PHYSICAL REVIEW B VOLUME 24, NUMBER 8 15 OCTOBER 1981

Analytic (unitarity-preserving) approximation for the electronic structure of amorphous systems

Vijay A. SinghSolar Energy Research Institute, Golden, Colorado 80401

and Department ofPhysics, Uni Uersi ty of Colorado, Boulder, Colorado 80309(Received 2 October 1980)

Several theories which claim to describe the electronic structure of amorphous systems yield unphysical resultssuch as negative density of states. In this report I present (i) a self-consistent theory which preserves the analyticbehavior of the Green's function and (ii) preliminary calculations based on this theory which, in contrast topreviously discovered analytic approximations, yields reasonable agreement with "exact" results. Extensions of thistheory to elementary excitations (magnons, phonons) in amorphous. systems is possible.

Over the past decade, many approximationschemes have been proposed to describe the elec-tronic structure of strongly scattering amorphoussystems. " These systems possess short-rangeorder, so the popular coherent-potential approxi-mation' (CPA) is not applicable to them. A tran-sition-element liquid metal is a good exampleof such systems. The approximations are builton the Green's-function formalism. It has oftenhappened that the approximation does violenceto the analytic properties of the Green's functionand this leads to nonphysical behavior such as anegative density of states, etc.' ' The analyticnature of the CPA has been investigated by variousauthors. ' " In contrast, the effective-mediumapproximation (EMA) of Roth, which is a CPA-equivalent theory for amorphous systems cannotbe proved to be analytic. Of all the proposed ap-proximations, the EMA is the only approximationwhich yields good agreement with "exact" calcu-lation'" involving a wide range of parameters.There is evidence, however, that it exhibits non-analytic behavior in some cases. ' ' Qn the otherhand, another self-consistent approximation, dueto Gyoroffy, and Korringa and Mills" (GKM) whichhas been shown to be analytic by Both, ' yieldsvery poor results. '

In this communication, I shall present an alter-native approximation which is analytic (unitaritypreserving). The discussion will be based on ang-band tight-binding model for the sake of sim-plicity. Extensions to multiband, muffin-tin,or abstract multiple-scattering formalisms arestraightforward. In fact, with a little effort, onemay even extend it to treat elementary excitations(magnons, phonons, etc. ) in amorphous systems. "

Following Roth, "we define the tight-bindingunaveraged Green's function Q„as

ESgg —Hgg Gg, = 5;,-.

Here, j, j, and g refer to site indices. H,-, and

$„.are the transfer and overlap intergrals, re-spectively. E sets the energy scale. The aver-aged Green's function in momentum space is giv-en by

nGk ~%t

E -nHg —Z —Z

where

Hg= H R -&S R g R e'"'RdR.

Z~= IIkGkMkdk 8m', (4a)

Pg-k' Mk Gk dk' 8m', (4b)

where

nM =nH +Zk k &k

and h(R) =g(R) —1, is the pair correlation func-tion. In the approximation I propose, both Z„and Z~ may be described by a single equation:

Z„+Z,-„= ~+nI k-k' uk. 'G-„,dk' 8~',

where

and

nMk =nHk +Z~k,

H„= H R -ES R g R '~' '"'Rd (8)

We also note that the density of states (DOS) isgiven by

n(E) = ——lm S-6-dk/8&'.k k

Here, n and g(R) are the number density and pairdistribution function (PDF), respectively.nature of the various proposed approximations' 'is determined by equations describing the self-energy Z„and Z,~. In the EMA of Roth'"

4852 1981 The American Physical Society

BRIEF REPORTS 4853

n (E)

0.30—

0.20—

0.10'

/'

n:"/I .

i I~ ~

Ijfl

~ ~

Legenda, = 1/3.0R, = 2.0

Fujiwara-Tanabe————EMA~——Proposed Approximation~ ~ ~ ~ ~ ~ ~ ~ ~ GKM

~ ~ ~ ~ ~ ~ ~

0.00 ' I

-5.0I

0.0 5.0 10.,0,

FIG. 1. The density of states n (8) in the various approximations, compared with the "exact" results (Bef. 17) fora nonorthogonal z-band tight-binding model. The EMA and GEM results have been calculated earlier by Aloisio et zL(Ref. 7).

This by Eq. (2) is non-negative if Im(Z~+Z») ~ 0.Since the structure factor [1+gk(k -k')] is non-negative, we note that in Eq. (6)

[1+~k(%—k')]iMk, i2o-0.

