New phenomenology from an old theory—The BCS theory of superconductivity revisited

I analyze the low temperature limit of the BCS theory of s-wave single-band superconductors, when the attraction band (defined as the single-particle energy interval in which the pairing interaction is manifested) may be asymmetric with respect to the chemical potential. I discuss equilibrium systems, taking consistently into account the variation of the energy and of the total number of particles with the populations of the quasiparticle energy levels and I show that the equation for the energy gap may have several solutions. When the chemical potential is the center of the attraction band (the standard BCS assumption), one solution for the energy gap is Δ0 (the standard BCS gap, at zero temperature) and another one is Δ0∕3. If the absolute value of the difference between the center of the attraction band and the chemical potential is less than 2Δ0, then there are still two solutions, with energy gaps in the center of the attraction band and quasiparticle imbalance, which is present even at zero temperature. If the total number of electrons is conserved (not exactly, as in Richardson’s method, but on average, in the thermodynamic sense) and the attraction band is asymmetric, then one of the two solutions mentioned above cannot be realized.