On the theory of localization in disordered systems

Solid State Communications. Vol. 17, pp. 1453—1457, 1975.

Pergamon Press.

Printed in Great Britain

ON THE THEORY OF LOCALIZATION IN DISORDERED SYSTEMS* F. Brouers Laboratoire de Physique des Solides,t Université Paris-Sud, 91405 Orsay, France and N. Kumar Institut de Physique, Université de Liege, Sart-Tilman 4000, Belgique (Received 3 June 1975; in revisedform 22 July 1975 by A.A. Maradudin)

A localization criterion is derived by using the self-consistent determination of the self-energy in the discussion of the convergence of the Green function renormalized perturbation expansion. This criterion depends on the dimensionality and the connectivity of the lattice. It yields localization for all states in one-dimension systems and tends to the Economon—Cohen criterion when the number of neighbours tends to infinity. Moreover, one can show that delocalization occurs when the width of the probability distribution of the self-energy is of the order of magnitude of the hopping integral.

1. INTRODUCTION

where n ) is a Wannier function centered at site n.

ALTHOUGH there has been a number of works done and published on the problem of localization of electrons in disordered systems (for a review see Thouless1), even in the simple Anderson model of cellular disorder,2 there is a large uncertainty in the critical value of the parameters which should lead to localization of the states. The various estimates differ by a factor of five or six. Therefore until now there has been no precise quantitative results about the position of the mobility edge and its variation with disorder.

The diagonal matrix element e~is a random potential characterized by a probability distribution p~()of width W. The second non-diagonal term is supposed to be non-random and the summation is restricted to first neighbours. In that case Vnm = V. In the ordered case (e = 0), this term gives rise to a band of width 2ZV where Z is the number of neighbours. The various approaches to the theory of localization lead to the conclusion that there exists a critical value of W, We,, such that for W> W~,the eigenstates at the middle of the band and therefore all the eigenstates are localized. The theory of localization in the Anderson model is based on the investigation of the perturbation series of the self-energy S 0 of the sitediagonal Green function

The Anderson2 model is given by the tight binding Hamiltonian

H

=

~ fl

*

n)e~(ii I

+

~

I~nmInXm

(1)

n*m

Partially supported by ESIS Programme.

=

~ Laboratoire associé au CNRS.

where 1453

~

=

E—e0—S0

(2)

1454

THEORY OF LOCALIZATION IN DISORDERED SYSTEMS

~

=

~

+

Vo~1

1

Von1

Vnn

n1*0

1

E

—

walk. Consequently, one can consider all Nth terms and Thouless6 have analyzed the oflengthNas2’5 identical. It is difficult to assess the validity Anderson of both approximations.

V~o

n

1

Vno +

Vol. 17, No. 11

...

n,n 2*O

E—

—

The Nth term of the series corresponds to random walks of length N starting from site and coming 2 is 0based on the back. Anderson’s original theory study of the perturbation series of S 0. When the series is convergent for almost all values of E in a certain range, then the states in that energy range are localized. When the series is divergent the states are supposed to be delocalized. As the terms appearing in the series are random quantities, the convergence is considered in a probabilistic way: for each value of e, one should find the probability that the series (3) converges from the knowledge of the probability distribution p~(e).There are two essential difficulties in solving that problem: (a) the same sites may be visited more than once and one cannot suppose the N factors in the Nth term as statistically independent, (b) the various walks corresponding to the Nth are not statistically independent since they have common factors. To overcome the first difficulty, Anderson proposed to renormalize the series (3) using Watson3 multiple scattering theory in order to obtain in the series only terms corresponding to non-repeating paths. The renormalized series can be written in the form

analytical properties of the renormalized series. The conclusions are that localized and non-localized states are characterized by different distribution of S~(E) when EE is lies complex its of imaginary part tends zero. If in theand range non-localized states,to the imaginary part of Im Sn(E) is finite even in the limit of Im (E) going to zero, since, for any particular distribution of site energies, the poles of the selfenergy are dense on the real axis. If E lies in the range of localized states, Im S~(E)tends to zero as Im E tends to zero, except at a (dense) set of points of measure zero. The average value Im (Sn(E)) is however finite since Im S~(E)has a vanishing small probability of being infinitely large. Outside the range of eigenstates of the system, Im S~(E)tends to zero except at a discrete set of points and its mean values tends to zero. Estimate of the critical W~can be made by replacing in the renormahized series every locators At the center of the band the G~~ I .~l(E)by its7). zero-order approximation condition convergence is (E ~ for (Ziman ~ 2eK (5) —

for a uniform distribution of potential between W/2 and W/2. An other estimate consists in replacing all energies in (2) by the CPA self-energy4’7 —

~CPA

5 0(E)

~

=

V0~G~°~V~0

and the convergence condition yields the

Cohen and Economou criterion

n1~O

+

~

VOn2 G~°’~’~V n2 n2n1 G° n1 Vn1 o+...

(4)

fl~~0 n 2~ ~ 1•O

where

G(0,n ni

~i_1)(E)

=

(E

—

n.

