Ga1−xAlxNsystem, Madelung, and strain energies: A study on the quality of cluster expansions

Abstract

First-principles calculations within the local-density formalism were used to study the accuracy of cluster expansion techniques to predict the energy band gaps and enthalpy of the pseudobinary system ${\mathrm{Ga}}_{1\ensuremath{-}x}{\mathrm{Al}}_{x}\mathrm{N}$, a technologically important alloy. The chosen pseudobinary system has the advantage of having small lattice mismatches, which minimizes the enthalpies of formation, and of being a semiconducting system with a direct band gap for any concentration $x$. Many different cluster expansion techniques were tested, some presenting clear advantages. The many cluster expansions were also compared against models of Madelung and strain energy, both long-range interactions. Though cluster expansions fail completely for the long-range Madelung interaction model, they behave remarkably well in the not so long-range strain model. The qualitative results for the strain model are similar to the results for the enthalpy and gap of the alloy system, thus giving us an assurance of our conclusions. Using only short-range interactions, all cluster expansions are clearly inadequate for the long-period orderings.