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The role of the chemical potential in the BCS theory

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 $[μ- \hbarω_c, μ+ \hbarω_c]$, where $μ$ and $ω_c$ are parameters of the model--$μ$ needs not be equal to the chemical potential of the system, denoted here by $μ_R$. The BCS results are recovered if $μ= μ_R$. If $μ\ne μ_R$ 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 $μ_R$ of the single-particle energy interval in which the pairing potential is manifested. For $μ_R - μ$ 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 $μ_R-μ$ at fixed $T$. The solution with the highest energy gap, which corresponds to the BCS solution when $μ= μ_R$, cease to exist if $|μ-μ_R| \ge 2Δ_0$ ($Δ_0$ 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 $μ_R - μ$.