Tilt and acoustic instabilities in ABX₄, A₂BX₄ and ABX₃ perovskite structure types: their role in the incommensurate phases of the organic–inorganic perovskites

An examination of the tilt modes and other low-frequency modes is made for an isolated, untilted perovskite layer, which maps very simply to the ABX₄ perovskites. A sheet of pure tilts exists at the Brillouin Zone boundary at {ξ, ½, ζ}. An instability is also found at all wavevectors which can be described as continuously varying from pure tilts to pure layer displacements as a function of the wavevector. Analysis is extended by considering the stacking of layers in the I-centered A₂BX₄-layer perovskites. The effect of freezing in the commensurate tilt required to generate the Cmca tilt system, pertinent to the modulated phases of propylammonium salts, is examined. The zero-frequency modes are restricted to two planes in the Brillouin Zone. All of the observed wavevectors associated with modulated phases, and the commensurately tilted propylammonium tetrachlorocadmate, are consistent with this calculation. The effect of full three-dimensional connectivity is briefly reviewed for the true ABX₃ perovskites. While pure tilt incommensurates appear to be hypothetically possible, they do not appear to have been observed to date.