Formation energies of neutral and charged O vacancies in MgO, ZnO, Al2O3, In2O3 and SnO2 were calculated by a first principles plane-wave pseudopotential method. Two kinds of polymorphs, i.e., an ordinary phase and a high-pressure or an hypothetical negative pressure phase, were chosen in order to see the effects of crystal structure. Super-cells composed of 54 to 96 atoms were employed, and structural relaxation around the vacancy within second nearest neighbor distances was taken into account. Defect levels were obtained from the difference in total energies of the neutral and charged super-cells that contain a vacancy. Ionization energies of the vacancy were calculated as the difference in the bottom of the conduction band and the defect levels. They were found to be proportional to band-gaps with a factor of approximately 0.5, which were prohibitively large for the n-type conduction.

Theoretical Formation Energy of Oxygen-Vacancies in Oxides. Tanaka, I., Oba, F., Tatsumi, K., Kunisu, M., Nakano, M., Adachim, H.: Materials Transactions, 2002, 43[7], 1426-9