Giant Gr\"uneisen parameter in a strain-tuned superconducting quantum paraelectric: A consequence of the vanishing ferroelectric phonon energy

Superconductivity and ferroelectricity are typically incompatible because the former needs free carriers, but the latter is usually suppressed by free carriers, unless their concentration is low. In the case of strontium titanate with low carrier concentration, unconventional superconductivity and ferroelectricity were shown to be correlated. Here, we report theoretically and experimentally evaluated Gr\"uneisen parameters whose divergence under tensile stress indicates that the dominant phonon mode that enhances the superconducting order is the ferroelectric transverse soft-mode. This finding rules out all other phonon modes as the main contributors to the enhanced superconductivity in strained strontium titanate. This methodology shown here can be applied to many other quantum materials.