Strange thermal transport properties of (Ca$_x$Sr$_{1-x}$)$_{3}$Rh$_{4}$Sn$_{13}$ below the superconducting transition temperature

The quasi-skutterudite superconducting material family (Ca$_x$Sr$_{1-x}$)$_{3}$Rh$_{4}$Sn$_{13}$ have a composition induced structural quantum phase transition that can be suppressed to zero temperature. \footnote{S. Goh et al., Phys. Rev. Lett. \textbf {114}, 097002 (2015).} The resulting soft phonon modes bring about many intriguing effects, like increasing electron-phonon coupling and enhanced superconductivity. \footnote{W. C. Yu et al., Phys. Rev. Lett. \textbf {115}, 207003 (2015).} At temperatures well below the superconducting transition temperature $T_c$, we expect the thermal conductivity of the sample to be dominated by phonons, which in the boundary scattering limit, is proportional to $T^3$ - the temperature dependence of the heat capacity. Here, we discuss the thermal conductivity measurement of six samples of (Ca$_x$Sr$_{1-x}$)$_{3}$Rh$_{4}$Sn$_{13}$ spanning the phase diagram, and show that the thermal conductivity exhibits a temperature dependence that is well below $T^3$ near the critical doping. Through a comparison with other thermodynamic measurements as well as theoretical calculations, we consider the consequences of scattering from soft phonon modes near the critical region for heat transport.