Magneto Conductance Sign Change

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PHYSICAL REVIE% 8 VOLUME 49, NUMBER 8 15 FEBRUARY 1994-IILevel broadening for localized electrons: Change of the magnetocondnctance sign

O. Entin-WohlmanSchool ofPhysics and Astronomy, Raymond and Beverly Sackler Eaculty ofExact Sciences, Tel Aviu University, Tel Auiu 69978, Israel

Y.Levinson and A. G. Aronov*Department ofPhysics, Weizmann Institute ofScience, Rehouot 76100, Israel

(Received 2 September 1993)The distribution of the hopping probabilities in the strong localization regime is calculated via the

independent-directed-path formalism, allowing for level broadening of the intermediate site energies ofthe hopping process. The distribution belongs to the Gaussian unitary ensemble even in the absence of amagnetic field, whereas in the absence of level broadening it belongs to the orthogonal ensemble. In thepresence of the field it leads to an orbital magnetoconductance quadratic in the field, changing from pos-itive to negative as the temperature is lowered and the number of paths increases.

I. INTRODUCTIONThe study of the magnetoconductance (MC) in the

weak-localization regime has shown that it is related tothe interference processes in such systems. There is nosimilar detailed understanding of the magnetotransportin the strong localization regime. However, descriptionsof the MC at low fields, which are based on the interfer-ence among the tunneling paths, seem to be successful inexplaining experimental data of highly localized samples.This observation has led to investigations of the conse-quences ensuing modifications in the interference pattern,e.g., due to spin-orbit scattering. This paper addressesthe interference-induced MC of strongly localized elec-trons when level broadening, due to phonon scattering, ofthe on-site energies along the tunneling paths is takeninto account.The low-temperature conductance in the strong locali-zation regime is dominated by thermal hopping. Whenelectronic correlations are not significant, it obeys Mott'svariable-range-hopping (VRH) law, g, -exp[( Tp/T)'~' +"], in two and three dimensions (d =2,3) with

hopping distance R typically on the order of several lo-calization lengths (depending on the temperature and theamount of disorder). The interference description, firstinvoked by Nguyen, Spivak, and Shklovskii, is basedupon the latter fact. They pointed out the importance ofthe interference among various paths associated with thehopping process, which are contained within a cylinder-shaped domain of length R and width (Rg)', where gis the localization length. They found that in the pres-ence of a magnetic field the modifications in the interfer-ence pattern considerably affect the hopping probability.Averaging numerically the logarithm of the hoppingprobability over many random impurity realizations, theyobtained under certain conditions a positive MC, linearin the field at weak fields.The picture of Nguyen, Spivak, and Shklovskii' refersto a single hopping process. Its relationship to the mac-roscopic conductance is based on the analysis of the con-ducting properties of a sample in the VRH regime in

terms of an equivalent resistor network. ' In thatdescription, any two sites between which the electronhops are connected by a conductance g,&g (I=gpy |/exp ([r t//g +e~@/kT] )

where go has units of conductance, r & is the distance be-tween the two sites, and e f2(~'E''~+~EI~+[6; E'I)), e;and e& being the initial and final site energies measuredfrom the Fermi level. In (1),y I denotes the renormaliza-tion of the effective overlap probability of the wave func-tions localized at the two sites, due to the interferenceeffect. The effective overlap results from electron tunnel-ing between the sites i and f and contains the effects ofthe interference among the various possible paths.The macroscopic conductance is determined by the"critical" conductances g, which connect sites whosespatial separation is R g and whose site energydifference is e))kT. Alternatively stated, the conductingproperties are dominated by hops of elementary conduc-tance g, which span a critical network throughout thesystem. ' Deep in the strong localization regime thecritical conductance may be obtained by treating sepa-rately the exponential factor and the interference factorin (1), since the scale of the r,, dependence of y,, is largecompared with g. (The change of the localization length,b,g, due to the interference is small compared with g. )The analysis of the exponential factor leads to Mott's lawgiving the essential temperature dependence of the con-ductance in the VRH regime, with R -g(TQ/T)' ande-kT(TQ/T)'~. The contribution of the interferenceterm is given' ' by exp[(lny)], i.e., by averaging overthe logarithm of the overlap probability.It follows that the interference aspect of the hoppingconduction is determined by the distribution of thewave-function overlaps in the highly disordered system.The latter is particularly sensitive to the effect of a con-stant magnetic field H. At weak-enough fields, such thatthe flux in the cylinder-shaped domain does not exceed pp(QQ hc /e is the quantum flux unit), the only orbitaleffect of the field is to multiply the overlap amplitude of

