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The quasi-skutterudite superconductors $A_3T_4$Sn$_{13}$ ($A$=Sr, Ca; $T$=Ir,Rh, Co) are highly tunable featuring a structural quantum critical point. Weconstruct a temperature-lattice constant phase diagram for these isovalentcompounds, establishing Ca$_{3}$Rh$_4$Sn$_{13}$ and Ca$_{3}$Co$_4$Sn$_{13}$ asmembers close to and far away from the structural quantum critical point,respectively. Deconvolution of the lattice specific heat and the electricalresistivity provide an approximate phonon density of states $F(\omega)$ and theelectron-phonon transport coupling function $\alpha_{tr}^2F(\omega)$ forCa$_{3}$Rh$_4$Sn$_{13}$ and Ca$_{3}$Co$_4$Sn$_{13}$, enabling us to investigatethe influence of the structural quantum critical point. Our results support thescenario of phonon softening close to the structural quantum critical point,and explain the enhancement of the coupling strength on approaching structuralinstability.

Soft phonon modes in the vicinity of the structural quantum critical point