Single-band model of substitutional disordered ternary alloys in the coherent-potential approximation

The single-band model Hamiltonian used by Velický, Kirkpatrick, and Ehrenreich (VKE) in their classic paper on the electronic theory of nondilute binary alloys is extended to the ternary-alloy case, and a number of results are derived. It is shown by direct calculation that the coherent-potential approximation (CPA) preserves the first seven moments of the density of states and the spectral density. Diagrammatic considerations put the highest moments at eight and seven, respectively. Exact information concerning the effective Hamiltonian leads via the Kramers-Kronig dispersion relation to a series of sum rules involving the self-energy. The localization theorem predicts the existence of two or three well-separated subbands for which the self-energy may have a pole between adjacent subbands. These singularities are also found in the CPA. Within the appropriate limits, all calculated quantities and various limiting behaviors (virtual crystal, dilute alloy, and the atomic limit), reduce to the binary-alloy results of VKE. A semielliptic reference density of states is used for a numerical presentation of the CPA description of the model ternary alloy. New numerical examples of the $\overrightarrow{\mathrm{k}}$-independent properties of the CPA are exhibited for the self-energy, the total density of states, and the partial densities of states, over a wide range of concentrations and scattering-potential strengths. Results are also displayed for the Bloch-wave spectral density. This $\overrightarrow{\mathrm{k}}$-dependent quantity provides information concerning the validity of describing the states of the disordered alloy in terms of quasiparticles with wave vector $\overrightarrow{\mathrm{k}}$.