0163-1829/94/49(8)/5165(7)/$06. 00 49 5165 1994 The American Physical Society


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5166 O. ENTIN-%'OHLMAN, Y. LEVINSON, AND A. G. ARONOVeach tunneling path by a magnetic phase factor. The in-terference pattern, and, consequently, the distribution ofthe overlaps, are then modified and the logarithmic aver-age is changed. This is the source of the orbita1 MC inthe VRH regime. ' lt should be noted that this descriptiondisregards effects occurring at higher fields(H ~H, =go/g ) such as the shrinking of the wave func-tions or modifications in the localization length.The distribution of the wave-function overlaps hasbeen calculated using various numerical and analytic ap-proximations. ' ' ' ' In particular it has been found thatit leads to positive MC in the presence and in the absenceof spin-orbit scattering, '" contrary to the situation inthe weak-localization regime. Recently, the relationshipbetween the distribution and the random-matrix theoryof transition strengths has been established. ' Here weemploy the independent-directed-path formalism to ob-tain the distribution of the overlaps when the on-site en-ergies at the intermediate impurity sites along the tunnel-ing paths may be broadened due to electron-phononscattering. The effect is introduced by adding a smallimaginary part to the site energies. We find that the re-sulting overlap distribution belongs to the Gaussian uni-tary ensemble (cf. Refs. 11 and 12) even in the absence ofa magnetic field. The reason is that this ensemble de-scribes systems in which time-reversal symmetry is bro-ken; in our case by level broadening. We use the overlapdistribution to calculate the logarithmic average and toobtain the effects of level broadening upon the MC in thestrong localization regime. The result is that for a smallamount of broadening the MC is positive. However, itsfunctional dependence upon the field at weak fieldschanges from linear (in the absence of the broadening" )to quadratic. Surprisingly enough, the effect of the levelbroadening becomes more pronounced as the number ofpaths increases, i.e., as the temperature decreases and thehopping distance becomes longer. Then the sign of theMC becomes negative.The formal considerations and details of the calcula-tions are given in the next two sections. Section IVpresents the results for the magnetoconductance andtheir discussion.

II. THE DISTRIBUTION OF THEWAVE-FUNCTION OVERI.APSThe interference factor y,& in the expression for the e1e-mentary conductance g/ [Eq. (1)] describes the renormal-ization of the tunneling probability due to all paths con-

necting the sites i and f. Here we adopt the picture ' ac-cording to which the important contribution to y,& ofpairs belonging to the critical network, deep in the stronglocalization regime, comes from all forward-orientedpaths within a cylinder-shaped region of length r,& andwidth (gr/)'~ This imp.lies that the paths consideredfor the hopping from the initia1 to the final sites areroughly of the same length, longer than the localization1ength. The microscopic model for the individual pathamplitude consists of an Anderson model, with disor-dered on-site energies taken from a symmetric distribu-tion of width 8'and hopping matrix element V. Then the

V= I' ll V k V 1W k (e,ek)/W 'where the product is over the L impurity sites along thepath. Thus the random part of the amplitude isrepresented by the last product in Eq. (2). As L is thelength of the path, the prefactor in (2) gives rise to theterm exp( r,//g) in the elementary conductance. Toa'ccount for level broadening of the intermediate site en-ergies, we add to the energies ek a small imaginary partI i, which is of a definite sign As a result, the randompart of the overlap amplitude takes the form J, +~Jz,where both J& and Jz are real random variables. It fol-lows that y,& is given by

