Generalization of the theory of the electron-phonon interaction: Thermodynamic formulation of superconducting- and normal-state properties

A thermodynamic formulation for the electron self-energy is given which is applicable when the electronic spectrum possesses structure on the scale of phonon frequencies, provided only that the ratio of phonon phase velocity to electron Fermi velocity is small. Electron-phonon, Coulomb, and electron-defect interactions are included on an equal footing and it is shown that their different frequency dependencies lead to specific effects on the Eliashberg self-energy: (a) the Coulomb interaction contributes nothing of essence to the normal-state self-energy (in this isotropic approximation) but retains its usual de-pairing effect upon the superconducting gap function, (b) defects affect superconducting properties primarily through a broadening of the electronic spectrum, and (c) phonons contribute a thermal shift and broadening as well as the mass enhancement. A generalization to intensive electron-phonon, electron-electron, and electron-defect interaction constants is necessary to redevelop an intuition into the effects of these interactions