Fluctuation effects in impurity band formation

JOURNALOFNON-CRYSTALLINE SOLIDS8--10(1972) 166--171© North-HollandPublishingCo.

FLUCTUATION EFFECTS IN IMPURITY BAND FORMATION

H. HASEGAWA Department of Physics, Kyoto University, Kyoto, Japan F. YONEZAWA

Department of Applied Physics, Tokyo Institute of Technology, Tokyo, Japan and M. NAKAMURA

Department of Physics, Kyoto University, Kyoto, Japan A concept of "pseudo-wave number" is introduced to describe the nature of electronic states in disordered crystals. In one dimension, the formation of an impurity band is interpreted as a growth of a pole of localized mode due to one impurity to a finite, measurable, closed contour in the complex wave number plane. Several formulas about the density of states, the correlation functions, and the transition probabilities related to the localization problem are given. An improvement of the impurity band density-of-states which includes potential fluctuations due to pairing is obtained.

1. Introduction The first theoretical investigation o f the f o r m a t i o n o f an impurity band in disordered crystals given by Lax and Phillips 1) is extended in two different approaches: One is a stochastic a p p r o a c h o f Frisch and Lloyd2), and the other a graphical perturbation method by Klauder a). In particular, Klauder's fifth approximation is a special case o f what is now k n o w n as the coherent potential approximation (CPA)4). A l t h o u g h the C P A is considered to be successful, it fails to give a detailed explanation o f one-electron properties; for instance, it provides only a p o o r shape o f density o f states at a band edge region. A purpose of the present paper is to revise this approximation in a completely analytic framework, so that it m a y afford a generalization of the m e t h o d b e y o n d the CPA.

2. Analytic interpretation of Klauder's method Consider the one-dimensional Lax-Phillips problem: dx 2

fi(x-xi ) O=wt),

2x o J

166

(1)

FLUCTUATION EFFECTS

167

where 2% ( > 0 ) denotes the strength of an attractive 6-potential whose centers are distributed randomly over the x-axis with density n per unit length. The constant % has a meaning of the inverse decay length of a bound eigenfunction of the single center problem:

j

(2)

~b (X) = const, x exp (-- % Ix -- %1).

(3)

- dx ~ - 2% ~ (x - x2) ~bb = Wbl/Jb, Wb = -- ~:0 2,

It may be characterized also by the singularity (a simple pole) of the scattering amplitude on the complex k-plane as ireo ~bk (x) = exp (ikx) + - exp (ik Ix k - ix o

%1 +

ikx0),

k = w~ = i ( - w) ~.

(4) (5)

Next, we proceed to discussion about the many-center problem. The average of the Green's function is in the CPA 1 G k (w) = w -

k 2 - tr ( w ) '

(Wk

= k2)'

(6)

where the self-energy ak (W) (k independent in CPA) is n times the diagonal element of the t-matrix a ( w ) = n t { z (w)} = - 2n%/[1 + 2XoZ (w)].

(7)

In this expression the quantity z(w) is the density-of-states function to be calculated from G~ (w) given in (6) as

z(w)=z o(w-a(w))=2n

1 ~ dk J w-k 2-a(w)"

(8)

-oo

The pseudo-wave number k, defined as the solution of w -

wk-

tr(w) = 0,

(9)

may be obtained from w =w k4- nt{z

- n%k

o(w~)} = k 2 q- k - i r e o

(10)

for any real value of w. Eq. (10) may be viewed as a map between the complex w- and k-planes (fig. 1). It is then easy to see the location of all the possible pseudo-wave numbers: They consist of three parts, i.e. (a) the contour F 1 (C), (b) the

168

H. HASEGAWA~ F, YONEZAWA AND M. NAKAMURA

I mk W = k 2-

2n2{'°k

k-,%

~Pl{c) = domoin encircled by E, Ic)

uppermost sheel ~-

Rek

Fig. 1. A map from the complex w-plane to the complex k-plane through eq. (10), in Klauder's approximation of the Lax-Phillips problem, eq. (1). contour F 2 (C), and (c) the imaginary axis. It is clear that the contour/"2 ( C ) is a deformation of the real axis corresponding to the original plane-wave states, and that the contour F~ (C) is a "growth" of the bound-state pole in the single center problem. The pseudo-wave numbers on these contours may have a non-vanishing real part, which corresponds to the allowed density of states. [The imaginary k-axis corresponds to the forbidden range of Re(w).] More precisely, one can show that the integrated density of states N(w) may be written to lowest order of the impurity density n as w

N(w)= j

Imz + (w')dw' = k°(w) + 0(w) ,

(ll)

-oo

where k°(w)=w ~ (w>0), k ° ( w ) - - 0 ( w < 0 ) and O(w) is the phase shift of a single scattering, arg t {z (w)). This relation represents the well-known Friedel's sum ruleS).

3. Correlation functions, transition probabilities, and the localization problem We discuss in our model the relation between the time-correlation and the transition probability in the limit t ~ ~ regarding two localized functions, since this is a subject of current interest6). Let us consider first the inner product (¢k', Ck (t)) and take the configuration average:

1 l" e x p ( - i w t )

((~tk,,~lk(t)))=6kk, 2~ilw~-k2~t~(W) dw ,J

-~

(k = real).

