Thermal resistivity of die ectric crystals due to four-phonon processes and optical modes

Abstract

The interaction rate of four-phonon processes at high temperatures is derived in terms of the Grüneisen constant and its dilational derivative. The interaction arises in part from quartic anharmonicities and in part from cubic anharmonicities to second order. The four-phonon relaxation rate varies as the square of the frequency and the square of the temperature. It is compared to the three-phonon relaxation rate and found to be weaker by at least a factor 30 at 1000 K. The anharmonic interaction between two acoustic phonons and one optical phonon does not only limit the lifetime of optical phonons, but also contributes to the relaxation rate of acoustic modes in thermal conduction. It is shown that this relaxation rate is similar in magnitude and temperature dependence to the relaxation rate due to processes involving three acoustic phonons. The frequency dependence is also similar except at low frequencies, since the interaction is forbidden at lowest frequencies. Neither of the interactions considered here can quantitatively explain the observation that the thermal resistance of many dielectric crystals at high temperatures varies more rapidly than linearly with temperature.