incompressible strips in dissipative hall bars as origin of quantized hall plateaus

Incompressible strips in dissipative Hall bars as origin of quantized Hall plateaus

Afif Siddiki and Rolf R. GerhardtsMax-Planck-Institut für Festkörperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany

(Received 8 June 2004; revised manuscript received 16 August 2004; published 23 November 2004)

We study the current and charge distribution in a two-dimensional electron system, under the conditions ofthe integer quantized Hall effect, on the basis of a quasilocal transport model, that includes nonlinear screeningeffects on the conductivity via the self-consistently calculated density profile. The existence of “incompressiblestrips” with integer Landau level filling factor is investigated within a Hartree-type approximation, and non-local effects on the conductivity along those strips are simulated by a suitable averaging procedure. This allowsus to calculate the Hall and the longitudinal resistance as continuous functions of the magnetic fieldB, withplateaus of finite widths and the well-known, exactly quantized values. We emphasize the close relationbetween these plateaus and the existence of incompressible strips, and we show that forB values within theseplateaus the potential variation across the Hall bar is very different from that forB values between adjacentplateaus, in agreement with recent experiments.

DOI: 10.1103/PhysRevB.70.195335 PACS number(s): 73.20.2r, 73.50.Jt, 71.70.Di

I. INTRODUCTION

The application of the quantized Hall effect1 (QHE) asresistance standard, and its importance for modern metrol-ogy, relies on the extremely high reproducibility(better than10−8) of certain quantized resistance values.2 This extremereproducibility points to an universal origin, which is inde-pendent of special material or sample properties. The pur-pose of the present paper is to propose and evaluate a quasi-local transport model that allows us to calculate, first, thepotential and current distribution in a two-dimensional elec-tron system(2D ES) under the conditions of the QHE, and,second, the longitudinal and the Hall resistance,RlsBd andRHsBd, in the plateau regimes of the QHE and in between.Whereas the resistance values between the QH plateaus willdepend on details of the used conductivity model, the exactlyquantized plateau values result from the existence of suffi-ciently wide “incompressible strips” along which the localconductivity vanishes, since occupied and unoccupied statesare separated by an energy gap(Landau quantization). Lo-calization assumptions,3 which played an important role inearly approaches to the QHE, are not included in our model.Our model is motivated by a recent experimental investiga-tion of the Hall-potential in a narrow Hall bar,4 and a criticalreexamination of a subsequent model calculation.5

Experimental information about the actual current and po-tential distribution in a Hall bar under QHE conditions hasbeen obtained recently by Ahlswede and coworkers4,6,7 witha scanning force microscope.8 The data were interpreted interms of “incompressible strips” with constant electron den-sity (corresponding to the filling of an integer number ofLandau levels),9–12 which are expected to develop in an in-homogeneous 2D ES as a consequence of its strongly non-linear low-temperature screening properties13 in a strong per-pendicular magnetic field. If the filling factor in the center ofthe sample was slightly larger than an integer, the Hall po-tential was found to drop completely across two strips, whilebeing constant elsewhere. With decreasingB, the stripsmoved towards the sample edges, just as one expects for the

incompressible strips in a sample, in which the electron den-sity decreases gradually from a maximum in the center to-wards the edges. If the center filling factor was slightly be-low an integer, a gradual potential variation was observed,either linear or a nonlinear, without clear indication for in-compressible strips.4

This interpretation was supported by subsequent theoreti-cal work of Güven and Gerhardts5 (GG), who extended theself-consistent Thomas-Fermi-Poisson approximation11–13

(TFPA) for the calculation of electron density profile andelectrostatic potential to a non-equilibrium situation with aposition-dependent electrochemical potential, determined bythe presence of an applied dissipative current through thesample. Electrochemical potential and current density werecalculated from a local version of Ohm’s law, with a localmodel for the conductivity tensor, determined by theposition-dependent electron density. The feedback of the cur-rent distribution on electron density and the measurable po-tential profile4 was included by the assumption of local equi-librium in the stationary nonequilibrium situation.

In agreement with the experiment,4 the calculation5 showsa linear variation of the Hall potential across the sample ifthere are no incompressible strips, e.g., for sufficiently hightemperature or if the magnetic field is so strong, that thelocal filling factor is everywhere in the sample less than two(spin-degeneracy is assumed and interactions which mightlead to the fractional quantized Hall effect are neglected).Also for center filling factors slightly larger than 2 or 4 thecalculation confirms the experiment, showing that the poten-tial drops across broad incompressible strips and is constantelsewhere. However, due to the use of the TFPA, GG(Ref. 5)obtain incompressible strips whenever the center filling fac-tor is larger than 2,9–12 and due to the strictly local conduc-tivity model these dominate the current and potential distri-bution, and lead to vanishing longitudinal resistance. Thusthe model assumptions of Ref. 5 lead to serious disagreementwith important aspects of the experiment.

The purpose of the present paper is to improve on themodel of GG (Ref. 5) so that, first, qualitative agreementbetween the calculated and the measured potential distribu-

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tion is achieved for all filling factor regimes, and, second,reasonable results forRlsBd andRHsBd are obtained. Follow-ing the lines suggested by GG,5 we investigate in Sec. II theconditions for the existence of incompressible strips, using aHartree approximation. In Sec. III we reexamine and weakenthe strictly local conductivity model, and show that a simplespatial-averaging procedure of the local conductivities cansimulate corrections expected from a Hartree calculation forthe equilibrium state and from a nonlocal transport calcula-tion. Transport results based on the self-consistent Born ap-proximation will be presented and discussed in Sec. IV. Inthe present work we will restrict our consideration on thelinear response regime, so that heating effects, which mightdestroy incompressible strips in the presence of highcurrents,5 can be neglected.

II. EXISTENCE OF INCOMPRESSIBLE STRIPS

A. Electrostatic self-consistency

Following Ref. 5, we consider a 2D ES in the planez=0, with translation invariance in they direction and anelectron densitynelsxd confined to the interval −d,x,d.The confinement potentialVbgsxd is determined by fixedbackground charges and boundary conditions on metallicgates. The mutual Coulomb interactions between the elec-trons are treated in a Hartree-type approximation, i.e., arereplaced by a potential energy termVHsxd which is deter-mined via Poisson’s equation by the electron density. Ex-change and correlation effects are neglected, and spin degen-eracy is assumed. Thus, the electrons move in an effectivepotential

Vsxd = Vbgsxd + VHsxd, s1d

VHsxd =2e2

kE

−d

d

dx8Ksx,x8dnelsx8d, s2d

where −e is the charge of an electron,k an average back-ground dielectric constant, and the kernelKsx,x8d solvesPoisson’s equation under the given boundary conditions.Kernel and background potential for the frequently usedmodel,9–12 that assumes all charges and gates to be confinedto the planez=0, are taken from Ref. 5,

Ksx,x8d = lnUÎsd2 − x2dsd2 − x82d + d2 − x8x

sx − x8ddU , s3d

Vbgsxd = − Ebg0 Î1 − sx/dd2, Ebg

0 = 2pe2n0d/k, s4d

where en0 is the homogeneous density of positive back-ground charges in the Hall bar. Other meaningful and trac-table boundary conditions are also possible.13

To perform explicit calculations, one needs a prescriptionto calculate the electron density for given effective potentialVsxd, which then together with Eqs.(1) and(2) completes theelectrostatic self-consistency. The self-consistent TFPA5,11–13

takes this prescription from the Thomas-Fermi approxima-tion (TFA)

nelsxd =E dEDsEdf„E + Vsxd − m!…, s5d

with DsEd the density of states (DOS), fsEd=1/fexpsE/kBTd+1g the Fermi function,m! the electro-chemical potential(being constant in the equilibrium state),kB the Boltzmann constant, andT the temperature.

