Origin of the tetragonal-to-hexagonal phase transitions in Fe-doped BaTiO$_3$

Based on detailed first-principles calculations, we investigate the tetragonal-to-hexagonal phase transition in Fe-doped BaTiO$_3$. Total energy calculations confirm a crossover from the tetragonal to hexagonal phases around 4\% Fe, in agreement with experimental observations, where comparative calculations show that neither CaTiO$_3$ nor SrTiO$_3$ exhibits similar behavior under equivalent substitution. Furthermore, three possible mechanisms are quantified: oxygen vacancies shift the crossover concentration from $\sim$4\% to $\sim$2\% through charge compensation, Jahn-Teller distortions impose a larger elastic penalty, both favoring tetragonal-to-hexagonal phase transitions; whereas the tolerance factor is reduced in comparison with that of pristine BaTiO$_3$ for reasonable Fe valence states, disfavoring the occurrence of the hexagonal phases. Detailed analysis on the electronic structure reveals that the charge redistribution induced by oxygen vacancy is strongly orbital dependent due to the local crystal structure distortions.