Theory of the Instrinsic Electronic Thermal Conductivity of Superconductors

Abstract

The analog to the Bloch equation for the case of thermal conduction in a superconductor limited by phonon scattering is derived by introducing an appropriate general form for the nonequilibrium part of the distribution function into the corresponding Boltzmann equation. This integral equation for the deviation function is solved numerically for different temperatures $T$ by replacing it by sets of simultaneous linear equations with dimensions up to 39. The limiting curve for the deviation function when $T$ approaches the transition temperature $T_c$ from below turns out to be identical to the curve which has been reported by Klemens for the normal state. With $T$ decreasing below $T_c$ the maximum of the deviation function rises and shifts to higher energies. The ratio of the thermal conductivity in the superconducting state to that in the normal state, $\frac{{\kappa }_s}{{\kappa }_n}$, plotted against $\frac{T}{T_c}$ is found to increase monotonically and to have a limiting slope of about 1.62 at $T_c$. Consideration of the energy dependence of the energy gap in the case of lead yields a sizable effect on the plot of $\frac{{\kappa }_s}{{\kappa }_n}$ vs $\frac{T}{T_c}$.