Role of oxygen vacancy in the spin-state change and magnetic ordering in SrCoO$_{3-delta}$

We present the first-principles investigation of the structural, electronic, and magnetic properties of SrCoO$_{3-delta}$ ($delta=0, 0.25, 0.5$) to understand the multivalent nature of Co ions in SrCoO$_{3-delta}$ along the line of topotactic transition between perovskite SrCoO$_{3}$ and brownmillerite SrCoO$_{2.5}$. From the on-site Coulomb interaction $U$-dependent ground state of stoichiometric SrCoO$_{3}$, we show the proximity of its metallic ferromagnetic ground state to other antiferromagnetic states. The structural and magnetic properties of SrCoO$_{3-delta}$ depending on their oxygen-content provide an interesting insight into the relationship between the Co-Co distances and the magnetic couplings so that the spin-state transition of Co spins can understood by the change of $pd$-hybridization depending on the Co-Co distances. The emph{strong} suppression of the $dpsigma$-hybridization between Co $d$ and O $p$ orbitals in brownmillerite SrCoO$_{2.5}$ brings on the high-spin state of Co$^{3+}$ $d^{6}$ and is responsible for the antiferromagnetically ordered insulating ground state. The increase of effective Co-Co distances driven by the presence of oxygen vacancies in SrCoO$_{3-delta}$ is consistent with the reduction of the effective $pd$-hybridization between Co $d$ and O $p$ orbitals. We conclude that the configuration of neighboring Co spins is shown to be crucial to their local electronic structure near the metal-to-insulator transition along the line of the topotactic transition in SrCoO$_{3-delta}$. Incidentally, we also find that the textit{I2mb} symmetry of SrCoO$_{2.5}$ is energetically stable and exhibits ferroelectricity via the ordering of CoO$_{4}$ tetrahedra, where this polar lattice can be stabilized by the presence of a large activation barrier.