In most superconducting materials including metals such as Al, Pb and Nb and metallic compounds such as Nb3Sn, the electron pair is induced by the phonon-mediated interaction between electrons. This interaction is simplified in the BCS theory and is replaced by a direct attractive force between electrons. Here we discuss a theory in which electron–phonon interaction is considered from the beginning. In so doing an approximation to the Coulomb repulsion between electrons is also taken into account. In band theory, which serves as the foundation of the theory of electrons in solid state physics, one considers the Schrödinger equation with periodic potential of ions resting on the crystal lattice together with the mean-field potential that incorporates to some extent the Coulomb interaction between electrons. As a result, one-electron states expressed by Bloch functions are obtained as the eigenfunctions. These states are specified by the band index, the wave number k in the first Brillouin zone and the spin α, and will be denoted simply by k, α. Furthermore, the spin index α is dropped in many cases since we will not treat a system with magnetism unless so stated. It is assumed in the following that the energy eigenvalue ξk is already known through the band theory.

Electron–phonon system

The motion of ions is treated as a lattice vibration. The normal mode is specified by the wave number q and the polarisation σ, and will be denoted simply by q.