BINARY ALLOYS IN THE STRONG-SCATTERING LIMIT : MOMENT ANALYSIS

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BINARY ALLOYS IN THE STRONG-SCATTERING LIMIT : MOMENT ANALYSIS

S. Kirkpatrick

To cite this version:

S. Kirkpatrick. BINARY ALLOYS IN THE STRONG-SCATTERING LIMIT : MOMENT ANAL- YSIS. Journal de Physique Colloques, 1972, 33 (C3), pp.C3-247-C3-250. �10.1051/jphyscol:1972338�.

�jpa-00215072�

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BINARY ALLOYS IN THE STRONG-SCATTERING LIMIT

:

MOMENT ANALYSIS

(*)

S. KIRKPATRICK

James Franck Institute, The University of Chicago, Chicago, Illinois 60637 and IBM Thomas J. Watson Research Center (**)

Yorktown Heights, New York 10598

Rhum6. - Un hamiltonien d'un modkle simple decrivant les electrons dans un alliage binaire est etudiC a I'approximation de la diffusion forte par plusieurs techniques numeriques directes.

On obtient des informations sur des propriktes du systeme infini tridimensionnel, telles que lar- geurs de bande, queues de bandes et la structure residuelle de la densite d'etats. Une structure anormale de la densite dyetat est observee pres du centre de la bande dans un modkle a deux dimensions. On en donne une explication possible.

Abstract. - ^Asimple model Hamiltonian for electrons in a binary alloy is studied in the strong- scattering limit by several direct numerical techniques. Information is obtained about such average properties of the infinite, three-dimensional system as band widths, band tails, and the remnants of critical point structure in the density of states. Anomalous structure in the density of states is observed near the center of the band, in a two-dimensional case. A possible explanation is given.

This paper presents a summary of numerical studies of the density of electronic states in a simple model strongly-disordered band, using techniques which do not rely on either average or approximate long-range order. Most of the results presented describe three-dimensional systems. However, in the dis- cussion of special anomalies in the density of states (the final section of this paper) a two-dimensional example has been introduced because the anomalous structure is far more prominent in two dimensions than in three.

The system treated is a narrow, singly-degenerate band in the presence of an arbitrary concentration of repulsive, short-ranged scatterers. The use of a tight-binding representation facilitates calculation of moments of the disordered band to very high order.

Evaluation of the moments, and properties derived from them, does not require the introduction of k- space, but depends on the details of the average environment within a short distance of an atom in the material.

From the moments, the density of states was obtained directly by two techniques : an infinite continued fraction representation of the Green's function [I] ; and a polynomial expansion of the density of states [2].

The predictions of a self-consistent effective field theory (the coherent potential approximation, or CPA [3], [4]) are found to be in excellent overalI agreement with the numerical results as long as the atoms contributing electrons to the band outnumber the scatterers. Tails of the density of states were

(*) Research supported by ARO(D) and ARPA.

(* *) Present address.

calculated within a phenomenological model, fitted to the high-order moments. This model gives a successful quantitative description of the fluctuations causing band tailing. A full description of this work will be submitted for publication elsewhere.

The Single Band Model. - We consider a narrow band made up of orbitals from only one constituent, say the A atoms, of a disordered material. A possible Hamiltonian to describe the band is

where the sum over atoms i and j is restricted to nearest neighbors, b is a (constant) overlap matrix element, and PA is a projection operator which selects orbitals

I

i

>

on A sites only. Disorder can be introduced into this model by varying the atomic arrangement, as in liquids and amorphous materials, or by allowing the fraction, x, of atoms which are A's to vary. In the latter case, the excluded sites will cause a narrowing of the band. In both cases, fluctuations in the local environment cause critical points in the density of states to be smeared out and disappear, while the band edges may develop tails. The calculations des- cribed below consider the second situation, in which the A atoms are positioned at random on some fraction, x, of the sites of a simple cubic lattice.

The even order moments of the density of states in the disordered band are defined by

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972338

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C3-248 S . KIRKPATRICK

where the integration extends over the (finite) range of possible eigenenergies of HA. Odd moments vanish in the case considered, as is shown below. The density of states is simply related to the site-diagonal part of the averaged one-electron Green's function

where

and

I

⁰

>

represents an orbital on an arbitrarily chosen origin atom. Substituting the large z expansion of either form of (4) into (2) yields a relation between FA and the moments

from which is obtained

The second form of (6) is the most useful. Since HA does nothing except connect nearest neighbors, each term in (HA)2p contributing to (5) can be considered as a path of unit steps on the lattice, including only A sites, and returning to site 0 at the 2 pth step.

