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Coherent-Potential and Average T-Matrix Approximations for Disordered Muffin-Tin Alloys

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Coherent-Potential and Average T-Matrix Approximations for Disordered Muffin-Tin Alloys PHYSICAL REVIEW B VOLUME 20, NUMBER l0 15 NOVEMBER l 979 Coherent-yotential and average f-matrix ayyroximations for disorderefi muffin-tin alloys. I. Formalism A. Bansil Department of Physics, Northeastern University, Boston, Massachusetts 02115 (Received 28 November 1978) The average density of states (p(Z)) and the component charge density associated with an A (B) atom in the alloy, (p„&s&(E)), are discussed for the disordered alloy A„B, „within the framework of the muf6n- tin Hamiltonian. A new version of the average t-matrix (ATA) is developed. The structure in the spectral density function, (p(k,E)P, in the coherent-potential approximation (or the new ATA) is seen to result from not only the Bloch-type states in the medium of coherent-potential effective atoms tzp (or the average t- atoms (t)) but also from non-81och-type impurity levels arising when a single A or B atom is embedded in an otherwise perfect effective medium. The proposed ATA equations would allow a simple yet reliable treatment of many aspects of the electronic spectrum of disordered transition and noble-metal alloys. I. INTRODUCTION hand, the difficulties are formal in nature: Al- though the currently used ATA expression for the It has become clear in recent years that, in average total density of states (p(E)) appears to order to obtain a realistic description of the elec- give reasonable results in all cases studied so tronic structure of disordered metallic alloys, far, the corresponding expressions for the com- the atomic potentials must be treated within the ponent densities (p„(E)) and (pn(E)) yield negative framework of the muffin-tin model, as is usually results in many cases and are not reliable. ' done for the corresponding perfect crystals. In In this series of two papers, we address some particular, the simple one- and two-band tight- of the questions that have arisen in the application binding model Hamiltonians are not adequate for of the CPA and ATA to the disordered muffin-tin dealing with the detailed experimental information alloys. The present paper discusses the relevant which is now becoming available, pertaining to the formal aspects. A new set of ATA equations electronic spectrum of transition and noble-metal (designated as ATA2 for convenience) is consider- alloys. ' ' ed. The comparison of the ATA2 spectrum with The application of the multiple-scattering theory the currently used form of the ATA' ' (referred techniques to the disordered muffin-tin alloys has to as ATA1) is presented. However, the bulk of proceeded rapidly. In this connection, two of the the results for the CPA and ATA2 and their com- most commonly used approximation schemes have parison is undertaken in the second following been the coherent-potential (CPA) and the average paper. ' t-matrix (ATA) approximations. " Both belong to We emphasize that a fully self-consistent calcu- the class of the so-called single-site approxima- lation of the electronic spectrum of the disordered tions. However, of the two only the CPA treats alloy requires not only the self-consistent treat- the disorder self-consistently and is to be pre- ment of disorder but also that of the atomic po- ferred. The attractiveness of the ATA derives tentials. The CPA treats just the disorder self- mostly from its relative simplicity in application consistently in the sense of a single-site approxi- to realistic mode1. s.4 ' In spite of the significant mation. The problem of self-consistency of atomic progress made with regard to the application of potentials has been handled to date only via semi- the CPA and the ATA to the muffin-tin alloys, empirical models"' in which the effect of dis- difficulties have persisted with each of these order on atomic potentials is taken into account schemes. The CPA formalism is well develop- by adjusting one or another feature of the constit- ed, ~' but its practical implementation to the uent atomic potentials to force agreement between muffin-tin Hamiltonian has not been undertaken theory and one particular experiment. Our ability until recently, ""owing to the difficult and re- to do accurate calculations of the component peated Brillouin-zone integrations necessary to charge density (p„,»(E)) has important conse- solve the CPA self-consistency equations. Addi- quences in this regard. This point is discussed tional difficulties involving the free-electron poles further with the example of Cu„Ni, „ in the article must also be faced in order to evaluate the com- that follows. " ponent density