NUMERICAL ASPECTS OF THE TWO-SITES COHERENT POTENTIAL APPROXIMATION

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JOURNAL DE PHYSIQUE Colloque C4, supplkment au no 5, tome 35, Mai 1974, page C4-153

NUMERICAL ASPECTS OF THE T WO-SITES COHERENT POTENTIAL APPROXIMATION

K. MOORJANI (*)

Applied Physics Laboratory, The Johns Hopkins University 8621 Georgia Avenue, Silver Spring, Maryland 20910, U. S. A.

T. TANAKA

Department of Physics, The Catholic University of America Washington, D. C. 20017, U. S. A.

M. M. SOKOLOSKI

Harry Diamond Laboratories, Washington, D. C. 20438, U. S. A.

and S. M. BOSE

Department of Physics, Drexel University Philadelphia, Pennsylvania 19104, U. S. A.

RbsumB. - Nous discutons brihvement une theorie auto-coherente d'amas dans les alliages desordonnes. Sont compris dans la theorie le desordre non diagonal et les diffusions de paires. La communication presente des resultats numeriques et une mkthode pour la rksolution des difficultks de convergence dans le cas de fortes diffusions.

Abstract. - A two-site coherent potential approximation including both off-diagonal randomness and pair scattering is briefly outlined. Numerical results on disordered binary alloys are obtained and a procedure for resolving convergence difficulties in the split-band case is presented.

The single-site coherent potential approximation (SS-CPA) has provided a simple and elegant approach to the study of disordered substitutional alloys [I, 21.

A number of extensions of SS-CPA to include the effects of scattering from a pair of atoms have also been discussed [3-91. The need to include off-diagonal randomness within the framework of CPA has been pointed out [lo] and a number of attempts in this direction have recently appeared in the literature

[5-7, 11-15].

We have previously formulated a two-site coherent potential approximation which incorporates both off-diagonal randomness (ODR) and pair scattering [5-71. The inclusion of ODR in the formulation makes it suitable for application to the problem of magnetic disorder [16, 171 also, and treating diagonal randomness in the pair approximation is essential whenever

(*) Supported in part by Naval Ordnance Systems Command, Contract N00017-72-C-4401 Task A13B.

the fluctuations in the single-site energies are large or in the discussion of the high impurity concentration limit when the impurity clustering effects are impor- tant. The theory treats z-nearest neighbours of a given site on equal footing, is invariant under the interchange of A and B atoms and in the absence of ODR and pair-scattering reduces to the usual SS- CPA.

Numerical calculations based on the present method indicated that for the split band limit, convergence difficulties are encountered in the minority band [6].

A thorough description of similar difficulties has been recently given by Nickel and Butler [18] in their study of three-site case (in the absence of ODR) for a one- dimensional solid. They showed that the origin of these difficulties lies in the non-analytic behavior of the averaged Green's function and further conjec- tured that this feature will generally exist in higher order approximations in the coherent potential formulation. In the present paper, a particular example

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Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974427

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C4- 1 54 K. MOORJANI, T. TANAKA, M. M. SOKOLOSKI AND S. M. BOSE of the strong scattering case is discussed and a possible

resolution of the numerical problems by an iterative method is indicated.

The formulation of the model considered here has been discussed before [6, 71 and we will only briefly outline the basic assumptions underlying our approach.

We consider the model Hamiltonian of a tight binding alloy

where the site energies cl as well as the overlap inte- grals W,, are allowed to be random. el takes on the values E A or EB with corresponding probabilities x and (1 - x), the respective concentrations of the A and B components of the alloy. Similarly Wlm is equal to WAA, WAB (= WBA) or WBB with the probabilities x2 x(l - x) and (1 - x)' respectively.

In the spirit of the coherent potential approach, we introduce a configurationally independent Hamil- tonian X , and assume that it has the form

The summation over m in both Hamiltonians is restricted to the nearest neighbours of I. These assumptions are essential if one wants t o carry out numerical work and restrict the computations involved to a manageable size.

The problem can now be treated as a pair of nearest- neighbour atoms (I, m) embedded in an effective medium defined by the effective potential C, and the effective bandwidth El. Both C, and C , need to be determined self-consistently. The procedure is straigh- forward though quite laborious and the details have been published elsewhere [7]. The scattering potential

r

⁼^X^-^X,is easily decomposed in the diagonal and the off-diagonal parts

r,

and T, respectively to obtain the corresponding scattering T-matrices. The total T-matrix T is the sum of independent as well as correlated scattering from I', and T2. Self-consis- tency condition requires that on the average there be no further scattering from the pair of nearest-neighbour atoms (I, m). The diagonal and the off-diagonal matrix elements of the configurationally averaged T-matrix,

<

^T

> ,,

^and

<

T

> ,,

respectively, equated to zero then provide the two coupled non-linear equations which determine 1, and C , in terms of concentration x, the normalized band splitting parameter 6 = (E* - EB)/2 z WAA, ( E ~ = - E ~ ) and the normalized bandwidths

P

⁼WBB/WAA and y = WAB/ WAA of the pure B and the A-B lattice res- pectively.

