Distortion modes in halide perovskites: To twist or to stretch, a matter of tolerance and lone pairs

Using first-principles calculations, we show that $\mathrm{Cs}B{X}_{3}$ halides with $B=\text{Sn}$ or Pb undergo octahedral rotation distortions, while for $B=\text{Ge}$ and Si, they undergo a ferroelectric rhombohedral distortion accompanied by a rhombohedral stretching of the lattice. We show that these are mutually exclusive at their equilibrium volume although different distortions may occur as functions of lattice expansion. The choice between the two distortion modes is in part governed by the Goldschmidt tolerance factor. However, another factor explaining the difference between Sn and Pb compared with Ge and Si is the stronger lone-pair character of Ge and Si when forced to be divalent as is the case in these structures. The lone-pair chemistry is related to the off centering. While the Si-based compounds have not yet been synthesized, the Ge compounds have been established experimentally. As a final test of the importance of the tolerance factor we consider $\mathrm{RbGe}{X}_{3}$, which has smaller tolerance factor than the corresponding $\mathrm{CsGe}{X}_{3}$ because Rb is smaller than Cs. We find that it can lower its energy by both rotations or rhombohedral off-centering distortions but the latter lower the energy slightly more efficiently.