Comparison of the Average-t-Matrix and Coherent-Potential Approximations in Substitutional Alloys

Abstract

This paper will review techniques useful in theoretical analyses of the electronic structure of disordered substitutional alloys. The coherent-potential approximation (CPA) and the average-$t$-matrix approximation (ATA), which have received a great deal of attention recently, are compared for two model Hamiltonians. A reconsideration of the ATA shows that the unphysical results usually attributed to it are in fact the consequence of unnecessary further approximations. The two Hamiltonians in question consist, respectively, of a single-band model and a two-band model relevant to the description of transition-metal alloys. In the first case the ATA is found to provide a correct description of the qualitative features of the density of states for a wide variety of scattering strengths and concentrations. In the two-band model, which has been previously applied in connection with the optical properties of Ag-Au alloys, the results of the ATA and the CPA are essentially identical over a wide range of energies. This agreement can be shown to persist as long as the constituent $d$ subbands lie within the broad $s$ band of the alloy. This is precisely the situation that obtains in many of the transition-and noble-metal alloys. This result is of particular importance since the ATA is far easier to implement numerically than the CPA. In fact, the ATA may be regarded as a good first approximation in an iteration scheme leading to the self-consistent CPA solution.