Theory of the Electronic Thermal Conductivity of Superconductors with Strong Electron-Phonon Coupling

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Abstract
The electronic thermal conductivity of superconductors is discussed without recourse to the quasi-particle approximation. The Kubo formula for the thermal conductivity is used as the starting point. This is first examined in the Hartree-Fock approximation in the Nambu space. It is shown that in the Eliashberg approximation of neglecting the momentum dependence of the electronic self-energy, the calculation of the thermal conductivity is reduced to a quadrature, involving however the complex energy gap and renormalization functions which are solutions of the Eliashberg gap equations at finite temperatures. These equations are given in an Appendix. The problem is also considered in the ladder approximation in the Nambu space, and a generalized Boltzmann equation is derived which includes corrections to the Hartree-Fock approximation corresponding to the replacement of the scattering by the transport cross section. It is shown that the standard Boltzmann equation for superconducting quasiparticles is obtained in the weak-coupling limit. No numerical calculations are performed in this paper, but a clear scheme for such calculations is outlined. Reasons for believing that such calculations will explain the anomalous drop in the electronic thermal conductivity of superconducting lead are given.