Spin-texture induced by oxygen vacancies in Strontium perovskites (001) surfaces: A theoretical comparison between SrTiO$_3$ and SrHfO$_3$

The electronic structure of SrTiO$_3$ and SrHfO$_3$ (001) surfaces with oxygen vacancies is studied by means of first-principles calculations. We reveal how oxygen vacancies within the first atomic layer of the SrTiO$_3$ surface (i) induce a large antiferrodistortive motion of the oxygen octahedra at the surface, (ii) drive localized magnetic moments on the Ti--3$d$ orbitals close to the vacancies and (iii) form a two-dimensional electron gas localized within the first layers. The analysis of the spin-texture of this system exhibits a splitting of the energy bands according to the Zeeman interaction, lowering of the Ti--3\emph{d$_{xy}$} level in comparison with \emph{d$_{xz}$} and \emph{d$_{yz}$} and also an in-plane precession of the spins. No Rashba-like splitting for the ground state neither for ab initio molecular dynamics trajectory at 400K is recognized as suggested recently by A. F. Santander-Syro \emph{et al.} \cite{Santander-Syro2014}. Instead, a sizeable Rashba-like splitting is observed when the Ti atom is replaced by a heavier Hf atom with a much larger spin-orbit interaction. However, we observe the disappearance of the magnetism and the surface two-dimensional electron gas when full structural optimization of the SrHfO$_3$ surface is performed. Our results uncove