Theory of superconductivity mediated by Rashba coupling in incipient ferroelectrics

Experimental evidence suggests 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 microscopic 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 $p$-wave state which shows a stronger pairing instability than the $d$-wave state. The bare Rashba coupling constant is estimated for the lowest band of doped SrTiO$_3$ with the aid of first-principles computations. Extrapolating the estimation to the multi-band regime we show that it can explain bulk superconductivity. For densities where the Meissner effect has not been observed an extra softening due to inhomogeneities is needed to explain the zero resistance state.