Lattice Modes in Ferroelectric Perovskites: PbTiO3

A complete study of the lattice-dynamical behavior of the ferroelectric tetragonal perovskite PbTi${\mathrm{O}}_3$ has been carried out using Raman spectroscopy. The temperature dependence of all the long-wavelength mode frequencies are determined. We show that a damped-harmonic-oscillator model with a frequency-independent damping coefficient is sufficient to explain the observed shape of the soft $E\left(1\mathrm{}\mathrm{TO}\right)$ mode in the ferroelectric phase at all temperatures below $T_c=493\mathrm{}$ °C. Moreover, by combining our experimental mode-frequency data with pyroelectric measurements of the change in spontaneous polarization $P_s$ with temperature, we obtain values of $P_s\left(25\mathrm{} {}_{}{}^{\circ }{}_{}{}^{}\mathrm{C}\right)=81\mathrm{}$ μC/${\mathrm{cm}}^2$ and $P_s\left(T_c\right)=42\mathrm{}$ μC/${\mathrm{cm}}^2$. This is in excellent agreement with a recent direct experimental measurement. The temperature dependence of all the mode strengths has also been determined to $T_c$. These results are used to extract the temperature dependence of the clamped dielectric constants. In PbTi${\mathrm{O}}_3$ the dielectric constant along the ferroelectric axis ${\epsilon }_c$ is determined primarily by the lowest frequency $A_1\left(1\mathrm{}\mathrm{TO}\right)$ mode at all temperatures to $T_c$, in contrast to BaTi${\mathrm{O}}_3$ where in the ferroelectric phase the lowest mode determines only $\approx 25\%$ of ${\epsilon }_c$. In PbTi${\mathrm{O}}_3$ the dielectric constant perpendicular to the $c$ axis ${\epsilon }_a$ is also determined by the lowest $E\left(1\mathrm{}\mathrm{TO}\right)$ mode at all temperatures. The possibility of observing critical effects near $T_c$ in the soft-mode data of PbTi${\mathrm{O}}_3$ has been examined. These effects are not observed. Also, it is shown that very careful fitting of the soft-mode temperature dependence to a functional form containing a minimum number of separately determined parameters is required before critical effects can be invoked in first-order phase transitions of the displacive type.