Soft mode theory of ferroelectric phase transitions in the low-temperature phase

Historically, the soft mode theory of ferroelectric phase transitions has been developed for the high-temperature (paraelectric) phase, where the phonon mode softens upon decreasing the temperature. In the low-temperature ferroelectric phase, a similar phonon softening occurs, also leading to a bosonic condensation of the frozen-in mode at the transition, but in this case the phonon softening occurs upon increasing the temperature. Here we present a soft mode theory of ferroelectric and displacive phase transitions by describing what happens in the low-temperature phase in terms of phonon softening and instability. A new derivation of the generalized Lyddane-Sachs-Teller (LST) relation for materials with strong anharmonic phonon damping is also presented which leads to the expressionɛ₀/ɛ_∞= |ω_LO|²/|ω_TO|². The theory provides a microscopic expression forT_cas a function of physical parameters, including the mode specific Grüneisen parameter. The theory also shows thatωTO∼(Tc-T)1/2, and again specifies the prefactors in terms of Grüneisen parameter and fundamental physical constants. Using the generalized LST relation, the softening of the TO mode leads to the divergence ofϵ₀and to a polarization catastrophe atT_c. A quantitative microscopic form of the Curie-Weiss law is derived with prefactors that depend on microscopic physical parameters.