Conductivity tensor of a two-dimensional electron gas in a strong magnetic field in higher orders of perturbation theory. Quantum hall effect

0. G. BALEV and I. I. BOIKO: Conductivity of a Two-Dimensional Electron Gas 721

phys. stat. sol. (b) 133, 721 (1986)

Subject classification: 73.40

Institute of Semiconductors, Academy of Sciences of the Ukrainian SSR, Kiev1)

Conductivity Tensor of a Two-Dimensional Electron Gas in a Strong Magnetic Field in Higher Orders of Perturbation Theory

Quantum Hall Effect

BY 0. G. BALEV and I. I. BOIKO

The components of the conductivity tensor 2 are considered for the case of a strong magnetic field H . The investigation of the higher orders in perturbation theory up to the fourth order in static scattering potential shows that for an electric field E under the condition E<< E,-

3 [ h ~ H ~ / 2 m ~ ~ ~ ] ~ 1 ~ the Hall conductivity ozy = 2 { 1 + O[exp (-E&!P)]} and the transverse

conductivity oyy - exp ( -Ei /E2); here N , is the carrier concentration. In virtue of the extreme small collisional contribution to the conductivity the picture of the quantum Hall effect is similar to that for an ideal noninteracting two-dimensional electron gas.

eN c H ,

npOBeEeH paC'IeT KOMnOHeHT TeH3Opa npOBOHBMOCTB ^o B CBJILHOM MarHBTHOM none H .

(DO 9eTBepTOrO ITOpfliTICa BKJIIO9HTeJILHO) nOHa3aJI0, 9 T O B 3JIeKTpM9eCICBX ITOJIfiX E , rne BbInOJIIICHO YCJIOBBe E < E,, 3 [heH3/2m2~3]112, XOJIJIOBCKaII ITpOBOA&IMOcTb

klccnexoBasne BbICUIBX ~ O P R H K O B Teopmi BO3MyKqeHHfi no pacceBsaIoKqeMy noTeHqMany

PN c

IJ, oZr, = -,- { 1 + 0 [exp ( -E;/E2)]} , a nonepewafi II~OBOABMOCTL oyy-exp (-E:/E2); a ~ e c ~

'v, - IiOH4eHTpaUMR 'IaCTLIII. npeneJILHaH MaJIOCTL CTOJIKHOBMTeJlbHLIX BKJIaHOB IIpIIROAHT K KapTMHe KBaHTOBOrO 3??@2KTa XOJIJIa, CBOfiCTBeHHOfi HAeaJlbHOMy (He &iCIILITLIBaIo~eMy paCCefiHI?fi) ABYMepHOMY ra3y 3JIeKTPOHOB.

1. Introduction

Consider the electrons and the external electric field E to be arranged in the xy plane, the magnetic field M being directed along z-axis. I n experiments for a two-dimensional electron gas (ZDEG) on quantum Hall effect (&HE) were noted rather wide regions of H and gate voltage V , in MIS structures where the ratio of diagonal t o nondiagonal components of the conductivity tensor loyy/oZyl ;=: lo-', while the component G . ~ ~ within an accuracy of about is an integer multiple of e2/(2nh) [l, 21. Usually the situation G!,!, - 0 is related to the totally occupied Landau levels ; if one level is partially occupied, the negligible value of transverse conductivity is connected with passing of the Permi level through localized states [3, 51. However, many essential features of the &HE (e.g. a set of plateaus on o z y ( H ) and ozy( V,) dependences) are also realized for ZDEG in the absence of scattering [6]. The negligible effect of scattering by a static potential on the ZDEG conductivity is deduced from the calculations of the transverse conductivity performed to the second order in perturbation theory. In a quantizing magnetic field M and low electric field E the conductivity oyv due to scattering by a static poten-

1) Kiev, USSR.

46 physic& (b) 133/2

722 0. G. BALEV and I. I. BOIKO

tial turns out to be exponentially small, proportional to exp ( -const/E2) (see [7, 81). It was shown in 191 that within a linear response approximation the conductivity oUU vanishes.