Hence, Eq. (6) may be self-consistently chosento be non-negative. One can also show that Eq.(6) will have a unique solution (fixed point theor-em"). A detailed proof will be provided else-where but the reader is referred to the works ofRoth' and Haydock, Heine, and Kelly' for similararguments.

If, in Eq. (7) we set ~k = H„, we recover theGyoroffy-Korringa-Mills approximation'4 (GKM).The poor, structureless results of the GKM (seeFig. 1) can be attributed to the lack of correlation[i.e., g(R)] and self-energy (Z») effects in thedefinition of M&. In contrast, my proposed ap- .

proximation includes both effects. One can, infact, construct a hierarchy of analytic approxi-mations ranging between the GKM and the approx-imation proposed in this work. The square rootof the PDF is taken in Eq. (8) to avoid over-mul-tiplication of the diagonal term (Z~) by a PDF.The case where [g(R)]'~' is replaced by g(R)would be an interesting numerical exercise. Aneven more interesting exercise would be to takethe converged EMA solutions [Eq. (4)] and iterateit just once in Eq. (6).

Having established an analytic approximationone may ask how accurately would it represent

an amorphous system. Recently, Fujiwara andTanabe" performed an essentially exact calcula-tion for the DOS of an s-band model of amorphousiron. They employed the negative eigenvaluecounting method" for the nonorthogonal case shownin Fig. 1. The transfer integral was given by

H(R) = —V(1+R/a, ) exp(-R/a, ),=0, R&R,

with H(R)=$(R)/( —p). The results of the GKM,EMA, and the present approximation are com-pared with their "exact" results.

%e have shown recently that the EMA is basedon a sound decoupling procedure. ' It also has awell-defined diagrammatic analysis. '" The pres-ent approximation will be found lacking in someof these features. However, (i) it is analytic(unitarity preserving), (ii) it compares wellwith "exact" results in a preliminary investiga-tion, and (iii) Eqs. (4) and (6) make it clear thatthe numerical difficulty involved in solving theproposed approximation is less severe than theEMA. A fuller numerical and theoretical investi-gation of the proposed approximation and its vari-ations will be carried out in the future.

I acknowledge many useful discussions withProfessor L. M. Both. This work was partlysupported by NSF Grant No. DMR-75-18104 andSEBI Subcontract No. HS-0-9188-4.

~For a discussion of these approximations, see L. M.Both, Phys. Rev. B 9, 2476 (1974).

2For a comprehensive derivation of these approxi-mations, see Vijay A. Singh and L. M. Both, Phys.Bev. B 22, 4089 (1980).

P. Soven, Phys. Rev. 156, 809 (1967).B. G. Nickel and W. T. Butler, Phys. Rev. Lett. 30,373 (1973).

L. Schwartz, Phys. Rev. B 21, 522 (1980); 21, 535(1980).

BRIEF REPORTS

Vijay A. Singh and L. M. Roth, Phys. Rev. B 21, 4403(1980).

M. Aloisio, Vijay A. Singh, and L. M. Roth, Bull. Am.Phys. Soc. 25, 242 (1980); J. Phys. F (in press).L. M. Roth, Phys. Rev. B 22, 2793 (1980).L. Huisman, L. Schwartz, D. Nicholson, and A. Bansil(unpublished), and private communication.R. L. Mills and P. Ratanavaraksa, Phys. Rev. B 18,5291 (1978).E. Muller-Hartman, Solid State Commun. 12, 1269(1973); F. Ducastelle, J. Phys. C 7, 1795 (1974);L. Schwartz and A. Bansil, Phys. Rev. B 18, 1702(1978).J. S. Faulkner and G. M. Stocks, Phys. Rev. B 21,3222 (1980).

Vijay A. Singh and L. M. Both, J. Appl. Physics 49,(3), 1642 (1978).B. L. Gyoroffy, Phys. Rev. B 1, 3290 (1970); F. J.Korringa and R. L. Mills, Phys. Rev. B 5, 1654 (1972).L. M. Both, Phys. Rev. B 11, 3769 {1975);J. Phys.F 6, 2267 (1976).R. Haydock, V. Heine, and M. J. Kelly, J. Phys. C 5,2845 (1972). A rigorous proof of the herglotizity ofthe proposed approximation is based on a nontrivialextension of the work of J. A. Shohat and M. Tamarkin,Problem of Moments (American Mathematical Society,New York, 1943).T. Fujiwara and Y. Tanabe, J. Phys. F 9, 1085 (1979).P. Dean, Proc. R. Soc, London, Ser. A 254, 507(1960).