Hybrid functional studies of the oxygen vacancy in TiO 2

The electronic and structural properties of the oxygen vacancy $({V}_{\text{O}})$ in rutile ${\text{TiO}}_{2}$ are studied using generalized Kohn-Sham theory with the Heyd, Scuseria, and Ernzerhof (HSE) hybrid functional for exchange and correlation. The HSE approach corrects the band gap and allows for a proper description of defects with energy levels close to the conduction band. According to the HSE calculations, ${V}_{\text{O}}$ is a shallow donor for which the $+2$ charge state is lower in energy than the neutral and $+1$ charge states for all Fermi-level positions in the band gap. The formation energy of ${V}_{\text{O}}^{2+}$ is relatively low in $n$-type ${\text{TiO}}_{2}$ under O-poor conditions but…