Modeling micro- and nanoscale phonon transport using an improved spherical harmonic approximation to the Boltzmann transport equation

The phonon Boltzmann transport equation (BTE) is an accurate and reliable tool for modeling phonon transport from nano- to macroscale. However, its numerical solution remains computationally intensive, even under the relaxation time approximation. Inspired by the mathematical equivalence between the phonon BTE and the photon radiative transport equation, an approximate method, which succeeds in modeling thermal radiative transport, is proposed to capture the non-diffusive feature of phonon transport. In this framework, the diffusive component of the phonon intensity is calculated using the first-order approximation of the spherical harmonic expansion (P1 approximation) of the intensity, while the ballistic component is traced back to the boundary conditions. Transient and steady-state heat conduction problems in both thin films involving interfaces and a rectangular box are used to demonstrate the merits of this framework, as compared to other prevailing methods. In particular, analytical solutions are derived to explicitly demonstrate the time retardation effect, thereby resolving the unphysical instantaneous propagation inherent in the diffusion equation.