Thermal Conductivity of Superconducting Lead-Indium Alloys

The thermal conductivity in the mixed state $K_m$ of a series of lead-indium alloys containing from 3 to 21 at.% In was measured as a function of temperature and magnetic field. The temperature dependence of the thermal conductivities in the superconducting and normal states $K_s$ and $K_n$, respectively, was also measured in the range from 1.35 to 7.5 K. In the mixed state, the main emphasis was on the region near the upper critical field $H_{c2\mathrm{}}$ where Caroli and Cyrot found theoretically that the electronic thermal conductivity of dirty type-II superconductors varies linearly with applied magnetic field. In order to compare with their theory it was necessary to separate the electronic and lattice thermal conductivities, and to analyze the lattice contribution in terms of the scattering by boundaries, point defects, and conduction electrons. The field dependence of the lattice thermal conductivity was then theoretically calculated and subtracted from the experimentally measured field dependence of the total thermal conductivity. The experimental results are in good agreement in the dirty limit but large deviations are observed as the indium concentration is reduced. The phonon mean free path due to scattering by point defects was found to be in reasonably good agreement with the Klemens theory. The upper critical fields $H_{c2\mathrm{}}$ obtained from $K$-vs-$H$ curves were compared with the theory of Helfand and Werthamer. From the critical fields, the product of the electronic mean free path and the residual electrical resistivity was found to be 0.66×${10}^{-11\mathrm{}}$ Ω ${\mathrm{cm}}^2$. The coherence length in the pure limit ${\xi }_0$ was also computed from the data and a value of 1060 Å was found.