THE BREAKDOWN OF CLUSTER COHERENT POTENTIAL APPROXIMATIONS IN DISORDERED SYSTEMS

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THE BREAKDOWN OF CLUSTER COHERENT POTENTIAL APPROXIMATIONS IN DISORDERED

SYSTEMS

P. Leath

To cite this version:

P. Leath. THE BREAKDOWN OF CLUSTER COHERENT POTENTIAL APPROXIMATIONS IN DISORDERED SYSTEMS. Journal de Physique Colloques, 1974, 35 (C4), pp.C4-99-C4-101.

�10.1051/jphyscol:1974417�. �jpa-00215608�

JOURNAL DE PHYSIQUE Colloque C4, supplkment au no 5, Tome 35, Mai 1974, page C4-99

THE BREAKDOWN OF CLUSTER COHERENT POTENTIAL APPROXIMATIONS IN DISORDERED SYSTEMS (*)

Physics Department, Rutgers University New Brunswick, New Jersey, USA (**)

and

Department of Theoretical Physics, Oxford University, Oxford, England

RBsumC. - Nous presentons et discutons un theorbme qui donne les conditions necessaires et suffisantes pour que l'approximation du potentiel coherent Four IES yaircs et les arras ait les lirnites correctes pour un cristal virtuel et pour des bandes dparees. Cette condition, que le potentiel coherent ou d'energie propre soit diagonal pour les amas (pas d'616ments inter-amas), est inconsis- tante avec l'invariance translationnelle de la fonction de Green moyenne sauf pour I'approximation du potentiel coherent ordinaire, ce qui indique que l'expansion en amas habituelle n'est plus valable.

Si I'on rnaintient I'invariance translationnelle, des non-analyticites serieuses similaires a celles indiquks par Nickel et Butler apparaissent dans la limite des bandes separkes. D'autre part, si I'on abandonne I'invariance translationnelle et ses effets, on peut obtenir des formules analytiques adequates qui illustrent certains effets d'amas. Un exemple d'un tel modele deja paru dans la littkrature est discutk.

Abstract. - A theorem is detailed and discussed which gives the necessary and sufficient conditions for pair and cluster coherent potential approximations to have the correct virtual crystal and split band limits. This condition, that the self-energy or coherent potential be cluster diagonal (no inter-cluster elements), is inconsistent with the translational invariance of the average Green's function except for the single-site or ordinary coherent potential approximation, thus indicating a breakdown of the cluster expansion, as generally conceived. If one insists upon transla- tional invariance, serious non-analyticities of the sort reported by Nickel and Butler occur in the split band limit. On the other hand, if one is willing to sacrifice the translational invariance and its related effects, one can obtain properly analytic formulae which illustrate correctly certain cluster effects. An example of such a model in the literature is discussed.

1. Introduction. - Recently, Nickel and Butler [l]

and previously, Capek [2] have reported the onset of strong and spurious non-analyticity, appearing in numerical calculations for various generalizations of the coherent potential approximation which include scattering by pairs of defects, when the impurity scattering potential A becomes of the order of the unperturbed bandwidth W. This non-analyticity gene- rally occurs as branch points on the first quadrant of the physical sheet of the calculated Green's function in the complex energy plane and hence t o unphysical solutions of the generalized CPA equations. This spurious behavior in the split band regime, further- more, seemed t o be the rule rather than the exception, since it was produced by several, seemingly well conceived theories.

(*) Work supported in part by the National Science Founda- tion, USA.

t S. R. C. Senior Visiting Fellow at Oxford University during part of 1972-73.

(* *) Permanent address.

Since such non-analyticities d o not occur in the ordinary (single-site) CPA [as has been shown recently by Miiller-Hartmann [3]], the obvious question remain- ing is what property of the pair and, presumably, higher cluster CPA is responsible for the analytic breakdown. Since the breakdown occurs only in the split band regime and since these formulae generally had been derived from propagator expansions (or perturbation expansions about the weak scattering limit) the answer is thus clearly that these various generalizations did not, in fact, contain the appro- priate split band (or atomic) limit.