B. Hartree approximation

A less restrictive approximation would be to insertVsxdinto Schrödinger’s equation,

F 1

2mSp +

e

cAD + VsxdGFlsr d = ElFlsr d, s6d

with Asr d a vector potential describing the magnetic fieldB=s0,0,Bd= ¹ 3A, and to calculate the density from theeigenenergiesEl and functionsFlsr d,

nelsr d = ol

uFlsr du2fsEl − m!d. s7d

Exploiting the symmetry of our system, we may use the Lan-dau gauge,Asr d=s0,Bx,0d, and factorize the wave func-tions, Flsr d=Ly

−1/2 expsikydfn,Xsxd, with Ly a normalizationlength,X=−l2k a center coordinate,l =Î" /mvc the magneticlength, and vc=eB/ smcd the cyclotron frequency. TheSchrödinger equation then reduces to

F−"2

2m

d2

dx2 +m

2vc

2sx − Xd2 + VsxdGfn,Xsxd = EnsXdfn,Xsxd,

s8d

and the electron density becomes

nelsxd =gs

2pl2onE dXf„EnsXd − m!

…ufn,Xsxdu2, s9d

wheregs=2 takes the spin degeneracy into account and thesum over X has been replaced by an integral,Ly

−1oX⇒ s2pl2d−1edX.

C. Thomas-Fermi approximation (TFA)

If the potential Vsxd varies slowly on the scale of themagnetic lengthl, its effect on the lowest Landau levels(LLs) can be treated perturbatively, with the lowest orderresult

EnsXd < En + VsXd, En = "vcsn + 1/2d. s10d

On the length scale relevant for the variation ofVsxd, theextent of the Landau wavefunctions may be neglected,ufn,Xsxdu2<dsx−Xd. Then the Hartree result for the electrondensity, Eq.(9), reduces to the TFA, Eq.(5), with the LandauDOS

DsEd =1

pl2on=0

`

dsE − End. s11d

To evaluate the self-consistent TFPA we follow Ref. 5.First we fix the sample width 2d and the density of positive

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background chargesn0, and thereby according to Eq.(4) thebackground potentialVbgsxd and the relevant screening pa-rameter asc;pa0/d, with a0= k"2/ s2me2d the screeninglength (2a0=9.79 nm for GaAs). Next we choose the actualwidth 2b of the density profile atT=0 andB=0, and solvefor uxuøb the linear integral equation5

Vsxd − Vbgsxd =1

ascE

−b

b dx8

dKsx,x8dfm0

! − Vsx8dg s12d

[with m0!=Vs−bd=Vsbd] to which the self-consistent TFPA

reduces in this limit. From the corresponding density profilenelsx;B=0,T=0d=D0fm0

!−Vsxdg, with D0=m/ sp "2d theDOS of the 2D ES atB=0, we calculateVsxd for uxu ød, theaverage densitynel=e−d

d dxnelsx;B=0,T=0d /2d with the cor-responding Fermi energyEF= nel/D0, and, for later reference,the Fermi energyEF

0 =nels0;B=0,T=0d /D0 corresponding tothe electron density at the center.

In the following we will consider only symmetric densityprofiles and takeb, or equivalently the depletion lengthd−b, as a free parameter, that fixes the density profile andthe electro-chemical potentialm0

! at B=0 (where the tem-perature dependence is weak). In real samplesm0

! may bedetermined by an electron exchange between the 2D ES andits surrounding, which may be possible at high but not at lowtemperatures. A restriction that fixesm0

! will also determinethe value ofb.

Next, we fix the value of the magnetic field and start witha high temperature to calculate the electron density fromEqs. (5) and (11) self-consistently with Eq.(2), using thepreviously calculated potentialVsxd as initial value. Finallywe lower T stepwise until the required low temperature isreached, and iterate(using a Newton-Raphson approach) ateach temperature until convergence is achieved.

The solid lines of Fig. 1 show results ford=1.5 mm and

n0=431011 cm−2 (which impliesEbg0 =4.38 eV) obtained for

501 equidistant mesh points, −d=x0,xn,xN=d sN=500d.The density profile was fixed by choosingb/d=0.9, whichyields nel=2.931011 cm−2 and nels0;B=0,T=0d=3.5831011 cm−2, and thus, withD0=2.831010 meV−1 cm−2 forGaAs,EF=10.37 meV andEF

0 =12.75 meV. We prefer to useEF

0 (rather thanEbg0 ) as a reference energy, since it has the

same order of magnitude as the cyclotron energies of inter-est. For finiteB we define an effective center filling factorn0=2pl2nelsx=0;B=0,T=0d=2EF

0 /"vc.The result obtained for the TFPA(solid lines in Fig. 1)

shows the expected well developed incompressible stripswith constant electron density at local filling factornsxd=2.For the largerB value we obtain wide density plateaus at0.32ø uxu /dø0.46, in each of which we find at 36xn values,with high accuracy,nsxnd=2. For the lowerB value the in-compressible strips are much narrower, however we obtainthe high precision valuesnsxnd=2 still at five neighboringxn

values.Typical results of the TFPA are summarized in Fig. 2,

which shows, as a gray scale plot, the filling factor profile forvarying magnetic field, with horizontal lines correspondingto a fixed B value. At sufficiently largeB field, the localfilling factor nsxd is everywhere in the Hall bar less than 2,and the 2D ES is completely compressible. At somewhatlower B s"vc/EF

0 <1d the center of the sample becomes in-compressible with local filling factornsxd=2, while nsxdgradually decreases outside the incompressible center andfalls off to zero in the depletion regions at the sample edges.With further decreasingB, the filling factor in the centerincreases and incompressible strips withnsxd=2 move to-wards the sample edges and become narrower. At sufficientlylow B values, incompressible strips with local filling factornsxd=4 occur, first in the center, and then move towardsthe edges. They then coexist with incompressible strips ofnsxd=2, which exist near the edges and are narrow, but,within the TFPA, still have a finite width. For low enough

FIG. 1. (Color online) Electron density profiles for two values ofthe magnetic field("vc/EF

0 =0.94 and 0.65) and different approxi-mations: Thomas-Fermi(solid lines), Hartree(dashed), and quasi-Hartree(dash-dotted). The insets show the enlarged plateau regionsfor both cases.a=0.01,kBT/EF

0 =0.0124. Kinks in the upper insetindicate mesh points.

