Octahedral tilting instabilities in inorganic halide perovskites

Dynamic instabilities, stabilized by anharmonic interactions in cubic and tetragonal halide perovskites at high temperature, play a role in the electronic structure and optoelectronic properties of halide perovskites. In particular, inorganic and hybrid perovskite materials undergo structural phase transitions associated with octahedral tilts of the metal-halide octahedra. We investigate the structural instabilities present in inorganic $\mathrm{Cs}M{X}_{3}$ perovskites with Pb or Sn on the metal site and Br or I on the $X$ site. Defining primary order parameters in terms of symmetry-adapted collective displacement modes and secondary order parameters in terms of symmetrized Hencky strain components, we unravel the coupling between octahedral tilt modes and macroscopic strains as well as the role of $A$-site displacements in perovskite phase stability. Symmetry-allowed secondary strain order parameters are enumerated for the 14 unique perovskite tilt systems. Using first-principles calculations to explore the Born-Oppenheimer energy surface in terms of symmetrized order parameters, we find coupling between octahedral tilting and $A$-site displacements is necessary to stabilize $Pnma$ ground states. Additionally, we show that the relative stability of an inorganic halide perovskite tilt system correlates with the volume decrease from the high-symmetry cubic phase to the low-symmetry distorted phase.