Extracting E versus k effective band structure from supercell calculations on alloys and impurities

The supercell approach to defects and alloys has circumvented the limitations of those methods that insist on using artificially high symmetry, yet this step usually comes at the cost of abandoning the language of $E$ versus $stackrel{Pvec}{k}$ band dispersion. Here we describe a computational method that maps the energy eigenvalues obtained from large supercell calculations into an effective band structure (EBS) and recovers an approximate $E(stackrel{Pvec}{k})$ for alloys. Making use of supercells allows one to model a random alloy A${}_{1ensuremath{-}x}$B${}_{x}$C by occupying the sites A and B via a coin-toss procedure, affording many different local environments (polymorphic description) to occur. We present the formalism and implementation details of the method and apply it to study the evolution of the impurity band appearing in the dilute GaN:P alloy. We go beyond the perfectly random case, realizing that many alloys may have nonrandom microstructures, and investigate how their formation is reflected in the EBS. It turns out that the EBS is extremely sensitive in determining the critical disorder level for which delocalized states start to appear in the intermediate band. In addition, the EBS allows us to identify the role played by atomic relaxation in the positioning of the impurity levels.