Sblid State Communications, Vol. 51, No. 5, pp. 301-303, 1984. Printed in Great Britain.

0038-1098/84 $3.00 + .00 Pergamon Press Ltd.

ON THE SELF-CONSISTENT THEORY OF LOCALIZATION

T.N. Antsygina, L.A. Pastur and V_A. Slusarev

Institute for Low Temperature Physics and Engineering, UkrSSR Academy of Sciences, Kharkov 310164, USSR

(Received 1 March 1984 by I~.A. Kaner)

A modification of W. G6tze's self-consistent approach is proposed, which predicts a complete localization in the one- and two-dimensional cases. The modification is based on a choice of such a form of the time depend- ence for the two-particle Green function, which takes an apparent account of time-reversal invariance.

THE PROBLEM of localization in the disordered systems and the resulting kinetics properties is one of the most difficult and simultaneously important in the solid state physics. Although certain progress has been achieved in this field, particularly in the recent years, the derivation of sufficiently effective formalism to describe this and related phenomena encounters serious difficul- ties resembling those in the theory of the critical phenomena. The attempt is therefore natural to con- struct an analog of the mean field approximation, which, as is well known in the theory of the phase transitions, gives sufficiently good qualitative, and often even quan- titative, description of the system behaviour in the vicinity of the phase transition.

A version of the similar theory was proposed in [1-3] . These works, however, possess a certain short- coming - their self,consistent equations do not predict the complext localization in the one-dimensional and two-dimensional cases [3], which disagrees with current widely-accepted concepts [4, 5]. This is accounted for by the violation of time-reversal invariance in the approximation scheme [1]. Such a property must be an indispensable characteristic of the current correlation function due to the fact that exact wave functions of the particle in a random potential may always be chosen real. The question has already been discussed in literature [6, 7].

This paper describes a scheme similar to that proposed in [1 ,2] , but taking an explicit account of the symmetry mentioned. The importance of this symmetry follows also from the calculations of quan- tum corrections to the transport coefficients [5], which show that latter are contributed not only by standard diagrams of the kinetic theory but also by the so-called cooperons existing just due to time-reversal invariance. We show that ff this invariance is taken into account, a complete localization occurs in the dimen- sions 1 and 2. In the Otree-dimensional case, as in [2], the Anderson transition and the same critical exponents for d.c. conductivity are obtained [see equation (8)].

It should be noted that the complete localization at d = 1, 2 was first obtained in [7] on the basis of the self-consistent approach. The appropriate self-consistent equations [7] were derived in terms of the perturbation theory with subsequent self-consistence (closing) of the equations obtained by the diagrammatic technique. Our approach, similar to that in [1, 2], is based on the concepts of the kinetic theory and the corresponding self-consistent equation [see equation (6)], though of a different form, gives the results, which at d = 1, 2 coincide with the results of [7], and at d = 3 with those of [1, 2] despite a somewhat different mathematical mechanism of appearance of our results. Note that, unlike [2], we consider the lattice model (the Anderson model), within which the calculation is of a more trans- parent nature.

The Hamiltonian of the system in consideration is

H = H o + H t

where

k f

Here a~, af are the creation and annihilation operators in the site f of the d-dimensional lattice (simple cubic for the sake of certainty)

air = N -a/2 )-'.aze an, e(k) = T ~ ( 1 - - c o s k ), f 8

and Uf is the random potential, which is assumed to be weakly correlated so that its correlation function decreases sufficiently fast. We shall write down the current as an operator with the Fourier transform of

= ~ k ÷ ( q / 2 ) a k - ( q / 2 ) , k

~6 v(k) = Dk T a~ sink&

301

302

To investigate the kinetic properties and the localization phenomenon, we shall introduce the two-particle correlation function

F(f l , f2 , fa , f4) = ; dt f dXe '~°' 0 0

where the symbol ( . . . ) denotes both the thermodynamic averaging and the random potential averaging. The some- what unusual time.dependence of the operators in this function, as compared to the conventional one, is due to the explicit form of the time-reversal invariance whose importance was explained above. As in [2], we assume that the correlation function of equation (1) obeys the model kinetic equation

SELF-CONSISTENT THEORY OF LOCALIZATION Vol. 51, No/5

may lead to false divergencies [ 11 ]. The reasoning for this is given in [7].

