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We present a neutron scattering study of phonons in single crystals of(Pb$_{0.5}$Sn$_{0.5}$)$_{1-x}$In$_x$Te with $x=0$ (metallic, butnonsuperconducting) and $x=0.2$ (nonmetallic normal state, butsuperconducting). We map the phonon dispersions (more completely for $x=0$) andfind general consistency with theoretical calculations, except for thetransverse and longitudinal optical (TO and LO) modes at the Brillouin zonecenter. At low temperature, both modes are strongly damped but sit at a finiteenergy ($\sim4$ meV in both samples), shifting to higher energy at roomtemperature. These modes are soft due to a proximate structural instabilitydriven by the sensitivity of Pb-Te and Sn-Te $p$-orbital hybridization tooff-center displacements of the metal atoms. The impact of the soft opticalmodes on the low-energy acoustic modes is inferred from the low thermalconductivity, especially at low temperature. Given that the strongestelectron-phonon coupling is predicted for the LO mode, which should be similarfor both studied compositions, it is intriguing that only the In-doped crystalis superconducting. In addition, we observe elastic diffuse (Huang) scatteringthat is qualitatively explained by the difference in Pb-Te and Sn-Te bondlengths within the lattice of randomly distributed Pb and Sn sites. We alsoconfirm the presence of anomalous diffuse low-energy atomic vibrations that wespeculatively attribute to local fluctuations of individual Pb atoms betweenoff-center sites.

Electron-phonon coupling and superconductivity in the doped topological-crystalline insulator (Pb$_{0.5}$Sn$_{0.5}$)$_{1-x}$In$_x$Te