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Evolution of the electronic and lattice structure with carrier injection in BiFeO$_3$

We report a density functional study on the evolution of the electronic and lattice structure in BiFeO$_3$ with injected electrons and holes. First, the self-trapping of electrons and holes were investigated. We found that the injected electrons tend to be localized on Fe sites due to the local lattice expansion, the on-site Coulomb interaction of Fe $3d$ electrons, and the antiferromagnetic order in BiFeO$_3$. The injected holes tend to be delocalized if the on-site Coulomb interaction of O $2p$ is weak (in other words, $U_\mathrm{O}$ is small). Single center polarons and multi-center polarons are formed with large and intermediate $U_\mathrm{O}$, respectively. With intermediate $U_\mathrm{O}$, multi-center polarons can be formed. We also studied the lattice distortion with the injection of carriers by assuming the delocalization of these carriers. We found that the ferroelectric off-centering of BiFeO$_3$ increases with the concentration of the electrons injected and decreases with that of the holes injected. It was also found that a structural phase transition from $R3c$ to the non-ferroelectric $Pbnm$ occurs, with the hole concentration over 8.7$\times10^{19} cm^{-3}$. The change of the off-centering is mainly due to the change of the lattice volume. The understanding of the carrier localization mechanism can help to optimize the functionality of ferroelectric diodes and the ferroelectric photovoltage devices, while the understanding of the evolution of the lattice with carriers can help tuning the ferroelectric properties by the carriers in BiFeO$_3$.