LOCAL ENVIRONMENT EFFECTS IN DISORDERED ALLOYS

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LOCAL ENVIRONMENT EFFECTS IN DISORDERED ALLOYS

F. Brouers, F. Ducastelle, F. Gautier, J. van der Rest

To cite this version:

F. Brouers, F. Ducastelle, F. Gautier, J. van der Rest. LOCAL ENVIRONMENT EFFECTS IN DISORDERED ALLOYS. Journal de Physique Colloques, 1974, 35 (C4), pp.C4-89-C4-94.

�10.1051/jphyscol:1974415�. �jpa-00215606�

JOURNAL DE PHYSIQUE

Colloque C4, suppldment au no 5, Tome 35, Mai 1974, page C4-89

LOCAL ENVIRONMENT EFFECTS IN DISORDERED ALLOYS

F. BROUERS

Laboratoire de Physique des Solides, BLtiment 510, 91405 Orsay, France F. DUCASTELLE

0.

N. E. R. A., 92320 Chatillon, France F. GAUTIER

Laboratoire de Structure Electronique des Solides, Universitt Louis-Pasteur, 67000 Strasbourg, France

J. VAN DER REST Laboratoire de Physique des Solides, Universitt de Li6ge, 4000 Sart-Tilman, Belgique

R6sum6. - On dtcrit une approximation permettant de calculer simplement la diffusion produite par un amas compact immerge dans un milieu donne. On en deduit une methode de calcul de la densite d'ttats dans les alliages desordonnts qui generalise I'approximation du potentiel coherent (CPA). On analyse cette nouvelle approximation et on la discute dans le contexte des recentes tentatives de generalisation de la CPA.

Abstract.

- A

simple method to describe the scattering produced by compact clusters embed- ded in a given medium is derived. It is applied to the calculation of the density of states in disordered alloys, which gives an extension of the coherent potential approximation (CPA). This new approxi- mation is studied and discussed in the framework of the recent attempts to extend the CPA.

1. Introduction.

-

It is now widely recognized that in a disordered system, most of the physical properties may be explained from the knowledge of its local properties within some finite range. For example, numerous magnetic properties of alloys are well explained by assuming that the magnetic state of an atom depends mainly on the chemical nature of its neighbours [I]. In this context, it is then important to be able to describe correctly the scatter- ing properties of clusters embedded in a given medium.

When dealing with the scattering by single centres, the answer is simply given through a phase shift or t-matrix analysis. Formally, there is no major diffi- culty in extending these formalisms to clusters of a higher order, but in practice the structure of for example the cluster t-matrix soon becomes very intricate.

It is the purpose of this paper to give a simple

by the shell of its Z first neighbours and embedded in an uniform medium. These equations are appro- ximated in such a way that the local density of states on the central site depends only on the occupation of this site and of the average occupation of the shell.

This approximation is then used to build up a self- consistent alloy theory by writing that the averaged local density of states on the central atom must be identical to the medium density of states. In section 3, we compare our approximation with that of Brouers, Cyrot and Cyrot-Lackmann [2] which is based on similar ideas. Although the connections are not obvious, some arguments are given in favour of our theory, which is confirmed by some numerical results.

Finally in section 4 we give a more general discussion of this kind of approximation in the framework of the recent attempts to extend the coherent potential approximation (CPA).

method to describe the scattering produced by corn-

2 Scattering for a ,,luster

[3]. -

Let pact clusters within a tight-binding approximation.

us start from the usual tight-binding hamiltonian : Although extensions to other disordered svstems are -

possible, we shall work here on the usual disordered

32 =

1

i

>

^{E i}

< ⁱ I + I i > fl <

j

I . (2.1) alloy model. In section 2, we derive the scattering

^{i i # j}

equations for a cluster formed by an atom surrounded We neglect the off-diagonal randomness and keep

7

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974415

C4-90 F. BROUERS, F. DUCASTELLE, F. GAUTIER AND J. VAN DER REST

only first-neighbours hopping integrals P

;

ci is the random atomic level, equal to

E,

or

^8,

depending on the chemical nature of the atom at site

^i.