(Only even paths return to 0 on the simple cubic lattice.) Evaluation of ,UZ~, thus reduced to a fairly simple path enumeration problem was carried out exactly through order 2 p = 12, and by a Monte Carlo procedure for selected orders through 80.

Infinite Continued Fractions. - A useful representation of FA(z) is the continued fraction

where the e's and q's can be obtained from the known moments [5]. If the first m coefficients of (7) are obtained in this way, and all higher ones set equal to zero, the resulting FA, expanded as in (5), will be correct to order z-'"-'

,

but its imaginary part, for z + E

+

ⁱO+, will consist of up to 2 m delta functions.

For a band of finite width, however, both the e's and the q's tend to constant limits [I]. The asympto- tic part of such a continued fraction may be summed to infinite order. Treating the high-order coefficients of (6) in this way gives FA(z) a cut along the real axis, with square root singularities in the amplitude at the end- points, as is appropriate for band edges in a perfect 3-dimensional system.

Figure 1 shows the unperturbed density of states p(O'(E) (for x ⁼1). Since p(O) is symmetric, only positive energies are shown. The solid line represents the best infinite continued fraction approxin~ation

ENERGY

FIG. 1. -Comparison of the density of states for a tight- binding s-band with two approximations to its infinite continued

fraction representation.

obtained, using twenty exact coefficients, while the light dashed line was calculated with only seven, the same number as could accurately be obtained for an arbitrary x. The best fit, in figure 1, gives a quantita- tively accurate description of the critical point, yet the paths contributing to the largest moments used extend, on the average, only four sites from the origin.

The more << practical ⁾⁾ treatment in figure 1 is in good overall agreement with p(O), indicating clearly the presence and rough location of the critical point, yet almost all the paths considered remain within 3 sites of the origin. Knowledge of the details of a rather small volume of this system, therefore, is suffi- cient to characterize this cc long-ranged ⁾⁾property.

The infinite continued fraction representation may be applied to the dilute band if we disregard the contribution of the tails in choosing the constant coefficients with which to terminate the expansion.

Figure 2 compares the density of states obtained in this way with the predictions of the CPA, over a range of concentrations in which the residual critical point structure vanishes. The center part of each plot in figure 2 was insensitive to small variations in the effective width chosen in terminating the continued fraction, and should therefore be reliable. The wiggles

.a

-...

^CPA

---EXACT - CTD FRACTION

I

I I I

G

1 x z . 7

O .2 .4 .6 .8 1.0 ENERGY

FIG. 2. - Density of states in the dilute single band. Concentra- tion of allowed sites is indicated at the right of each graph.

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in the outer part of each curve, however, are spurious.

A Legendre polynomial expansion [2] was found to be strongly convergent in this region, and agreed very closely with the CPA predictions. The two calcu- lations taken together furnish a direct test of the effective field theory in this strong-scattering regime, for which no simple exact treatments apply. The good agreement implies that fluctuations about the single CPA effective field do not produce important correc- tions to the density of states over most of the majority band.

Band Tails. - In the tails of the band, however, two different sorts of fluctuations have a significant effect. States deep in the tails are associated with relatively rare fluctuations in which large regions of nearly pure A material are found. Lifshitz [6] first argued that the density of states a t an energy E away from the pure A band edge was given by the probabi-

lity of finding a pure A region (of volume (n2/2 E ) in ~ ~ ~ our model) large enough to have an eigenstate at

that energy. For x close to 1, this is given by

Fluctuations about the mean effective field dominate the near tail. We describe these by adopting Kane's [7]

semiclassical picture in which the band edge undergoes local fluctuations about some mean energy V,, a phenomenological parameter, with variance 0 2 ( ~ ) . The result is

Eggarter and Cohen [lo] related a2 to fluctuations in the number of forbidden sites in a finite sampling region by introducing a local effective field, depen- dent upon the local concentration. Their sampling volume is also (n2/2 E)312, as in (8), but is given by an uncertainty principle argument that is physically justified in the limit x + 1. Extending their results to arbitrary concentration, and inserting a parameter c to absorb errors, one finds

The cutoff (lob) is introduced to keep o2 fixed within the band, where the arguments of (10a) no longer hold. If c = 1, the theory reduces to the original Eggarter-Cohen model, and has the correct Lifshitz limit as E -+ 0.

An analysis of the calculated higher-order moments shows that the tail is not exponential, but indeed

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FIG. 3. - Band tails, as given by fitting (9) to moments.

