Thermal Conductivity of Complex Dielectric Crystals

Abstract

The theory of the thermal resistivity of dielectric crystals at ordinary and high temperatures in terms of anharmonic three-phonon interactions is reformulated. The resistivity is similar in form to that obtained by Leibfried and Schl\"omann, but larger by a factor of 6.8. The theory is then extended to crystals where the unit cell contains many atoms. To include all three-phonon interactions one sums over all harmonics of the reciprocal-lattice vectors in an extended-zone representation. This sum increases the scattering rate. However, the matrix elements of the three-phonon processes are reduced in the case of large unit cells, because coherence is lost in the Fourier transform of the different bonds in each cell. A simplified model is chosen, and in this case the latter effect cancels the former, so that the anharmonic relaxation rate is substantially independent of the number of atoms per unit cell. However, the zone boundaries affect the phonon dispersion curves and reduce the group velocity of most modes. Using a model proposed by Slack, in which only the acoustic phonons of the fundamental zone contribute to the conductivity, and invoking the independence of the relaxation time with cell size here derived, the conductivity varies as the inverse cube root of the number of atoms per cell. The conductivity varies inversely with temperature, even if the phonon mean free path is shorter than the cell dimensions, because the major contribution to the anharmonic interaction comes from the highest harmonics of the fundamental reciprocal-lattice vectors.