1y;~ = g (J+iJzi )exptPiI (3)where the sum runs over all forward-directed paths con-necting the sites i and f and P& is the phase acquired fromthe magnetic field along the path l. ' ' In (3), M is thenumber of oriented paths, exponential in the path lengthL in the VRH regime. 'In calculating the distribution of the overlaps, P(y), wefollow the approach of Ref. 5 in which correlationsamong paths are neglected and the real parts of the pathamplitudes, Jj&, are taken from a normal distribution ofzero mean and variance (J i &. As for the imaginary partJz& which represents the level broadening, we assume it tobe taken from a normal distribution offinite mean, (Jz &,and variance

(4)Thus it is seen that in our picture the level broadeningmodifies the interference among the tunneling paths, andinterferes with the effect of the magnetic field.With those considerations, it is straightforward to ob-tain P (y). One first derives the joint probabilityP (JJz), where from Eq. (3)

y +J1J,= g (Jcosgi Jz, sing, ), &M1(Jiinyi+Jzicosyi) .v'M

The required distribution P (y ) is thenP(y)= fdJ, dJzP(J, ,Jz)5(y J,Jz),

in which the magnetic phases of the individual paths, Pi,appear as parameters. When the integration in Eq. (6) iscarried out, these phases appear in the forms1 g expi, Pig, +igz,

and

effective hopping matrix element along a single path isgiven by'L


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49 LEVEL BROADENING FOR LOCALIZED ELECTRONS: . . . 51671 y exp~20I pl+ ~p2 ' (8)

In addition, the parameters of the J& and the J2 distribu-tions appear in P (y) in the forms(9)

when a1=a2 =0 and cr+ =o' = (J1 ), the distribution(11) reduces to that obtained before (cf. Refs. 5, 11, and12}.Finally, at high fields such that the magnetic fluxthrough the hopping region is on the order of the quan-tum flux unit A, a &, and a2 approach zero. In that limit,the distribution (11) takes the formfor the sum and the difference of the variances, andr=M(J )' (10) F(s)= 11+so'+which characterizes the effective amount of levelbroadening in the hopping process.The distribution of the overlaps, P (y ), is most con-veniently presented in terms of its Laplace trans-form ""F(s)=f dy exp( sy)P(y)=exp[ 1p(s)]/F7(s), (11)0where

which is formally the same distribution as in the casewhere level broadening is not considered, ""except forthe appearance of 0+. It should be noted that highervalues of the magnetic field are beyond the scope of ourmodel, as explained above.III. THE CALCULATION OF THELOGARITHMIC AVERAGE

sa, b, sa2b21+sb1 1+sb2F7(s)=[(1+sb, )(1+sb2 ) ]'i~ .

Here we have introduced the following notations:b1=0'++Aa', by=0'+ A(T

where

(12)

(13)

As explained above, the conductance, and hence themagnetoconductance, deep in the strong localization re-gime, is determined by the logarithmic average of the in-terference term, i.e., by

(lny )=J dy P (y)lny . (18)0In this section we calculate the logarithmic average byusing for P(y} in Eq. (18} the inverse Laplace transformofF(s), Eq. (11). This leads to the expression

A =p+p +1 (14) (lny )= I F(s)ln( ys),s2m'l L s (19)The remaining two parameters in the distribution (11),a,and az, are proportional to the amount of level broaden-ing 1 [Eq. (10)] and are given by

g1+ lg ~ [21M2rI19$+P'1( }1 )2)]z 2 212 b 91+ l2+ [2P29192+Pl(91 92)]I 2 2 1 222b2

(15)

srFo(s}=exp 1+s sr+

We first consider the overlap distribution in the ab-sence of a magnetic field. In that case, A =1,p&=q&=1,and @2=F2=0. Consequently, the Laplace transform ofthe distribution takes the form1s, 2 &0&i,2 (20)

