Convergence properties of the supercell approach in the study of local defects in solids

The “supercell” scheme is applied to the study of local defects in MgO (Ca substitution, cation and anion vacancies) and bulk silicon (carbon substitution). The trend of the quantities of interest (defect formation energy, geometrical relaxation, charge distribution around the defect) as a function of the supercell size is explored; when neutral defects are considered, supercells containing 50 to 100 atoms are large enough to allow for most of the nuclear and electronic relaxation and to produce a negligible interaction between defects in different cells. These conclusions apply both to ionic and covalent host crystals. Present day ab initio quantum mechanical periodic computer programs can handle cells of such a size at a relatively low cost and high numerical accuracy. When charged defects are considered (vacancies in MgO), the supercell scheme must be modified in order to avoid Coulomb divergencies, but the usually adopted correction, which consists in introducing a compensating uniform background of charge, generates spurious higher order electrostatic interactions, which are far from being negligible. The resulting defect formation energies show very slow, if any, convergence trends and “a posteriori” semiclassical corrections proposed in the literature do not represent a general solution to the problem. On the other hand, other properties, such as atom relaxation and charge distribution, show a much faster convergence than energy with respect to the cell size.