Oxygen deficiency and migration mediated electric polarization in Fe,Co-substituted SrTiO$_{3-\delta}$

We use density functional theory (DFT) calculations to show that oxygen vacancies ($v_\mathrm{O}$) and mobility induce noncentrosymmetric polar structures in SrTi$_{1-x-y}$Fe$_{x}$Co$_{y}$O$_{3-\delta}$ ($x=y=0.125$) with $\delta = \{0.125, 0.25\}$, enhance the saturation magnetization and give rise to large changes in the electric polarization $\vert\Delta P\vert$. We present an intuitive set of rules for SrTiFeCoO$_{3-\delta}$ (STFC), which are based on the interplay between (Co/Fe)-$v_\mathrm{O}$ defects, magnetic cations coordination and topological vacancy disorder. STFC structures convey layered crystals with sheets of linear organized O$_{4,5,6}$-coordinated Fe-Co pairs, sandwiched with layers of O$_{5}$-coordinated Ti. Co,Fe-$v_\mathrm{O}$ defects are the source of the crystal distortions, cations off-centering and bending of the oxygen octahedra, which added to the charge redistribution mediated by $v_\mathrm{O}$, the cations electronegativity and valence states trigger an effective electric polarization. Oxygen migrations for $\delta=0.125$ provides us with $\vert\Delta \mathbf{P}\vert$ $>\sim10 \mu$C/cm$^2$ due to a quantum-of-polarization differences between $\delta=0.125$ structures. Increasing the deficiency to $\delta=0.25$ yields $\vert\Delta \mathbf{P}\vert$ whose O-migration resolved polarization for $\delta=0.25$ is $>\sim3 \mu$C/cm$^2$ in the worst case scenario. Magnetism is dominated by the Fe,Co spin states for $\delta=0.125$ while there is a raid of Ti magnetic moments ($\sim1\mu_{B}$) for $\delta=0.25$. Magnetic and electric order parameters change for variations of $\delta$ or oxygen migrations for a given deficiency. Our results capture characteristics observed in the end-members of the series SrTi(Co,Fe)O$_{3}$, and suggest the existence of a broader set of rules for oxygen deficient multiferroic oxides.