Let us now consider a cluster formed by a central atom 0 surrounded by its shell S of

Z

first neighbours and embedded in a homogeneous medium characte- rized by an effective local potential C. The correspond- ing Green function at the centre of the cluster G :

: (= < 0 1

^GOS

I 0 >)may conveniently be written as

:

where A ~ O is the Green function of the medium only perturbed by the potentials ci

- C

of the shell. Using then the t-matrix

T S

of the shell, we get

:

where Goo

= --

G(0) is the diagonal Green function of the medium, and the prime means that the summation is restricted to sites belonging to the shell S. In the single band model used here, Goi and Gjo are equal, and depend only on the distance R between first neighbours. Thus (2.3) can also be written as

:

where r is an averaged propagator on the shell

:

Thus T: depends only on the occupation of site i, which is consistent with eq. (2.8) where this assump- tion was implicitly made. The solution for <

T S

>

is now given by (2.8) and (2.9)

:

The final form of of GE is obtained from (2.4) and (2.2). An important point to notice is that

G::

depends only on the occupation of the central atom and of the averaged occupation of the shell, and of the propaga- tors G(O), G(R) and r which can be viewed as pro- pagators establishing the 0 - 0 , 0-shell, and shell- shell couplings respectively. Moreover in a single band model G(R) and r are easily related to G(0) (see Appendix)

Now, the multiple scattering equations for To get a self-consistent alloy theory, we can now

TS

⁼

C' ^T-S. impose that the averaged local Green function

G::

1J

i

is identical to the Green function G(0) of the medium

:

are given by

: -

G(0)

=

1

-

P(0, S) G: : (2 .14)

0 , s

(2.5)

where P(0, S) is the probability of occurrence of the cluster ( 0 , S). This condition was also used by Butler where ti is the usual single site t-matrix

:

[4],

Brouers et al. [2] and by Tsukada [5]. These ti

=

(ei

-

C) [I

-

(ci

- C)

C(O)]-I (2.6) theories will be compared in next sections.

The complete solution of (2.5) would require the solution of Z coupled equations involving all matrix elements of G on the shell. At this stage, we approxi- mate T: in the r. h. s. of (2.5) by its averaged value over the ( Z

-

1) sites different from i

where

<

T S

>

= -

z 1

( n ,

T,S +

^n,

T;) ^(2.8)

3.

Comparison with the approximation

by

Brouers et

al.

-

As pointed out in section 2, the main advan- tage of our approximation is that it enables rather easy three-dimensional calculations. A similar theory is that proposed recently by Brouers

et

al. [2]. As the two theories may seem rather different, it is of some interest to look for their possible interrelations.

The theory by Brouers et al. (hereafter called BPA as in the original paper) starts from the following form for G::

:

G,O,S

= ( z

- e0

-

A ; ) - ' . (3.1)

n ,

and n, being the numbers of A and B atoms on the Then, the self-energy

A:

is expanded using the Watson- shell, and

T & ~ )

is the value of

T:

when the site

i

is Feenberg method [6] (see also the appendix), and the occupied by an A(B) atom. In this way, we have expansion is truncated at some level, the correspond- performed an average over all the configurations ing approximation being exact on a Caylee tree. The of the shell having the same

n A

and n, values. Inserting key equations are

:

(2.7) into (2.5) yields

:

7; =

ti[l + T ( Z <

^T'

>

^-

T')] (2.9)

LOCAL ENVIRONMENT EFFECTS IN DISORDERED ALLOYS C4-91

and

The self-consistent condition determining

C

is still given by eq. (2.14), G: : being now expressed from (3. I), (3.2) and (3.3).

The BPA provides an exact description of the scattering produced by the cluster in the case of a Caylee tree

;

on the other hand the same is true for our approximation for in eq. (2.5) all the inter-shell matrix elements G, (k

= i )

are then identical and therefore equal to

T ;

as a consequence eq. (2.9) is exact. We conclude that both theories givi an exact treatment of the scattering on a Caylee tree

( I ) .

The linear chain is a Caylee tree with Z

=

2

;

therefore in this case both theories reduce to the theories of Butler [4] and Tsukada [5].

Let us now try to compare more precisely our theory with the BPA. To this end, we calculate A; as given by our approximation. We find

:

The two first terms on the r. h. s. of (3.4) are the CPA terms, the third one being the correction due to the'

-

shell. The appearance of a factor G(R)2

-

f in this G(O)

last term is significant. It can be shown imdeed that this last term vanishes on a Caylee tre2: (sZe Appen- dix). Another simple relation (also deived in the appendix) between Green functions, vhlicl on a Caylee tree, is

:

where almost all quantities have been expressed in terms of their values on a Caylee tree (whence the explicit presence in the formula of the hopping inte- gral p). Our present approximation lies at the other extreme

;

indeed, whereas the BPA is obtained from topological approximations on the lattice, our approxi- mation is essentially a statistical one (see eq. (2.7)).

Furthermore it is also exact in another case, namely when all the atoms of the shell are identical (n,(,,

=

2).

Thus, in some way the BPA can be viewed as includ- ed in our approximation. The best check of this is provided by the numerical computations. It is clear from figures I and 2 that the fine structures associated

X z 0.1 -,,CPA

6 = w ...

^BPA

-

^{our result}

ENERGY

FIG. 1. - Impurity band in a simple cubic lattice ; 6/W = 1 ; x = 0.1. Comparison of our results with those obtained within

the CPA and the BPA.

