Structural instabilities in Aurivillius compound Bi$_4$Mn$_3$O$_{12}$ from First Principles

Layered perovskite oxide materials are good candidates for the potential synthesis of natural multiferroic materials. One approach is to choose a ferroelectric host and to incorporate a magnetically active species. The Aurivillius layered perovskites are chosen because most are ferroelectric. Described by the formula [Bi$_2$O$_2$][A$_{n-1}$B$_n$O$_{3n+1}$], they are formed by stacking Bi$_2$O$_2$ slabs with $n$ perovkitelike blocks. A ferroelectric prototype compound is Bi$_4$Ti$_3$O$_{12}$ (BIT), where bismuth also occupies the $A$ sites. Using first-principles calculations, we investigate here the three-layer Aurivillius Bi$_4$Mn$_3$O$_{12}$ (BIM) that results of substituting all Ti$^{4+}$ B-site cations in the BIT lattice by Mn$^{4+}$ cations. We report the structural instabilities in the high-symmetry tetragonal structure (space group symmetry $I4/mmm$). We find an unstable $E_u$ phonon mode, which mainly involves movements of the Bi ions in the perovskite $A$ sites with respect to the TiO$_6$ octahedra. This instability, also observed in non-magnetic BIT and associated to the in-plane electronic polarization, suggests the presence of ferroelectricity in BIM. We also explore different collinear spin orderings of the magnetic Mn atoms and its effect on the structural instabilities.