Kohn-Luttinger Correction to Tc in a Phonon Superconductor

Abstract

Weak coupling theory predicts the critical temperature of a phonon superconductor to be $T_c = 1.13 e^{-3/2} ω_D e^{-{1/λ}}$, where $ω_D$ is the Debye frequency, $λ$ is the dimensionless electron-phonon coupling constant, and the factor $e^{-3/2}$ comes from fermionic self-energy and frequency dependence of the interaction. Other corrections are small either in $ω_D/E_F$, by Migdal's theorem, or in $λ$. However, this formula assumes that $ω_D \ll E_F$, where $E_F$ is the Fermi energy. We obtain $T_c$ in the dilute regime, when the Fermi energy is smaller than $ω_D$. We argue that in this situation Migdal's theorem is no longer valid, and Kohn-Luttinger-type corrections to the pairing interaction must be included to obtain the correct prefactor for $T_c$.