Phonon-mediated superconductivity in low carrier-density systems

Motivated by the observation of superconductivity in SrTiO3 and Bi, we analyze phonon-mediated superconductivity in three-dimensional systems at low carrier density, when the chemical potential mu is comparable to or even smaller than the characteristic phonon frequency omega(L). We consider the attractive part of the Bardeen-Pines pairing interaction, in which the frequency-dependent electron-phonon interaction is dressed by the Coulomb potential. This dressing endows the pairing interaction with momentum dependence. We argue that the conventional Migdal-Eliashberg approximation becomes invalid when mu <= omega(L), chiefly because the dominant contribution to pairing comes from electronic states away from the Fermi surface. We obtain the pairing onset temperature, which is equal to T-c in the absence of phase fluctuations, as a function of mu/omega(L). We find both analytically and numerically that T-c increases as the ratio mu/omega(L) becomes smaller. In particular, in the dilute regime, mu -> 0, it holds that T-c alpha omega(L) (Ry/omega(L))(eta), where Ry is the Rydberg constant and eta similar to 0.2.