Phonon mediated superconductivity in low carrier-density systems

Abstract

Motivated by the observation of superconductivity in SrTiO$_3$ and Bi, we analyze phonon-mediated superconductivity in three-dimensional systems at low carrier density, when the chemical potential $μ$ (equal to Fermi energy at $T=0$) is comparable to or even smaller than the characteristic phonon frequency $ω_L$. 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 (ME) approximation becomes invalid when $μ\leq ω_L$ 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 $μ/ω_L$. We find both analytically and numerically that $T_c$ increases as the ratio $μ/ω_L$ becomes smaller. In particular, in the dilute regime, $μ\rightarrow 0$, it holds that $T_c\proptoω_L\left(\frac{Ry}{ω_L}\right)^η$, where $\text{Ry}$ is the Rydberg constant and $η\sim 0.2$.