Mean free path spectra as a tool to understand thermal conductivity in bulk and nanostructures

A rigorous derivation is given for the thermal conductivity accumulation function and mean free path (MFP) spectrum of an isotropic bulk material. The key physical insight is to express the kinetic theory integral in terms of MFP rather than frequency. Extending this framework to incorporate boundary scattering in nanostructures leads to an integral equation that transforms a material's bulk MFP spectrum into the size-dependent thermal conductivity of the nanostructure. The kernel of this transform represents the boundary scattering rule for the particular type of nanostructure. The principal benefit of this transform is that it requires only a single function, the material's bulk MFP spectrum, or equivalently its accumulation function. Explicit knowledge of the material's dispersion relation and frequency-dependent bulk MFPs is not needed, nor is a summation over polarizations, because the bulk MFP spectrum already contains this information in exactly the form required to evaluate boundary scattering. The utility of this framework is demonstrated through a case study of six models for the phonon thermal conductivity of silicon: three analytical, one gray, and two numerical.