Lattice dynamics of cubic PbTiO$_3$

The lattice dynamics of cubic PbTiO$_3$ has been investigated using inelastic neutron scattering. We found four kinds of soft modes in cubic PbTiO$_3$: (1) the TO modes toward the $\Gamma$ point, (2) the TA $\Lambda_3$ mode toward the $R$ point, (3) the TA $\Lambda_3$ mode around the midpoint (1/4,1/4,1/4), and (4) the TA branches in the entire range. Moreover, the TO $\Sigma_4$ branch becomes flat away from the zone center. The steep dispersion of the TO modes toward $\Gamma$ is isotropic and confined to the region $\xi<0.2$. The temperature dependence of the $\Gamma_{15}$ mode up to 1173 K is explained by a combination of the Lydanne-Sachs-Taller relation and the Curie-Weiss law. In contrast, the TA $\Lambda_3$ modes at the midpoint and $R$ point are weakly temperature dependent. The coexistence of the soft $\Gamma_{15}$ and $R_{25}$ modes is in agreement with the predicted phonon instability. The midpoint softening suggests the tendency toward forming a fourfold periodicity along the [1,1,1] direction. The energy of the TO $\Delta_5$ branch for cubic PbTiO$_3$ is considerably higher than that for Pb(Zn$_{1/3}$Nb$_{2/3}$)O$_3$. This indicates that the TO modes are dominated by the B-site atom motion.