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 expression $\varepsilon_{0}/\varepsilon_{\infty}=|\omega_{LO}|^{2}/|\omega_{TO}|^{2}$. The theory provides a microscopic expression for $T_c$ as a function of physical parameters, including the mode specific Gr\"uneisen parameter. The theory also shows that $\omega_{TO} \sim (T_{c}-T)^{1/2}$, and again specifies the prefactors in terms of Gr\"uneisen parameter and fundamental physical constants. Using the generalized LST relation, the softening of the TO mode leads to the divergence of $\epsilon_0$ and to a polarization catastrophe at $T_c$. A quantitative microscopic form of the Curie-Weiss law is derived with prefactors that depend on microscopic physical parameters.