The role of the chemical potential in the BCS theory

•We revisit the BCS theory of superconductivity.•The pairing energy band may be asymmetric with respect to the chemical potential.•The superconductor–normal metal phase transition may be of the first order.•The quasiparticle population may be asymmetric in equilibrium.•A feature like the “superconducting dome” appears.

We study the effect of the chemical potential on the results of the BCS theory of superconductivity. We assume that the pairing interaction is manifested between electrons of single-particle energies in an interval [μ−ħωc,μ+ħωc][μ−ħωc,μ+ħωc], where μμ and ωcωc are parameters of the model—μμ need not be equal to the chemical potential of the system, denoted here by μRμR. The BCS results are recovered if μ=μRμ=μR. If μ≠μRμ≠μR the physical properties change significantly: the energy gap Δ is smaller than the BCS gap, a population imbalance appears, and the superconductor–normal metal phase transition is of the first order. The quasiparticle imbalance is an equilibrium property that appears due to the asymmetry with respect to μRμR of the single-particle energy interval in which the pairing potential is manifested.For μR−μμR−μ taking values in some ranges, the equation for Δ may have more than one solution at the same temperature, forming branches of solutions when Δ is plotted vs. μR−μμR−μ at fixed TT. The solution with the highest energy gap, which corresponds to the BCS solution when μ=μRμ=μR, ceases to exist if |μ−μR|≥2Δ0 (Δ0 is the BCS gap at zero temperature). Therefore the superconductivity is conditioned by the existence of the pairing interaction and also by the value of μR−μμR−μ.