Analytic solution of the BCS gap equation in D dimensions (D = 1, 2, 3), at finite temperatures

摘要

The Bardeen-Cooper-Schrieffer (BCS) gap equation is solved analytically in one, two and three dimensions, for temperatures close to zero and T_c. We work in the weak coupling limit, but allow the interaction width v ≡ hω_m/E_F to lie in the interval (0, ∞) Here, hω_m is the maximum energy of a force-mediating boson, and E_F denotes the Fermi energy. We obtain expressions for T_c and ΔC, the jump in the electronic specific heat across T = T_c, in the limits v 1 (the usual phonon pairing) and v > 1 (non-phononic pairing). This enables us to see how T_c scales with the mediating boson cut off. Our results predict a larger jump in the specific heat for the case v > 1, compared to v 1. We also briefly touch upon the role of a van Hove singularity in the density of states.