Therefore, 0&a &s&. Because of the exponential factorin F(s), the singular points correspond to essential singu-

in which lny=0. 577 is the Euler constant and the con-tour of integration is depicted in Fig. 1. Equation (19) isderived from Eq. (18) as follows. The contour L whichdefines the inverse Laplace transform ofF(s) is a straightline between ~ and +a(x} in the half planeRes &0 of the complex s plane, for which all the singular-ities of F(s) lie on its left side. Those singularities [seeEqs. (11) and (12)] are branch points at s =s, z, where

1 11+s(o.++a ) 1+s(o+ )1/2

(16)s plane

and behaves as 1/s in the limit s~ ~. This implies thatthe inverse Laplace transform P (y ) is independent ofy asy approaches zero, i.e., the distribution belongs to theGaussian unitary ensemble. ' We note that this feature isdirectly related to the imaginary component of the over-lap amplitude, J2. where (J2 )=0 and (z )=0, theLaplace transform would have approached s 'ns~ ao, which in turn leads to P(y)-y ', i.e., to a dis-tribution of the Gaussian orthogonal ensemble. Second,we note that in the absence of the level broadening, i.e.,-Sp s1

FIG. 1. The contour of integration in the s plane.


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O. ENTIN-%OHI. MAN, Y. LEVINSON, AND A. G. ARONOVlarities. Note that the argument of the square root [Eq.(12)] is defined such that R (s)& 0 at s &0, with a cut con-necting the points s, and sz (see Fig. 1). In addition,the term ln( ys) in Eq. (19) is defined in the complex splane with a cut along Res )0.In order to obtain an expression for (lny ) which in-cludes less parameters, and to remove the pole s =0 fromthe branch point of the logarithm [see Eq. (19)], wechange the integration variables as follows. Let

The contour C can now be transformed into the con-tour C~ +C (see Fig. 2), where C~ is a circle of a large ra-dius R disconnected at the cut, and C is along the cut.On Cz one may replace g(g) by unity and R (g) by g.The integral can then be easily calculated, yieldingln[cR /) ]. The integral along C can be reduced to a sim-ple one by noticing that for g=g, +tgz, with g, &0 and(z~+0,

s =(cg+d) (21) ln (g, +tgz+e) n (g,tgz+e) =2mt . (28)such that s =s, and s =sz become (=1 and (=0,respectively. This implies that The integral along C then becomes

C =b& b2 &0 d =62&0The cuts in the g plane are shown in Fig. 2, with

(22) R dx Biexpv'x (x + 1) x + 1 (29)c=d/c )0 . (23)

In terms of the new variable g, the contour L becomes acircle,

Using now the relation (for R 1)dx v'E+&e+ I

~ x(x+1) (30), e.=-+ 14a c (24) we finally obtain( ) c +e+1+v Ewhich can be transformed into an arbitrary contour C,encompassing counterclockwise the cut (0,1) (see Fig. 2).Carrying out the above transformations we obtain dx&x (x +1) 8)1exp x+1

(lny )= f exp[/(g)]ln g+Edg c2nt c R (g) ywhere

(25)

(26)

(31)In the last integral we have replaced the upper bound by~, as the integral converges for x~~. This expressionfor the logarithmic average is used in the next section todetermine the e6'ects of the level broadening upon theMC.

anda)b)B,= B =C 2 a2b2 (27)

p lane

Note that R (g) & 0 for g &. [This follows fromR (s)& 0 for s &0.]

IV. DISCUSSION OF THE MAGNETOCONDUCTANCE%'ithin the interference model, the MC is given by' '

g, (&)g, (0)MC= =1+exp((lny )(lny )o) . (32)g, (0)In order to characterize the magnetic-6eld dependence ofthe MC, we assume' that for any path with a phase Ptthere is also a symmetric path with a phase Pt (a validassumption in the model of Nguyen, Spivak, andShklovskii' in which the number of paths is large). Thenpz=g2=0 and consequently a, =0 and

a2= I q)/b~ .It follows that [see Eq. (27)]

(34)r=0, B cand the expression for the logarithmic average takes theform

FIG. 2. The contours of integration in the g plane.

