GENERALIZATIONS OF THE COHERENT POTENTIAL APPROXIMATION

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GENERALIZATIONS OF THE COHERENT POTENTIAL APPROXIMATION

L. Schwartz

To cite this version:

L. Schwartz. GENERALIZATIONS OF THE COHERENT POTENTIAL APPROXIMATION. Jour-

nal de Physique Colloques, 1974, 35 (C4), pp.C4-71-C4-74. �10.1051/jphyscol:1974411�. �jpa-00215602�

GENERALIZATIONS OF THE COHERENT POTENTIAL APPROXIMATION (*)

L. M. SCHWARTZ

Department of Physics, Brandeis University, Waltham, Massachusetts 02154, U. S. A.

R4sum6.

- Nous considerons le probleme d'etendre l'approximation de potentiel coherent pour traiter les effets de

integrales de transfert alCatoire dans le modele de reliure serree d'un alliage binaire et

ordre de courte extension dans les metaux liquides.

Abstract. - We

consider the problem of extending the coherent potential approximation to treat the effects of

random transfer integrals in the tight binding model of a binary alloy and

short range order in liquid metals.

1.

Recent authors have devoted a great deal of attention to a particularly simple model of the substitutional alloy A, B,-,. A single orbital

I

> is associated with each of the N lattice sites

and the electronic properties of the system are then described by the Hamiltonian

where the prime indicates that only nearest neighbors are included in the summation. The crucial feature of this model is that the size of the individual scatterers is essentially zero. Thus the local energy levels

may take either of the two values

6 or

0, but the transfer integrals

are always periodic. On the basis of a comparison with various exact results concerning the energy spectrum of the Hamiltonian (1.

it has generally been agreed that an excellent description of this model is provided by the coherent potential approximation (CPA) [I]-[4]. There are, however, many physical systems in which the disorder inherently extends over finite distances and it is therefore natural to ask whether the CPA can, in some sense, be extended to treat this new effect.

The two problems with which we will be concerned are (1) the effects of random inter-site coupling in eq.

[i. e. of allowing

to depend on the type of atoms occupying

and

short range order in liquid metals. The first of these problems is obviously of interest in connection with the electronic structure of alloys [e. g. CuZn] whose constituent bandwidths are significantly different. In addition, several authors have stressed the importance

(*) Work

supported

part

Grant No GH-35691

the

National

Science Foundation.

of including analogous effects in the calculation of the vibrational spectra of disordered alloys [S]. Regarding the short range order problem, it should be empha- sized that this effect is always present in liquids, its simplest manifestation being the characteristic oscilla- tions of the radial distribution function. Indeed, it is essentially meaningless to speak of a liquid metal without short range order since, at the very least, the density correlations will always guarantee that neigh- boring atoms do not overlap appreciably.

2.

- To begin our discussion we review the central features of the CPA. Given the Hamiltonian (1. l), the equilibrium properties of the alloy are described by the Green's function G(z) or, alternatively, by the self-energy Z(z).

These operators are functions of the (complex) energy z and are defined by the equations

where < ... > denotes an ensemble average over the possible configurations of the alloy. The CPA has generally been developed within the framework of multiple scattering theory. The individual scatterers are then viewed as being embedded in an effective medium whose choice is open and can be made self-consistently.

The physical condition corresponding to this choice is simply that if the part of the medium belonging to a given site is removed and replaced by the true atomic potential then, on the average, there should be no further scattering. The effective medium so determined is then equivalent to the CPA self-energy. Alternatively, the CPA may be derived from an exact self-consistent expansion of Z(z) in terms of the impurity (A) concen- tration x, the scattering'strength 6, and the Green's function G(z) [4]. In this approach the CPA is found to retain all contributions to Z(z) that involve only the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974411

C4-72 L. M. SCHWARTZ

single site matrix element F(z)

< n

G(z) I n >.

C(z) is then a simple scalar and is given by

where the renormalized cumulant Q,(x) is a polynomial in x of degreep but can, in fact, always be written as a symmetric function of x and

1

x [6]. The second of eq. (2.2) follows from the first with the aid of the generating function for the cumulants Qp(x).

The fact that Q,(x) contains contributions through xP leads to what are generally referred to as multiple occupancy corrections. In the language of scattering theory these corrections correspond to the statement that in a proper description of an impurity embedded in an effective medium, the part of the medium belong to the impurity site must first be removed. It is precisely these higher order contributions that allow us to interpret the CPA as a mean field theory of the disordered alloy. Indeed, a systematic analysis of the moments y of the exact electronic density of states

where s(k)

C' eik.Rn, allows us to evaluate exactly the matrix eleGents tA(k, kt) = < ^{k I tA(z)} I kt > of the single impurity scattering matrix tA(z) 171. [In operator notation tA(z)

[I -

G0(z)]-' uA, where

Ho)-' is the Green's function of the host crystal.] t(z) provides a complete description of the scatterer's effect on the electronic spectrum of the system. For example, the locations of any localized impurity states are determined entirely by the singula- rities of t(z) in the complex z plane. However, more important for the present discussion is the fact that the low concentration behavior of the macroscopic alloy is determined by the diagonal matrix element t(k, k). Indeed, to lowest order in x the self-energy reduces to

C(k, z) = < k I C(z) 1 k > - x[tA(k, k)] , (3.3) and each of the discrete impurity states will give rise [in the dilute alloy] to sub-band of finite width.

