Urbach tail for ferroelectric materials with an order-disorder-type phase transition

We present a model describing the Urbach tail of exciton absorption spectra in ferroelectric materials with order-disorder-type phase transition. In addition to the thermal effect of exciton-lattice interactions, the internal Stark effect by the local electric field due to the displacement of protons is taken into account. The random distribution of the direction of local electric field induces a random polarization of the lowest exciton state. This results in an additional randomness of the exciton band in the disordered phase. The anomalous temperature dependence of the steepness parameter for the Urbach tail observed in ${\mathrm{PbHPO}}_4$ is reproduced by a numerical simulation within the present model.