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Terahertz lattice and charge dynamics in ferroelectric semiconductor Sn$_x$Pb$_{1-x}$Te

The symmetry breaking induced by the ferroelectric transition often triggers the emergence of topological electronic states such as Weyl fermions in ferroelectric-like metals/semimetals. Such strong coupling between the lattice deformation and electronic states is therefore essentially important for the control of novel topological phases. Here, we study the terahertz lattice and charge dynamics in ferroelectric semiconductor SnxPb1-xTe thin films hosting versatile topological phases by means of the terahertz time-domain spectroscopy. With lowering the temperature, the resonant frequency of transverse optical phonon shows the significant softening and upturn. This temperature anomaly of lattice dynamics directly indicates the displacive-type ferroelectric transition. The resulting phase diagram suggests the enhancement of ferroelectricity in the films due to compressive strain compared with the bulk crystals. The soft phonon induces the large DC and terahertz dielectric constant even in metallic state. Furthermore, we find that the Born effective charge of soft phonon mode is enhanced at around the compositions showing the band gap closing associated with the topological transition.