Model for Lattice Thermal Conductivity at Low Temperatures

TLDR

Phonon scattering is modeled by frequency‑dependent relaxation times assuming isotropic, dispersion‑free crystal vibrations with no distinction between longitudinal and transverse modes. The study develops a phenomenological model to calculate lattice thermal conductivities at low temperatures. The model incorporates point‑impurity, normal, umklapp, and boundary scattering with normal‑process contributions derived from the Boltzmann equation and Herring’s relaxation time, and computes conductivity via numerical integration. For germanium, the model predicts a T^–3/2 dependence in normal material and T^–2 in single‑isotope material between 50–100 K, and numerical integration from 2–100 K yields conductivities that agree well with experimental data.

Abstract

A phenomenological model is developed to facilitate calculation of lattice thermal conductivities at low temperatures. It is assumed that the phonon scattering processes can be represented by frequency-dependent relaxation times. Isotropy and absence of dispersion in the crystal vibration spectrum are assumed. No distinction is made between longitudinal and transverse phonons. The assumed scattering mechanisms are (1) point impurities (isotopes), (2) normal three-phonon processes, (3) umklapp processes, and (4) boundary scattering. A special investigation is made of the role of the normal processes which conserve the total crystal momentum and a formula is derived from the Boltzmann equation which gives their contribution to the conductivity. The relaxation time for the normal three-phonon processes is taken to be that calculated by Herring for longitudinal modes in cubic materials. The model predicts for germanium a thermal conductivity roughly proportional to ${T}^{\ensuremath{-}\frac{3}{2}}$ in normal material, but proportional to ${T}^{\ensuremath{-}2}$ in single-isotope material in the temperature range 50\ifmmode^\circ\else\textdegree\fi{}-100\ifmmode^\circ\else\textdegree\fi{}K. Magnitudes of the relaxation times are estimated from the experimental data. The thermal conductivity of germanium is calculated by numerical integration for the temperature range 2-100\ifmmode^\circ\else\textdegree\fi{}K. The results are in reasonably good agreement with the experimental results for normal and for single-isotope material.