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Superconductivity and ferroelectricity are typically incompatible because theformer needs free carriers, but the latter is usually suppressed by freecarriers, unless their concentration is low. In the case of strontium titanatewith low carrier concentration, unconventional superconductivity andferroelectricity were shown to be correlated. Here, we report theoretically andexperimentally evaluated Gr\"uneisen parameters whose divergence under tensilestress indicates that the dominant phonon mode that enhances thesuperconducting order is the ferroelectric transverse soft-mode. This findingrules out all other phonon modes as the main contributors to the enhancedsuperconductivity in strained strontium titanate. This methodology shown herecan be applied to many other quantum materials.

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