Strong Coupling Theory of the Superconductivity and Ferroelectric Quantum Criticality in metallic SrTiO₃.

Doped SrTiO₃ (STO) is one of the most studied examples of quantum ferroelectric metals where superconductivity (SC) coexists with ferroelectric (FE) order. To date, there is no consensus on the mechanism leading to SC in STO. Pristine STO is paraelectric naturally close to a ferroelectric quantum critical point (QCP). Strain or chemical substitution drives STO through the QCP to the FE phase, which manifests itself in the softening of the transverse optical (TO) phonon mode, providing a possible mechanism for SC. In this work, we show that a linear coupling of electrons to the soft TO mode, generated by a dynamical Rashba effect, plays the dominant role in giving rise to SC in STO. We solve the strong coupling Eliashberg theory for quantum ferroelectric metals, where the microscopic parameters are extracted from the experimental data. The fermionic and bosonic self-energies allow us to address the critical region, where strong fluctuations are present. Our calculation reveals that the linear order can qualitatively explain the experimentally measured superconducting phase diagram, while an additional nonlinear (quadratic) coupling term is essential for an agreement with its finer features.