Real-space description of semiconducting band gaps in substitutional systems

The goal of ``band-gap engineering'' in substitutional lattices is to identify atomic configurations that would give rise to a desired value of the band gap. Yet, current theoretical approaches to the problems, based largely on compilations of band structures for various latice configurations, have not yielded simple rules relating structural motifs to band gaps. We show that the band gap of substitutional AlAs/GaAs lattices can be usefully expanded in terms of a hierarchy of contributions from real-space ``atomic figures'' (pairs, triplets, quadruplets) detemined from first-principles band-structure calculations. Pair figures (up to fourth neighbors) and three-body figures are dominant. In analogy with similar cluster expansions of the total energy, this permits a systematic search among all lattice configurations for those having ``special'' band gaps. This approach enables the design of substitutional systems with certain band-gap properties by assembling atomic figures. As an il...