Ab initiocalculations for the neutral and charged O vacancy in sapphire

The energetics, lattice relaxation, and the defect-induced states of a single O vacancy in $\alpha -{\mathrm{Al}}_2{\mathrm{O}}_3$ are studied by means of supercell total-energy calculations using a first-principles method based on density-functional theory. The supercell model with 120 atoms in a hexagonal lattice is sufficiently large to give realistic results for an isolated single vacancy (□). Self-consistent calculations are performed for each assumed configuration of lattice relaxation involving the nearest-neighbor Al atoms and the next-nearest-neighbor O atoms of the vacancy site. Total-energy data thus accumulated are used to construct an energy hypersurface. A theoretical zero-temperature vacancy formation energy of 5.83 eV is obtained. Our results show a large relaxation of Al (O) atoms away from the vacancy site by about 16% (8%) of the original Al-□ (O-□) distances. The relaxation of the neighboring Al atoms has a much weaker energy dependence than the O atoms. The O vacancy introduces a deep and doubly occupied defect level, or an $F$ center in the gap, and three unoccupied defect levels near the conduction band edge, the positions of the latter are sensitive to the degree of relaxation. The defect state wave functions are found to be not so localized, but extend up to the boundary of the supercell. Defect-induced levels are also found in the valence-band region below the O $2s$ and the O $2p$ bands. Also investigated is the case of a singly occupied defect level (an $F^{+}$ center). This is done by reducing both the total number of electrons in the supercell and the background positive charge by one electron in the self-consistent electronic structure calculations. The optical transitions between the occupied and excited states of the $F$ and $F^{+}$ centers are also investigated and found to be anisotropic in agreement with optical data.