Lattice dynamics, structural stability, and phase transitions in incommensurate and commensurateA2BX4materials

Numerous materials with the general formula ${\mathit{A}}_2$ ${\mathit{BX}}_4$, where A and B are cations and X is the anion, are isomorphous to β-${\mathrm{K}}_2$ ${\mathrm{SO}}_4$ (space group Pnam) at high temperatures. A considerable number of them exhibit a structural instability leading to an incommensurately modulated phase with identical superspace symmetry. ${\mathrm{K}}_2$ ${\mathrm{SeO}}_4$ is the archetypical example. From the analysis of the incommensurate structures, the polarization vector of the unstable frozen mode can be determined, being similar in all investigated compounds. We report here a comparative energetic study and lattice-dynamics analysis of a set of compounds in this family. The set of materials ${\mathrm{K}}_2$ ${\mathrm{SO}}_4$, ${\mathrm{Rb}}_2$ ${\mathrm{SeO}}_4$, ${\mathrm{Cs}}_2$ ${\mathrm{SeO}}_4$, ${\mathrm{Cs}}_2$ ${\mathrm{ZnCl}}_4$, ${\mathrm{Cs}}_2$ ${\mathrm{ZnBr}}_4$, ${\mathrm{K}}_2$ ${\mathrm{CrO}}_4$, ${\mathrm{K}}_2$ ${\mathrm{SeO}}_4$, ${\mathrm{K}}_2$ ${\mathrm{ZnCl}}_4$, ${\mathrm{Rb}}_2$ ${\mathrm{ZnCl}}_4$, and ${\mathrm{Rb}}_2$ ${\mathrm{ZnBr}}_4$, includes compounds with and without an incommensurate phase.