Now, a proposed generalization of the CPA to include the effects of off diagonal disorder must (at the very least) satisfy three criteria. The theory should be invariant under the exchange of A and B atoms, exact to first order in x and y in the appropriate low concentration limits, and should reduce to the CPA if the transfer integrals are again periodic. While these reveals that the relevant expansion parameter for the

requirements certainly do not determine a unique CPA is Z - I where Z i s the number of nearest neighbors.

description of the alloy, it is important to note that the that are both sym- they

satisfied by the theories of off- metric under the exchange of A and B atoms and

diagonal disorder [8]. In particular, these theories are correct to lowest order in

and

in the appropriate

not compatible with the exact result (3.3) [7].

dilute limits, only the CPA retains all higher order

Accordingly, we present here only the simplest consis- contributions [to the moments ,up] through first order in

tent equations. In this spirit let us assume that the

In other words, the essential point is not that

A-B transfer integral is given by the average value the CPA reproduces the exact values of the first eight

h,,

(hAA + hBB)/2. Without any further approxima- moments, but rather that it almost gives the correct

tions the alloy Hamiltonian can then be written as values of the higher moments ri. e. that the corrections

to the CPA moments are 0(i2/Z2) instead of simply H

H, + C ^{I n} > 6 <

I +

O(x2)]. This characterization of the CPA as an

approximation that retains the principle contribution

(

+ C ' 6 , [ j n > < m l + l m > < n l ] to every moment will be especially useful in connection

with both the problems of random transfer integrals

= H B + C v,A.

and short range order.

3.

diagonal disorder. - To begin, we consider In eq. (3.4) the disordered part of

sum of (finite ranged) scattering potentials associated an isolated A impurity at the origin of an otherwise

with each of the A atoms. These equations are, how- perfect B lattice. The Hamiltoniam is simply

ever, symmetric under the exchange of A and B atoms H = H o + I O > 6 < O I + and H can, in fact, also be written as

-I-

x'61[10>

< n I + I n > < O I ] @ . l a ) H = H A + z ^v:,

n e R

E

Ho

(3. lb) where

is the impurity potential for a B

atom in an A host.

where 6,

(h,, - h,,) and h,, and h,,

Z - ' are

Having written H as in eq. (3.4), we consider the the host-impurity and host-host transfer integrals

following self-consistent equation for C(k, z) respectively. Now, the fact that the potential uA is

separable in the momentum representation, i. e. that

~ ( k ,

C ^Q,(x) < ^k i uA[c(z) uAIp-' 1 k > ,

uA(k, k') < k

I k >

61[~(k) + ~(k')] + ^{6 ,}

(3 -2) (3.5)

where the matrix elements of

are defined by (3.2).

In view of the symmetry of both H and the cumulants Q,(x), it is clear that eq. (3.5) is invariant under the interchange of the host and impurity designations.

In the absence of off-diagonal disorder, vA(k, k')

6, and [since F(z)

< k I G(z)

k >] we regain the CPA result (2.2a). In addition, as the concentration x

0, Qp(x)

x (all p) and eq. (3.5) reduces properly to the dilute limit (3.3). The approximation (3.5) thus satisfies the three requirements outlined above.

Equally important is the fact that eq. (3.5) can be shown to provide a generalization of the CPA in the sense of the mean field parameter

i. e. that this approximation reproduces the principal contribution to every moment of

Given values of x, 6 and 6, eq. (3.5) can be summed analytically and then iterated to self consistency. The results of detailed numerical calculations will be discussed in a forthcoming publication [9].

4. Liquid metals.

In a binary alloy the atomic constituents are arranged at random on a periodic lattice. By contrast, in a liquid the disorder is structural and the spatial configuration of the scatterers no longer exhibits any long range order. Despite the lack of long range order, it is generally assumed that density correlations in the liquid play an important role over distances on the order of several inter-atomic spacings and that, at the very least, they prevent any overlap of the atomic potentials. It is precisely this fact [i. e. that the scatterers are non-overlapping] that provides a formal link between the liquid metal and substitutional alloy problems. In treating either system multiple occupancy effects must be taken into account, and it is this feature of the CPA that should be generalized to the liquid.

We suppose that in

specific configuration of the liquid the scattering centers are located at the N points R, ... R,. Denoting the individual atomic potentials by

- R,, I)

vRn(r), the single particle Hamiltonian is simply

H

C v ( I

- Rn I) ^{(4. la)}

n

= H ,

+ j n(R) ouR(r) d3R .

l b ) Here Ho

represents the free electron kinetic energy and n(R) = 6(R - R,) is the configuration

n

dependent density operator. The available information about the spatial arrangement of the atoms is contained in the sequence of correlation functions < n(R) >,

< n(R) n(R1) >, <

(R) n(Rf) n(Rn) > ... [where the angular brackets now represent an average over the possible configurations of the scatterers]. Of this sequence, however, only the quantities < n(R) > = n (the mean density) and < n(R) n(R') > - n2 g(( R- R'() [the pair correlation function] are readily available from experiment.