That the ordinary CPA has the appropriate split band limit was shown first by Onodera and Toyozawa 141. This author in a previous paper [5]

showed that not only did the CPA have the correct split band limit but that it could be derived, in fact, from the perturbation expansion in the interatomic hopping elements. The purpose of this brief report is to detail and discuss the implications of a recently proven theorem [6] o n the conditions required to

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974417

C4-100 P. L. LEATH

correctly obtain both the split band and virtual crystal limits in a cluster CPA.

We consider the tight-binding electronic hamil- tonian

X ^{= Z,} V(n) a: a, + ^En,, W(n - m) a: a,, (1) where a: creates an electron in the Wannier state about site n in the single energy band of interest, where the hopping matrix element W(n - m) is assumed independent of the kinds of atoms on sites n and m, and where the diagonal site energy V(n) takes on a value v with probability P(v), which is independent of n. The discussion given here also would be valid for additive off-diagonal disorder as discussed below.

2 Propagator expansion. - In the propagator expan- sion, the configuration-averaged Green's function G ⁼ (E - X)-I is expanded in powers of ( V - C), the deviation of X from the translationally invariant effective medium hamiltonian,

EM = z,,,[Z(n, m) + W(n - m)] a: a,, (2) where Z(n, m) is the so-called coherent potential or self-energy. The expansion of < G > becomes

where GM ⁼ (E - EM)-' this equation can be rearrang- ed exactly into the form

+ GM < ^t, G', f,, G', t, > GM + ...

₍₄₎

where t, is the sum of all scattering by a cluster of n atoms and where G)M is the medium propagator connecting dzfferent n-clusters. The generalized CPA approximation is to make < G > ^1. GM by decou- pling the average t-matrix product < t, t, ... t, > into products of averages < ^tn > < t, > ... < t, >, and setting < t, > to zero. This condition is conveniently written in the nxn matrix form [7],

where, for example,

where 1, 2, ..., n are the sites in the n-cluster. These nxn matrices are clearly the (nxn) block of a larger matrix representing the entire crystal. These equations must be solved simultaneously with the Dyson equation.

3. Hopping expansion. - In the hopping expansion (or locator expansion), the configuration-averaged

< G > is expanded in powers of the interatomic hopping W,

where g(n) = (E - V(n))-I is the local-site Green's function and where U , the interactor, gives the fully renormalized hopping. If we now consider an effective medium with Green's function

and

where a will be called the locator, Urn is the effective interactor. Then eliminating W from eq. (7b) using (8b), we obtain the expansion

which is formally analogous to eq. (3) of the propa- gator expansion. Thus rearrangements of the expansion to include all multiple scatterings by a cluster (as in (4)) and decoupling and setting to zero the cluster

cc ?:matrix >> gives an equation analogous to the generalized CPA eq. ( 3 ,

w

-

where g,, a, and 6, are nxn matrixes as in eq. (7).

Thus, this approach is formally analogous to the CPA approach as has been previously shown dia- grammatically by Leath [5, 61 and by an argument like the one given here by Ducastelle [7], in the single-site case.

4. Equivalence. - In order to see the effect of the physical equivalence of the two approaches, we formally set GM ⁼ G, which gives

where C and a are the full coherent potential and locator matrices for the crystal. Then, we replace - -

on, g,, and U, in (10) in terme of ^{E ,} v , C, and G,, using eq. (8) and (ll), and the definition of in. After considerable nxn-matrix algebra, as given in [6], we find that the necessary and sufficient condition that eq.

(10) is precisely the CPA eq. (5) is that the full C matrix must not have any off-diagonal elements C(n, m) connecting a site within the cluster [for which

< < ^{> =} 01 to the rest of the crystal. This cluster

THE BREAKDOWN OF CLUSTER COHERENT POTENTIAL APPROXIMATIONS C4-101

diagonality of the coherent potential C (and of o) is the basic theorem discussed here.

The proof of the theorem also obviously carries through in precisely the same manner in the case of off-diagonal disorder if the perturbation potentials due to each impurity are additive ; in this case not only C but also V turns out to be required to be cluster diagonal.