FIG. 2. (Color online) Gray scale plot of filling factor versusposition x and magnetic fieldB, calculated within the TFPA. Theregions of incompressible strips withnsxd=2 andnsxd=4 are indi-cated. For sufficiently largeB sVc;"vc.EF

0d the system is com-pressible(indicated by “CS”), while for the lowerB values includedin the figure it always contains incompressible strips(“IS” ). Thedashed horizontal line refers to Fig. 5 below;a=0.01, kBT/EF

0

=0.0124.

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temperature, this type of behavior continues at still lowervalues ofB, where further incompressible strips with succes-sively higher filling factors evolve from the center and movetowards the edges, coexisting with the edge-near incom-pressible strips of lower local filling factors.

D. “Quasi-Hartree” approximation

The dashed lines in Fig. 1 are calculated in the Hartreeapproximation. We started again atB=0, T=0 and inserted ineach of the following iteration steps the previously calcu-lated potentialVsxd into the eigenvalue problem of Eq.(8),took each mesh pointxn as center coordinateX and diago-nalized the problem in the space spanned by the eightlowest-energy unperturbed Landau-Hermite functions

fn,X0 sxd =

exps− fx − Xg2/2l2dÎ2nn ! Îpl

Hnsfx − Xg/ld, s13d

whereHnsjd is thenth order Hermite polynomial. The result-ing energy eigenvalues and functions were used to calculatethe electron density according to Eq.(9). The overall appear-ance of the Hartree results for the density profiles and alsothe wide plateaus for the higherB value (see lower inset ofFig. 1) are in good agreement with those of the TFA. Thenarrow plateaus, obtained in the TFA for the lowerB value,are now however smeared out. As is clearly seen in the upperinset of Fig. 1, the Hartree result for the filling factornsxdcrosses the value 2 with a finite slope.

The essential qualitative difference between the Hartreeapproximation and the TFA is the neglect of the extent of thewavefunctions in the latter. So we interpret the smearing-outof narrow incompressible strips in the Hartree approximationas being due to the finite width of the wave functions. Tocheck this idea, we consider a “quasi-Hartree” approximation(QHA) in which, instead of solving the problem of Eq.(8),we replace the wave functions by the eigenfunctions(13) ofthe unconfined Landau problem and take the energy eigen-values from Eq.(10). The latter would be correct in the senseof a lowest order perturbation approximation with respect tothe effective confining potentialVsxd, if Vsxd would be alinear function of position over the extent of the unperturbedwavefunctionfn,X

0 sxd. The numerical effort with this QHA ismuch less than that required for the full Hartree approxima-tion, since no numerical calculation of energy eigenvaluesand functions is necessary. Density profiles calculated withinthis QHA are also shown in Fig. 1 as dash-dotted lines. It isseen that the results are very similar to those of the fullHartree calculation, in particular also the results for thesmearing-out of the incompressible strips. Apparently thesmearing effect of the QHA is even stronger than that of thefull Hartree approximation. This is understandable, since theHartree wave functions are asymmetrically squeezed inspace regions of a rapid variation ofVsxd, and therefore havea smaller spatial extent than the unperturbed Landau wavefunctions.

In Fig. 3 we compare the widths of incompressible stripswith nsxd=2 for several approximations. The line labeledCSG is the analytical half-plane result of Chklovskiiet al.,9

anic=16Îa0d0/pÎnic / fn0

2−nic2 g, with d0,150 nm a depletion

length,nics=2d the filling factor of the incompressible strip,and n0=2EF

0 /"vc the effective filling factor.11 This resultagrees well with the corresponding self-consistent result ofRef. 11 and for sufficiently lowB values(note the invertedBscale in Fig. 3) also with our present TFPA result for samplesof finite width. For smallB values, the width decreases pro-portional toB, and remains finite throughout the figure.

The result calculated from our QHA is also included inFig. 3. As for the TFPA, we have determined the width of theincompressible strips by a simple extrapolation, using threepoints next to a plateau to determine a plateau edge. Forwide plateaus(large B values), the QHA width is onlyslightly smaller than that calculated within the TFA. How-ever, with decreasingB the QHA width decreases faster andgoes to zero at a relatively large magnetic fields"vc/EF

0

<0.7d, far before the incompressible strips withnsxd=4 de-velop in the center of the sample.

These results require a modification of Fig. 2. Within theHartree-type approximation, the width of the incompressiblestrip with filling factor 2 shrinks more rapidly with decreas-ing B and vanishes at"vc/EF

0 <0.7. Between thisB valueand the value"vc/EF

0 <0.5 no incompressible strips exist inthe sample. At still lowerB values there is aB-interval inwhich only incompressible strips with local filling factor 4can exist. This modification required by the QHA is indicatedin Fig. 7 below.

In view of the following it may be interesting to note thatthe essential effects of the Hartree-type approximations canbe simulated in a very simple way. If we first perform acalculation within the TFPA and then take a spatial average,e.g., nsxd=s2ld−1e−l

l dx8nsx+x8d, of the filling factor profilensxd, we will smear-out incompressible strips of a width lessthan 2l, while incompressible strips with a width larger than2l will survive. With l of the order of the magnetic length,we will obtain filling factor profilesnsxd very similar to thoseobtained in the Hartree approximation.

FIG. 3. (Color online) Magnetic-field dependence of the widthof the n=2 incompressible strips, for three different approxima-tions: the analytical result of Ref. 9(CSG), the TFPA(TFA), and thequasi-Hartree approximation(QHA). Note the invertedB scale. In-set: ratio of the incompressible strip width to the magnetic length inQHA.


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III. INCOMPRESSIBLE STRIPS AND DISTRIBUTIONOF DISSIPATIVE CURRENT

A. The local conductivity model

We now describe the effect of an applied current on ourHall bar system, following again the approach of Ref. 5. Inthe presence of a dissipative currentI, the electrochemicalpotentialm!sr d will become position dependent, and its gra-dient E= ¹m! /e will drive the current densityj sr d. We as-sume the linear local relation(Ohm’s law)

j sxd = ssxdEsxd, ssxd = s„nelsxd…, s14d

with a position-dependent conductivity tensorssxd, whichhas the same form as for a homogeneous sample, howeverwith the homogeneous density replaced by the local electrondensitynelsxd. Due to the translation invariance in they di-rection, which is indicated in Eq.(14), and the equation ofcontinuity, the componentsjx andEy of current density andelectric field, respectively, must be constant,5

jxsxd ; 0, Eysxd ; Ey0. s15d

For the other components one finds

j ysxd =1

rlsxdEy

0, Exsxd =rHsxdrlsxd

Ey0, s16d

in terms of the longitudinal componentrl =rxx=ryy and theHall component rH=rxy=−ryx of the resistivity tensorr=s−1. For a given applied currentI =e−d

d dxjysxd one obtainsfor the constant electric field component along the Hall bar

Ey0 = IFE

−d

d

dx1

rlsxdG−1

, s17d

and for the Hall voltage across the sample

VH =E−d

d

dxExsxd = Ey0E

−d

d

dxrHsxdrlsxd

. s18d

With the usual normalization of the resistances to a square-shaped conductor, this yields for the Hall and the longitudi-nal resistance

RH =VH

I, Rl =

2dEy0

I. s19d

Here we see the essence of the local model. Any reason-able model for the conductivity of a high-mobility 2D ES atzero temperature will give simple results for the conductivitycomponents at even-integer filling factors(where no elasticscattering is possible):

slsn = 2kd = rlsn = 2kd = 0, s20d

sHsn = 2kd =1

rHsn = 2kd=

e2

h2k. s21d

Thus, if an incompressible strip of finite width exists in thesample, the integral in Eq.(17) diverges andEy

0 and, there-fore, the longitudinal resistanceRl is zero. At low tempera-tures,kBT!"vc, rlsn=2kd and, therefore,Rl will be expo-

nentially small and relevant contributions to the integralcome only from the incompressible regions.

