The Flux Derivatives Formalism: the Guyer-Krumhansl equation from the Boltzmann equation for general semiconductors

We present a method, the flux derivatives formalism (FDF), to derive a hydrodynamic heat transport equation (Guyer-Krumhansl equation [1]) similar to Navier-Stokes for general semiconductors. This method is valid for material where momentum-preserving collisions dominate and also for kinetic materials dominated by resistive collisions. Therefore, it can be applied to semiconductors like silicon or germanium [2]. In constrast to the common belief, this kind of materials are well-described by the Guyer-Krumhansl equation (GKE), which generalizes the Fourier’s law.

This contrasts with other approaches, where efforts have been addressed to use Fourier’s law with effective coefficients depending on the physical situation [3,4] or through the phonon Boltzmann transport equation (BTE) [2,3], whose complexity only allows to face simple geometries. Our model allows the understanding of the experimentally observed Fourier’s law breakdown at small length and time scales by using geometry independent parameters calculated from ab initio, resulting in a predictive model for arbitrary complex geometries [5] or materials.

Furthermore, it supplies a connection between microscopic (phonons) and mesoscopic (temperature, heat flux, ...) variables with an explicit solution for the nonequilibrium phonon distribution function [1]. This not only allows to interpret in terms of phonons the transport equation, but also properly derive boundary conditions. These results give the possibility to go further in the study of the thermal boundary resistance (TBR) without attributing other phenomena to it and deepen the understanding the hydrodynamic nature of the thermal transport.