The non-linear coupled equations for

60 (= COIZWAA) and 0 1 (= CIIZWAA) are solved by an iterative procedure in which the band edges are approached from the large energy side

with virtual crystal values for the two quantities as the initial guesses. The density of states p(E) as a function of energy is then obtained from the computed values of a, and o , . We have carried out extensive numerical calculations on various alloys with the body-centered cubic lattice structure [7]. The density of states for the ordered bcc lattice, shown in figure 1, exhibits

ENERGY

FIG. 1.

-

The density of states for an ordered body-centered cubic lattice.

the well-known van Hove singularities at the band edges and the center of the band. The effects of random alloying on p(E) for the alloy A,.,B,., with 6 = 0.2 (corresponding to overlapping minority and majority component bands) and a set of values for the bandwidths (P, y) are illustrated in figure 2. As expected,

FIG. 2. - The density of states for a random alloy A o . ~ B o . ~ .

the random alloying removes the van Hove singularities and extra structure due to the presence of minority component appears in the vicinity of

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NUMERICAL ASPECTS OF THE TWO-SITES COHERENT POTENTIAL APPROXIMATION C4-155

However, no fine structure due to pair scattering is found in the absence (not shown) or presence (Fig. 2) of ODR. Such a structure has been reported for calculations on one-dimensional system even in the case of overlapping minority and majority bands [19].

The peak due to the minority band (Fig. 2) does not lie exactly at - 2 6 but slowly shifts toward higher energy with increasing values of the set

(P,

y), while the main peak shifts towards lower energies. The presence of the ODR also modifies the total bandwidth which is larger than the virtual crystal value.

Increasing the value of 6 shifts the minority component band towards lower energies till 6 > 1 when the minority band splits off from the majority band. In this strong scattering limit, we first consider the case when

fi

= y = 1 (no ODR). This facilitates the comparison between the results on pair scattering with those of the SS-CPA. As alluded to above convergence difficulties are encountered in this split- band case. Specifically, in the vicinity of a non- analytic point two different solutions are obtained depending on whether the point is approached from the left or from the right. However, convergences throughout the minority band and unique values for p(E) can be obtained by adding a small imaginary part A to the energy. Starting with the computed values o,(A) and o,(A) as the new initial guesses, A is now slowly reduced to iteratively obtain a, and o, for A --+ 0. The density of states calculated by this procedure (labeled TS-CPA) is shown in figure 3, and

-

^TS-CPA

---

^SS-CPA

E N E R G Y

FIG. 3. - The density of states for a random Ao.9Bo.r alloy in

split-band limit. The vertical scale on the right is for the main band and that on the left is for the minority band.

compared with the values based on the SS-CPA.

Major differences between the two approximations are encountered in the minority component band.

The sidebands in the TS-CPA result from the bonding and anti-bonding states of the twosite cluster imbedded

in an effective medium [9]. Note that the bandwidth of the minority band is somewhat larger in the TS- CPA compared to the SS-CPA. Also the density of states is conserved as one would expect. This however is not found to be the case when spurious solutions are obtained due to convergence problems.

The structure in the density of states of the minority band must be related to the behavior of the two self- energies o, and a,. The pair scattering leads to the presence of structure in Re o, and Im a. [7] and, moreover, the Re o, and the Im o, (Fig. 4) exhibit

- 1 2 0 ~ _ . .-.L I... L. . . L- L . - - i - / - 0 1 4

- 2 5 2 4 - 2 3 2 2 2 1 - 2 0 1 9 1 8

E N E H G Y

FIG. 4. - The real and imaginary part of effective bandwidth parameter ol over the energy range of the minority band.

a behavior compatible with the dispersion relations and with the interpretation of associating the sidebands as originating from the bonding and anti-bonding states of a molecule. Since the bonding state has a symmetric wave function, o, will have a large imaginary part near the center of the Brillouin zone and a small imaginary part near the edge. The converse statement applies to the anti-bonding state. Hence one would expect Im a, to change sign as seen in figure 4. Note that the SS-CPA would imply Re o1 ⁼0.125 and Im a, = 0 throughout the entire energy range.

Bose and Foo have analyzed a one-dimensional disordered alloy in the three-site CPA [20]. They find that the convergence difficulties for the split band case disappear in the presence of ODR. Our preli- minary calculations on pair scattering plus ODR in the three-dimensional case tend to confirm this result.

We also find further structure in the minority band when ODR is introduced. The exact reason for this behavior is not entirely clear and the details are being investigated.

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K. MOORJANI, T. TANAKA, M. M. SOKOLOSKI AND S. M. BOSE

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