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∞= |ωLO|2/|ωTO|2. The theory provides a microscopic expression forTcas 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 atTc. A quantitative microscopic form of the Curie-Weiss law is derived with prefactors that depend on microscopic physical parameters.