(lny )= ln21n,(v'ad+1+ v'e)y+ 1exp Bto rv'1+r


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49 LEVEL BROADENING FOR LOCALIZED ELECTRONS: . . . 5169In the absence of level broadening, 8 =0 ande=(1A)/2A. Equations (32) and (35) then yield

MC=&Ia'. (36)This reproduces the results of Refs. 11 and 12, and showsthat under this condition the MC is positive. At weakfields we have [see Eqs. (7), (8), and (14)]3=12h, g =1where h is the magnetic field in dimensionless units0 '

H,h2= 1M IH, =$0/R&Rg=H, (T/To) ~ &&H, .

(37)

(38)It, follows [see Eq. (36)] that in the absence of levelbroadening, the MC at weak fields is linear in the Geld,i.e., MC ~ 2 ~ h ~.To find the weak-field behavior of the MC in the pres-ence of level broadening, we define

M =exp (N )Rp (44a)N'ln(1/N ), N 1A[in{%)]'~', N &&1 .N)= ' (44b)

(lny )(lny ) ~J o= 21n2(8Jz) . tn(J)&2

Btr&(t+1)

Hence, the number of paths becomes exponentially largeas the hopping distance R increases, that is, as the tem-perature tends to zero. It follows that the sign of the MCmay change with temperature. This point will be dis-cussed further below.Next we consider the efFect of the level broadeningupon the zero-field logarithmic average. From Eq. (35}we find

MC=Ah, A,= (lny )o .Bh (39} &0. (45)Then, from Eqs. (22), (23), and (35) we obtain

+~=2+ exp2 21/2

Xexp (40}

This result is examined in two limits. (i) When the J2 dis-tribution is almost symmetricM( J,&'2& fiJ, &, (41)

the last term in Eq. (40) may be neglected and A, &0. (ii)%hen the J2 distribution is strongly asymmetricM (J, )'2(5J, ), (42)

N=(p Ng')' (43)where N is the density of hopping centers and p is thescattering length of each of them. One then has

one may neglect the second term in (40) and obtainA, (1. Thus the level broadening changes the function-al dependence of the MC at small fields from ~h~ to h .The crossover from ~h~ to h occurs at H =H that is,the linear dependence is smeared out by the levelbroadening. The sign of the MC at sma11 fields, i.e., thesign of A. [see Eqs. {39)and (40)], depends on the degree ofasymmetry of the J2 distribution and the number ofpaths. The latter can be obtained following the expres-sions derived in Ref. 7. One defines a parameter N,

That is, the level broadening enhances the interferencecontribution to the conductance at zero Geld. This resultis quite surprising as one would have expected thebroadening at the intermediate states to destroy phasememory. The enhancement of the zero-field conductancecan be explained as follows. Since we consider a sym-metric distribution for the real part of the tunneling am-plitude (J& }, then in the absence of an imaginary part(J2=0) there is a significant cancellation of amplitudes ofopposite signs. This cancellation is destroyed in the pres-ence of J2, i.e., when the transition amplitudes are com-plex. The effect is larger as the J2 distribution becomesmore asymmetric. %e emphasize that the derivation ofEq. (45) assumes a &0, i.e., (J, ) & (5J2z). When thebroadening is stronger and the inequality is reversed, oneexpects the phase memory to be lost.To deduce the overall behavior of the MC up to fieldssuch that the Aux through the hopping region is largerthan $0, ' i.e., H H, (but H Hhigher fields arebeyond the scope of our model), we consider thedifference

& lny )lny &, , (46)where the first term is the logarithmic average at 0~ (x&.According to the discussion in Sec. II [see Eq. (17)] thisterm is

(lny )=ny+lno +, (47)and it is independent of the asymmetry of the J2 distribu-tion. It should be noted that this result is valid as long as

