Lattice dynamics of incipient ferroelectric rutile TiO2

Phonon dispersion curves in rutile Ti${\mathrm{O}}_2$ have been calculated with a rigid-ion model and a shell model. The shell model closely reproduces experimental data. The hardening of soft ferroelectric ${\Gamma }_1^{-}$ and ${\Gamma }_5^{+}$ TO modes with increasing temperature, which is observed experimentally, is found to be due to a small simultaneous decrease in effective ionic charge and oxygen polarizability. The decrease in both quantities can be understood in terms of fourth-order inter-ion and oxygen core-shell anharmonic interactions.