Nature of localized states induced by an oxygen vacancy in SrTiO$_3$

Combining the density function theory and model analysis, we study the nature of localized states induced by an oxygen vacancy (OV) in SrTiO$_3$. The key effect caused by an OV is that it introduces a local mixing between $3z^2-r^2$ and $4s$, $4p$ orbitals of Ti next to the vacancy. According to our analysis, the OV-induced localized state is highly one-dimensional and is mainly composed of Ti $3z^2-r^2$ orbitals along the Ti-OV-Ti axis (defined as the $z$-axis) and Ti $4s$, $4p$ at the OV site. Ti $t_{2g}$ based localized states are ruled out. We suggest the $4s$, $4p$ components of the OV induced localized state should be reflected in the x-ray adsorption spectroscopy when Ti $4s$, $4p$ are served as final states. Our analysis and conclusion apply to general transition metal perovskites.