presented theaverage inverse compressibility and its relative fluctuati

g-

d

s-

s,ee-e,ur

-.-

-as

a-

s

as a function of the inverse system size in Fig. 6. It is clthat the HF approximation reproduces essentially exactues of the average inverse compressibility for all values ot.The relative fluctuations obtained in the HF approximatiare also in excellent agreement with the exact results for bt50.0 and 5.0, while the HF approach is found to overemate the relative fluctuations fort50.5. Furthermore, theoverestimation tends to increase with the system size.cordingly, the appearance of the maximum relative flucttions at small butfinite t in large systems may be attributeto the overestimation in the HF approximation~see the re-sults forN550 in Fig. 2!, and it is presumably the case ththe relative fluctuations in fact decrease monotonically wthe hopping strength. One possible origin of the overestimtion is the error in the HF ground-state energy: Unlike taverage inverse compressibility, in the calculation of whthe error is averaged over the disorder configuration andcomes vanishing, the relative fluctuations contain the squof the error, which remains finite.

We have also performed numerical calculations onN3N lattices forN54, 6, and 8. The disorder average hbeen taken over 5000 realizations forN54 and over 1000realizations forN56 and 8. As in 1D systems, the HF approximation together with free boundary conditions has beemployed. Surprisingly, as shown in Fig. 7, it is found ththe general behaviors of the average inverse compressiband its relative fluctuations in two dimensions are largsimilar to those in the 1D system of sizeN54M in the

ilitthlu

eeeve

rnthro

la

n

e

2r

thse

ve

utng

rseonxi-ob-ss-a

ar-etheed

ze,

iththe

in-ing

iththeex-

he

ncein

tion


periodic-boundary conditions@see Fig. 4#: As the hoppingstrengtht is increased, the average inverse compressibincreases slightly and saturates to a finite value whilerelative fluctuations generally decrease to a nonzero vawhich is consistent with the behavior observed forN54 inRef. 12. The similarity to the 1D systems of sizeN54M canbe attributed to the degeneracy at the Fermi level in a frelectron system. In 2D free-electron systems with the frboundary conditions, the single-particle energies are giby

enm522tS cos2pn

N111cos

2pm

N11D ,

with n andm being integers from 1 toN. Accordingly, thereare N degenerate states with (n,m)5(1,N),(2,N21), . . . ,(N,1) at the Fermi level. This degeneracy in tuyields both the finite-average inverse compressibility andnonvanishing relative fluctuations even in the free-electlimit, similarly to the case of 1D systems of size 4M in theperiodic-boundary conditions. The slight increase of the retive fluctuations for smallt in the system ofN58 may alsooriginate from the overestimation in the HF approximatioIn particular, Fig. 7 reveals that fort50, the HF resultsreproduce the universal relative fluctuations12 within the er-ror bars. For larget, on the other hand, it is shown that thsaturation value of the relative fluctuations decreases withN,which is consistent with the prediction in Refs. 9 and 1These observations are also demonstrated by the invecompressibility distributionF(x), presented in Fig. 8 forseveral values oft. For t50, the distribution displays bothasymmetry and universality in its shape, irrespectively ofsystem size. The increase of the hopping strength tendmake the distribution more or less symmetric and narrowas in one dimension. In contrast with the 1D case, howe

T.

M.

G.

C.

,

.

yee,

e--n

en

-

.

.se-

etor,r,

the width of the distribution does not reduce to zero bremains finite even for very large values of the hoppistrength.

IV. SUMMARY

In summary, we have investigated numerically the invecompressibility and its fluctuations both in one dimensiand in two dimensions, employing the Hartree-Fock appromation to treat electron-electron interactions. It has beenserved that the relative fluctuations of the inverse compreibility decrease with the hopping strength, which providessignature of a metal-insulator transition in the sample. Pticularly in the one-dimensional system with half filling whave found two regimes according to the behaviors ofrelative fluctuations with the system size: In the localizregime where the system sizeN is larger than the localizationlengthj the relative fluctuations are independent of the siwhereas forj sufficiently larger thanN corresponding to thediffusive regime the relative fluctuations scale linearly wN. It has also been pointed out that the degeneracy atFermi level affects crucially the behavior of the averageverse compressibility and its fluctuations for large hoppstrengths. The comparison of the Hartree-Fock results wthe exact ones for small systems has confirmed thatHartree-Fock approach produces rather accurate resultscept for the overestimation of the relative fluctuations in tintermediate range of the hopping strength.

