Effect of the Superconducting Energy-Gap Anisotropy on the Thermal Conductivity of Tin

Measurements of the normal- and superconducting-state thermal conductivities ($K^n$ and $K^s$) were made on one pure and eleven cadmium-doped tin single crystals in the temperature range $1°<T<\mathrm{}4\mathrm{}°$K. Cadmium concentrations ranged from 5×${10}^{-3\mathrm{}}$ to 0.3 at.%. The crystal orientations are tightly grouped in the near-perpendicular direction (direction of heat flow approximately perpendicular to the tin tetrad axis). The fractional change of the lattice conductivity, ${K_g}^s$, in the superconducting state and an effect in the electronic part of the conductivity ${K_e}^s$, reflecting the anisotropy of the superconducting energy gap, are studied as a function of impurity concentration. The variation of the size of the anisotropy effect with electron mean free path is found to be in reasonable agreement with the theoretical predictions of Ulbrich et al. and the ultrasonic attenuation results of Claiborne and Einspruch. However, the magnitude of the anisotropy effect in tin is found to be approximately twice as large as that predicted by theory. Unsuccessful attempts to bring the theoretical calculations into agreement with experiment by altering the angular features of the energy-gap function are discussed. The variation of $\alpha$, the temperature coefficient of the electron-phonon scattering term in the thermal resistivity expression, was studied as a function of impurity concentration and found to increase monotonically with concentration. Also the anisotropy of $\beta$, the temperature coefficient of the impurity scattering term, is shown to be similar to that of the residual resistivity. From the measured values of the residual resistivity, ${\rho }_0$, and $\beta \text{'}\mathrm{s}$ obtained from the normal-state thermal conductivities, the Lorentz number $L_0$ was found to be $L_{\exp }=2.49\mathrm{}\pm 0.08$ for these samples.