—

5(O,n

ni_i))_1

and s~1?’n~...n~_i) is the self-energy for a site 0 if in the self-avoiding paths from n, to n1, the sites n0, n1, ..,n~_1 have been removed (or what is equivalent the potentials e~,~ .Cnf_I have been put = )~ In that renormalized factors site in every termcannot occur only once and the series, renormahized energies be expected to be statistically independent however one assumes that they are. To overcome the second difficiutly, Economou and Cohen4 have argued that for large N the Nth terms are strongly correlated and that the majority of the same sites are visited by each .

ZV/IE~—~(E~)I = I

(6)

for the mobility edge. Anderson in his original paper argued that because of the long tail of the probability distribution the largest term of any order is dominant and this gives rise to a factor of the order log (W/J) on the right hand side of(5).8 The much simpler convergence Economou and Cohen9 lead argumentsat of however theZiman, middle of the band to localization conditions closer to “experimental” numerical works. Herbert and Jones’°and Economou and Cohen9

Vol. 17, No. 11

THEORY OF LOCALIZATION IN DISORDERED SYSTEMS

1455

have argued that these terms cannot possibly be independent of one another and that the second estimate is therefore better than the first one.

Nth order terms are strongly correlated i.e. they are identical and we make the further approximation = (.~mi}(f’) (8) GO,nl...rhi_1 ~G

To investigate the origin of this discrepancy, 11 have proposed another Abou-Chacra etal. approach. Instead of considering high order terms in the renormalized perturbation series of S to determine the radius of convergence, they consider only

where m 1, m2 .mKN are the nearest neighbour sites of n 1 belonging to the sequence 0, n1, n 1 and K~
the second order term but calculate the renormalized denominator self-consistently. This formulation is exact on a Cayley tree but approximate for a real lattice. The equation is the first term of (4) 2 Si=~ (7) 1V1 j

mIm2”mKN

. .

...,

—

—

where K is the connectivity constant defined as the Nth root of the number of all possible N-step selfavoiding paths on the lattice as N K is generally oftheorderofZ—2and —* °°.

Lc~—S

1

The self-consistency problem is to find the probability distribution for S~,which, when used on the right of the equation will yield the same probability distribution for S. The stability of localized states is examined by looking for a solution of the self-consistency equation in which Im S,(E) is proportional to Im E. When a solution exists theexists statesthe arestates localized andsuch when no such solution are assumed to be non-localized. At the middle of the band, the critical value W~obtained by this method are close to those of the first Anderson2 theory. Abou-Chacra et al. method has been developed later12 to obtain an expression of the mobility edge E~(W)when not all states are localized, 2. SELF-CONSISTENT PROBABILITY DISTRIBUTION OF THE SELF ENERGY AND CONVERGENCE CRITERION The purpose of this letter is to show how the idea of determining self-consistently the probability distribution of the self-energy S can be applied to the study of the convergence of the renormalization series, and yields an improvement of the Economou and Cohen criterion by introducing explicitly the dimensionality and the connectivity of the lattice into the expression of the mobility edge. We consider here the case of a Cauchy distribution where the calculation can be done exactly.

_________

E

with

ni

1’~~}(F) =

(11)

—

ni

_______

5

—

1m nj~nj, lmi} E

1} n

~n1

—

(12)

Under thes~conditions, statistical series converges 13 the (Kolmogoroff criterion) if with probabilityReunity flog TPT(T)dT < 0 (13) where pT(T) is the probability distribution of T. This probability distribution can be calculated from (11) if the probability distribution of S and are known. The probability distribution p~(e)is given by 1 z~ P~() = ~ + (14) ~

~‘

(where we have chosen the Cauchy distribution), and

ps(s) is calculated self-consistently from (12).

In the localization region, except for a set of zero measure the self-energy is real on the real axis. IfS is supposed to be real, the self-consistent determination of p(S) is performed simply and exactly if one assumes that the probability distribution of SItmI} is the same as that of S~2i}, in (12): “~

/

~

p 5(S)

=

j.

...

J -~

To express the Nth term of the series (4), we follow the arguments of Licciardello and that the 4”6 We make the assumpticn Economou.’

1

Gk~2i}(E) =

6

(s \

KN

V2

)

________

—

j=1

x flp(e~)p~(S])de~dS]

(15)

1456

THEORY OF LOCALIZATION IN DISORDERED SYSTEMS

—1—-f ~

=

(IdeS

the two equations (19) and (26) are identical. From (21), one has

dS’

2

x

KN

x exp

{—

it

V~/(E— e — S’)] } p~(e) Ps ~(S1))

[

.(S’)=

2

~~ +

~

~‘

I

ir (S’

—

AS’ S~)2+ A~’ (17)

the self-consistency yields KNV2 So + iA~=

—

If we define ~y= As/V, e

+

—

Ely, s

=

—

—

(e2—X)X+72(e2—X)

if

X

(e

=

—

KN\ K2 + KN)

(K2

=

2s)2.

