Oxygen vacancies in strontium titanate: A <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>DFT</mml:mi><mml:mo>+</mml:mo><mml:mi>DMFT</mml:mi></mml:math> study

We address the long-standing question of nature oxygen vacancies in strontium titanate, using a combination density functional theory and dynamical mean-field (DFT+DMFT) to investigate particular effect vacancy-site correlations on electronic properties. Our approach uses minimal low-energy subspace including Ti-$t_{2g}$ orbitals plus an additional vacancy-centered Wannier function, provides intuitive physically transparent framework study local electron-electron interactions excess charge introduced by vacancies. estimate strength screened interaction parameters constrained random phase approximation find sizeable Hubbard $U$ parameter for vacancy orbital. main finding, which reconciles previous experimental computational results, is that ground state either with double occupation localized defect or singly-occupied one electron transferred conduction band. The balance between these two competing states determined both Ti sites, Ti-Ti distance across vacancy. Finally, we contrast case doping SrTiO$_3$ via La substitution, show latter well described simple rigid-band picture.