Hardening and softening of soft phonon modes in ferroelectric thin films

The renormalized phonon spectrum beyond the random phase approximation in ferroelectric thin films is obtained using a Green’s function technique and the transverse Ising model including anharmonic spin-phonon and phonon-phonon interactions. The temperature and film thickness dependence of the soft phonon mode energy ω is calculated for the case of positive and negative anharmonic spin-phonon interaction constants R and for different surface constants R_s. For R>0 the phonon mode energy increases with increasing temperature, T→T_C, whereas for R<0 the phonon mode decreases with increasing temperature. For the case of smaller surface constant compared to the bulk, R_s<R_b, we obtain that ω of the thin film is larger in comparison to that of the bulk, i.e., we obtain hardening of the soft mode. In the other case, R_s>R_b, the soft phonon mode energy of the thin films is smaller compared to the bulk, i.e., we obtain softening of the phonon mode. The first case could explain the behavior of the phonon mode, soft mode hardening in SrTiO₃ and (Ba,Sr)TiO₃ thin films obtained experimentally, whereas the second case is responsible for PbTiO₃ thin films. The dependence of the phase transition temperature on film thickness and different surface spin-spin J_s and spin-phonon interaction constants R_s is discussed, too. The spin-phonon interaction enhances the phonon damping of the thin film. The damping increases near T_c for the two cases R>0 and R<0. It is larger for thin films in comparison to the bulk. Taking into account the influence of an external electric field we obtain hardening of the soft phonon modes for R>0 and R<0.