A comprehensive study was made of the vacancy in bulk silicon in all its charge states from 2+ to 2-, by using a super-cell approach within plane-wave density-functional theory, and the various contributions to the well-known finite size errors associated with calculating formation energies and stable charge state transition levels of isolated defects with periodic boundary conditions were systematically quantified. Furthermore, it was found that transition levels converge faster with respect to super-cell size when only the Γ-point was sampled in the Brillouin zone, as opposed to a dense k-point sampling. This arose from the fact that defect level at the Γ-point quickly converges to a fixed value which correctly described the bonding at the defect center. The calculated transition levels with 1000-atom super-cells and Γ-point only sampling were in good agreement with available experimental results. Two simple and accurate approaches were also demonstrated for calculating the valence band offsets that were required for computing the formation energies of charged defects: one based upon a potential averaging scheme and the other using maximally-localized Wannier functions. Finally, it was shown that the latter provided a clear description of the nature of the electronic bonding at the defect center that verifies the canonical Watkins model.

System-Size Convergence of Point Defect Properties: the Case of the Silicon Vacancy. F.Corsetti, A.A.Mostofi: Physical Review B, 2011, 84[3], 035209