of states (p„t»(E)) [i.e., the elec- An outline of the present paper is as follows: tronic charge density associated with an A (or 8) Section II considers the exact equations for (p(E)) atom in the alloy]. With the ATA, on the other and (p„&n&(E)). The formula for (p(E)) is well 20 4025 1979The American Physical Socic:ty 4O26 A. BAN SI I 20 known. ' " The expression for (p„,» &(E)), which is fective atoms. The physical content of the new the focus of our discussion, has also been sug- ATA2 decoupling is that it treats the single im- gested in an earlier paper. " However, several purity contributions better than ATA1. We show delicate features inherent to this definition were that if (&& is replaced by ac~ in the ATA2 expres- not recognized at that time. We clarify the exact sions for (p(E)) and (p„,»(E)&, the CPA spectrum meaning of our definition of (p„&»(E)& and its will result. In this sense, ATA2 can be viewed as relationship to the more conventional definition in the lowest-order approximation to the CPA, and which this quantity is defined as the integral of the ATA2 spectrum may be improved systemati- the charge density over a signer-Seitz unit cell at cally by simply replacing (w& by successively the center of which an A (or B) atom is placed. ' In better approximations to 7~p. Finally, using the fact, our definition of (p„&»(E)& may be more ap- illustrative example of Cuo 75Ni, », the spectral propriate for considering transfer of charge bet- density and densities of states in ATA1 and ATA2 ween the constituent atoms of an alloy, because are compared in order to elucidate some of the this definition does. not artificially carve out the important differences between these two approxi- Wigner-Seitz unit cell as the basic region of space mations. to which the transfer of the electronic charge is restricted. In any event, our conclusion is that II. EXACT EQUATIONS even though the two viewpoints of charge transfer T-he substitutionally disordered binary alloy are somewhat different, they are li'Rely to yield A„B, „will be discussed within the framework of similar results for physically relevant quantities the one-electron Hamiltonian sensitive to the environment of the A and & atoms in the alloy. If =p'/2m++ o"'s&(r). (2.1) Section III specializes to the CPA. The expres- sions for (p(E)&c~ and .»(E)&c~ have already Here the crystal potential j.s assumed to be given (p„, ' been discussed in Ref. 13. While the free-electron as the sum of nonoverlapping spherically sym- metric muffin-tin potentials v"'s '(r}= v" ~s '( singularities cancel exactly in (p(E}& owing to ~ r— the translational symmetry of the average alloy, —R ~) centered on atomic sites (8 ). The A and these singularities are present in an essential way B atoms are assumed to occupy the sites (R~) ran- so that the in the expression for (p„,s&(E)& . (The techniques domly, probability that a given site is used for handling this prob].em are outlined in occupied by an A, (or B) atom is proportional to its —= Appendix A.) The focus of our discussion in this concentration &(: [or y (1-x)]. section concerns the symmetry properties of the The electronic density of states p(E) (per atom, CPA equation. First, we show that the CPA equa- per spin) is given most conveniently in terms of tion is symmetric to the interchange Cc~(E)- the Green's function Cc&(E), where Ccp(E) denotes the matrix of cotan- Q(E) =- (E -H) ' (2.2) gents, of CPA phase shifts. The solution with by the relation Im[Cc&, (E)](0 satisfies the unitarity constraint and is physically relevant. Second, we show that p(E) = —(vN)-'Im T r[8 (E')] (2.3a) for cubic systems, 7~~ is diagonal if only s, p, d phase shifts (i.e., I ( 2) are used for pure con- (w)() tm jd «()(r,—r;K'). (2.3b) stituents. While this statement is by no means surprising, an explicit proof is useful and indi- Here E =—E+s0 and iV xs the total number of atoms con- cates how CPA can be implemented most in the system. By using the coordinate representa- veniently for /) 2 and for other symmetries. tion explicitly, Eq. (2.3b) expresses p(E) as an The ATA formalism is taken up in Sec. IV. integral over the charge density, which is pro- This discussion concerns the manner in which the portional to Im9(r, r; E'). restricted averages of the path operators An exact expression for the average density of decoupled in of the (T .&o.„&»& are terms corre states (p('E)) can be obtained by using the Lloyd' s sponding path operators T„'„", for an ordered crystal formula" for the density of states of an arbitrary of effective atoms t, «=(t&. The currently used assembly of muffin=tin atoms. This well-known form of the ATA decoupling"o (referred to as result is" ATA1) and the proposed ATA2 decoupling are dis- cussed.
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