Ambient Pressure Structural Quantum Critical Point in the Phase Diagram of (Ca$_x$Sr$_{1-x}$)$_3$Rh$_4$Sn$_{13}$

The quasi-skutterudite superconductor Sr$_3$Rh$_4$Sn$_{13}$ features a pronounced anomaly in electrical resistivity at $T^*sim$138 K. We show that the anomaly is caused by a second-order structural transition, which can be tuned to 0 K by applying physical pressure and chemical pressure via the substitution of Ca for Sr. A broad superconducting dome is centred around the structural quantum critical point. Detailed analysis of the tuning parameter dependence of $T^*$ as well as insights from lattice dynamics calculations strongly support the existence of a structural quantum critical point at ambient pressure when the fraction of Ca is 0.9 (i.e., $x_c=0.9$). This establishes (Ca$_x$Sr$_{1-x}$)$_3$Rh$_4$Sn$_{13}$ series as an important system for exploring the physics of structural quantum criticality without the need of applying high pressures.