Theory for dirty superconductors. I. Strong-coupling equations

A strong-coupling theory for dirty bulk superconductors is developed, which in particular constitutes a unifying description of most disorder effects considered so far separately within strong coupling or Bardeen-Cooper-Schrieffer–type formalisms. This is achieved by calculation of all six Hartree-Fock self-energy contributions (four exchange and two direct ones) within an exact eigenstate formalism, which yields disorder renormalization of vertices formally exactly in terms of electron-correlation functions. Compared with standard dirty-limit theory, this extends the region of validity substantially toward strong disorder. The most important points are that Hartree diagrams are crucial, and that one more renormalization function arising from the normal self-energy has to be introduced. A systematic procedure for calculation of the disorder-induced vertex corrections is given as well as a detailed discussion of how previous approaches are derived as special cases of the present theory. Among these are the increase of λ due to transverse phonons found by Keck and Schmid, and various effects discussed recently in connection with the $T_c$ degradation observed in high-$T_c$ superconductors. A preliminary discussion of new effects includes a novel diffusion enhancement of λ and $T_c$-decreasing effects due to the normal self-energy. All these effects are found to be of comparable magnitude. The structure of the theory favors disorder-induced $T_c$ degradation for strong coupling, and enhancement of $T_c$ for weak coupling.