Phonon-mediated superconductivity in low carrier-density systems

Abstract

Motivated by the observation of superconductivity in ${\mathrm{SrTiO}}_{3}$ and Bi, we analyze phonon-mediated superconductivity in three-dimensional systems at low carrier density, when the chemical potential $\ensuremath{\mu}$ is comparable to or even smaller than the characteristic phonon frequency ${\ensuremath{\omega}}_{L}$. 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 $\ensuremath{\mu}\ensuremath{\le}{\ensuremath{\omega}}_{L}$, 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 $\ensuremath{\mu}/{\ensuremath{\omega}}_{L}$. We find both analytically and numerically that ${T}_{c}$ increases as the ratio $\ensuremath{\mu}/{\ensuremath{\omega}}_{L}$ becomes smaller. In particular, in the dilute regime, $\ensuremath{\mu}\ensuremath{\rightarrow}0$, it holds that ${T}_{c}\ensuremath{\propto}{\ensuremath{\omega}}_{L}{(\frac{Ry}{{\ensuremath{\omega}}_{L}})}^{\ensuremath{\eta}}$, where Ry is the Rydberg constant and $\ensuremath{\eta}\ensuremath{\sim}0.2$.