ON NONANALYTICITIES OF THE AVERAGE GREEN'S FUNCTION IN DISORDERED BINARY ALLOY

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ON NONANALYTICITIES OF THE AVERAGE GREEN’S FUNCTION IN DISORDERED BINARY

ALLOY

S. Bose, E-Ni Foo

To cite this version:

S. Bose, E-Ni Foo. ON NONANALYTICITIES OF THE AVERAGE GREEN’S FUNCTION IN DISORDERED BINARY ALLOY. Journal de Physique Colloques, 1974, 35 (C4), pp.C4-95-C4-98.

�10.1051/jphyscol:1974416�. �jpa-00215607�

JOURNAL DE PHYSIQUE

Colloque C4, supplkment au no 5, Tome 35, Mai 1974, page C4-95

ON NONANALYTICITIES OF THE AVERAGE GREEN'S FUNCTION IN DISORDERED BINARY ALLOYS

S. M. BOSE and E-Ni FOO

Department of Physics and Atmospheric Science Drexel University

Philadelphia, Pennsylvania, 19104, USA

RBsumB. -

Nous presentons des resultats numeriques fondes sur la theorie generaliske de la CPA (a) pour les amas. Comme Nickel et Butler si nous prenons des integrales de recouvrement homogenes nous trouvons que la CPA

(cas du triplet

une dimension) ne donne pas de solu- tions analytiques pour la fonction de Green rnoyenne. Cependant si nous choisissons des integrales de recouvrement essentiellement differentes, nous obtenons des solutions analytiques. Par cons&

quent, nous supposons que le modele utilise dans le cas des integrales de recouvrement homogenes donne des solutions non analytiques parce qu'il n'est probablernent pas physique.

Abstract. -

Numerical results based on the generalized cluster theory of the CPA (a) are presented. Like Nickel and Butler, we find that for homogeneous off-diagonal elements CPA

(a three-site cluster in one-dimension) gives rise to nonanalytic solutions for the average Green's function. However, if the hopping integrals are substantially different from one another, analytic solutions are obtained for the average Green's function.

1. Introduction.

In a recent paper Nickel and Butler [I] have presented numerical results based on two different extensions of the single-site coherent potential approximation (CPA) to a three-site cluster in a one dimensional system. Assuming the hopping integrals to be homogeneous, they find that the average Green's function has nonanalyticities in the form of branch cuts in the complex energy plane. They suggest that this nonanalyticity may be the result of arbitrary truncation of the T-matrix equation or some other basic problem inherent in a multiple-site CPA theory.

In order to shed more light on this important problem and to study the effects of inhomogeneous hopping integrals on the density of states, we have carried out numerical calculations by applying the generalized cluster theory

of the CPA (a) both in the presence and absence of the random-off-diagonal elements. In the generalized CPA

theory, recently developed by Foo, Bose and Ausloos, the effects due to all possible scatterings from clusters consisting of all atoms resi- ding within the ath shell of the central atom are taken into account in a self-consistent manner. In principle, the errors in this CPA

theory can be made arbitrarily small by increasing the size of the cluster, i. e. by making

arbitrarily large. However, to carry out numerical computation one is forced to introduce approximations and truncate the T-matrix by retaining only scatterings among atoms of a finite size cluster, i. e. by making

finite. In this paper we present numerical calculations based on CPA (1) (a

three-site cluster in one dimension). Although CPA (1) is not correct even to order

in the dilute limit in the presence of random-off-diagonal elements, we choose to work with it because this approximation is sufficient to settle certain important aspects of the problem in a relatively simple manner.

2. Theory.

Let us review in brief the formalism that we have used in this paper. The one electron Hamiltonian for a disordered binary alloy A,-,B, can be represented in the tight binding approximation as

where

represents the atomic energy level a t site n and h,, represents the hopping integrals between the nearest neighbor sites n and m. In the present theory both

and h,,,'s can be randomly distributed, i. e.

can be either

or

depending on whether the site

is occupied by an atom of type A or B and h,,,,,'~ can take values hAA, hBB or hAB depending on whether the pair of sites n and m are occupied by two atoms of type A, two atoms of type B or one atom of type A and one atom of type B, respectively.

The Green's function corresponding to the above Hamiltonian is

I

where E is the energy of the system.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974416

C4-96 S. M. BOSE AND E-Ni FOO

Various static properties of the system are given

by the configurational average of the Green's function.

For instance, the electronic density states is given by

where the angular brackets denote configurational average.

In the spirit of CPA we now introduce a configu- ration-independent one particle Hamiltonian in the most general form as

where Co and Cn-,, are the diagonal and off-diagonal coherent potentials to be determined by using appro- priate self-consistency conditions. The summations over n and

go over all lattice sites of the crystal and hence Cn-, represents the effective hopping integrals between two sites nand m of arbitrary separation. The Green's function corresponding to the effective Hamil- tonian

is related to

by the relationship

-

-

G

G + GTG (6)

where the T-matrix is obtained from

where the scattering potential is given by

The matrix components of X,,, are determined by requiring that

< T > = O (9)

which implies that

Since < T > is translationally invariant, the crite- rion given by eq. (9) implies that

< Tn, >

< T,,-,>

< T,>

for alla (1 1) where a describes the lattice distance between sites 1 and n. This criterion of course leads to an infinite set of coupled nonlinear equations which make the problem intractable. However, since the requirement

< T, >

0 becomes less important for large a's, one can truncate the hierarchy of equations at a finite

value of a without seriously affecting the accuracy of the calculation.