FIG. 4. - Density of states for a dilute single band in a finite 2 d square lattice of 1 500 atoms.

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C3-250 S. KIRKPATRICK falls off slightly faster than a simple Gaussian, in

agreement with the phenomenological form (9, 10).

The parameter c proved to be only slightly less than 1 for the majority band cases studied. The tails thus found are shown for three cases in figure 3. The square root character of the band edge, still evident at x = .9, is eroded by the much larger tail seen at x = .5.

We note with interest that the CPA band edge, as indicated with arrows in figure 3, is a good approximation to the observed average band narrowing, Vo.

This observation, plus the success of the CPA in describing the bulk of these majority bands, suggests that such effective-field theories, when extended to include local fluctuations, will prove extremely power- ful in treating disordered systems far more complex than our simple model.

Anomalous Structure. - The anomalous sharp peak in the density of states observed by Kohn [9] for a model essentially identical to (1) deserves comment here. We have observed such spikes in unpublished numerical calculations of the density of states of a dilute single band on a planar square lattice, shown in figure 4. In these calculations, various concentrations of A atoms were placed at random on a finite lattice of 1 500 sites. The Sturm-sequence method of Dean and Bacon [lo] was used to obtain the density of states for each sample.

The strongest spike shown, that of figure 4a, for a concentration of. 28

%

allowed sites, contains roughly . l l states per atom, or more than one-third of the states in the dilute band. The spikes in cases (b), (c) and (d) contain .09, .035, and approximately .0l states/

atom, respectively. In each case, the weight is greater than can be accounted for by enumerating small isolated clusters with an eigenstate at zero energy.

A class of localized states which appears to explain the spikes in figure 4 has been identified 1111. These states have the interesting property that they are found not only in large isolated clusters of A atoms but also in A regions not bounded by non-A atoms.

Because their wave functions resemble standing waves, vanishing at nodes on one or more atoms, only the

atoms on which the wave functions are non-vanishing need be bordered non-A atoms to produce localization.

An enumeration of these localized states is in rough agreement with the observed weights of the spikes found in the 2 d cases [l I]. A similar enumeration for the 3 d analogues of these states predicts a spike with a total weight of about .03 stateslatom at the center of a dilute band of 50 percent A concentration, twice the contribution from isolated small clusters but still less than half the size of the spike observed by Kohn [9].

Legendre polynomial expansions of the 3 d density of states (such as those shown in Fig. 5) are consistent

.8

.6 P ( E )

.4 .2

0

.2 .4 -6 .8 1.0

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FIG. 5. - Successive approximations to p(E) (3 d case) obtained

by Legendre polynomial expansion based upon exact moments.

The approximations ps and plz use moments up to f i e and plz,

respectively. Contributions from isolated clusters of one or two A-atoms have been subtracted off.

with this estimated spike weight. A spike at zero energy with weight 0.015 states/atoms has already been subtracted off from figure 5, yet the successive approximations plotted are more slowly convergent at the center of the band than elsewhere, suggesting that a small anomalous contribution remains. We suspect, therefore, that the anomalous contribution to the density of states observed by Kohn [9] is a real bulk effect, but that its strength has been exagge- rated by the approximate statistics necessary in his

<< super cell ⁾⁾method.

References [I] GORDON (R. G.), Review o f Chemical Physics, 1969,15,

79, and earlier work referred to therein. I am indebted to Volker Heine for pointing out this work to me.

[2] DOMB (C.), MARADUDIN (A. A.), MONTROLL (E.) and WEISS (G.), Phys. Rev., 1959, 115, 18, 24.

[3] SOVEN (P.), Phys. Rev., 156,809, 1967.

[4] VELICKY (B.), KIRKPATRICK (S.) and EHRENREICH (S.), Phys. Rev., 1968, 175, 747.

[5] HENRICI (P.), Proc. Symp. Appl. Math., 1963, 15, 159.

[6] LIFSHITZ (I. M.), Usp. Fiz. Nauk, 1964, 83, 617 [English trans. : Sov. Phys. Usp., 1965,7,549].

[7] KANE (E. O.), Phys. Rev., 1963,131,79.

[8] EGGARTER (T. P.) and COHEN (M. H.), Phys. Rev.

Letters, 1970, 25, 807, Phys. Rev. Letters, 1971, 27, 129 and to be published.

[9] KOHN (W.), proceedings of this conference.

[lo] DEAN (P.) and BACON (M. D.), Proc. Roy. SOC., 1964, A 283, 64.

[ll] EGGARTER (T. P.) and KIRKPATRICK (S.), unpublished.