Coulomb repulsion among the many electrons in a metal is in a balance, which can be toppled by even a weak electron-phonon attractive interaction. Therefore neglecting the Coulomb term from the BCS reduced Hamiltonian has little effect on $T_c$. This is shown by a field-theoretic argument, an analysis based on the Bogoliubov model potential and a direct numerical calculation. Detailed knowledge about electrons and phonons for various materials can be incorporated into the BCS theory through a refined treatment of the self-consistent gap equation. Consequently the universal ratio 3.5 in the BCS theory is replaced by a range of values varying from 3.51 for Ga to 4.76 for Hg. It is found that the phonon cutoff frequency is much lower than the Debye frequency. Extraordinarily high $T_c$ could be expected if all phonons were involved in pairing electrons in a BCS superconductor.

• Received 18 November 2004

DOI:https://doi.org/10.1103/PhysRevB.71.134512