The perturbation theory calculations are merely correct when the expansion is a power series in a small parameter. Since in zeroth and first orders in interaction nu,,, = 0, while corrections to the ideal value of crGV are absent in first and second orders, we must investigate the conductivity in higher orders of perturbation for examining if their contribution is essential. Indeed, as was shown in [lo], the first nonvanishing contribution to ovU appears in the fourth order in the interaction.

conipo- nents up to the fourth order in perturbation theory. We consider the electrons being scattered by a random static potential ; electron-electron interaction is neglected, for sitnplicity an interaction of the spin momentum with H i s also ignored. In absence of a scattering potential we suppose the state of the electron system to be uniform and stable. The investigation of expressions obtained for conductivity shows that in quantizing magnetic fields the approximation of a response linear in E is inadmissible for 2I)EG and shows that in a weak electric field the terms of third and fourth orders decrease in powers V being also exponentially small similar to the value ovv calculated in the second order in V . Thus we conclude that the effective collision frequency of the 2DEG scattering on a static potential a t small electric field is extremely small. The typical value of a seatotering potential V, must be merely small in comparison with the interval between Landau levels. Then &HE (nearly ideal plateaus of crz?{) is realized similar to that in collisionless 2DEG moving in crossed electric and magnetic fields.

In what follows we carry out the calculation for the conductivity tensor

2. Electrical Current in n-th Order of Perturbation Theory

The Hamiltonian of two-dimensional electron motion in crossed fields H I [ Ox E: I I Oy including interaction with a random potential V ( r ) is given by

where r = {z, y}. The scattering potential is assumed to have the form N N

j=1 j=1 V ( r ) = c U ( r - Tf) - <c U ( r - r , ) ) ,

where N is the total number of scattering centres in the xy plane, U ( r - r,) is the energy of the interaction between the electron and the centre located a t the point r l ; angular brackets denote the average over positions of scattering centres. The eigenvalues of the operator Ho are

where Q = eH,/mc, W: = 191 (n, + +), the a-state is specified by the set of quantum numbers n, and kza. In the following all matrix elements will be calculated with the eigenfunctions of H,.

A

The equation of motion of the single-electron density matrix 6 is

i% --iR - + [go + V ( r ) , @I = 0 . at

Conductivity Tensor of a Two-Dimensional Electron Gas in a Strong Magnetic Field 723

Applying the Laplace transformation over time to (4) and denoting R(co) =

= J eiWt c( t ) dt one can present the solution of (4) in a power series in V ,

A

00

0 00

Ra,(o) = C R$)(w) . n=O

Let the scattering potential V be absent for t < 0 and be adiabatically switched on for

where v + +O. A t t = 0 the density matrix is suggested to be diagonal; we denote c(t = 0) = &. Then

Here W is the energy of the lowest electric subband, [ the Fernii energy, &.p the Kron- ecker symbol. The concentration of particles of BDEG, N,, is connected with by the condition 2 C eaa(t) = N,L2, where L is the linear dimension of the system in x- and

y-directions (the factor two is due to the spin). The choice of f a correspondsto the equi- libriuni distribution of electrons in the frame of reference moving with the collisionless drift velocity in crossed fields,

a

(in this reference frame the electric field is absent).

formula The higher-order terms in ( 5 ) are related to the lower-order ones by the recurrence

w + i x

1 do’p((t) - w’) x (n)

R a P ( C O ) = 2Zh(OJ + <a,a) 1 - m + i x

x [ V a y R y ) ( O l r ) - P - l ) a)r (w ’ ) V y ~ l . Here

0 I m o > x > O , (oba - LO, = + vUz(kZ6 - k z a ) ,

Let us show that RZ”)(m) = 0. According to (6) x

= - c RC)(W) = 0 . a

Then we have

N J 2 = 2 C eaa(t) = 2 C eOaa 2 C f a - a a a

46 *


The average current density of ZDEG in the zy plane is w + G 2e

jz,&) = 7 C (6&,)pa I , d o ciwt rap(^)) , 2 d a,@ - w +ax

where ^v is the velocity operator. One can rewrite this expression as

Here 6 = x or y. I n order to obtain (9) we used the property (7) and the denotation w: = vE - (eE,/mS) d ~ , ~ ; one must point out that (Gt)ba = [ ( 6 t ) p a ] ~ = ~ .

