phases of inorganic halide perovskites with B=Ge*

Pb-free halide perovskites have recently attracted interest. In Phys.Rev.Mat.2,063605(2018) we proposed the use of Si & Ge as B sites cation and found that these perovskites undergo an off-centering of the group-IV atom leading to a rhombohedrally distorted perovskite with band structure similar to those of the cubic ones. However, for some of these materials, like RbGeX3, a monoclinic structure has been found experimentally.
In this work, we show that the monoclinic structure and the rhombohedral ferroelectric structure are closely related. In fact, if one considers these structures as GeX3 tetrahedrons rather than corner sharing octahedrons, we find that the monoclinic and rhombohedral structure to have parallel and oppositely oriented tetrahedral units. Thus the monoclinic structure can be understood as an anti-ferroelectric arrangement of the GeX3 dipoles. They have a completely different band topology with a wide band gap compared to that of the cubic ones yet are energetically close to each other. We understand this energy difference using a spin-1/2 hamiltonian for dipole interactions, favouring the antiferroelectric alignment of the GeX3 dipoles, and additional Ge-X bonding due to parallel tetrahedron orientation causing the ferroelectric alignment.