Theoretical investigation of oxygen vacancy clustering at SrTiO$_3$ surfaces

Oxygen vacancies have been found, both in experiment and in theory, to create a twodimensional electron gas at SrTiO$_3$ surfaces. We use density functional theory to search for the most stable structural arrangements of multiple oxygen defects. We find that there is a tendency of oxygen defects to cluster. While individual oxygen defects occupy Ti $t_{2g}$ orbitals and contribute to a metallic state, additional defects closeby form localized states in the former SrTiO$_3$ bulk gap. We discuss the relevance of these findings for photoemission experiments.