The Coherent Potential Approximation Method

The aim of the Coherent Potential approximation is to calculate configurationally averaged properties of a random material in a self consistent way. Essentially one can describe the ran- dom metallic alloy by a lattice of effective potentials in such a way that the average motion of an electron through the effective medium is approximatively the same like through the actual material. This means that if one wishes to describe the system using an(periodic) co- herent potentialVCP A_{, the Green’s function corresponding to this coherent potential should}

be equivalent to the true ensemble-averaged Green’s function of the alloy.

The Coherent Potential approximation (CPA) was introduced simultaneously by Soven [83] in connexion with disordered electronic systems, and by Taylor [84] in connection with the lattice dynamics of mass disordered alloys. The CPA belongs to the class of mean-field the- ories, in which the properties of the entire material are determined from the behaviour at a localized region, usually taken to be a single site (cell) in the material. In order to create the configuration of the random substitutional binary alloy of compositionAxB1−x it is as-

sumed that the site occupancies are uncorrelated and that the probability that a particular lattice site iis occupied by element A isxA = x and likewise for B isxB = 1−x. For this

disordered system, one may assume that there are only two distinct types of potential, VA

andVB_{, corresponding to the two elements A and B of the material.}

This coherent potential VCP A _{(which in general is a complex energy-dependent quantity)}

is constructed by replacing at any single site in the effective medium the individual con- stituent potentials of the alloy, given asVA_orVB_{, in such a way that no further scattering is}

produced on average.

This medium can be chosen in some physically and intuitively reasonable manner in such a way that averages over the occupation of a site embedded in the effective medium should yield quantities indistinguishable from those associated with a site of the medium itself. Be- cause a translational invariant medium produces no scattaring of a wave, it is assumed that the scattering off of a real atom embedded in the CPA medium must vanish on the average. This condition, schematic presented in Fig. 2.1 has the following mathematical expression:

c c c x c c c c c c c c c c c c c c c c c c c c c c = A + x_B B A

Figure 2.1: The schematic representation of the CPA condition. Label ’c’ stands for ’effec- tive atoms’ of the coherent medium and the sites labelled ’A’ and ’B’ are occupied by the constituent atoms A or B with relative probabilityx=xAand respectively1−x=xB.

τC =xAτA+xBτB , (2.178)

where τc _{is the scattering path operator corresponding to this hypothetical ordered CPA}

medium and τA _{or τ}B _{describes the total scattering due to a single atom of type A or B,}

respectively, which is embedded in the effective coherent-potential medium. Equivalently, the site-diagonal part of the Green’s function of a real atom embedded in the CPA medium, averaged over the possible occupations of a single site, should be equal with the correspond- ing Green’s function of the medium itself.

The averages one performs in the CPA involve only the occupation of a single site and con- sequently the CPA is a single-site (SS) approximation, this averaging procedure neglects scattering off of clusters of atoms, which may be important in some cases.

Our first aim is to obtain approximations for the average < G > over the previously de-

G(E, ~r, ~r0_{)for an electron moving in the field of a collection of muffin-tin scatterers can be}

written in the form (see also Eq. (2.155)) :

G(~r, ~r0_{, E) =} X ΛΛ0 Zi Λ(~ri, E)τΛΛii 0(E)Zi × Λ0(~r 0 i, E)− X Λ Zi Λ(~ri<, E)JΛi×(~ri0_>, E) (2.179)

when~rand~r0_{are both in the neighbourhood of theith scatterer, so the vectors~rand~r}0_may

fall inside theith muffin-tin sphere, but they must not be in any other sphere. Zi _andJi _are

the regular and irregular solutions of the Dirac equation for single-site potentialVi _{(see also}

Eqs. (2.129) and (2.130)). We will refer to Eq. (2.179) as the site-diagonal (SD) expression of the Green’s function.

If the vector~ris in the neighbourhood of theith scatterer and~r0 _{is in the neighbourhood of}

thejth scatterer, the Green’s function may be written in the form: G(~r, ~r0, E) = X ΛΛ0 Z_Λi(~ri, E)τ_ΛΛij 0(E)Z j× Λ0(~r 0 j, E). (2.180)

This equation (2.180) is the non-site-diagonal (NSD) Green’s function expression. Doing the following step in the CPA description, is now needed to calculate the average of such a Green’s functions, the averaging being over the ensemble of all alloy configurations that can be formed by distributingxAN atoms of type A andxBN atoms of type B over the lattice

sites.

