Ferroelectricity in oxyfluoride Aurivillius phase Bi$_2$TiO$_4$F$_2$ from first-principles

Using first-principle calculations, we restrained the m=1 oxyfluoride Aurivillius Bi$_2$TiO4F$_2$ to adopt three ordered key configurations, such as tetragonal $I4/mmm$ phases with fluorine anions distributed either at the octahedral apical sites or at the octahedral equatorial ones and orthorhombic $Pma2$ phase with fluorine anions distributed over both apical and equatorial sites. We explored the energetics of the metastable phases and their potential for ferroelectricity and found that the least favorable configuration is equatorial sites for fluorine and the most favorable one is the mixture between apical and equatorial sites for fluorine. Phonon calculations performed in the tetragonal $I4/mmm$ with fluorine at the apical sites showed a strong c-axis polar instability $\Gamma_{3}^{-}$ that co-exists with the ab-plane polar instability $\Gamma_{5}^{-}$, commonly found in Aurivillius oxide phases. These two polar instabilities lead to $Pc$ phase that displays in-plane and out-of-plane spontaneous polarizations as large as 44 $\mu$c/cm$^{2}$ and 35 $\mu$c/cm$^{2}$, respectively, opening new prospective design in thin films Aurivillius FeRAM devices.