Cluster Theory of the Electronic Structure of Disordered Systems

Abstract

The equation of motion for the averaged Green's function in an alloy couples the latter to the Green's function for which the average is restricted so that the composition of one atom is held fixed. The average Green's function may be regarded as the Green's function for a zero-atom cluster, and it is coupled to the Green's function for a one-atom cluster. There is thus an infinite hierarchy of equations of motion in which the $n$-atom functions are coupled to the ($n+1$) atom functions. The coherent potential approximation (CPA) of Soven corresponds to truncation in the equation of motion of the one-atom function. We have generalized the coherent potential theory to a theory of $n$-atom functions with truncation in the equation of motion of the ($n+1$) atom function ($\mathrm{CP}\stackrel{\ensuremath{\rightarrow}}{\mathrm{n}}$). The formalism is developed, and specific formal results are reported. In particular, the existence of localized states in the band tails can be demonstrated, but the transition region from localized to extended states is beyond the reach of a cluster theory. The theory provides a systematic basis for quantitative improvement over the CPA, and allows for a discussion of the effects of randomness in the off-diagonal elements of the Hamiltonian. The cluster hierarchy is formally solved to provide a multiple-scattering expansion of the average Green's function, where terms involving one, two, etc., atom scattering are grouped together. This expansion can be used to generate recently proposed generalizations of the CPA, but when used in conjunction with the self-consistent $n$-atom functions of the $\mathrm{CP}\stackrel{\ensuremath{\rightarrow}}{\mathrm{n}}$, it provides the best approximate averaged Green's function for which the lowest-order corrections involve the scattering from compact ($n+1$) atom clusters.