Transformation of equation (3) similar to [2] by replacing the mean values of the form

(UqU~, + It, It, . , , t + i k ) ak+tq/2)~Jak-tq/2)~)ak-(q/2)(--~

by the mean values

(UqU~')(a~+tq/2)(2)ak-tq/2)(2)a~'-t~/2)~2 +iX)

(')) X ak ,+ lq , /2 ) - - ~ + iX

taking into account that (UqU~,) = DqSa, q, gives

q qkqk ) (4, 1 E D q E~k,q) l~(k , ,q )F + ' ~ , k - - ~ , 2' N q kit'

r ( 6 o ) =

v2(k)Of/ae) k

(26o + ~f, -- el2 -- el3 + eto) F(fl , f2, f3, f4)

+ ir(6o)(2 - gft.f2 - 6t3,t, -/ i t , , t3 - 6f~.t, + 5e,,f~

+6f , , f , )F( f l , f2 , f3 , f4) = ff(fl ,f2,f3,f4), #

~b(f,, f2, fa, f4) = 2i f dX (a~(O)at2(O)a~3(iX)at,(iX)), o (2)

where r(6o) is the effective current relaxation kernel to be found. To do this, we shall assume that the rates of current relaxation found from the above equation and by the non-equilibrium statistical operator method [8, 9] coincide and obtain

dt f dX , ®,, - +ix q.q ' 0 0

r(6o) = ,

f dX (j(O) j(iX)) (3) u , o

Oq - ~ ~k I~ (k, q) a~+ tq/2) ak- tq/2),

q v q

This way of finding the relaxation kernel as a quantity proportional to the force-force correlation function dates back to Kirkwood [10]. In the above formulae we take into account the time dispersion in r since it is essentially important to consider the localization though neglect the spatial dispersion. One should be careful while taking into account the latter, since this

where f(e) is the Fermi distribution function. While solving the set of equations (2) and (4) we

substitute all the energetic denominators with 2iP in all the integral terms:

26o + 2iF + e(kl) -- e(k~) - e(k3) + e(k,) ~ 2i t . (5)

In the three-dimensional case the reasons for this substi- tution are given by the fact that the damping in the vicinity of the mobility edge is not small. It is not the case for the space dimensionalityd = 1, 2. However, the singular behavior of the kinetic characteristics in this case is determined by the region of small momenta and therefore in the integral terms of equation (5) which do not lead to singularities, this substitution is again possible. In other words we assume that in the situation considered the expansion in small q may be performed (see [2]), where q is either the transferred or total momentum. In our approach this fact reflects the known in the diagrammatic technique situation [5, 7], when the diagrams singular with respect to this parameter (diffusion and cooperon channels) are of the main importance to the localization effects. Besides, we do not take into account the effect of the random potential upon the equal-time mean values. The validity of this approximation was assumed without discussion [1-3]. We would like to point out the fact that it is reasonable either for restricted dimensionality (e.g. [4, 5]) or for temperatures much higher than the band width and the random potential amplitude (with an arbitrary relation between the latters).

V61.51, No. 5

In the high-temperature limit equation (4) can be simplified as follows:

cos k 8 Do F = 290 ~ T 2 t - - - (6)

3N k,8 --i6o +--~- ~ ( 1 - - c o s k l) Y ' 6i

since according to equation (2)

e 2 n

o = - - [-- i o + r(~,) l-1, (7) m

then the critical value of the parameters responsible for the Anderson transition from the side of the conducting region can be found from the condition for r(O) to turn into infinity. It follows from equation (6) that at d = 3 this boundary is determined by the condition

2Do 1 = 1),

where w is the Watson integral, w = 1.51. Hence, assu- ming a uniform distribution of Us in the range (-- W/2, I¢/2), the critical value of the parameter x = I~/T is

x c ~ 6,

and the dependence of d.c. conductivity on x in the vicinity o f x c is of the same form as in [2] :

od.o. ( x o - x ) . ( 8 )

SELF-CONSISTENT THEORY OF LOCALIZATION 303

In the one- and two-dimensional cases equation (6) has the solution F(to) which is turned into infinity at co -~ 0 and any values of the parameters. Because of equation (7) this means the absence of d.c. conductivity, i.e. the complete localization of all the states.

Note that in terms of the diagrammatic technique the whole effect is in our case due to the contribution from the Cooper channel.

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Introduction to Theory o f Disordered Systems. Nauka, Moscow (1982).

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