First-principles based Monte Carlo modeling of oxygen deficient Fe-substituted SrTiO$_3$ experimental magnetization

Ferroics based on transition-metal (TM) substituted SrTiO$_{3}$ have called much attention as magnetism and/or ferroelectricity can be tuned by using cations substitution and defects, strain and/or oxygen deficiency. C. A. Ross et al. [Phys. Rev. Applied 7, 024006 (2017)] demonstrated the SrTi$_{1-x}$Fe$_{x}$O$_{3-\delta}$ (STF) magnetization behavior for different deposition oxygen-pressures, substrates and magnetic fields. The relation between oxygen deficiency and ferroic orders is yet to be well understood, for which the full potential of oxygen-stoichiometry engineered materials remain an open question. Here, we use hybrid-DFT to calculate different oxygen vacancy ($v_{o}$) states in STF with a variety of TM distributions. The resulting cations' magnetic states and alignments associated to the $v_{o}$ ground-states for $x=\{0.125,0.25\}$ are used within a Monte Carlo scope for collinear magnetism to simulate the spontaneous magnetization. Our model captures several experimental STF features i.e., display a maximum of the magnetization at intermediate number of vacancies, a monotonous quenching from $\sim{0.35}\mu{_{B}}$ for small ${\delta}$, and a slower decreasing of such saturation for larger number of vacancies. Moreover, our approach gives a further insight into the relations between defects stabilization and magnetization, vacancy density and the oxygen pressure required to maximize such ferroic order, and sets guidelines for future Machine Learning based computational synthesis of multiferroic oxides.