In accordance with (5) the current density can be expanded into a power series in the scattering potential,

A , .

Here the zero-order term is given by

where LzTn' is the Laguerre polynom, qH = (m I QI / f ~ ) * / ~ . After sufficiently long time t > Y-* as well as t 5 jv,qHI

rent density becomes independent of time and reduces to the expression for the cur-

x I d2q( V(q ) I ( b q ) a y p$-*) - (bq),p Pg-')]) . (13)

Here F$ = fada,; the recurrent relation between W ) and W-l ) is evident from (13). Owing to the properties of velocity matrix elements (Gt)pa/cupa == (i&)pa/wja.

We use the identity

Conductivity Tensor of a Two-Dimensional Electron Gas in a Strong Magnetic Field

t he top sign refers to i&. Then (13) can be rewritten as

725

3. General Expressions for the Lowest Four Orders (0) - In the beginning we consider the terms of first order. At n = 1 the quantity P,, d,,

stands in the right-hand side of (15). After summation over k,, the combination qg,,S(qz) 6(q,) appears under the integral over q in accordance with (12). Therefore, jg!, = 0 and

;(I) = 0

for arbitrary distribution of the scattering centres (under averaging z(l) vanishes also due to the condition ( V ( q ) ) = 0).

.(2) 2ie In the second order in perturbation theory

c d2q d2q’(b,)13, (bn’)aB mp %,z( V ( q ) V ( q ’ ) ) *

(16) s s (47~2L)~ hmQ 3 2 , V = f

Here fs, = fa - fa. The correlation functions ( V ( q ) V ( q ’ ) ...) are represented in the Appendix. Substituting (A2) in (16) and taking into account the symbolic form

in 6(a) , 3 a + i O a

- 1

we find

Here nI = NL-2 is the concentration of scattering centres,

For the third order in V using (A2) we obtain the expression

where


For the fourth order of perturbation theory by means of (A2) we obtain

4. Investigation and Simplification of the Obtained Expressions

Kow we examine the expressions obtained. From the very beginning we single out the group of terms which do not contain 8-functions of energy; only principal-value integrals are contained there. These terms involve quantities A(1), B(l), D(I) and we shall demonstrate now that they do not give any contribution to l?(2), 6(3), and l?(4), respectively.

One can see from (18) to (23) that the wave vectors constitute the following combination :

This combination appears naturally after averaging over possible locations of scattering centres (see Appendix). However, (24) is also valid for any fixed arrangements of centres (and any number of them up to the single centre). For the x-component of (2-1) this follows immediately from the Kronecker symbol in (12) and the structure (b,)%,j ... (bg-l))t,, closed up with indexa. After certain transformations of this structure in the formulae for currents the sum appears

q + q‘ + ... + q(n -1 ) = 0 . (24)

We apply to the currents operators which transpose indices and wave vectors 11 - 1

k=O ( l / n ) C ($n )k where

$ % = ~ o l t - B ” ’ ... t L Y ; q t q ’ - q ’ ’ t . . . t q ( n - - l ) , q } *

Then the expressions for these currents are converted in a way that the combination ( l / n ) (pY,, + qb,, + ... + p:;’)) appears instead of pl/,z as a factor to 8(q + q’ i + ... + qcffl-l)). Hence by virtue of (24) we come to jgil = 0. Therefore, the remaining ternis in currents jg; surely contain combinations of the form f,p8(wap) = fap 8[to,p 4- + u,(k,, - kzp)] . From this expression one can observe that the currents $,’; are equal t.0 zero when an electric field is absent (then

0

= 0) .


