Efficient cluster expansion for substitutional systems

Abstract

We demonstrate a cluster expansion technique that is capable of accurately predicting formation energies in binary substitutional systems—even for those with large atomic relaxations. Conventional cluster expansions converge rapidly only in the absence of atomic relaxations, and they fail for long-period lattice-mismatched superlattices. When combined with first-principles total-energy methods, our method allows for very fast calculations for structures containing hundreds or thousands of atoms. The convergence and effectiveness of the cluster expansion are enhanced in two ways. First, the expansion is recast into reciprocal space, which allows for the inclusion of all important pair interactions. Second, a reciprocal space formulation for elastic strain energy is introduced, allowing accurate predictions for both long- and short-period superlattices. We illustrate the power of the method by performing a cluster expansion that requires total-energy calculations for only 12 simple input structures, with at most eight atoms per unit cell. We then correctly predict the formation energies of relaxed long-period superlattices, low-symmetry intermixed superlattices, structures with varied compositions, substitutional impurities, and a im1000 atom/cell simulation of the random alloy.