Superconducting- and Normal-State Thermal Conductivity of Impure Tin

Low-temperature measurements of the normal- and superconducting-state thermal conductivities were made on ten tin specimens, one of which was pure (99.996%) and nine of which were lightly doped (up to 1 at.%) with mercury, lead, or bismuth. The ratios $\frac{K^s}{K^n}$ of the superconducting- to normal-state thermal conductivity are used to analyze the data. The normal-state thermal conductivity is assumed to consist of a fractionally small lattice component ${K_g}^n$ consistent with the "universal-curve" formalism of Lindenfeld and Pennebaker, added to a much larger electronic component of the Wiedemann-Franz type. The super-conducting lattice thermal conductivity ${K_g}^s$ is assumed to be simply related to ${K_g}^n$ in a manner roughly independent of impurity concentration. Proceeding in this manner, it is shown that the variation of $\frac{K^s}{K^n}$ with changing electronic mean free path is consistent with a normal-state lattice conductivity having a temperature dependence similar to that observed by other investigators on other alloy systems. Furthermore, if one quantitatively adopts the "universal-curve" formalism, it is seen that the analysis yields a temperature-dependent ratio of lattice conductivities, $\frac{{K_g}^s}{{K_g}^n}$ which is consistent with the theory of Bardeen, Rickayzen, and Tewordt. The thermal conductivities of the pure and the three lowest impurity samples are mostly electronic, and thus it is possible to compare their $\frac{K^s}{K^n}$ ratio with the theoretical $\frac{{K_e}^s}{{K_e}^n}$ ratio of Kadanoff and Martin, calculated for an isotropic gap. The pure-sample data fit the theory with a value of $3.3k_B{T}_c$ for the superconducting energy gap. However, a value of $3.9k_B{T}_c$ is found for the gap for the three impure samples.