Importance of oxygen vacancies for the two-dimensional metallic state at the surface of SrTiO$_3$

We analyze by means of density functional theory (DFT) the electronic structure of various oxygen-deficient SrTiO$_3$ surface slabs. We find a significant surface reconstruction after introducing oxygen vacancies and we show that the charges resulting from surface-localized oxygen vacancies --independently of the oxygen concentration-- redistribute in the surface region and deplete rapidly within a few layers from the surface suggesting the formation of a two-dimensional electron system (2DES). We also investigate possible oxygen-vacancy clustering effects and discuss our results in the context of recent angle-resolved photoemission spectroscopy observations of a highly metallic 2DES at the (001) vacuum-cleaved surface of SrTiO$_3$.