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The quasi-skutterudite superconductor Sr$_3$Rh$_4$Sn$_{13}$ features apronounced anomaly in electrical resistivity at $T^*\sim$138 K. We show thatthe anomaly is caused by a second-order structural transition, which can betuned to 0 K by applying physical pressure and chemical pressure via thesubstitution of Ca for Sr. A broad superconducting dome is centred around thestructural quantum critical point. Detailed analysis of the tuning parameterdependence of $T^*$ as well as insights from lattice dynamics calculationsstrongly support the existence of a structural quantum critical point atambient pressure when the fraction of Ca is 0.9 (i.e., $x_c=0.9$). Thisestablishes (Ca$_x$Sr$_{1-x}$)$_3$Rh$_4$Sn$_{13}$ series as an important systemfor exploring the physics of structural quantum criticality without the need ofapplying high pressures.

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