Electron-phonon mediated superconductivity in dilute systems

Reports of superconductivity in systems with very low carrier densities are becoming increasingly prevalent. Examples include strontium titanate, bismuth, and twisted bilayer graphene. Here, we investigate this general issue by studying superconductivity in the Bardeen-Pines model of an electron gas interacting with an optical phonon with frequency ω_L, as function of carrier concentration. The pairing interaction in the model is a combination of screened Coulomb repulsion and an attraction due to phonons, dressed by electron polarizability. We first focus on the attractive part of the interaction and argue that processes away from the Fermi surface cannot be neglected at low electron density. Contrary to expectations from BCS theory, we obtain a large increase of the critical temperature T_c as the system enters the dilute region, with a power-law dependence of T_c on ω_L. Next, we explore how the superconducting instability and the frequency dependence of the pairing function get modified once we add the Coulomb repulsion.
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