Anharmonic thermal resistivity of dielectric crystals at low temperatures

The intrinsic thermal resistance of dielectric crystals at low temperatures due to three-phonon Umklapp processes has been treated theoretically, taking account of the restrictions imposed by wave-vector selection and frequency conservation rules, as well as the known phonon-dispersion curves. Since low-frequency phonons cannot interact directly with zone-boundary phonons, the transfer of excess momentum takes place in two steps: a normal process with modes of intermediate frequency, followed by an Umklapp process of intermediate-frequency phonons with phonons near the zone boundary of two different polarization branches with different frequencies. The intermediate frequency is given by that frequency difference. The second step can also be scattering by point defects, so that the apparent Umklapp resistance can be modified by point defects, which scatter the intermediate-frequency phonons much more strongly than the thermal phonons. The important processes in different frequency ranges depend on the dispersion curves and on the residual point defects; various cases are discussed. The theory has only one adjustable parameter, the Grüneisen constant. The point-defect scattering strength is deduced from the departures of the thermal conductivity in the boundary scattering regime at very low temperatures from a ${\mathit{T}}^3$ dependence. The theory has been compared to measurements on LiF, NaF, diamond, Si, and enriched ${}_{}{}^{74}{}_{}{}^{}\mathrm{Ge}$. Reasonably good agreement with experimental data can be obtained without invoking an adjustable exponent in the Umklapp resistance.