Thermal phonon boundary scattering in anisotropic thin films

Boundary scattering of thermal phonons in thin solid films is typically analyzed using FuchsSondheimer theory, which provides a simple equation to calculate the reduction of thermal conductivity as a function of the film thickness. However, this widely-used equation is not applicable to highly anisotropic solids like graphite because it assumes the phonon dispersion is isotropic. Here, we derive a generalization of the Fuchs-Sondheimer equation for solids with arbitrary dispersion relations and examine its predictions for graphite. We find that the isotropic equation vastly overestimates the boundary scattering that occurs in thin graphite films due to the highly anisotropic group velocity, and that graphite can maintain its high in-plane thermal conductivity even in thin films with thicknesses as small as ten nanometers. 1 ar X iv :1 50 8. 00 61 1v 1 [ co nd -m at .m es -h al l] 3 A ug 2 01 5 Thermal transport in thin solid films with thicknesses from tens of nanometers to micrometers is a topic of considerable importance, with such films being used in applications ranging from quantum well lasers to LED lighting. In these thin films, the phonon mean free paths (MFPs) are comparable to the film thickness, resulting in boundary scattering that reduces the thermal conductivity and inhibits heat dissipation. This thermal management problem is presently an important challenge in many devices such as GaN transistors. Heat transport along the in-plane direction of thin films is typically described using the Fuchs-Sondheimer equation, which is an analytical solution of the Boltzmann transport equation (BTE) for thin films with partially specular and partially diffuse walls. This equation was originally derived for electron transport and later extended to phonon thermal transport assuming an average phonon MFP, enabling the calculation of thermal conductivity as a function of the film thickness. Mazumder and Majumdar used a MonteCarlo method to study the phonon transport along a silicon thin film including phonon dispersion and polarizations. While the Fuchs-Sondheimer equation is in wide use, it cannot be applied to anisotropic transport in its typical form because it assumes the crystal of interest is isotropic. However, there are many situations in which boundary scattering occurs in thin anisotropic films, with the most familiar example being thin graphite films. Such films have been studied experimentally, and few-layer graphene films have been investigated as in heat spreaders for GaN transistors. Provided that the films are sufficiently thick that phonon dispersion modifications in the cross-plane direction can be neglected, mathematically describing thermal transport in anisotropic thin films requires a Fuchs-Sondheimer equation that is valid for any crystal, regardless of its anisotropy. Surprisingly, despite the simplicity of the derivation, no such equation has been reported to the best of our knowledge. Here, we report the generalization of Fuchs-Sondheimer theory to crystals with arbitrary anisotropies. We find that highly anisotropic solids with small group velocities along certain crystallographic directions experience minimal boundary scattering because the thermal conductivity reduction depends only on the component of the MFP normal to the boundary rather than the overall MFP. As a result, thin films of anisotropic crystals like graphite maintain their high thermal conductivity even as the film thickness becomes very small. This observation has important implications for heat spreading in electronic devices and the thermal conductivity of graphite foams.