Four-Phonon Interactions among Acoustic and Optic Modes in Strontium Titanate

Abstract

A Hamiltonian used by Silverman to discuss electric susceptibility in the paraelectric phase of a ferro-electric material is here adapted to the calculation of the temperature dependence of thermal conductivity. First-order time-dependent perturbation theory is used to calculate relaxation times for longitudinal acoustic (L.A.), transverse acoustic (T.A.), and the lowest transverse optic (T.O.) branch, assuming the scattering is by four-phonon processes. These processes are selected as being the lowest order in which the temperature-dependent band of T.O. phonons of low wave number can cause appreciable scattering of other modes. In strontium titanate, it is found that four-phonon processes can indeed cause such large scattering of L.A. and T.O. modes that these modes may be absent from the heat current about 60°K. It is not, however, the participation of unstable, temperature-dependent modes which is responsible, but rather the fact that a degeneracy between L.A. and T.O. branches permits appreciable interaction of acoustic with optic phonons of low wave number. On the assumption that T.A. modes are the carriers, a thermal-conductivity-versus-temperature curve which is slightly convex downward is predicted between 32 and 70°K, in qualitative agreement with experiment. Because of the neglect of frequency dispersion, so that phonon frequencies depend linearly on wave number, the analysis is inapplicable to high temperatures. Thus only conjectures can be made as to the cause of the very slow decline of thermal conductivity with temperature above 70°K.