—

KN+K2 =

—~-

generalizing the Economou and Cohen criterion4 which for the Cauchy distribution (14) yields e = At the middle of the band the critical ‘y is given Ye

(22) e—2s

=

2K—Z

KV $ (~_~ 5)~~dsd

1 where

it

(23)

2 + A~,

A7

= =

(24)

If

—

2s)2 -(e

—

5)KV 2’ (E—S0)+(A~+ A5)

KV(E

e

The most interesting feature of our expression (31) in that it contains the connectivity of the lattice. For instance, in a one dimensional lattice, one gets = 0 since Z = 2 and K = 1. This is in agreement with the well known result that all states in the linear chain are localized however weak the disorder is. This important result is not contained in (33). Comparing (33) to (32), approach one can see thata our results based on a probabilistic gives smaller critical width

Ye’

So) (ES)2 + (A + A)2~

s)2

(33)

(A~+ A

—

(25)

Introducing this distribution in (14) and (22), one gets for the mobility edge condition (e

z.

—

AT

(T— T0)

(32)

whereas the Economou—Cohen gives =

The probability distribution p~(T)is then determined from

—

(30)

by

=

~

z

since in that case the mobility edge corresponds to the band edge and (29) can be written as e = ~/z2 ~2 [Z/(2K —Z)]2 (31)

(21)

s.y

=~

(28)

(20)

4KNX

The width A~can be calculated from

PT(T)

—

5

=

0/V and I.~s A5 IV, one gets for s a fourth order equation 2 + y2(e —s)s = KN(e 25)2 s(e s)(e 2s) (19) which can be written as —

e2

Substituting in (20), one obtains an expression for the mobility edgc ___________________________ e= ~ +KN)/K]2 —y2[(K2 + KN)/(K2 KN)]2. (29) In the limit of zero disorder we must have

(18)

~

=

dt (16)

p~(e) —

Vol. 17, No. 11

+

—

s

‘y2(e s

—

s)2

K2 KN

=

K2(e

—

2s)2. (~6) (27)

Similar conclusions were reached recently by Licciardello and Economou14 for a rectangular distribution taking account within the CPA of the site restrictions in the renormalized series. For a rectangular distribution, one should obtain with our method a critical value WC/V = 2ey. This gives We/V = 7 for the square lattice (K = 2.64), We/V = 9.6 for the diamond lattice (K=2.88) and W~/V 18.5 for the simple cubic lattice (K = 4.68) which are in very close agreement with the direct numerical estimates available and quoted in Licciardello and Economou.14

Vol. 17, No. 11

THEORY OF LOCALIZATION IN DISORDERED SYSTEMS

Another interesting result arises from the con-

For three dimension lattices K

sideration of the width of the self-energy probability distribution As. From (30) and (27) one has at the mobility edge s

=

A8

=

and then from (22)

Z

—

Z

K e

Z—K 2K—Z

1457

(34)

~Z and A5 / V 1. One can therefore conclude that the localiied states become unstable when the width of the self-energy (supposed to be real in the localized region) probability distribution becomes of the order of magnitude of the hopping matrix element.

(35)

REFERENCES 1.

THOULESS D.J.,Phys. Rep. 13c, 95 (1974).

2. 3.

ANDERSON P.W.,Phys. Rev. 109, 1492 (1958). WATSON K.M.,Phys. Rev. 105, 1388 (1957).

4.

ECONOMOU E.N. & COHEN M.H.,Phys. Rev. Lett, 25, 1445 (1970).

5.

ANDERSON P.W., Comments Solid State Phys. 2, 193 (1970).

6.

THOULESS D.J., J. Phys. C.’ Solid State Phys. 3, 1559 (1970).

7.

BROUERS F.,J. Phys. C: Solid State Phys. 4,773(1971).

8.

ZIMAN J.M.,J. Phys. C: Solid State Phys. 2,1230(1969).

9.

ECONOMOUE.N.&COHENM.H.,Phys. Rev. B5,2931 (1972).

10.

HERBERT D.C. & JONES R., J. Phys. C: Solid State Phys. C4, 1145 (1971).

11.

ABOU-CHACRA R., ANDERSON P.W. &THOULESS D.J.,J. Phys. C: Solid State Phys. 6, 1734 (1973).

12.

ABOU-CHACRA R. & THOULESS D.J., J. Phys. C: Solid State Phys. 7, 65 (1974).

13.

ATHREYA K.B., SUBRAMANIAN R.R. & KUMAR N., Curr. Sd. 41,867 (1972); FELLER W., Introduction to the Theory of Probability and Application, Vol. I, (1967), Vol. II. Wiley, NY (1972).

14.

LICCIARDELLO D.C. & ECONOMOU E.N., Solid State Commun. 15, 969 (1974).

15.

SHANTE V.K.S. & KIRKPATRICK S., Adv. Phys. 30, 325 (1971).

16.

LICCIARDELLO D.C. & ECONOMOU E.N.,Phys. Rev. BlI, 3697 (1975).