The integral in Eq.(18) has the same type of singularity.If only incompressible strips with the same valuensxd=2k ofthe local filling factor exist, this singular integral is just the2k-fold of the integral in Eq.(17), so that we get the quan-tized result RH=h/ s2ke2d. At zero temperature one canevaluate the singular integrals by first introducing a cutoff,e.g., by replacingrlsxd with rl

esxd=maxfe ,rlsxdg, then calcu-lating the integrals, and finally removing the cutoffse→0d.This yields exact quantization of the Hall resistance, and thecorresponding calculation at very low temperatures yieldsexponentially small corrections. If incompressible strips offinite widths with different values ofnsxd exist, e.g., due to amanipulation of the background potential, other values forthe Hall resistance may be possible. But, as we have learnedfrom the Hartree-type approximations in the previous sec-tion, such a situation will not occur in our simple translation-invariant Hall bar geometry. From these arguments we ex-pect in the resistance-versus-B curves plateau regions offinite widths, where the resistances have the well knownquantized values.

These considerations are quite general and do not dependon details of the conductivity model. On the other hand, ifwe want to calculate the resistances between the plateau re-gions, we need to specify a conductivity model, and the re-sults will depend on details of this model. We will presentsuch detailed results in Sec. IV below.

B. Limitations of the local model

In Sec. II we have shown that, within a Hartree-type ap-proximation, incompressible strips of a width smaller thanthe extent of typical wavefunctions cannot exist. As a conse-quence, incompressible strips with a given filling factor 2kdo exist only in a finite interval of magnetic fields. For lowerB values, thensxd profile crosses the value 2k=nsx2kd withfinite slope at some pointx= x2k. At zero temperature, theintegrals in Eqs.(17) and (18) become singular sincerlsx2kd=0. Whether the singularity is integrable or not de-pends on the filling-factor dependence of the longitudinalconductivity, slsnd. For the SCBA model, to be consideredbelow, it is integrable, for the Gaussian model considered byGG it is not. But should we worry about such sophisticatedquestions depending on details of the conductivity model?We think we should not, for the following reasons.

All quantities that are related by Eq.(14), the currentdensity, the conductivity and the gradient of the electro-chemical potential, represent local values of physical vari-ables, which are defined by macroscopic statistical argu-ments. In principle, they have to be calculated as averagevalues over sufficiently small subsystems, which neverthe-less should contain many electrons. We can not expect thatthe local relation(14) still holds on a length scale of theorder of the mean distance between the constituents of our2D ES, or, equivalently, of the order of the Fermi wavelengthlF. On such a length scale one should consider a nonlocalversion of Ohm’s law instead. This would, however, make


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things much more complicated, and we will not enter suchproblems.

In order to simulate qualitatively the expected effects of anon-local treatment, we start as before with a local model forthe conductivity tensor, take the spatial average over a lengthscale of the order oflF, e.g., withl=lF /2 as

ssxd =1

2lE

−l

l

djssx + jd, s22d

and use still the local version(14) of Ohm’s law, but nowwith the averaged conductivity tensor(22). As a conse-quence, the resistivity components occurring in Sec. III A

have to be calculated by tensor inversion ofssxd.This simple simulation of nonlocal effects has several ap-

pealing aspects. First, ifslsxd vanishes at an isolated positionx= x2k, the averagedslsxd.0 will be positive in the neigh-borhood ofx2k, and the integrals in Eqs.(17) and (18) willnot be singular. Intervals of vanishingslsxd will exist only ifwe start before averaging with sufficiently wide incompress-ible strips (wider than 2l). Finally, for high-mobility sys-tems, the Hall conductivity is given to a very good approxi-mation by the free-electron valuesHsxd=se2/hdnsxd. Thus,the averaged Hall conductivitysHsxd will be given by theaveraged filling factor profilensxd. As mentioned at the endof Sec. II, this averaged profile will agree qualitatively withthe Hartree profile, if we start with the TFPA profilensxd andaverage overl, l (l the magnetic length). Since for the largemagnetic fields of our interestl &lF, there is actually noneed to perform the time consuming Hartree calculation, ifwe finally want to calculate the averaged conductivity tensor(22).

To summarize, our approximation scheme that simulatesboth, the effect of finite width of the wavefunctions in thethermal equilibrium calculation, and nonlocal effects on thetransport, is as follows. First we calculate the density profilensxd within the self-consistent TFPA, replacing the bare Lan-dau DOS, Eq.(11), by the collision broadened DOS that isconsistent with the model used for the conductivity tensorssnd. [Note that both the broadened DOS andssnd are takenfrom the impurity-disordered,homogeneous2D ES.] Nextwe calculate the position-dependent conductivity tensorssnsxdd, evaluating thessnd of the homogeneous system atthe valuen=nsxd. Then we perform the averaging of Eq.(22)and, finally, follow the calculations described in Sec. III Awith the conductivity tensor replaced by the averaged one. Incontrast to Ref. 5 we restrict our calculations here to thelinear response regime and do not investigate the feedback ofthe applied current on the electron density and the electro-static potential, that is mediated in principle by the position-dependent electrochemical potentialm!sr d in the presence ofa dissipative current.

C. Self-consistent Born approximation

In principle we could use the conductivity models of Ref.5 in order to calculate explicit examples. We prefer, however,to take the transport coefficients from the self-consistentBorn approximation(SCBA),14–16 which allows for consis-

tent models of longitudinal and Hall conductivities, and forthe consideration of anisotropic scattering by randomly dis-tributed finite-range impurity potentials. We assume that therelevant scatterers are charged donors distributed randomlyin a plane parallel to that of the 2D ES, with an areal densitynI, and we approximate the impurity potentials by Gaussianpotentials

vsr d =V0

pR2expS−r2

R2D , s23d

with a rangeR of the order of the spacing between 2D ESand doping layer.