' (J', &&8J,& (48)Assuming the magnetic phases, p~, to be Gaussian vari-ables, we find g, -exp[ /2]. Then the exponential


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5170 O. ENTIN-VfOHLMAN, Y. LEVINSON, AND A. G. ARONOV 49factor in Eq. (48) is Mil, . Requiring this factor to besmall yields [see Eqs. (44)]

2H'&, f(N) . (49)At the same time, we also have H &H, . Combining thesetwo requirements together, we obtain the temperaturerange at which we may consider high fields:' 1/2 ~f(g) .T pAs for the second term in (46), we calculate it in thetwo limiting cases (41}and (42). We find that for an al-most symmetric level-broadening distribution

2 1/2

(50)

1+ 1(lny )1ny )0=n 0+

' 3/21 (51)0'+ +0

Under condition (41) this difference is positive and hencein this case the MC is positive for all fields, though small-er than in the case where the broadening is neglected.The situation is changed when the level-broadening dis-tribution is substantially asymmetric [Eq. (42}]. We showin the Appendix that in this case (lny )0=lnI' and conse-quently

30

MC-1+ (52)yISince for a'+-e, condition (42) [cf. Eq. (10)] impliesthat I & a+, it follows that a very asymmetric distribu-tion of the level broadening renders the MC to be nega-tiue at all fields.As it is pointed out above, the amount of asymmetry ofthe J2 distribution [Eqs. (41) and (42)] depends upon thetemperature. This dependence has two origins: the rela-tionships between the number of paths and the hoppingdistance [Eqs. (44)] and the temperature dependence ofthe level-broadening parameters. The level width I k of agiven intermediate state of energy ek is determined by thefastest transitions, i.e., those occurring between nearestneighbors with a spontaneous phonon emission. This im-plies that I k depends weakly upon the temperature.Consequently, one expects J, and J2 to decrease as thetemperature decreases, but not exponentially. On theother hand, the number of paths increases exponentially

I

FIG. 3. A ring model for the magnetoconductance.as the temperature is reduced. As a result, the J2 distri-bution is less symmetric at lower temperatures. Hence,the orbital MC of strongly localized electrons can be neg-ative at low temperatures, and changes into positive athigher temperatures. We emphasize that we have notconsidered the effect of electron correlations, whichshould be important at very low temperatures.It is instructive to compare our results for the effect oflevel broadening with those obtained by considering asimple ring model (see Fig. 3), used in Ref. 7. In thissimplified picture, there are only two paths. Therefore,the total transition probability becomesy(4)=~(Ji++iJz+ )e" +(J +iJz )e (53)

XP(J,+,J, ,Ji+,J2 )ln y 0 (54)Let us first confine ourselves for simplicity to the limitingcase in which the path amplitudes are purely imaginary,i.e., J*, =0. In this case, (b, lny ) is exactly the same asin Ref. 7, when one replaces the real amplitudes by theimaginary ones. Moreover, it is shown in Ref. 7 thatwhen the distribution is symmetric, the MC is positive,while when it is asymmetric, MC&0. This finding is inaccordance with our conclusions regarding the relation-ship between the sign of the MC and the asymmetry ofthe J2 distribution.The ring model with real transition amplitudes gavethat the sign of the MC is related to its periodicity. Fora period of $0/2, the MC is positive, while for $0 periodi-city it is negative. Let us, therefore, examine the MC of aring with complex transition amplitudes, in the casewhere the distribution of J & is symmetric, and the distri-bution of Jz is arbitrary. One then finds from Eq. (54)

where 4 is the Aux through the ring, measured in units of$0, and J are the path amplitudes (see Fig. 3). To findthe MC we need to calculate