Sote that for 2DEG in a quantizing magnetic field ( t z 191 > T) the consideration of the response by means of linearization of the structure fap S(wap) in the electric field, as was performed in [ l l ] for the three-dimensional case, is inadmissible. Indeed, min I ex Ak,] = I 91 > T / h and a small parameter for expanding the currents in a power series in electric field does not exist. Therefore, the consideration of the 2DEG conductivity in a quantizing magnetic field provided from the origin within the framework of a linear approximation in the electric field response is not valid.

For simplicity of calculations in the following we shall suggest V(q) = V ( q ) ; practically i t is quite enough that V(q) should be even on qv.

It is easy to obtain

ZY = 0 . (25)

This arisesfrom (18) where q, is a factor to a quantity even in qY (the quantities A‘,; are independent of qv ; after integration over q’ the combination I (bq).p[ even in q, due to (12 ) appears).

is different from zero among the second-order terms. As a result Only the current

Applying the transposition operator {a s p ; q 4 -q; q’ s -q”} to (20) we find that, j!j3),3 and jf),2 annihilate. Finally we obtain

5. Dependence of Conductivity on Electric Field

Consider (26) in the ultra-quantum limit when the lowest Landau level is arbitrarily occupied. Insertion of (12) and (19) into (26) yields

m


Introduce the characteristic field

In the case E < Eo the main contribution to (29) gives the item with p

Everywhere below we shall assume

E<EO (this electric field region is of most interest).

the form If the electrons populate more than one Landau level the expression for u'y",' takes

where Az(qv, 4%) - V 2 is a smooth function of its arguments. For this case, as well as in the ultra-quantum limit, necessarily the factor appears

hi91 heH3 exp (--) 2mv: = exp (---) 2m2c3E2 = exp (-$)

Here and furtheron we suppose

If the inequality (32) is satisfied, the factor exp ( -Ei/E2) practically determines all dependences of CT:; on E and H . The conductivity ufi appears to be exponentially small; such results were obtained previously for the nondegenerate 2DEG in [7] and for high degenerate gas in [S]. Now we start to consider the higher-order contributions in the expansion of the current density in the scattering potential. First we investigate the conductivity @. According to (27)

'(3)

g ( 3 ) = 1- h c I d2Q I d2q' V ( q ) V(p") W I Q + Q' I 1 4.v x Ev &BY

x (bq)aB (b-n-n')B, ( b q , ) y a S(0JaB) w%J fa, * ( 3 4 ) Eixploring the combination ytapy=(bq)ap (b-n-q,)B, (bq,),a we have in accordance with

(12)

Conductivity Tensor of a Two-Dimensional Electron Gas in a Strong Magnetic Field 7 29

where A is the angle between the vectors q and q’. Prom the expression thus obtained one can see the characteristic small exponent t o appear in (34),

Another small exponential factor exp ( -Et/3E2) arises after integration over the angle of the quickly oscillating exponent exp (-qq‘ ei4/2q&). Eventually we have

Examine now the expression for the transverse conductivity ~$2. According t o (27)

Consider the following SUIYJ

1 a3

I B a ( Q 1 , 4 2 ) = c c -- (bq,)Py ( b q 2 ) y a * ny=O kzy may

Using (12) and carrying out the summation over k,, we have

For the sake of certainity we suppose n, > ng. Using the identity ex I eluz dz

e2ir121 - 1 1 - . o - - . -.. ~ -

and the summation forniula following from c (bqlJPY (bq& = (bqL+q2)Ba, Y

00

C fp-@L;-p(u) L$-B(v) = (1 + f)”-” e-ufLb--B /3 ( u + + vf-’ + uf) 9 p = o

we find after integration over z

(37)

'i 30 0. G . BALEV and I. I. BOIKO

Here y is the incomplete y-function, L = qZ2w,/\ Q\ , the coefficients C, are determined by the expression

