Local polarization in oxygen-deficient LaMnO$_3$ induced by charge localization in the Jahn-Teller distorted structure

The functional properties of transition metal perovskite oxides are known to result from a complex interplay of magnetism, polarization, strain, and stoichiometry. Here, we show that for materials with a cooperative Jahn-Teller distortion, such as LaMnO$_3$ (LMO), the orbital order can also couple to the defect chemistry and induce novel material properties. At low temperatures, LMO exhibits a strong Jahn-Teller distortion that splits the $e_g$ orbitals of the high-spin Mn$^{3+}$ ions and leads to alternating long, short, and intermediate Mn--O bonds. Our DFT+$U$ calculations show that, as a result of this orbital order, the charge localization in LMO upon oxygen vacancy formation differs from other manganites, like SrMnO$_3$, where the two extra electrons reduce the two Mn sites adjacent to the vacancy. In LMO, relaxations around the defect depend on which type of Mn--O bond is broken, affecting the $d$-orbital energies and leading to asymmetric and hence polar excess-electron localization with respect to the vacancy. Moreover, we show that the Mn--O bond lengths, orbital order and consequently the charge localization and polarity are tunable via strain.