W is the half-band width of the pure metal and x the concentra- tion of B atoms. Notations nB refer to contributions due to particular configurations of the shell surrounding a B atom.

Combining these relation$ wrth those given by

^0.4-

x

= 0.05

eq. (2.12) and (2.13), it beemes possible to get the

Ln

6=

1.25 w

BPA from our approximation. This means that the g ----

^CPA

two theories are two d~fferent approximations for 5

^{03- B PA}

which some relations

odq/

valid on a Caylee tree have -

^{our result}

been used at different levels. It is quite obvious that

^O

a lot of other a p p r o ~ h a t i o n s can be built up in a similar way

^{( 2 ) .}

Nevertheless, one may point out that

I \

properties of the Czylee tree can be used more or less explicitly. The BPA represents an extreme situation

01 - (1) As other ap$toltimations have been made before, this

does not mean of Course that these theories are exact on a

Caylee tree. 0 I I I

02 0.4 0.6 0.8 10 1.2

(2) The recent dpproximation put forward by Jacobs 171 can

be included in this category. It is in fact closely related to the FIG. 2. - Impurlty band in a simple cubic lattice ; 6/W = 1.25 ;

BPA. x = 0.05.

C4-92 F. BROUERS, F. DUCASTELLE, F. GAUTIER AND J. VAN DER REST

with the impurity bands are more well-marked in our approximation. The agreement is also rather good between our results and the exact numerical results of Payton and Visscher obtained for the equi- valent phonon problem 181 (Fig. 3).

FIG. 3. - Simple cubic phonon frequency spectrum obtained from the present method compared with Payton and Visscher

histogramme spectrum.

As far as the numerical computations are concerned, another point deserves a comment. It might seem that our theory depends independently on G(0) and Z, the other Green functions being obtained from (2.12) and (2.13). In fact, from the numerical computations, it appeared important to use bare Green functions G(0) giving first moments consistent with the Z value.

In practice, when using model densities of states, they were fitted in such a way that the second moment

p, = ZP2

is preserved.

4.

Discussion.

- Let us now briefly discuss the connections between our approximation and other recent theories. Numerous attempts have been done these last years to go beyond the CPA. The first attempts have been to include pair, triplet, ... effects in a self-consistent way, by introducing non-local matrix elements of the effective potential

[9].

While formally appealing, these theories lead to rather heavy numerical computations. Moreover a severe objection has been raised by Nickel and Butler [lo] about the analytical properties of these approximations. In the strong disorder case (6

= E,

-

^{E A} %

b), branch points appear for the self-energy in the complex plane, and the density of states can no more be unambiguously determined. The same behaviour was observed for

some other cluster approximations similar to ours.

On the other hand the molecular extension of the CPA of Tsukada

1111

(see also Leath [12]) seems to be analytic [lo]. This last approximation is a more consis- tent extension of our approximation

;

the scattering is exactly treated, including off-diagonal self-energies, and the self-consistency is required for all matrix elements of the Green function or of the t-matrix.

Since then Miiller-Hartmann has shown that, as expected, the CPA is analytic [13]. This result can in fact be extended to the molecular generalization of Tsukada [14]. Another important property of these approximations is that they provide positive condi- tional densities of states [14].

Numerical computations by Nickel and Butler [lo]

and ourselves indicate that the approximate self- consistent equation used here leads to non-analyticities for strong disorder. But for moderate values of 6 a single solution is obtained and it can be shown that the positivity of the local densities of states (proportional to

-

Zm G: :) is preserved. The limi- tation to these values of 6 is not very important in view of the applications to usual alloys [I], and it is clear from the figures that in this case our approxima- tion provides a substantial improvement over the CPA.

As a conclusion, let us say that our approximation, though limited to moderate disorder has the obvious advantage to be easily tractable for realistic three- dimensional systems, which is not the case for a lot of recent alloy theories. More generally, it can be of some use in other problems where the scattering properties of clusters are needed.

5. Appendix.

^-

The renormalized perturbation expansion of the self-energy A: occurring in eq. (3.1) is now well-known and has been used by many authors, mainly dealing with the problem of the localization of states in disordered systems [15]. It can nevertheless be useful to give here a compact way of deriving this series.

Let us write the Green function G as

:

where Ho is the off-diagonal part of the hamiltonian X given by (2. I), and g the so-called locator

:

(A. 1) is equivalent to

:

with

LOCAL ENVIRONMENT EFFECTS IN DISORDERED ALLOYS C4-93

When all g i are the same (in a pure metal g i

=

go for all

i),

U may be considered as the generating func- tion of the paths on the lattice, the matrix elements of H; being directly related to the n-steps paths

:

Fixing gi

= 0

(or

E~ =

co) is equivalent to excluding site

^i,

and the generating operator Ui of the paths avoiding i is

:

ui

⁼

U (where gi

=

0) (A.