That implies s1 = -na, s2 = ng. Below, in the integration over q and q', the vicinity of the saddle-point where

zul = 0 and w2 = 0 will be essential. Taking into account the small value of the argu- ment of the y-function in the vicinity of the saddle-point, and the fact that the main contribution to the sum over s is given by the term with s = sl, we can writ,e

*

That yields

After integration over q' in (36) with the aid of the saddle-point met..od we a-tain for smooth potential V(q) ,

For a quickly changing potential V(q) , e.g. a Coulomb potential, one must replace V ( 0 ) by V ( l / g ) . (39) contains the same combination fa@ 8(to,p) I (bn).pI2 which enters GI;:) (see (26) and (19)) and determines its value. It follows from (39) that

Comparison of (39) with (26) gives

where N is the number of Landau levels be!ow the Fermi level. Therefore, the third- order terms are exponentially small like the terms of second order.

Consider now the fourth-order terms. According to (28) and (23) the expression for GL;) = ji4)/Eu involves the product of two 8-functions of energy and one combination having the form of (37). The expresssion for o!,",' = jg)/Ev contains either a product of one 8-function of energy and two combinations having the forin of (37) or a product of three 8-functions of energy. The main contribution to crd is given by the terms

Conductivity Tensor of a Tu-o-Dimensional Electron Gas in a Strong Magnetic Field 731

of the first group. Carrying out the operations indicated in (28) we obtain

where A!$(q,) - U4 is a smooth function of qz. That iniplies

c g - exp ( -B"). E;

The examination of the expression for ~$7 demonstrates that terms linear in the scattering centre concentration nr involve the same factor exp (-4Ei/3E2) as c!$. The terms connected with the part of the correlation function proportional to n: contain the factor exp (-Ei/E2). Thus, for E < E,, in the fourth order of perturbation theory an essent,ial difference appears between the problems with single- and numerous-scattering centres. In the general case

S o w we discuss the results obtained. The expressions contain a t least one 8-function of the difference of energies for two states taking part in the transition,

f l , 8[IQl (n, - n,) + %(k21 - ks2)I . It is clear that the elastic scattering is accompanied by a simultaneous change of n and k,, i.e. transitions occur between different Landau levels. For In, - n2/ = 1 the change of the wave vector is lAk,l = I S/wzI. In the case of small electric fields ( E < E,) the value lAk,/ is much greater than the reverse magnetic radius qII, and transitions are accompanied by a large change of the y-coordinate of the oscillator centre: lay1 =

= q$ /Ak,l > qi ' . For distances which are large in comparison with 9%' the wave functions of the oscillator decrease exponentially and the matrix elements for transitions on large distances are exponentially small.

Our calculations carried out up to the fourth order in perturbation give the possibil- ity for a guess for arbitrary higher orders. lt is very likely that the typical contribution to conductivity in n-th order contains the coinbination

P n f a p 8(wa~) (bql)aij. (bq2)pv ... (bqJ im . The iiiaximuni value of the product of these matrix elements corresponds to the case when k,,, ... ) k,, are within the interval (k,,, kza) :

I hxi3 h:y k q i &a

1 I I I I I I 4

As ;t consequence of (12) and the energy conservation law (w, = mB) the matrix elettient (b&p generates the factor exp [ - (k$, -- k,p)2/4q&] = exp ( -Qz/4q$2.':) = = exp ( --Eg/2E2). The greatest factor from other matrix elements corresponds to an equidistant distribution of points k,p) k,,, ... ) k,,,, k,, and has the form exp [ - (kze - - klp)2/4q$(n - I)]. A t n > 2 an additional, exponentially small factor arises also after the integration of quickly oscillating factors in matrix elements & ) $ , T ~ , (see (12)) over qj , l , qTz, ... ) qqn. As a result one can assert that in the n-th order of perturbation theory the conductivity tensor components contain the exponentially small factor exp ( -qE;/E2) where q 2 n/2(n - 1). The actual calculations carried in the third and the fourth orders show that q 2 1.