(hlny) =f dJ,+dJ, dJ2+dJ2

(b, lny) =Co f dJi+dJi f dJ2+dJ2 P[(Ji+ ),(Ji ),J2+,J2 ]ln 1+C,sin C2sin~@+0 2 (55)where C, and C2 are real combinations of J~ and J2. Inparticular, C, vanishes when J2 =0. It follows from thisequation that for nonzero J2, the periodicity of the MCis always $0, while its sign depends upon the details of theJ2 distribution. The ring-model results are, therefore, inaccordance with our conclusions. Moreover, this exam-ple shows that even in the most simplified situation, add-

ing an imaginary part to the transition amplitude changesessentially the resulting magnetoconductance.ACKNOWLEDGMENTS

We are grateful to A. Aharony and A. B. Harris forhelpful discussions. The research was partially supported


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49 LEVEL BROADENING FOR LOCALIZED ELECTRONS: . . . 5171by the fund for basic research administered by the IsraelAcademy of Sciences and Humanities, and by the UnitedStates-Israel binational science foundation (BSF).A.G.A. is grateful to the Landau-Weizmann program forfinancia1 support.

APPENDIX: THK LOGARITHMIC AVERAGEFOR A STRONGLY SYMMETRIC DISTRIBUTIONHere we derive (Iny ) at zero field for a strongly asym-metric Jz distribution, i.e., under the condition (42}. Tothis end we write the integral in Eq. (35) in the form

The first term here gives in[@'a+ 1+Pe~+2 ln2nv . (A2}

Et(Bv) E&Bz ')=lnv+lnyB, (A3)where Et is the exponential integral. ' Combining (A2)and (A3), the integral in Eq. (35), for B/s 1, becomes

In the second term, under condition (42) [see Eq. (34)] wemay expand (1+t) '~ . Up to corrections of order Bthe second integral in (Al) gives

. 'dr I . 'dr 1lim exp Btv~0 v r v I+t v t 1+t ln ,(v'a+1+&a)+InyB .(Al) It follows that when (42) holds, (lny )o=lnl .

(A4}

'Permanent address: I. F. Ioffe Physical-Technical Institute,194021 Saint Petersburg, Russian Federation, and Dept. ofCondensed Matter, ICTP, Trieste, Italy.'V. I. Nguyen, B. Z. Spivak, and B. I. Shklovskii, Pis'ma Zh.Eksp. Teor. Fiz. 41, 35 (1985) [JETP Lett. 41, 42 (1985)];Zh.Eksp. Tear. Fiz. 89, 1770 (1985) [Sov. Phys. JETP 62, 1021(1985)].B.I. Shklovskii and B.Z. Spivak, l. Stat. Phys. 38, 267 (1988).A. Miller and E.Abrahams, Phys. Rev. 120, 745 (1960).4V. Ambegaokar, B. I. Halperin, and J. S. Langer, Phys. Rev. B4, 2612 (1971).5U. Sivan, O. Entin-Wohlman, and Y. Imry, Phys. Rev. Lett. 60,1566 (1988);O. Entin-Wohlman, Y. Imry, and U. Sivan, Phys.Rev. B40, 8342 (1989).W. Schirmacher, Phys. Rev. B41, 2461 (1990).

B. I. Shklovskii and B. Z. Spivak, in Hopping Transport inSolids, edited by M. Pollak and B. I. Shklovskii (North-Holland, Amsterdam, 1991).8J. L. Pichard, M. Sanquer, K. Slevin, and P. Debray, Phys.Rev. Lett. 65, 1812 (1990).E. Medina and M. Kardar, Phys. Rev. Lett. 66, 3187 (1991);Phys. Rev. B46, 9984 (1992).H. L. Zhao, B. Z. Spivak, M. P. Gelfand, and S. Feng, Phys.Rev. B 44, 10760 (1991).' Y. Meir, N. S. Wingreen, O. Entin-Wohlman, and B. L.Altshuler, Phys. Rev. Lett. 66, 1517 (1991).' Y. Meir and O. Entin-Wohlman, Phys. Rev. Lett. 70, 1988(1993).t3M. Abramowitz and I. A. Stegun, Handbook ofMathematicalFunctions (Dover, New York, 1964)~

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Magneto Conductance Sign Change