Two-fluid theory of thermal conductivity of dielectric crystals

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Abstract
A theory of lattice thermal conductivity is formulated in which phonons are divided into propagating and reservoir groups. At high temperatures, the groups are differentiated according to whether they occupy nondispersive modes below or dispersive modes above the U-process combination frequency threshold. Below this threshold, only N processes can be completed and it is assumed that there is no entropy production. Dissipation occurs in the rapidly relaxing reservoir modes above the threshold, and the heat current is carried by the more slowly relaxing phonons in the nondispersive modes. Appeal to the variational principle yields the result that the anharmonic relaxation rate for propagating modes is determined uniquely by the scattering into the reservoir modes. The theory carries over toward low temperatures with extrinsic resistivity providing reservoir relaxation in nondispersive modes as U processes die out. An explicit expression for the required anharmonic scattering rate for transitions to reservoir modes is adapted from existing acoustic absorption theory and combined with direct extrinsic scattering in the conventional way to provide the total relaxation time for the thermal conductivity. Minimum zone-boundary frequencies, maximum frequencies of nondispersive branches, a reservoir relaxation time, a Grüneisen gamma, and parameters for boundary and imperfection scattering are used as adjustable parameters to fit experimental thermal conductivities over broad temperature ranges for crystals of four major classes. Good to fair agreement is obtained for reasonable values of these parameters in all cases. Low-temperature relaxation times for N-process scattering into the extrinsic reservoir are in accord with conventional results except that they depend upon the purity of the specimen.

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