Systematically exact integrated density-of-states Lloyd’s formula for disordered alloys with short-range order

Abstract

We derive an analytic expression for the configurationally averaged integrated density of states $⟨N⟩$, or Lloyd's formula, within the dynamical cluster approximation (DCA) to dynamical mean-field theory, or so-called nonlocal coherent potential approximation (NLCPA) within the Korringa, Kohn, and Rostoker multiple-scattering theory. Our generalized Lloyd's formula includes atomic correlations over a finite cluster in which the NLCPA/DCA systematically incorporates multisite configurational effects, including short range order (SRO). The Lloyd's formula is necessary and sufficient for constructing analytically a variational total-energy formalism within density functional theory (DFT) that contains SRO and is systematically exact with increasing cluster size. We show also that our $⟨N⟩$ expression is stationary with respect to changes in the NLCPA/DCA effective medium, a critical requirement for establishing a stationary DFT for alloys with disorder. We then demonstrate the correctness of the Lloyd's formula by comparing the density of states obtained from the numerical derivative of our formula and from the direct calculation from Green's function for disordered fcc CuAu and bcc NiAl, with and without SRO and different sized cluster environments.