Damping of the softE-symmetry phonon in lead titanate

Abstract

The frequency and temperature dependence of the phonon damping function $\gamma (\omega ,T)$ of the soft $E$-symmetry phonon in PbTi${\mathrm{O}}_3$ has been determined using Raman scattering. From 40°K to room temperature the frequency of the lowest $E$ mode, ${\omega }_0\mathrm{}\left(T\right)\mathrm{}$, is nearly constant and permits the temperature dependence of $\gamma$ to be measured at constant frequency. Below room temperature, $\gamma ({\omega }_0,T)$ is proportional to the absolute temperature $T$ and extrapolates to zero damping at $T=0\mathrm{}$. The frequency dependence of $\gamma (\omega ,T_{\mathrm{rm}})$ was determined at room temperature using near-forward Raman scattering from polaritons. New results are obtained by reanalyzing previously reported data. The damping is found to increase for decreasing polariton frequencies down to about 53 ${\mathrm{cm}}^{-1\mathrm{}}$. Below the ferroelectric phase transition at 766°K, it is shown that the apparent divergence of $\gamma \left({\omega }_0\mathrm{}\right(T\left)\right)$ as $T\to T_c^{-}$ results mainly from this frequency dependence and a nearly linear dependence on temperature. At fixed frequency, $\gamma (\omega ,T)$ is proportional to the absolute temperature although smaller nonlinear contributions cannot be ruled out. This linear temperature dependence indicates that the dominant damping mechanism for the soft $E$ mode is due to cubic anharmonicity in the lattice potential. These results are interpreted in terms of the model described by Cowley for phase transitions arising from anharmonic decay. A review of experimental results concerning the temperature dependence of the damping of soft modes in related materials is included.