SELF-CONSISTENT PHONON TREATMENT OF SECOND-ORDER DISPLACIVE FERROELECTRIC

The high-temperature Curie-Weiss law of electric susceptibility χ can be explained by quartic interactions between a k∼ 0 optic mode, whose normal mode co-ordinate q0 is proportional to the polarization, and acoustic vibrations of the lattice. These interactions will either stabilize q0 at low temperature, preventing a ferroelectric transition, or else such a transition will occur. To obtain coexisting paraelectric and ferroelectric phases at 0 °K, with distinct χ vs. T curves for each phase as observed by Saifi and Cross in SrTiO3, one must invoke anharmonic interactions between q0 and low-lying, temperature-dependent optic modes. These interactions are treated by the self consistent phonon method, which introduces an effective harmonic Hamiltonian Hh and calculates the Helmholtz function F to first order in H-Hh. The approximation is optimized by minimizing F with respect to the renormalized optical mode frequencies appearing in Hh, for which a system of integral equations is obtained by the minimization procedure. The condition that there are enough low-lying optic modes to produce coexisting phases leads to a prediction of weak optic dispersion at 0 °K. Under these circumstances, the self-consistent phonon equations can be approximated by a pair of coupled cubic equations. From the solution of these, we conclude that F has three minima at 0 °K in SrTiO3, corresponding to paraelectric and two oppositely polarized ferroelectric states. 180° domain walls must be few in number and immobile under a field, except in the presence of a high dislocation concentration. © 1969 Springer-Verlag.