Lattice Thermal Conductivity at Low Temperatures

The theoretical models of Callaway and Klemens are compared to experimental results in (a) silicon and diamond, (b) solidified argon and neon, and (c) KCl and NaF, which are taken from the literature. Three sources of scattering are postulated: phonon-phonon processes, isotopic point defect scattering, and boundary scattering, except in NaF where isotopic scattering is absent. The Callaway model, in all cases, gives a good fit and excellent agreement is obtained for diamond and solidified argon and neon. In the vicinity of the conductivity maxima the Callaway model gives a better fit with the experimental curve. The disagreement between the results of the two models is most pronounced in solidified argon and neon where the Debye temperature is low and Umklapp processes are predominant at the conductivity maximum. In diamond, where the Debye temperature is quite high and Umklapp processes are unimportant at the temperatures of interest the results of both models are more or less similar. The Casimir model for finding the characteristic length for the boundary scattering fails to explain the observed results in solidified argon and neon. This indicates the existence of microscale fluctuations in the composition of the solid. In NaF the characteristic length has to be decreased by a factor of about 5. For diamond the value of the parameter $A$, which gives the best fit with experiment, is a hundred times the value calculated from Klemens' expression for $A$. This supports the earlier suggestion that there occur clusters of point defects in diamond. Good agreement with the experimental results beyond the conductivity maximum is obtained by taking (${B}_{1}+{B}_{2}$) to be temperature independent in all the cases.