Several authors have attempted to generalize the CPA to the liquid metal problem [lo]-[13]. Their various approaches, while not identical, have in common two features

(1) the electronic properties are determined by three input units

the scattering potential v(r), the atomic density n, and the radial distribution function g(R), and (2) in the random limit g(R)

1 the electron self-energy [again defined as G(z)

[z

-

is determined by the equations

and

oR

[I -

GI-' vR . (4.2b) We emphasize, however, that the different approxima- tions are not equivalent if g(R)

1. Formally, they may be said to correspond to different decouplings of the higher order correlation functions in terms of just

and g(R). What has not generally been realized, however, is that this decoupling must be done in such a way as to properly include the multiple occupancy corrections. Unfortunately, in this connection the random limit of eq. (4.2) is essentially irrelevant. In the random liquid the scatterers are allowed to overlap and there are no multiple occupancy corrections.

Nevertheless, it is possible to formulate an exact perturbation theory for the aon-random liquid and, by studying the form of the moments A, of the self- energy, to derive a sequence of exact results that are directly related to the multiple occupancy correc- tions. To understand these results in their simplest form let us take the following simple model for vR(r)

6 : l r - R l < r m ,

V R ( ~ )

0 : ( P - R ( > r,, (4.3) where r, denotes an effective muffin tin radius. Having adopted this model it is possible to calculate exactly the leading contribution from a given order of pertur- bation theory (say the pth) to the moment A,. The result is simply

Ap - ^{6PQp(x) , (4.4)}

where Qp(x) again denotes the remormalized cumulant and

x

n 4 nr:,/3 (4.5) is the dimensionless density parameter. Eq. (4.4) are exact and their derivation [13] depends only on the fact that the short range order prevents any overlap of the scatterers.

Eq. (4.4) provide an important test of the validity

of the different proposed generalizations of the CPA

to the liquid metal. It is not difficult to show that

each of these approaches is correct to first order in x

and, in addition, that none of them is exact to any

higher order. However, all of the approximations do

contain certain terms of every order in x. Now, in

each order of perturbation theory Eq. (4.4) isolate

C4-74 L. M. SCHWARTZ

just those terms of higher order in

that are most important for the moments of

It is clearly of interest then to inquire whether a given approximation can in fact reproduce these results and, to our knowledge, only the equations of Schwartz and Ehrenreich [I

are consistent with this requirement.

In the approach of ref. [l

the second of eq. (4.2) is replaced by

o, =

u,[l

G(u,

;,)I ^- , ( 4 . 6 a ) where

[ I - g ( l R - R ' I ) ] o R t d 3 ~ ' . ( 4 . 6 b )

The quantity o", is seen to subtract from the full self energy the contribution due to all other sites, and in this sence represents the part

the efective medium to be associated with an atom held fixed at R. In view of the structural similarity between ( 4 . 6 ~ ) and the CPA equation (2.2b), the fact that eq. ( 4 . 6 ) correctly describe the multiple occupancy effects is not very surprising.

The work on off diagonal disorder reported in this paper has been carried out in collaboration with H. Fukuyama and H. Krakauer. Conversations on the liquid metal problem with W. Butler, J. S. Faulkner and B. Gyorffy are also appreciated.

References

[I] SOVEN, P., Phys. Rev. 156 (1967) 809. SHIBA, H., Progr. Theor. Phys. 46 (1971) 77 ; 121 VELICK+, B., KIRKPATRICK, S. and EHRENREICH, H., Phys. BROUERS, . F.. . Solid State Commun. 10 (1972) 757 : . , ,

Rev. 175 (1968) 747. BLACKMAN, J. A., ESTERLING, D. M. and BERK, N. F., Phys.

[3] SOVEN, P., Phys. Rev. 178 (1969) 9936. Rev. B 4 (1971) 2412.

[41 SCHWARTZ, L. and SIGGIA, E., Phys. Rev. B 5 (1972) 383. [91 KRAKAUER, H., SCHWARTZ, L. and FUKUYAMA, H., Phys.

151 KAPLAN, T. and MOSTOLLER, M., Phys. Rev. (submitted for

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publication).

[61 YONEZAWA, F., Progr. Theor. Phys. 40 (1968) 734. 1101 GYORFFY, B., Phys. Rev. B 1 (1970) 3290.

[7] SCHWARTZ, L., KRAKAUER, H. and FUKUYAMA, H., phys. [Ill SCHWARTZ, L. and EHRENREICH, H., Ann. Phys. 64 (1971)

Rev. Lett. 30 (1973) 746. 100.

[8] Foo, E.-N., AMAR, H. and Aus~oos, M., Phys. Rev. B 4 [I21 ROTH, L., Phys. Rev. (submitted for publication).

(1971) 3350 ; [13] SCHWARTZ, L., Phys. Rev. B 7 (1973) 4425.