Metallicity-driven polar transitions in topological epilayers

Polar metals, materials that exhibit both electric polarization and high conductivity, can also host topological phases. Because free carriers strongly suppress distortive polar order and change the Fermi level, controlling charge dynamics is crucial for simultaneously tuning ferroelectric and topological phases in the same material. Here, we explore the experimental conditions that enable access to these phases in bismuth-doped Pb$_{1-x}$Sn$_x$Te epilayers. For samples in the topological phase at $x = 0.5$, we use terahertz time-domain spectroscopy to evaluate their complex permittivity as a function of temperature. We observe a non-monotonic variation in carrier concentration with bismuth doping, indicating a change in carrier type. By tracking the transverse optical phonon mode, we identify a ferroelectric phase transition when distortive polar order emerges below a critical temperature that depends on carrier concentration. We show that bismuth doping controls the metallicity-dependent order parameters in the softening and hardening phases. Our work demonstrates a tunable platform for engineering exotic states of matter that integrate metallicity, ferroelectricity and topology.