Strong-Coupling Superconductivity. I

The pairing theory of superconductivity is extended to treat systems having strong electron-phonon coupling. In this regime the Landau quasiparticle approximation is invalid. In the theory we treat phonon and Coulomb interactions on the same basis and carry out the analysis using the nonzero-temperature Green's functions of the Nambu formalism. The generalized energy-gap equation thus obtained is solved (at $T=0\mathrm{}$ °K) for a model which closely represents lead and the complex energy-gap parameter $\Delta \left(\omega \right)$) is plotted as a function of energy for several choices of phonon and Coulomb interaction strengths. An expression for the single-particle tunneling density of states is derived, which, when combined with $\Delta \left(\omega \right)$, gives excellent agreement with experiment, if the phonon interaction strength is chosen to give the observed energy gap ${\Delta }_0$ at zero temperature. The tunneling experiments therefore give a detailed justification of the phonon mechanism of superconductivity and of the validity of the strong-coupling theory. In addition, by combining theory and the tunneling experiments, much can be learned about the electron-phon interaction and the phonon density of states. The theory is accurate to terms of order the square root of the electron-ion mass ratio, ${\mathrm{\sim }10}^{-2\mathrm{}}$-${10}^{-3\mathrm{}}$.