732 0. G . BALEV and I. I. BOIKO

6. Discussion. Quantum Hall Effect

Investigate now the Hall conductivity as a funciton of the magnetic field H and the gate voltage VG in a MIS structure. Changing VG we alter the energy W of the lowest electric subband. Since for E < E,, the scattering correction to axy is extremely small, one can take

In accordance with ( 7 ) the concentration N , depends on the Fermi energy 5 only, not on the field E and the value of the scattering potential. We suggest the value 5 to be given by the position of the Fermi level in the bulk. Consider the Hamiltonian fi = &(E = 0 ) + V ( r ) . Let Ex be eigenvalues of A-states. Then N , can be written as follows:

N,L2 = 2 c / ( E n + w - C) , where f is the Fermi function. Therefore, (43) takes the form

Let the spectrum of energies kA be composed of bands with a width of the order of V,,, which are separated by gaps of the order of tz 191. The number of states in each band equals &L2/2n. The change of the values VG or H leads to a shift of the bands of allowed states relative to the Fermi level. When a broadened Landau level (really one spin sublevel) passes through the Fermi level, the Hall conductivity oxy changes by (ec/HZL2) (&L2/2n) = e2/2nh. After that a plateau of aXy arises and a further change occurs when the Fermi level passes through the next broadened Landau level. A s a result the dependence of Hall conductivity on V , and H is step-like with plateaus dif- fering in height by the value e2/2ntz. The width of transition regions is related to the plateau width as V,, to h 191.

According to our calculations the value ayy is exponentially small for any V , and H . The disappearance of transverse conductivity ayy for the case when the Fermi level passes through a broadened level was observed experimentally for the first Landau level in [ 2 ] . At the same time ayy was not always small in experiments. Isolated humps were observed when the Fermi level falls within the limits of the broadened Landau level (see, e.g., [2]) . According to 1121 for a partly occupied Landau level a phase transition can occur due to electron-electron interaction leading to the formation of charge density waves; our calculations are not applicable for this case.

The absence of scattering contributions to conductivity allows to explain some properties of &HE, e.g. the plateaus of Hall conductivity ax. and the extremely small value of transverse conductivity ayV.

Note that in the considered case of exceptionally small scattering contributions we have found axv to be always strictly proportional t o the concentration of particles of 2DEG (see (43)). It follows hence that localized states of electrons are practically absent .

Usually QHE is explained by another model. In that model Landau levels are broadened due to the interaction of electrons with the static random potential V ( r ) ; simul- taneously the significant part of states of the broadened Landau level refer to localized states (see, e.g., [3,4, 13 to 151). In these papers the step-like dependence of axy on magnetic field or gate voltage is connected with the change of the concentration of


2DEG. The dependence of oxu on N , is essentially non-linear in a wide interval. The edge of the plateau corresponds to the situation when the Fermi level passes through the boundary between localized and delocalized states. The plateau itself corresponds to the position of the Fermi level beyond the region of delocalized states. Even for a large number of localized electrons (up to 97%, see [ 2 , 151) the conductivity ozy a t the plateau equals the “ideal” value eN,c/HZ where N , is an integer multiple of q:j/2n. The fact that a small part of electrons provides the ideal Hall conductivity is not trivial and requires an explanation.

We examine the peculiarity of conductivity in the frame of reference moving rela- tively to the crystal in x-direction with the velocity v, =cE,IH,. In this frame of reference the electric field is absent. Let N , = Nl + N2 where index “1” refers to delocalized and index “2” to localized electrons. In the laboratory frame of reference in the presence of an electric field,

Here VI is the velocity of scattering centres, the line above denotes averaging over all delocalized electron states. For simplicity we consider the Fermi level to be situated between broadened Landau levels. In the moving system the velocity of scattering centres is v;, = -vx, while the average velocity of delocalized electrons is

~- j , = eNewx = e(N, + N 2 ) v, = eNl v,,(E, + 0; vI = 0) .

