The structures and energies of oxygen-deficient rutile ${\text{TiO}}_2$ were calculated using density-functional theory. The electronic interactions are described within the $\text{LDA}+U$ formalism, where on-site Coulomb corrections are applied on the $3d$ orbital electrons of Ti atoms $\left(U^d\right)$ and $2p$ orbital electrons of the O atoms $\left(U^p\right)$. We show that the $U^d$ parameter affects only the character of the conduction band and values higher than 7 eV produce an unphysical description of the electronic interactions. The results are dramatically improved, when correlation corrections are introduced additionally on the $\text{O}2p$ orbitals by employing the $\text{LDA}+U^d+U^p$ approach, and we observe systematic shifts for both the valence and conduction bands. This combined approach produces a corrected energy band structure and the band-gap energy is in very good agreement with experimental data. Using this approach for the oxygen-deficient structure, in the case of a neutral oxygen vacancy we find that the electrons from the Ti nearest to the vacancy become localized and induce defect states within the band gap, in very good agreement with experiment and other theoretical calculations. Additionally, these defect states are strongly limited on the three Ti atoms surrounding oxygen vacancy. The doubly positively charged oxygen vacancy is found to be more stable than neutral oxygen vacancy in most range of Fermi level and the vacancy-formation energies show significant dependence on the deposition conditions, i.e., Ti- or O-rich environments.

• Received 6 January 2010

DOI:https://doi.org/10.1103/PhysRevB.82.115109