Theory of lattice thermal conductivity: Role of low-frequency phonons

Abstract

The lattice thermal conductivity arises from contributions by phonons of all frequencies. The mean free path l(ω) is limited mainly by three-phonon interactions, and l _i(ω)∝ω⁻² T ⁻¹ where ω is the phonon frequency, and T is the absolute temperature. Since the spectral specific heat varies as ω ², the spectral thermal conductivity is independent of frequency, and low frequencies play a larger role than they do in the heat content. The effect of additional scattering processes due to defects must be compared to intrinsic scattering, not just at the highest frequency, but over the full spectral range. This enhances the resistance due to grain boundaries and large obstacles, and reduces the effect of point defects. Some typical examples are discussed. The role of low-frequency phonons may be even further enhanced if longitudinal low-frequency phonons have their interaction with other phonons reduced by wave vector conservation. Such modes would then contribute substantially to the overall thermal conductivity, and this contribution would be sensitive to grain size and to large-scale defects. However, the mean free path must be consistent with ultrasonic attenuation data. This enhanced sensitivity may be observable.