ACKNOWLEDGMENTS

This paper was supported in part by the Basic ScieResearch Institute Program, Ministry of Education, andpart by the Korea Science and Engineering Foundathrough the SRC Program.

v.

B

tored

isze

l.

tt.

tt.

1See, e.g.,Mesoscopic Phenomena in Solids, edited by B. L. Alt-shuler, P. A. Lee, and R. A. Webb~North-Holland, Amsterdam,1991!.

2See, e.g., S. Washburn and R. A. Webb, Adv. Phys.35, 375~1986!, and references therein.

3A. M. Chang, H. U. Baranger, L. N. Pfeiffer, K. W. West, andY. Chang, Phys. Rev. Lett.76, 1695 ~1996!; J. A. Folk, S. R.Patel, S. F. Godijn, A. G. Huibers, S. M. Cronenwett, C.Marcus, K. Campman, and A. C. Gossard,ibid. 76, 1699~1996!.

4U. Sivan, R. Berkovits, Y. Aloni, O. Prus, A. Auerbach, andBen-Yoseph, Phys. Rev. Lett.77, 1123~1996!.

5F. Simmel, T. Heinzel, and D. A. Wharam, Europhys. Lett.38,123 ~1997!.

6S. R. Patel, S. M. Cronenwett, D. R. Stewart, A. G. Huibers,M. Marcus, C. I. Duruo¨z, J. S. Harris, Jr., K. Campman, and AC. Gossard, Phys. Rev. Lett.80, 4522~1998!.

7M. L. Mehta, Random Matrices~Academic Press, New York1991!.

8R. A. Jalabert, A. D. Stone, and Y. Alhassid, Phys. Rev. Lett.68,3468 ~1992!.

9R. Berkovits and B. L. Altshuler, Phys. Rev. B55, 5297~1997!.10Ya. M. Blanter, A. D. Mirlin, and B. A. Muzykantskii, Phys. Rev

Lett. 78, 2449~1997!.

.

11R. O. Vallejos, C. H. Lewenkopf, and E. R. Mucciolo, Phys. ReLett. 81, 677 ~1998!.

12A. A. Koulakov, F. G. Pikus, and B. I. Shklovskii, Phys. Rev.55, 9223~1997!.

13In one dimension it is generally believed that the metal-insulatransition does not take place with all states being localizwhile it is still under controversy in two dimensions. In thpaper the termmetal merely denotes the sample with the sismaller than the localization length.

14H. Maurey and T. Giamarchi, Phys. Rev. B51, 10 833~1995!.15C. M. Naon, M. C. von Reichenbach, and M. L. Trobo, Nuc

Phys. B485, 665 ~1997!; A. R. Goni, A. Pinczuk, J. S. Weiner,B. S. Dennis, L. N. Pfeiffer, and K. W. West, Phys. Rev. Le70, 1151~1993!; B.Y.-K. Hu and S. Das Sarma,ibid. 68, 1750~1992!; S.-R. E. Yang and G. C. Aers, Phys. Rev. B46, 12 456~1992!.

16G. S. Jeon, M. Y. Choi, and S.-R. E. Yang, Phys. Rev. B54,R8341~1996!.

17M. E. Raikh, L. I. Glazman, and L. E. Zhukov, Phys. Rev. Le77, 1354~1996!.

18P. D. Kirkman and J. B. Pendry, J. Phys. C17, 4327~1984!; M.Kappus and F. J. Wegner, Z. Phys. B45, 15 ~1981!.

fluctuations of the inverse compressibility in disordered electron systems

Documents