Defect-Induced Electronic Structures and Formation Energies of Vacancy Complexes in SrTiO$_{3}$

Recently defect induced ferroelectricity in SrTiO$_{3}$ has been reported at room temperature. Strontium-oxygen vacancies were suggested as a possible source of electric polarization regarding to the existence of mid-gap states. To understand the detailed electronic structures induced by defects and their formation energies, we carried out density-functional-theory calculations for various defects such as Sr, Ti, O, Sr-O, Sr-O- O vacancies. We employed the LDA+$U$ method as implemented in the VASP code to describe the d-orbital occupation at the Ti- site due to the presence of oxygen vacancy. A complex of Sr-O-O vacancies is found to contribute to the localized electronic states in the band gap and its formation energy is small enough to form easily under the poor oxygen limit. We conclude that the vacancy-complex defects play a crucial role in determination of the physical properties of SrTiO$_{3}$ thin films.