An important aspect of the SCBA is that, similar to the“lowest order cumulant approximation” used in Ref. 5, itallows us to treat the transport coefficients and the collisionbroadening of the LLs in a consistent manner. The relevantSCBA results for the transport coefficients and the collisionbroadening of a homogeneous 2D ES are summarized in theappendix. Consistency with the transport coefficients re-quires that we replace in the self-consistent TFPA calcula-tions thed-functions of the Landau DOS(11) by the semielliptic spectral functions(A5). In addition to the rangeR,the impurity strengthV0 and concentrationnI determine thesequantities via the relaxation timet0 defined by the energy" /t0=nIV0

2m/"2. In strong magnetic fields, this energy com-bines with the cyclotron energy to

G2 = 4nIV02/s2pl2d = s2/pd"vc"/t0, s24d

whereG is the width of the LLs in the limit of zero-rangescattering potentialssR→0d. We find it convenient to char-acterize the impurity strength by the dimensionless ratiog=G /"vc at the strong magnetic fieldB=10 T and define(for GaAs), therefore, the strength parameter

gI = fs2nIV02m/p "2d/s17.3 meVdg1/2. s25d

Figure 4 shows the effect of collision broadening on thedensity profile at strong magnetic fields. The sample param-eters ared=1.5 mm, n0=431011 cm−2 and b/d=0.952, re-sulting in nel=3.3731011 cm−2 and EF=12.02 meV, EF

0

=13.51 meV. Data are shown fort=kBT/EF0 =0.01, three val-

ues of the magnetic field["vc/EF0 =0.6, 0.8, and 0.95, corre-

sponding to central filling factorsns0d=3.33, 2.5, and 2.1,respectively], and for three sets of the impurity parametersRandgI. It is seen from Fig. 4 and Table I that, for sufficientlysmall collision broadening(small gI and, eventually, largeR), incompressible strips still may exist, but that their widthdecreases with increasing broadening of the LLs. Table Ishows, for several sets of impurity parameters, the relativewidths gn of the lowest LLs and the zero field mobilities.Data for the second setsR=10 nm, gI =0.1d are not includedin Fig. 4, since they could not be distinguished from thetraces for the first, high-mobility set. From the insets of Fig.4 it is evident that incompressible strips can only survive, ifthe gap between the broadened LLs remains broad enough. Alarge collision broadening(low-mobility set no. 4) has asimilar effect as a spatial averaging(long-dashed lines in Fig.4) or an elevated temperature(see Fig. 8 below), and maysmear out the incompressible strips.


195335-6

IV. RESULTS AND DISCUSSION

A. Effect of spatial averaging

In the following we combine the TFPA with the SCBA.For the TFPA calculation of the density profile we replacethe delta functionsdsE−End in the Landau DOS, Eq.(11), bythe SCBA spectral functionsAnsEd of Eq. (A5). Then wecalculatessxd evaluating the SCBA formulas(see appendix)at the local valuesnsxd of the filling factor, and perform theaveraging according to Eq.(22). With this averaged conduc-tivity tensor we calculate the current distribution, the electro-chemical potential profile, and the resistances. The averagingprocedure, which is motivated by the importance of nonlocaleffects at small distances[especially near lines withslsxd=0], eliminates the effect of narrow incompressible strips,which are known to be an artifact of the TFPA(see Sec. II).

The effect of spatial averaging, introduced to simulatenonlocal effects on the scale of the Fermi wavelength, isillustrated in Fig. 5. It shows, for a magnetic field valuecorresponding to a central filling factorns0d=4.18, the fillingfactor profile calculated with the SCBA broadened DOS, to-gether with the conductivity profiles, Fig. 5(a). Here we have

assumed a short-range potential(leading to the rather lowmobility mB=0=6.4 m2/Vs), in order to obtain a noticeabledeviation of thesHsxd trace from that for the filling factornsxd. Clear incompressible strips with the quantized valuesfor sHsxd and vanishingslsxd are visible wherensxd assumesthe integer values 4 and 2. The effect of spatial averaging onthe conductivities is demonstrated in(b) and (c). The wides,90 nmd plateau defined bynsxd=4 shrinks due to the av-eraging(to ,50 nm) but clearly survives, as is shown in Fig.5(b) for slsxd, and holds similarly for the plateau ofsHsxd.On the other hand, the plateau behavior near the narrows,25 nmd strip defined bynsxd=2 is completely smearedout, andslsxd does no longer vanish in this region, Fig. 5(c).This has, of course, drastic consequences for the current dis-tribution, which is dominated by the strips with vanishingslsxd, i.e., vanishingrlsxd, see Eq.(16). Without averaging afinite fraction of the total current would flow through theincompressible strips withnsxd=2 (on both sides of thesample). With the averaged conductivity tensor, the total cur-rent must flow through the incompressible strips withnsxd=4 [assuming that thereslsxd=0 holds exactly].

The mechanism illustrated in Fig. 5 is, of course, alsoeffective at other values of the magnetic field. At largerBwith ns0d*2, broad incompressible strips withnsxd=2 willexist near the center of the Hall bar, and the spatially aver-aged conductivities will show clear plateau behavior. WithdecreasingB, the incompressible strips move from the centertowards the sample edges and shrink. If the strip width be-comes of the order of 2l or smaller, the averaging accordingto Eq. (22) will destroy the plateau behavior of the conduc-tivities andslsxd will no longer vanish near the strips. Thenthe current density still may have a(finite) maximum nearthe strips withnsxd=2, but a finite amount of current willspread over the bulk of the sample, and the global resistances

FIG. 4. (Color online) Filling factor nsxd<hsHsxd /e2 versusposition in the left half of a symmetric high-mobilitysR=20 nm, g I =0.1d Hall bar of widthd=1.5 mm, for three values ofthe magnetic field,"vc/EF

0 =0.6, 0.8 and 0.95, and without(l=0,solid black lines) and with (l=20 nm, long-dashed green lines)averaging according to Eq.(22). The insets show the plateau re-gions (incompressible strips) enlarged and include in addition tworesults for larger collision broadening but no averagingsl=0d.Other specifications in the text.

TABLE I. Relative width gn=Gn/"vc of the Landau levelsn=0, 1, 2 at"vc/EF

0 =0.6, and mobilitymB=0 at B=0, T=0 in units ofm2/Vs, for four sets of model parametersR, gI.

No. R [nm] gI g0 g1 g2 mB=0

1 20 0.1 0.07 0.06 0.05 747.5

2 10 0.1 0.11 0.08 0.07 75.1

3 10 0.3 0.34 0.24 0.21 8.34

4 10 0.5 0.56 0.40 0.35 3.00

FIG. 5. (Color online) (a) Filling factor and conductivity profilesfor the left half of a symmetric sample withd=1.5 mm, n0=431011 cm−2, b=0.9d, calculated within the TFPA-SCBA, forR=0.1 nm andgI =0.1,"vc/EF

0 =0.48, andkBT/EF0 =0.01.(b) and(c)

repeat the data of(a) (solid lines) near the incompressible stripswith filling factors nsxd=4 and nsxd=2, respectively. The dash-dotted lines demonstrate the effect of spatial averaging, accordingto Eq. (22), with l=30 nm.


195335-7

will no longer have the quantized valuesRH=h/2e2 and Rl=0.

