  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.
