Spin polarized electronic states and spin textures at the surface of oxygen-deficient SrTiO$_3$*

We investigate the electronic structure and spin texture at the (001) surface of SrTiO$_3$ in the presence of oxygen vacancies by means of {\it ab initio} density functional theory (DFT) calculations of slabs. Relativistic non-magnetic DFT calculations exhibit Rashba-like spin winding with a characteristic energy scale $\sim 10$ meV. However, when surface magnetism on the Ti ions is included, bands become spin-split with an energy difference $\sim 100$ meV at the $\Gamma$ point. This energy scale is comparable to the observations in SARPES experiments performed on the two-dimensional electronic states confined near the (001) surface of SrTiO$_3$. We find the spin polarized state to be the ground state of the system, and while magnetism tends to suppress the effects of the relativistic Rashba interaction, signatures of it are still clearly visible in terms of complex spin textures.