Figure 6 shows typical results for the dependences of theglobal resistances on magnetic field and averaging length, ascalculated from Eq.(19). At high magnetic fields,"vc.EF

0,we have everywhere in the samplensxd,2, and no incom-pressible strips exist. The filling factor and consequently theconductivities and the current density vary slowly across thesample. Thus, the spatial averaging has little effect and,within the accuracy of the figure, the results with and withoutaveraging agree. For slightly lower magnetic fields,"vc&EF

0, broad incompressible strips exist near the center of thesample and, for all the considered averaging lengthsl, stripsof finite width with slsxd=0 andsHsxd=2e2/h survive. As aconsequence, the resistances are quantized, independently ofthe width of these strips.

For still lower B values the situation becomes more com-plicated. Within the TFPA, incompressible strips exist for alltheseB values. Without spatial averaging,slsxd vanishes onthese strips and, as a consequence,Rl =0, and, for"vc/EF

0 .0.5, RH=h/2e2, as shown by the traces forl=0.For "vc/EF

0 ,0.5 there are two types of incompressiblestrips, withsHsxd=2e2/h or 4e2/h, and, without averaging,both contribute according to their widths toRH, while stillRl =0. The fluctuations in theRH curve forl=0 result fromour TFPA calculation on a finite meshsN=500d, which yieldsdiscontinuous changes of the widths of the incompressiblestrips with changingB. This unsatisfactory picture, obtainedfor l=0, results from the model calculation of Ref. 5.

The introduction of the spatial averaging improves thissituation dramatically and leads to qualitatively correct re-sults. With decreasingB the incompressible strips withnsxd=2 become narrower. As the width becomes smaller than 2l,slsxd no longer vanishes, the integrals in Eqs.(17) and (18)and thusRl become finite. This happens at higherB values ifl is larger, and the resistances near the low-magnetic-fieldedge of the quantum Hall plateau depend strongly onl.While slsxd may have a sharp minimum near the strips withnsxd=2 if the width of these strips is only slightly smaller

that 2l, this minimum, and also the corresponding maximumof the current density, will smoothen out as the width of thestrips becomes much smaller than 2l. Then the total resis-tances will become nearly independent ofl, as is seen in Fig.6 for 0.5,"vc/EF

0 &0.6. For "vc/EF0 &0.5, slsxd vanishes

only within the incompressible strips withnsxd=4, but not instrips with nsxd=2. As a consequence, we obtain again theexactly quantized results forRH andRl.

To visualize the intimate connection between the exis-tence of incompressible strips of finite width[now with con-stantsHsxd<e2nsxd /h], we show in Fig. 7 a gray scale plotof the spatially averaged filling factor profile for a relevantinterval of magnetic fields, together with the resulting resis-tances.

B. Effect of temperature and collision broadening

The spatial averaging procedure is essential to obtainquantum Hall(QH) plateaus of finite width for theRlsBd andthe RHsBd curves and to obtain the correct quantized valuesfor RHsBd on the plateaus corresponding to filling factorslarger than two. The width of the calculated QH plateausdoes, however, not only depend on the averaging lengthl,but also on the temperature and on the broadening of the LLsdue to the impurity scattering, since both affect the width ofthe incompressible strips. The effect of collision broadeningon the width of incompressible strips has been indicated inFig. 4. The temperature effect has been investigated in Ref.11, where it was shown that, in the absence of collisionbroadening, i.e., on the basis of the bare Landau DOS, Eq.(11), the incompressible strips have a finite width at zerotemperature. At finite, increasing temperatures, the widthshrinks(while the value of the filling factor remains exactlyconstant within the remaining strip) until at a sufficientlyhigh temperature(kBT&"vc/25 in Ref. 11) the width col-lapses to zero. A similar result is expected in the presence ofcollision broadening. In particular we expect that, within theself-consistent TFPA based on the SCBA DOS, the existence

FIG. 6. (Color online) Calculated Hall and longitudinal resis-tances versus scaled magnetic field"vc/EF

0, for different values ofthe averaging lengthl. The sample parameters ared=1.5 mm, n0

=431011 cm−2, b=0.952d, R=10 nm,gI =0.1, andkBT/EF0 =0.01.

FIG. 7. (Color online) Calculated Hall resistance(light solidline) and (scaled) longitudinal resistance(black solid line) versusmagnetic field, measured in units of"vc/EF

0. The underlying grayscale plot shows the averaged filling factor profile, as in Fig. 2. Thecrescentlike areas indicate the regions of incompressible strips withlocal filling factors 2(right) and 4(left).


195335-8

of an energy gap between two adjacent broadened LLs willalways lead to an incompressible strip, provided the tempera-ture is low enough. The necessary temperature will be thelower, the narrower the gap is. But we will not discuss thesequestions in further details.

The temperature effect on the calculated resistance curvesis shown in Fig. 8. As expected, the width of the QH plateausincreases monotonically with decreasing temperature, but ap-parently it has a finite limit forT→0. We have also includedthe high-temperature result forkBT/EF

0 =0.3 (since EF0

=13.5 meV, this meansT<47 K). At the low-B side of thefigure kBT,"vc, and the derivative of the Fermi functionoverlaps about two LLs. Since we consider only the lowestLandau levelssn=0,1,2d, our calculation is not correct inthis limit. Nevertheless it is interesting to compare this resultwith the Drude result, which should be valid if theShubnikov–de Haas oscillations are smeared out at highertemperatures.

In the Drude approximation we haverHsxd=vcttrrlsxd,with rlsxd=1/s0sxd, wheres0sxd=e2ttrnelsxd /m is indepen-dent of the magnetic field. Inserting this into Eq.(17), weobtain

I

Ey0 =

e2ttr

m2dnel =

e2

h

2EF

"/ttr. s26d

In Eq. (18) we take the integrand to bevcttr, but only foruxu,b, wherenelsxd is not exponentially small, and obtainVH /Ey

0=2bvcttr. With Eq. (19) we obtain the Drude result

RHD =

h

e2

b

2d

EF0

EF

"vc

EF0 , Rl

D =h

e2

"/ttr

2EF. s27d

The energiesEF and EF0 are calculated numerically from

the density profile atB=0, T=0, as is described above, andttr is calculated as described in the Appendix, withkF

=Î2pnel. The results are plotted as dashed straight lines inFig. 8. The large difference betweenRl

D and the peak valuesof RlsBd results from the fact that, at fixed impurity rangeR,

RlD is proportional togI

2 whereas the peak values, in theapproximation summarized in the Appendix, depend onlyweakly ongI (see Fig. 9).

Finally, Fig. 9 shows the effect of the Landau level broad-ening on the QH plateaus at a fixed, relatively low tempera-ture. The corresponding widths of the lowest LLs, in units ofthe cyclotron energy"vc, are given in Table II for the lowestand the largestB value shown in the figure. For the largestdamping, the LLs start to overlap for"vc/EF

0 &0.4 so thatthe n=4 QH plateau is not well developed. We note that theSCBA results summarized in the appendix are only valid athigherB values, where the LLs do not overlap.

