An amendment of the BCS theory of superconductivity
  Although the BCS theory of superconductivity is a well established theory, we have shown that the phenomenology predicted by this model is much richer than previously believed. By releasing the constraint that the attraction band is symmetric with respect to the chemical potential of the system, we observed that the energy gap may have more than one solution, the quasiparticle imbalance may appear in equilibrium, and the transition between the superconducting and the normal metal phases may be of the first order. The temperature of the superconductor-normal metal phase transition changes with the asymmetry of the attraction band and if we plot the phase transition temperature vs the chemical potential, we obtain a bell shaped curve, similarly to the superconducting dome, generally formed in high-Tc superconductors, but also in superconductors with narrow conduction bands.   While the pairing interaction is a microscopic characteristic of the system, determined by the effective interactions between constituent quasiparticles, the chemical potential is a macroscopic quantity, which can be changed by external conditions, like doping and pressure. Furthermore, if the conduction band of the system is narrow, then the attraction band is constrained to the conduction band and the chemical potential is not necessary in the center, as it happens in some of the bands in In-doped Pb$_z$Sn$_{1-z}$ and in MgB$_{2}$. For these reasons, the constraint that the attraction band is symmetric with respect to the chemical potential may be released. 
Dragos-Victor Anghel
cond-mat.supr-con