The ensemble average of the site-diagonal (SD) Green’s function can be written as:

< G(~r, ~r0_{, E)> = x} A X ΛΛ0 Z_Λi,A(~ri, E)< τΛΛii,A0(E)> Z i,A× Λ0 (~r 0 i, E) + xBX ΛΛ0

Z_Λi,B(~ri, E)< τ_ΛΛii,B0(E)> Z

i,B× Λ0 (~r 0 i, E) − X Λ

[xAZ_Λi,A(~ri<, E)J_Λi,A×(~r0

i>, E)

− xBZΛi,B(~ri<, E)JΛi,B×(~r

0

i>, E)], (2.181)

where Z_Λi,α(E, ~ri), respectivelyJ_Λi,α(E, ~ri), are the wave functions for the case when~ri is in

the ith muffin-tin sphere and an α atom is in that site, α being A or B. < τ_ΛΛii,α0(E) > is the

average over the subset of the ensemble for which the atom of typeα (A or B) is definitely

known to be on theith site. A similar average can be done for the non-site-diagonal (NDS)

Green’s function and the result of averaging looks like:

< G(~r, ~r0_{, E)> = x}2 A X ΛΛ0 Z_Λi,A(~ri, E)< τΛΛij,AA0 (E)> Z j,A× Λ0 (~r 0 j, E) + x2_B X ΛΛ0

Z_Λi,B(~ri, E)< τ_ΛΛij,BB0 (E)> Z

j,B×

Λ0 (~r

0

+xAxBX

ΛΛ0

Z_Λi,A(~ri, E)< τ_ΛΛij,AB0 (E)> Z

j,B× Λ (~r 0 j, E) +xBxAX ΛΛ0

Z_Λi,B(~ri, E)< τ_ΛΛij,BA0 (E)> Z

j,A×

Λ (~r

0

j, E). (2.182)

Here there is another kind of average for the scattering path operator, namely< τ_ΛΛij,αβ0 (E)>,

which is the restricted average over the subset of ensemble for which an atom of typeα(A

or B) is known to be on theith site and another atom of typeβ (B or A) is known to be on

the jth site. The next step is now to calculate those ensemble averages for the scattering

path operators. For this purpose, we will make use of the so-called single-site approxi- mation, which means that, if we calculate for example < τ_ΛΛii,α0(E) > we presume that the

effective scattering matrixtC_{(E)appears on every site except theith site andt}α_(E)appears

there. Equations (2.181) and (2.182) are exact, but they can be greatly simplified invoking the single-site approximation. This derivation ends up with the following result:

< τii,α(E)>=Dii_ατii_C , (2.183) where D_αii={I+τii_C£t_α−1−t−_C1¤}−1 (2.184) and τii_C = Ω (2π)3 Z τ_C(E, ~k)d3k . (2.185)

τC(E, ~k)is given by the matrix inversion:

τ_C(E, ~k) =h(tC₎−1_{−G(E, ~k)}i−1 _{. (2.186)}

We have to note that the average< τii,α_{(E) >is independent on the site indexiand that’s}

why we can name it< τ00,α_{(E)>. For the same reason, the matrixD}ii

αwill be named further

simpler, namelyD00_α.

In order to obtain the average< τij,αβ_{(E) >, one has to put t}α_{(E) on theith site, t}β_{(E) on}

thejth site andtC_{(E)on all the others. The result is} < τij,αβ(E)>=D00ατijCD˜

00

β (2.187)

where the matrixD00_α is defined in Eq. (2.184) andD˜00_β is given by:

˜ D00_β ={I+£t−1 β −t −1 C ¤ τii_C}−1 _(2.188)

The matrixτij_C is given by the expression: τij_C = Ω (2π)3 Z ei~k ~Rij_τ C(E, ~k)d3k (2.189)

Theτij_C is the same for thei-j pairs of sites separated by a vectorRij~ =Rj~ −Ri~ of the same

magnitude and direction.