The two high-mobility situations considered in Fig. 9 dif-fer only by the rangeR of the Gaussian impurity potentials.The larger range leads to slightly smaller level broadening,but to much lower longitudinal resistance(i.e., to muchhigher mobility atB=0).

C. Hall potential profile

The motivation of Ref. 5 and our present work came fromthe experimental investigation4 of the electrostatic potentialdistribution across a Hall bar under QH conditions, causedby an applied current. Ahlswede and co-workers4,7 observedthree types of potential distribution, depending on the fillingfactor regime. Type I was a more or less linear variation

FIG. 8. (Color online) Hall and longitudinal resistances versusmagnetic field, calculated for different temperatures,t=kBT/EF

0.Sample parameters:d=1.5 mm, n0=431011 cm−2, b=0.952d, R=10 nm andgI =0.1, l=30 nm.

FIG. 9. (Color online) Hall and longitudinal resistances versusmagnetic field, calculated for different values of the collision broad-ening. Sample parameters:d=1.5 mm, n0=431011 cm−2, b=0.952d, l=30 nm, andkBT/EF

0 =0.01 (i.e., T=1.57 K).

TABLE II. Relative widths of the lowest Landau levels,g nl

=Gn/"vc for "vc/EF0 =0.35, andg n

h=Gn/"vc for "vc/EF0 =1.10, for

the impurity parameters used in Fig. 9. First column: range in nm;last column: zero field mobility in m2/Vs.

R g I g 0l g 1

l g 2l g 0

h g 1h g 2

h mB=0

20 0.1 0.117 0.085 0.073 0.043 0.037 0.033 747.5

10 0.1 0.161 0.123 0.105 0.071 0.051 0.044 75.1

10 0.3 0.482 0.369 0.316 0.213 0.152 0.131 8.34

10 0.4 0.643 0.492 0.421 0.284 0.203 0.175 4.69


195335-9

across the sample and is observed if the filling factor in thecenter is smaller and relatively close to(but not too close to)an integern, i.e., n*ns0d*n−1/2. If the center filling isslightly larger than an integer,n,ns0d&n+1/2, type III isobserved, characterized by a constant potential in the centralregion and a rapid variation across(narrow) strips, whichmove with decreasingB towards the sample edges and havebeen interpreted as incompressible strips.4 Finally, type IIshows a rapid, nonlinear variation of the potential in thecenter region and is observed if the center filling factor isvery close to an integer.

In Ref. 5 it was shown that, in a local equilibrium picture,the changes of the electrostatic potential, caused by an ap-plied current, follows closely the current-induced variationof the electrochemical potentialm!, so that the resulting den-sity changes are small. In the present work we do not con-sider the feedback of the spatial variation ofm! on electro-static potential and density profiles(linear response). But weexpect from the results of Ref. 5, that, in the linear responseregime,m! should show the same position dependence as theelectrostatic potential would do, if the feedback were calcu-lated.

To calculate the Hall profile across the sample, we inte-grate Exsxd, Eq. (16), from the centerx=0 to the actualxvalue. Typical results as functions of positionx and magneticfield B are shown in Fig. 10. The normalization is chosenso that VHsB,x=dd / I =−VHsB,x=−dd / I =RH /2. One seesclearly that the plateaus of the quantized Hall effect(0.8&"vc/EF

0 &1 and 0.45&"vc/EF0 &0.5) coincide with poten-

tial variation of type III, caused by current density confine-ment to the incompressible strips. Moving from a plateauregion to smallerB values, the incompressible strips shrinkand finally vanish, and the current density spreads more andmore out into the bulk. This leads to the type I behavior(0.52&"vc/EF

0 &0.7 and"vc/EF0 &0.4). Immediately above

the integer values of the center filling factor[in our approxi-

mation assuming spin degeneracy nearns0d=2 and 4], wefind the rapid variation of the type II. This is in very niceagreement with the experiment. Without our spatial averag-ing of the conductivity tensor, we would have missed thetype I regions for"vc/EF

0 ,1, as has been observed in Ref.5.

D. Summary

The virtue of our approach is, that it allows us to calculateresistance traces with exactly quantized quantum Hall pla-teaus of finite width, and with reasonable values of the resis-tances between these plateaus. While these intermediate re-sistance values depend on the details of our conductivitymodel, the quantized plateau values do not. The reason forthis high accuracy and model independence of the plateauvalues is the fact that the latter are determined by the inte-grals in Eqs.(17) and(18) becoming singular across incom-pressible strips.

To obtain realistic widths of the QH plateaus, we had toconsider a mechanism that prohibits singular current flowalong very narrow incompressible strips. We have arguedthat small-scale nonlocal transport effects act into this direc-tion, and that consideration of the finite extent of wave func-tions will prohibit arbitrarily narrow ISs at low magneticfields, in contrast to the prediction of the Thomas-Fermi ap-proximation. We were able to simulate such nonlocal effectsby a simple spatial averaging procedure, with reasonable re-sults for Hall and longitudinal resistance as functions of themagnetic field. Also the resulting potential profile, and there-fore the current distribution across the sample, is in niceagreement with recent investigations.4 We consider this as astrong support for the relevance of our approach, notablybecause earlier approaches, which neglected dissipation, can-not explain the experiments, as has been discussed in Ref. 5.

Note that, for QH plateaus corresponding to filling factorsnù4, our results are qualitatively different form the conven-tional edge channel picture. The latter explains, for instance,the quantized conductance valueG=4e2/h as the sum of thecontributions of two spin-degenerate, quasi-one-dimensionalcurrent channels near each of the opposite sample edges, thustracing back the quantized Hall effect to the phenomenon of1D conductance quantization(in a situation where no back-scattering occurs).17 That is, the edge states, created by theLLs with quantum numbersn=0 andn=1, contribute both tothe current in the plateau regime of the QHE. Our results, onthe contrary, indicate that the total current flows along theincompressible strip with local filling factornsxd=4 (whereboth LLs n=0 andn=1 are occupied), whereas near localfilling factor nsxd=2 no incompressible strip and no contri-bution to the current exists.

Comparing our resistance curves with experiments,we notice that the high-field edge of a calculated plateauoccurs at a magnetic field, at which an incompressible strip(with an even integer value of the effective filling factorn0=2EF

0 /"vc) first occurs in the center of the sample. In ex-periments thesen0 values usually are found somewhere nearthe centers of the plateaus. We have good arguments that thisdiscrepancy is due to our neglect of long-range potential

FIG. 10. (Color online) Hall potential profile VHsxd=e0

xdx8Exsx8d across the sample, for varyingB/B0;"vc/EF0 and

constant applied currentI. Normalization:fVHsdd−VHs−ddg / I =RH;sample parameters:d=1.5 mm, n0=431011 cm−2, b=0.952d, l=30 nm, andkBT/EF

0 =0.01.