(12)

c The contour C of the integration in the second expression is the one indicated

FLUCTUATION

169

EFFECTS

m fig. 1 on the w-plane. Thus a time correlation between any two wave functions which can be e x p a n d e d i n ¢k'S is expressible by means of the above formula. As an example, for a 6-like localized function (which is unnormalized), ~bx-- ~ ( x - X), one gets the following

((Ox,, ~'x ( 0 ) ) = 2~i

(w) exp [iK (w)I J ( - X'l - iwt] dw,

(13)

c where K(w) in the exponential factor is the pseudo-wave number on the contours F l (C) and F2 (C) in the complex k-plane. Thus the imaginary part of k = K(w) represents the decay constant of the spatial correlation of the ~'x'S between the two distant points X and X'. The integration in (13) is conditionally convergent, and vanishes for t ~ oo at most like t - ~. On the other hand, the average (l(Ok', 0k(t))l 2) (the quantal transition probability from the plane-wave state k to k') is in general non-vanishing even in the t-~ oo limit, which will now be demonstrated. Following Economou and Cohen a), we write such a transition probability Pk~k' as

Pk~k' = lim ~-~o~

k, --

-

w+is-H

~)k'

k',

~

w-is-lf

k

dw

•

(14)

oc,

This does not vanish, if a correction factor to the product of each average, i.e. the so-called vertex correction, is properly taken into account. In the present example this correction can be included straightforwardly, yielding a formula as follows: =

1 ? Im G~- (w) im G~ (w) )

j

dw.

--cl2

Also, the relative probability for a transition ~x ~ ~x' may be written as or.

lim (l(Ox,, tPx(t)[ 25 = [

- 1 hn z + (w)

X

f l m {z+ (w) exp [ i R e K ( w ) I X -x \ lm z + (w) x e x p [ - 2 lm K ( w ) I X -

X'I] dw.

Xi]]~2 J (16)

Again, we can see an intimate relation between the imaginary part of K(w) and the spatial correlation of the Ox's. One may say that in the present one-dimensional example almost all eigenfunctions are spatially localized in accordance with the previous con-

170

H. HASEGAWA~ F. YONEZAWA AND M. NAKAMURA

jecture7). However, the meaning if Im K(w) should be different for the two different contours: A point located on F2 (C) corresponds more likely to a propagating or extended state than to a localized one, its characteristic quantity Im Krepresenting the inverse of a meanfree path. On the other hand, Im K on F 1 (C) whose value is nearly equal to Ko should represent a degree of localizationin view of the bound state of the single center problem, eq. (5). At present, it is not conclusive, since a comprehensive knowledge about the analyticity is still lacking. 4. Higher-order fluctuation effects

We wish to get an improvement of the CPA for the actual shape of the impurity band density of states whose trace is reflected by the contour F1 (C) in the k-plane. [This contour by means of the map (10) is nearly a circle, and must be far from realistic.] For this purpose let us recall the general expression for ok (w):

ak(W)=((t~k'V~kk))+(( t~k'V(I-PR) w+is-H1

(1-Pk) V~J*)) "

Clearly, the form (7) is a contribution of only selected terms taken from the perturbation expansion of (17), viz. all the single-site scattering part. It is then possible to consider the CPA G, (w) given in (6) as a free propagator, on which a renormalized perturbation of the above form ak (w) is performed. A lowest order double-site scattering effect may be obtained in second order of the potential V in the form

/

li

~,j, tE{z+(w)}2~

-

Gk,(w)exp[i(k,-k)(xj-xy)]dkl) oo

= nt2 {z+ (w)} z+ (w)/j'=,,2 ~ .... exp[i(K (w) - k)lxjl] + term(k~- k)), (18) where a certain stochastic assumption (e.g. Poisson distribution) should be made on the configuration average ( ) . Otherwise, this lowest order effect must vanish since the corresponding Feynman graphs are forbidden, and the possible double-site scattering terms start from the fourth order in V. All such terms have been collected S), given by co

+exp [iK (w) x] + coskx Aak(W) 2nEt+4z+3 [ t+z----(t+~z~2 e~p ~ i K ( w - ~ exp [3iK (w) x] dx. d

o

(19)

171

FLUCTUATION EFFECTS

I n fig. 2 a result of the numerical c o m p u t a t i o n based on the correction (19) is presented. A l t h o u g h this is a preliminary of the whole numerical p r o g r a m , the feature of the tailing at the b a n d edge seems fairly confident, suggesting a direction of i m p r o v i n g the present theory.

n=©.l

I

~2

-I

2

0

Fig. 2. A calculated density of states for the impurity band of the Lax-Phillips problem with the correction due to the second-order potential fluctuations based on eq. (19). The correction of the self-energy Aa~(w) is also indicated. Unit of the energy w is IwbJ= K0L References 1) 2) 3) 4) 5) 6) 7) 8)

M. Lax and J. C. Phillips, Phys. Rev. 110 (1958) 41. H. L. Frisch and S. P. Lloyd, Phys. Rev. 120 (1960) 1175. J. R. Klauder, Ann. Phys. (N.Y.) 14 (1961) 43. B. Velick3~, Kirkpatrick and H. Ehrenreich, Phys. Rev. 175 (1968) 747. H. Hasegawa and F. Yonezawa, J. Phys. Soc. Japan 26 Suppl. (1969) 74. E. N. Economou and M. H. Cohen, preprint. R. E. Borland, Proc. Roy. Soc. (London) A 274 (1963) 529. M. Nakamura and F. Yonezawa, to be submitted to Progr. Theoret. Phys. (Kyoto).