Treating interactions between polarons and oxygen vacancies in perovskite oxides

Interactions between polarons and oxygen vacancies in oxides, which cause them to modify one another's physical properties, are highly important for applications such as photovoltaics and ferroelectrics. While the difficulty in modeling polarons using density functional theory (DFT) calculations has been alleviated by the recent development of various techniques, including, e.g., the Hubbard-U parameter and finite-size corrections, the underlying physics of polaron interactions with defects remains unknown. Here, we demonstrate that the polaron-vacancy complexes in $\mathrm{PbTi}{\mathrm{O}}_{3}$ have a preferred orbital configuration, different from the orbital configuration of the bulk polaron, by exploring multiple nearby local minima using $\mathrm{DFT}+\mathrm{U}$. To address the issue of polaron property dependence on the Hubbard-U value, we determine the U value via enforcement of piecewise linearity, and we employ finite-size corrections. Three local minima with different electronic configurations are found by varying the initial conditions: (i) a polaron trapped in a $\mathrm{Ti}\text{\ensuremath{-}}3{d}_{eg}$ orbital on the first-nearest-neighbor Ti-ion of the oxygen vacancy ($eg$ complex), (ii) a polaron trapped in a $\mathrm{Ti}\text{\ensuremath{-}}3{d}_{t2g}$ orbital at the same position ($t2g$ complex), and (iii) electrons delocalized across several nearby sites and both spin channels, resulting in a semilocalized state. We find that the $eg$ complex is the most energetically favorable state, revealing a change in the orbital of the polaron when trapped by an oxygen vacancy, since the bulk polaron is found to be in a $t2g$ orbital. Furthermore, we demonstrate that great care must be taken to find the correct physical picture with $\mathrm{DFT}+\mathrm{U}$, since a small change in the initial conditions results in finding different local minima.