- .. ~ -, v Z ~ = vXi(E2; = 0 ; v;, = -w,) = v,~(E, + 0; V I = 0) - V , =

One can see that in the frame of reference where the electric field is absent, delocalized electrons move in the direction opposite to that of scattering centres. A t N 2 > N , (the number of localized electrons is much greater than that of delocalized ones) the absolute value of the velocity of delocalized electrons is much greater than the velocity of scattering centres. It is difficult to imagine a mechanism of interaction with the average frictional force being in the same direction as the average velocity of mobile electrons. A model with localized states does not overcome this difficulty. Note also that calculations of the conductivity in the model with localized states are usually carried out within the framework of a linear response on the basis of Kubo’s formula (see [3, 13, 141). Our calculations demonstrate that conductivity cannot be expanded in powers in the electric field a t E = 0. This feature was also noted for the Hall conductivity oXv in [161.

Appendix

Correlation functiom of the static potentials

The average over the positions of the scattering centres is defined as follows: L L

0 0

Assuming the value U ( r ) to be absolutely integrable and applying the Fourier transformation to ( 2 ) we have

where P(q) = J e--iqr V ( r ) d2r = V ( q ) A ( q ) ,

V ( q ) = J e-iqr V ( r ) d 2 r , n:

( C e--iqTi > . N A ( q ) = C e-*qTj -

j = 1 j=1

734 0. G. BALEV and I. I. BOIKO: Conductivity of a Two-Dimensional Electron Ga.s

References [I] K. v. KLITZING, G. DORDA. dnd M. PEPPER, Phys. Rev. 1,etters 45, 494 (1980). [2] M. ,4. PA~LANEY, D. C . TsoI, nnd A. C. GOSSARU, Phys. Rev. I3 25, 5566 (1982). [3] H. AOKI and T. AXDO, Solid State Commun. 38, 1079 (1981). 141 V. 31. PUDALOV and S. G. SEMENCHINSKII, roterkh. Fiz. Khim. Mekh. 4, 5 (1984). [5] R. E. PRANGE, Fhys. Rev. 13 23,4802 (1981). [(;I 0. V. KOSSTANTINOV, 0. A. MEZRIN, and A . YA. SHII;, Fiz. ’J’ekh. Poluprov. 3 7 , 1073 (1983). [7J B. A. TAVGER and M. SH. YERUKHIMOV, Zh. eksp. teor. Fiz. 51, 528 (1966). [ X I 0. G . RALEV and I. I. BOIKO, Ukr. fiz. Zh. 2!), 149 (1984). [!I] E. M, BASKIN, I,. I. MAG-ARILL, and &I. v. ENTIN, Zh. eksper. teor. Fiz. 7.5, 723 (1978).

[lo] Yu. A. BYCHKOV, S. V. IORDAKSKII, and G. M. ELIASHBERG, Zh. eksper. teor. Fiz., Pisma 34,

[ l l ] E. N. ADAM and T. HOLSTEIB, J. Phys. Chem. Solids 10, 254 (1959). [la] H. FEKUYAMA, P. 91. PLATZXAN, and P. M7. ANDERSOK, Phys. 1Zev. B 19, 5211 (1979). [13] N. A. Usov and F. R. I~LINICS, Zh. eksper. teor. Fiz. 83, 1322 (1982). [I13 N. A. USOV m d F. K. ULINICH, Zh. eksper. teor. Yiz. 86, b44 (1984). [I51 G. EBERT, K. r. KLITZING, G . PROBST, and I<. PLOOG, Solid State Commun. 44,95 (1982). [I61 KIX QIAN and D. J. THOULESS, Phys. Rev. I3 30, 3561 (1984).

496 (1981).

(Received June 24 , 1985)

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Conductivity tensor of a two-dimensional electron gas in a strong magnetic field in higher orders of perturbation theory. Quantum hall effect