In this paper we have carried out the numerical calculation by choosing a

1 which means that we have included all the scattering processes involving a three atom cluster in a one-dimensional system. The scattering potential due to such a three atom cluster can be written as

where

represents the center atom and

denotes an atom in the first shell. Also

V , r n = ~ n - C O and

,,,, - C l (13) where Co and C, represent the diagonal and first off- diagonal matrix elements of X,,,. The T-matrix equation [7] for such a three atom cluster becomes

One can thus solve for TI,, T-,, and Too from eq. (14). However since the configurational averaged

< Tlo >

< T-,, >, eq. (11) reduces to two coupled non-linear equations

< Too >

0 and < T I , >

0 (15) which can be solved for Zo and Zl. We can then cal- culate coo from the equation

-

1

=

E - I,

2 C, cos ka

=

[(E - Z0)2

4 c:]- ₍₁₆₎

from which we can determine the density of states

where 6 is an infinitesimally small positive number.

3.

Numerical

-We have solved the coupled eq. (15) for Eo and El by using a combination of the Newton-Raphson iterative scheme and the iterative average T-matrix approximation [3]. The iteration process has been carried out with the initial values

Note that for complex energy values eq. (18) gives

true solutions for Zo and E, in the limit of E + ^id

ON NONANALYTICITIES O F THE AVERAGE GREEN'S FUNCTION C4-97

(with either E or A or both) approaching infinity.

Thus to obtain a solution at E + ^i6, we start the iteration procedure with eq. (18) as the initial values at a complex E + ^id, (A, being a large number) where it is a reasonably acceptable solution. After several iterations, a solution Z(~'(E + id,) is obtained satisfying the criterion < T > <

for certain small values of

Then we reduce A, to A , and apply the iteration procedure until the convergence criterion is satisfied again. Carrying out this step by step reduction of A along with the employment of the above conver- gence criterion many times we obtain C, and C, for A

6 where 6 is arbitrarily small. If

is reasonably large and the rate of reduction of

(i. e. A,-,

A,) is reasonably small, we have found that a unique solution can be found for all alloy parameters and for all energy values.

The numerical results for the density of states p(E) for x

0.1,

-

and

are shown in figure 1. The solid curve is obtained by using the present three-site cluster theory and the histogram is obtained by using Dean's method to a

linear chain of 5,000 atoms. The agreement is excellent throughout entire host energy band region (E= -2 to 4). In the impurity band region ( E =

to -2) histogram shows several peaks corresponding to the various resonant modes associated with different impurity clusters. The three-site CPA theory repro- duces the two dominant peaks and another weaker peak reasonably well. Although we obtained unique solution for the entire energy range by carrying out the calculation by the above iteration scheme, we would encounter a nonanalyticity at E x

2.787 i. e.

we would get two different solutions indicated by the

FIG. 2. - Electronic density of states for x = 0.1, E A = - & B = 1 and ~ A= ~A B = B ~ A B = 3.5. At E N - 6.49 there is nonanaly-

ticity.

dashed lines if we calculated p(E) for E > - 2.787 by starting the iteration procedure with solutions for E <

2.787 as the initial values and vice versa. Note that apart from giving unique solutions on either side

FIG. I. -Electronic density of states for x = 0.1, & A = - E B = 1 and hA* = ~ B B= ~ A B = 1.5. Nonanalyticity exists at

E N

-

FIG. 3. -Electronic density of states for x = 0.1, &A = - EB = 1 and ~ A A = 1.5, ~ B B = 3.5 and ~ A = B 2.5. Solution is analytic in

the entire energy range.

C4-98 S. M. BOSE A N D E-Ni FOO

of the singular point, our iteration scheme enables us to pinpoint the nonanalytic point quite accurately.

Results in the impurity band region for the case x

1 and

are plotted in figure

We find that the density of state has again a nonanalyticity at E r - 6.49.

In figure

we plot the density of states for

E A = - E~ = 1

and

and

~ A A

+

2.5 .

h,, = - -

2

In this case observe the hopping integrals are signifi-

cantly different. We did not detect any nonanalyticity in the entire energy range in this case. These results are significant because when the hopping integrals are all equal to either

or

we encounter nonana- lyticity but the nonanalyticity disappears when they are a combination of

and

Thus we conclude that like Nickel and Butler, in the absence of random off-diagonal elements, we have found nonanalyticities in the density of states at certain energy values and alloy parameters. However, the density of states may become analytic if the hopping integrals are significantly different from one another.

References

[I] NICKEL, B. G . and BUTLER, W. H., Phys. Rev. Lett. 30 (1973) 373.

[2] Foo, E-Ni, BOSE, S. M. and Aus~oos, M., Phys. Rev. B 7 (1973) 3454.

[3] CHEN, A. R., Phys. Rev. B 7 (1973) 2230.