Structurally unstable III-Bi-O$_3$ perovskites are predicted to be topological insulators but their stable structural forms are just band insulators: A first principles study*

Several Bi oxides in the assumed cubic $Pm\bar{3}m$ perovskite structure have recently been identified as topological insulators or semi-metals by first-principles calculations. In these perovskites, Bi is at the octahedral site and the $A$ atom at the interstitial site is a column III cation, i.e., Al, Ga, In, Sc, Y, La. We use density functional total-energy calculations and crystal structure prediction to determine the energetically stable phases for these oxides. We find that these $Pm\bar{3}m$ $A$BiO$_{3}$ perovskites are topological insulators, confirming recent results obtained by our and other groups. However, switching the position of Bi and $A$ in the $Pm\bar{3}m$ perovskite produces trivial insulators or semimetals, as opposed to topological insulators. Indeed, symmetry-lowering via concerted tilting and internal deformation of the octahedra, stabilizes these Bi oxides, irrespective of the position of Bi, producing the stable $Pnma$ perovskite structure that is not a topological insulator. This illustrates that a simultaneous application of ``first-principles thermodynamics'' with first-principles electronic structure ($Z_{2}$ evaluation) is needed to establish stable topological insulators.