6)

which may be written

:

In a similar way, we can define generating operators for the paths avoiding several sites

i, j,

k, ...

:

Let us now apply this formalism to the diagonal Green function Gii

:

G

^{I I} =

g i + g , u i i g i . (A. 10) From (A. 7), we get

:

Gii

=

(2

- &i -

uii)-' (A. 11)

u:~ is then just the self-energy A i used in the main text.

The perturbation expansion for A i follows from (A. 8)

:

(A. 12) The renormalization of this expansion is achieved by expressing Ui as a function of uik on the r. h. s. of (A. 12)

:

and

A i =

Pikgk

k i i

1

- gk

U&

(A. 13) where

A'

-

uik

k - o I '

(A. 15)

Further renormalizations can be performed by using successive generating operators uik', etc ...

APPLICATION

^TO THE

CAYLEE TREE.

-

U: is the generating function of paths linking i and k and avoiding these sites in any intermediate step. Then, from the properties of the Caylee tree':

u;:

= /3

. ^{(A. 16)}

This is the relation used to derive the BPA mentioned in the main text. On the other hand, Uik can be related to U and then to G, by using eq. (A. 9) and (A. 3).

In a single band model, for any periodic lattice, we have

:

~ '

-

k

k i -

G(R) (A. 17)

~ ( 0 ) '

-

G(R)'

^'

As usual G(0) is the diagonal Green function, and G(R) its off-diagonal matrix element between first neighbours. Therefore, on a Caylee tree, the following relation is true

:

P

=

G(R) (A. 18)

~ ( 0 ) ~

-

G(R)'

^'

Coupled with the equation (2.12), (where we fix

Z =

0) this relation gives us the value of G(0) on a Caylee tree

:

Let now 0,

i, j

be a site and two other ones among its first neighbours

;

on a Caylee tree

:

UP.

=

0 .

I J

(A. 20)

Through straightforward calculations, this provides us with the relation

:

(A. 21) PROOF

OF EQUATIONS

(2.12)

AND

(2.13).

-

Eq. (2.12) of the main text is directly obtained from eq. (A. 1)

:

G(O)

= g

+ gP C Gio (A. 22)

i

The summation is restricted to first neighbours of 0

;

therefore

:

which is identical to eq. (2.12).

Eq. (2.13) would be obtained in a similar way by

applying (A. 1) to G(R).

C4-94 F, BROUERS, F. DUCASTELLE, F. GAUTIER AND J. VAN DER REST

References

[I] See for example in this conference : NICKEL, B. G. and KRUMHANSL, J. A., Phys. Rev. B 4 (1971) GAUTIER, F., BROUERS, F. and VAN DER REST, J., J. Phy- 4354.

sique 35 (1974) C4-207. DUCASTELLE, F., J. Phys. F : Metal Phys. 2 (1972) 468.

[2] BROUERS, F., CYROT, M. and CYROT-LACKMANN, F., Phys. SCHWARTZ, L. and EHRENREICH, H., Phys. Rev. B 6 (1972)

Rev. B 7 (1973) 4370. ^2923.

[3] BROUERS, F., DUCASTELCE, F., GAUTIER, F. and VAN DER B. G. and W. H., Phys. Letf. 30 REST, J., J. Phys. F : Metal Phys. 3 (1973) 2120. (1973) 373.

[4] BUTLER, W. H., Phys. Lett. ( A ) 39 (1972) 203. [ l l ] TSUKADA, M., J. Phys. SOC. Japan. 26 (1969) 684.

[5] TSUKADA, M., J. Phys. Soc. Japan 32 (1972) 1475. [12] LEATH, P. L., J. Phys. C 6 (1973) 1559.

[6] WATSON, K. M., Phys. Rev., 105 (1957) 1388. [13] MULLER-HARTMANN, E., Solid State Commun. 12 (1973) FEENBERG, E., Phys. Rev. 74 (1948) 206. 1269.

[14] DUCASTELLE, F., to be published in J. Phys. C : Solid [7] JACOBS, R. L., J. Phys. F : Metal Phys., 3 (1973) 933. state Phys. (1974).

[8] PAYTON, D. N. and VISSCHER, W. M., Phys. Rev. 154 [151 See for example:

(1967) 802. ANDERSON, P. W., Phys. Rev. 109 (1958) 1492.

[9] CYROT-LACKMANN, F. and DUCASTELLE, F., Phys. Rev. ECONOMOU, E. N. and COHEN, M. H., Phys. Rev. B 5

Lett. 27 (1971) 429. (1972) 2931.