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 $\mu$ (equal to Fermi energy at $T=0$) 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 (ME) approximation becomes invalid when $\mu \le {\omega }_L$ 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, $\mu \to 0$, it holds that $T_c\propto {\omega }_L{\left(\frac{Ry}{{\omega }_L}\right)}^{\eta }$, where $\text{Ry}$ is the Rydberg constant and $\eta \sim 0.2$.

10 pages, 6 figures

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• Phonon-mediated spin transport in quantum paraelectric metals