In order to obtain the ensemble averaged Green’s function within the single-site approxima- tion, Gc(E, ~r, ~r0_{) =< G(E, ~r, ~r}0_{) >, the ensemble averages < τ}00,α_{(E) >and < τ}ij,αβ_{(E) >}

have to be substituted into Eqs. (2.181) and (2.182). The site-diagonal (SD) and non-site- diagonal ensemble average Green’s functions are needed for most calculations of electronic properties in alloys.

It can be seen from the defining equations thatGc(~r, ~r0_{, E)describes a periodic system in the}

sense that

Gc(~r+Rn, ~r~ 0+Rn, E~ ) =Gc(~r, ~r0, E). (2.190)

No statement has been made so far in this derivation concerning the way that the effec- tive scattering matrix tC _{is defined. One has to note that the most remarkable feature of} Gc(~r, ~r0_{, E)is that the effective wave function for each site is different for the SD and NSD}

cases. The theory for electronic states in an alloy has been designed to arrive at an effective Green’s function rather than to get an effective wave function because every property of an alloy can be calculated usingGc(~r, ~r0_{, E). In the following section it will be shown how one}

can calculate electronic properties of alloys. Let’s consider first the average density of states for a random substitutional alloy, in the single-site approximation. This can be expressed in the formula: ρC(E) =−1 π= Z Ω Gc(~r, ~r, E)d3r (2.191)

Because only values of Gc(~r, ~r0_{, E) for which~r = ~r}0 _{enters in the expression, only the SD}

form of Green’s function is needed. Inserting Eq. (2.181), with the average scattering path operator given by Eq. (2.183) into this expression, leads to

ρC(E) =−1 π X α=A,B xαX ΛΛ0 =< τ_ΛΛ00,α0 (E)> Z Ω Z_Λα(~r, E)Zα× Λ0 (~r, E)d 3_{r , (2.192)}

whereα=A or B indicates that the site in question is occupied by an atom of type A or B. If we use the notation

Fαα_ΛΛ0(E) = Z Ω Z_Λα(~r, E)Zα× Λ0 (~r, E)d 3_{r (2.193)}

where the underline means a matrix with respect toΛ ={κ, µ}, one may expressρC(E)as ρC(E) =−1 π X α=A,B xα=T rFαα(E)D00_α(E)τ00_C(E). (2.194)

From the expression given for the average density of states (2.194) it is clear that one may resolve the total density of statesρC(E)into componentsρA(E)andρB(E). These quantities

may be thought of as the average density of states (per atomic cell) on an A- or B-type site in the alloy. Consequently the total density of states may be written as the sum of the concentration weighted component density of states thus

ρC(E) =xAρA(E) +xBρB(E), (2.195)

where the component density of states can be identified as

ρα(E) =−1

π=T r[F

αα_(E)D00

α(E)τ00C(E)] (2.196)

withα = A or B. The expressions for the total and component densities of states, which are

identical to those derived by Faulkner and Stocks [61], are given in a completely general form such that they can be used for both relativistic or non-relativistic calculations.

The charge densities ρA(~r) and ρB(~r) associated with a given atomic type can be obtained through a energy integration up to the Fermi energy, as follows:

ρα(~r) = −1 π Z EF −∞ =T r[Fαα(~r, ~r, E)D00_α(E)τ00_C(E)]dE (2.197) where Fαα_ΛΛ0(E, ~r, ~r) =Z_Λα(~r, E)Zα × Λ0 (~r, E). (2.198)

Another quantity, used by Gy¨orffy et al. [62] in order to formulate theories for soft X-ray emission, electron-photon interaction and other phenomena is so called density matrix, de- fined as

ρ(~r, ~r0_{E) =xAρA(~r, ~r}0_{, E) +xBρB(~r, ~r}0_{, E). (2.199)}

The components of density matrix are (αis A or B): ρα(~r, ~r0, E) =−1

π=T r[F

αα_{(~r, ~r}0

)D00_ατ00_C]. (2.200)

With the formulas forGc(~r, ~r0_{, E)available, the comparison with experimental data for den-}

of the best single-site theories for the description of random substitutional binary alloys. On the other hand, multiple-site scattering effects are implicitly included within the theory as the single-site approximation is based on the idea of a single scattering site immersed in an average ’effective’ medium. Consequently, it is appropriate to preceed developing calcula- tions combining the CPA with the KKR theory. The calculations done in this thesis use this approach in order to investigate the properties of random substitutional alloys.