5. Conclusion. - It is now noted that nowhere in this exercise have we assumed the translational inva- riance of < ^G > or C and that, in fact, we have proven that for generalized CPA equations with finite clusters of size n > 1 the translational invariance C does not exist if the equations are to reduce to those obtained from the hopping expansion and hence to have the correct split band limit. The broken transla- tional invariance follows immediately from the fact that the coherent potential can have non-zero, off-diagonal elements within a cluster but not between clusters.

With this cluster diagonality condition it is also clearly inconsistent to, for example, set to zero the t-matrix for pairs of atoms at all distances. In fact the only cluster approximation in the literative which meets these diagonality requirements is the cluster coherent potential approximation CCPA of Tsukada [8] which lets the cluster become a unit cell for a super lattice (i. e. the cluster is repeated through- out the crystal) where C has non-zero off-diagonal elements only within a cluster unit cell. This is clearly just an ordinary CPA but with several degrees of freedom within each unit. Some numerical results were carried out by Tsukada [8] on this model, but the results were not clearly revealed due to the usual difficulty with k-sums in cluster approximations.

Fortunately, very recently Butler [9] has found a very simple technique for calculating the CCPA in one dimension. His results were compared in detail with the essentially exact results obtained from the Schmidt integral equation technique [lo] and showed rapid convergence toward the correct spiked structure in the impurity band region, with excellent agreement for clusters of seven in the unit cell. Furthermore, the formulae were properly analytic in the split band regime. Apparently the insuring of the correct virtual

crystal and split band limits as described here, also insures the analyticity of the result since Ducastelle at this conference has reported a proof of proper analyticity in the cluster CPA of Tsukada [8].

Butler's method of calculation for linear chains (which he proved was equivalent to the CCPA) was by the continued fraction method where

is averaged over all possible occupations of a specified cluster imbedded within an effective medium. The equi- valence with the CCPA and the analyticity comes only when the subsidiary condition that the value of ^{G(2,1, E )} at the boundary is set equal to the effective medium ^G,, on the average (in contrast to several previous theories which had put the condition on the central site). It now seems likely that the similar method of continued fractions in shells about a central atom that has been developed recently [ l l ] for two and three-dimensional systems is likely to provide a similar simplification for real systems, if the proper boundary condition on the outermost shell of the cluster is used.

Finally, even though there exist numerical methods which will eventually give such basically local quanti- ties as densities of states accurately in the CCPA the broken translational invariance is fundamental and surely related to the onset of localization of the states in some way. This property will make the CCPA less useful in describing such quantities as G(k, w ) and the conductivity, especially near any mobility edges that might exist. The obvious way to avoid these difficulties is to sum infinite classes of terms involving clusters of all sizes.

Acknowledgments. - The author would like to acknowledge many useful conversations with Drs. R. J.

Elliott and B. Nickel and to express his gratitude to the Department of Theoretical Physics, Oxford Univer- sity for its hospitality during his stay.

References

[I] NICKEL, B. and BUTLER, W., Phys. Rev. Lett. 30 (1973) 373. [7] DUCASTELLE, F., J. Phys. C 4 (1971) L 75.

[21 CAPEK, V., Phys. Stat. Sol. (b) 43 (1971) 61 ; and Czech. [8] TSUKADA, J. Phys. Soc. Japan 26 (1969) 684 ; and 32 (1972)

J. Phys. B 21 (1971) 997. 1475.

131 MULLER-HARTMANN, E., Solid State Commun. 12 (1973) [91 BUTLER, W. H. Phys. Rev, 8 (1973) 4499, 1269.

[4] ONODERA, Y. and TOYOZAWA, Y., J. Phys. Soc. Japan 24 [lo]

H., Phys. Rev. lo5 425.

(1968) 341. [ l l ] HAYDOCK, R., HEINE, V., KELLY, M. 'J., J. Phys. C 5 [5] LEATH, P. L., Phys. Rev. B 2 (1970) 3078. (1972) 2845 ; and

161 LEATH, P. L., J. Phys. C 6 (1973) 1559. MOOKERJEE, A., J. Phys. C 6 (1973) 1340.