Electron-phonon coupling and a resonant-like optical observation of a band inversion in topological crystalline insulator Pb$_{1-x}$Sn$_x$Se

The optical reflectivity of Pb$_{0.865}$Sn$_{0.135}$Se and Pb$_{0.75}$Sn$_{0.25}$Se solid solutions was measured in the THz spectral region energetically corresponding to bulk optical phonon excitations and in the temperature range from 40 K to 280 K. The analysis of Pb$_{0.75}$Sn$_{0.25}$Se data performed within the dynamic dielectric function formalism revealed a new effect due to the electron-phonon coupling resulting in resonant changes of LO phonon frequency for energy gap equal to zero or to LO phonon energy. This effect is absent for Pb$_{0.865}$Sn$_{0.135}$Se that exhibits an open energy gap with trivial band ordering at all temperatures. These results show that reflectivity in the THz range constitute a versatile experimental method for precise determination of band inversion in narrow-gap topological materials. For Pb$_{0.75}$Sn$_{0.25}$Se the transition from trivial insulator to topological crystalline insulator phase takes place at temperature T$_0$ = (172 $\pm$ 2) K.