Quantum ferroelectric instabilities in superconducting SrTiO$_3$

We examine the effects of strain and cation substitution on the superconducting phase of polar semiconductors near a ferroelectric quantum phase transition with a model that combines a strong coupling theory of superconductors with a standard microscopic framework for displacive polar modes coupled to strain degrees of freedom. Our calculations reveal that the superconducting transition temperature $T_c$ is enhanced by proximity to the ferroelectric instability from the disordered side, while it is generally suppressed in the ordered phase due to its increase in dielectric stiffness and a reduction of critical fluctuations from dipolar induced anisotropies. The condensation of the pairing phonon excitations generates a kink in $T_c$ at a charge density that is generally lower than that of the quantum critical point (QCP) and where both superconducting and ferroeletric orders set-in. We apply our model to SrTiO$_3$ and find that the anti-adiabatic limit places the kink nearly at its QCP. As the QCP is pushed to higher charge densities with either tuning parameter, we find that the dome narrows and sharpens. Our model is in qualitatively and fair quantitatively agreement with the recent observation of overlapping ferroelectric-like and superconducting instabilities in n-doped Sr$_{1-x}$Ca$_x$TiO$_3$ and strain tunning of $T_c$ in n-doped SrTiO$_3$. We compare our results to previous models invoking order-disorder lattice dynamics to describe the pairing excitations.