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

Abstract

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 depairing 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. The change in the structure of the Eliashberg equation due to a nonconstant density of states (DOS) and the consequent interplay between static and thermal disorder is analyzed in detail, with a central feature being the change in frequency dependence of the self-energy compared to a constant DOS solution. The effect of DOS structure on the superconducting transition temperature $T_c$, which is manifested in the defect dependence of $T_c$, is analyzed in detail. Further it is proposed that an extension of the self-consistent Eliashberg approach be extended above $T_c$ to determine the normal-state self-energy and thereby the electronic contribution to thermodynamic quantities. Phonon broadening is shown to affect the spin susceptibility at finite temperature. Reinterpretation of several of the anomalous properties of $A15\mathrm{}$ compounds in terms of the present theory is suggested. Several aspects of the theory are compared to experimental data for ${\mathrm{Nb}}_3$Sn.