Effective electron-electron interactions and the theory of superconductivity

Electron pairing is examined from a viewpoint which treats electron-electron interactions first (both in a single band and two-band context) and only later adds in electron-phonon coupling. We report solutions to the Eliashberg equation for the one- and two-band interacting electron gas, in the absence of phonons and then with phonons included, using effective electron-electron interactions that are closely constrained by sum rules. No s-wave pairing is found for the one-band case in the absence of phonons but higher angular momentum pairing is possible. In some contrast, intrinsic s-wave pairing is found for the two-band case, and again nonzero angular momentum pairing may arise. With the subsequent inclusion of phonons, but treated on a completely equal footing with electrons, transition temperatures of several simple metals are determined, and found to agree to within 20% of measured values. For low density systems, significant deviations from the predictions of the McMillan expression assuming μ≈0.1 are found. An important example is Li where we obtain ${\mathrm{T}}_{\mathrm{c}}$≈0.4 mK in sharp contrast with previous approximations which give ${\mathrm{T}}_{\mathrm{c}}$∼1 K, and are not supported by experiment.