Theory of a Structural Phase Transition in Perovskite-Type Crystals

Structural phase transitions in perovskite crystals involving displacements of the oxygen octahedra due to the condensation of linear combinations of the triply degenerate ${\Gamma }_{25}$ modes at the $R$ corner of the Brillouin zone are discussed with the help of a model Hamiltonian. These distortions are essentially rotations. We consider separately the cases of rotations of the octahedra about a cube axis (SrTi${\mathrm{O}}_3$) and about a cube diagonal (LaAl${\mathrm{O}}_3$). The temperature dependences of the distortion angle and of the frequencies of the soft ${\Gamma }_{25}$ optical modes have been calculated. An approximate self-consistently determined free-energy expression is given from which the internal energy and the specific heat are derived. The theory is compared with the experimental results on the transition from the cubic to the tetragonal phase in SrTi${\mathrm{O}}_3$ and to the trigonal phase in LaAl${\mathrm{O}}_3$.