195335-10

fluctuations due to the randomly distributed ionized donors.We have simulated the “short-range” part of the Coulombpotentials of the remote donors by Gaussian potentials, butwe have neglected their overlapping long-range parts, whichlead to long-range potential fluctuations. We have evaluatedthe short-range disorder within the SCBA to calculate con-ductivities and LL broadening. We have seen that with in-creasing disorder scattering the level broadening increasesand, as a result, the widths of the QH plateaus shrinks. Onthe other hand, one knows from technical applications of theQH effect, that rather impure samples have usually espe-cially wide and stable QH plateaus. This points to the role oflong-range potential fluctuations, which become more impor-tant with increasing impurity concentration.

As a rough simulation of such long-range fluctuations, wehave added oscillatory terms to the confinement potentialand then repeated our calculations. We indeed find that suchmodulations can widen and stabilize the QH plateaus, andeventually even shift them to higher magnetic fields, depend-ing on the amplitude, the range, and possibly other details ofthe perturbation. This becomes understandable, if one con-siders the effect of such fluctuations on the existence and theposition of incompressible strips. Effects of long-range dis-order in an unconfined 2D ES on the longitudinal resistancebetween QH plateaus have already been discussed a decadeago.18 This discussion seems however not applicable to therather narrow samples of our present interest, since, first, theconfinement affects the self-consistently calculated potentialand thus the density distribution,13 and, second, the earlyassumptions about current-carrying and insulating regionsare not compatible with our results and the experimentalfindings.4

As mentioned in the Introduction, our approach does notinclude localization effects, which often are considered to beessential for an understanding of the QHE. Indeed, scalingpredictions obtained on the basis of localization theory(inthe limit of infinitely large samples) apparently have beenconfirmed by experiments.3 However, it seems questionablewhether these concepts are relevant for the microscopicallynarrow Hall samples of our present interest.4 Our calcula-tions demonstrate that an alternative origin of the QHE isconceivable: Screening effects(which are usually neglectedin localization theories) may lead in inhomogeneous,Landau-quantized samples to incompressible strips, in whichthe Fermi level lies in a Landau gap, so that in the stripregion no elastic impurity scattering is possible and, at lowtemperatures, the longitudinal conductivity(and thereby theresistivity) vanishes. As a consequence, the applied currentflows along the incompressible strips, which leads to vanish-ing longitudinal and quantized Hall resistance. For the nar-row Hall bars investigated in Ref. 4 the inhomogeneity dueto the electron-depletion regions near the edges is sufficientto explain the observed QH plateaus. Whether in macro-scopic samples inhomogeneities due to long-range potentialfluctuations or localization effects are more relevant for theQHE remains to be investigated.

ACKNOWLEDGMENTS

We gratefully acknowledge useful discussions with E.Ahlswede and J. Weis. This work was supported by the

Deutsche Forschungsgemeinschaft, SP “Quanten-Hall-Systeme” GE306/4-2.

APPENDIX: SCBA CONDUCTIVITIES

The low-temperature, high-field magnetotransport, deter-mined by elastic scattering of the 2D electrons by randomlydistributed impurities with scattering potentials of arbitraryrange, has been studied by Ando and co-workers.14–16 Theresults for the case of nonoverlapping LLs can be summa-rized as

n = gson=0

` E dEAnsEdfsE − md, sA1d

sl = gson=0

` E dEF−]f

]EGsxx

sndsEd, sA2d

sH =e2

hn − DsH, sA3d

DsH = gson=0

` E dEF−]f

]EGDsyx

sndsEd, sA4d

with the spectral functions of widthsGn,

AnsEd =2

pGn

Î1 −SE − En

GnD2

, sA5d

centered around the Landau energies(10), and

sxxsndsEd =

e2

h

p

2fGn

xxAnsEdg2, sA6d

DsyxsndsEd =

e2

h

p2

4

Gnyx

"vcfGn

yxAnsEdg3. sA7d

Assuming a single type of impurities with the Gaussian po-tential (23), these parameters can be expressed in terms ofthe integrals

sGns jdd2 = G2E

0

`

dx gns jdsxdexps− f1 + a2gxd, sA8d

where G2=4nIV02/ s2pl2d and a=R/ l, and the weight func-

tions

gns0dsxd = fLn

0sa2xdg2, gnsddsxd =

1 − x

2a2 gns0dsxd,

gns±dsxd =

xÎ2n + 1 ± 1

Ln0sa2xdLn−s171d/2

1 sa2xd

are determined by the associated Laguerre polynomialsLn

msxd. With these notations one obtains

Gn2 = sGn

s0dd2, sGnxxd2 = sGn

sddd2, sA9d


195335-11

sGnyxd4 = sGn

s+dd4 + sGns−dd4. sA10d

In the limit of short-range scattering potentials,a→0, onehasGn

2/G2=1, sGnxx/Gd2=n+1/2 andsGn

yx/Gd4=n+1/2. Withincreasinga these parameters decrease and remain fora*1 about an order of magnitude smaller than theira=0values.14,15

Some typical SCBA results are shown in Fig. 11. In gen-eral, sl and the correctionDsH to the free electron Hall

conductivity sH0 =e2n /h decrease with increasing range of

the scattering potentials.At zero temperature,DsH is proportional toG /"vc with a

factor depending only on the range parametera=R/ l. Thelongitudinal conductivityslsnd, on the other hand, dependsonly ona and not onG2=s2/pd"vc" /t0, i.e., depends not onthe impurity concentrationnI and strengthV0 entering theB=0 relaxation rate" /t0=nIV0

2m/"2. This is very differentfrom the B=0 conductivitys0=e2nelttr /m obtained for theimpurity model(23), which depends onnel=kF

2 /2p and viattr on both the potential strength and range. ForB=0 colli-sion broadening effects can be neglected, and one obtains forelastic scattering at the Fermi edge

"

ttr=

nIm

"2 E−p

p dw

2pfvqgq=kFf2s1 − coswdg1/2

2 s1 − coswd

="

t0fe−x

„I0sxd − I1sxd…gx=sRkFd2, sA11d

where the last equality holds for our impurity potential(23),with Fourier transformvq=V0exps−R2q2/4d, and Insxd is amodified Bessel function. This leads forRkF@1 to t0/ttr

<fÎ8psRkFd3g−1. With increasing temperature, the peak val-ues ofslsnd decrease and the minima at even integern val-ues are no longer exponentially small forkBT/"vc*0.1.This behavior of the SCBA results, shown in the middlepanel of Fig. 11, is similar to that of the Gaussian modeltreated in Ref. 5. At finite temperature, the longitudinal con-ductivity slsnd increases withg, i.e., with increasing scatter-ing strength(bottom panel of Fig. 11). This is as expectedfrom the Drude picture forvcttr @1: sl <s0/ svcttrd2

~1/ttr.

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FIG. 11. (Color online) SCBA results for longitudinalssld andHall ssHd conductivity, in units ofe2/h, versus filling factor, atfixed magnetic field for different values of impurity rangesa=R/ ld and strengthsg=G /"vcd, and of temperaturest=kBT/"vcd.For the two lower panels the correction −DsH to sH

0 =e2n /h isnegligible, andsH is not shown.


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incompressible strips in dissipative hall bars as origin of quantized hall plateaus

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