Density-functional-theory calculations for the silicon vacancy

The atomic configurations and formation energies of a silicon vacancy in the $+2$, $+1$, 0, $-1$, and $-2$ charge states have been computed using density-functional theory with norm-conserving pseudopotentials and a plane wave basis. Calculations were performed in simple cubic supercells using two different forms of exchange and correlation: the local-density approximation (LDA) and the Perdew, Burke, Ernzerhof formulation of the generalized-gradient approximation (GGA). Convergence with respect to Brillouin zone sampling was tested for all charge states, and effects due to electrostatic interactions between the periodically repeated vacancies were removed by extrapolating the formation energies obtained in 215-, 511-, and 999-atom supercells to an infinite sized supercell. In agreement with experimental results, the GGA yielded a configuration with $C_{2v}$ symmetry in the $-1$ charge state, whereas the LDA yielded $D_{3d}$ symmetry. Transition energies between the charge states were also computed. The experimentally observed negative-$U$ behavior of the donor states was reproduced in the GGA results, but not in the LDA results. Both the LDA and GGA predict negative-$U$ behavior for the acceptor states.