Theory of Rashba coupling mediated superconductivity in incipient ferroelectrics.

Experimental evidence suggest that superconductivity in SrTiO$_3$ is mediated by a soft transverse ferroelectric mode which, according to conventional theories, has negligible coupling with electrons. A phenomenological Rashba type coupling has been proposed on symmetry arguments but a microscopic derivation is lacking. Here we fill this gap and obtain a linear coupling directly from a minimal model of the electronic structure. We find that the effective electron-electron pairing interaction has a strong momentum dependence. This yields an unusual situation in which the leading s-wave channel is followed by a sub-leading triplet channel which shows a stronger pairing instability than the singlet d-wave state. The bare Rashba coupling constant is estimated for the lowest band of doped SrTiO$_3$ with the aid of first-principles computations and found to be much larger than previously thought. We argue that although for a uniform system the BCS coupling $\lambda$ is small, it can produce the right order of magnitude for $T_c$ in the presence of structural inhomogeneities.