Convergence of supercell calculations for point defects in semiconductors: Vacancy in silicon

The convergence of first-principles supercell calculations for defects in semiconductors is studied with the vacancy in bulk Si as a test case. The ionic relaxations, defect formation energies, and ionization levels are calculated for supercell sizes of up to 216 atomic sites using several k-point meshes in the Brillouin-zone integrations. The energy dispersion, inherent for the deep defect states in the supercell approximation, and the long range of the ionic relaxations are shown to postpone the convergence so that conclusive results for the physical properties cannot be obtained before the supercell size is of the order of 128--216 atomic sites.