Lattice Thermal Conductivity and Deviations from Matthiessen's Rule for Dilute Alloys of Tin with Cadmium

Abstract

The thermal and electrical conductivities of six single crystals of tin with cadmium (cadmium content ranging from 0.24 to 0.97 at.%) have been measured from 4.2 to about 80 K. The lattice thermal conductivity was deduced for three of these samples of cadmium content in excess of 0.70 at.%, oriented close to 78° with respect to the tetragonal axis. Values of the ideal thermal resistivity of pure tin were needed to effect the separation into lattice and electronic components at higher temperatures, and it was assumed that deviations from Matthiessen's rule for thermal resistivity were governed by measured electrical-resistivity deviations and the Wiedemann-Franz law. The lattice thermal conductivity was analyzed in terms of phonon scattering by electrons, point defects, and anharmonic phonon interactions. Below 12 K the three samples gave a lattice thermal conductivity proportional to $T^2$ and of magnitude $1.5\times {10}^{-4\mathrm{}}{T}^2$ W ${\mathrm{cm}}^{-1\mathrm{}}$ ${\mathrm{K}}^{-1\mathrm{}}$, rising to a maximum of about 0.045 W ${\mathrm{cm}}^{-1\mathrm{}}$ ${\mathrm{K}}^{-1\mathrm{}}$ near 40 K. Deviations from Matthiessen's rule for the electrical resistivity were obtained for all six samples. These deviations show a $T^2$ dependence at low temperatures. This is inconsistent with a twoband model but suggests phonon-assisted impurity scattering. A theoretical difficulty associated with this mechanism is discussed.