Modeling quasi-ballistic transient thermal transport with spatially sinusoidal heating: A McKelvey-Shockley flux approach

Abstract

Ballistic phonon effects, arising on length scales comparable to the mean-free-path, result in non-diffusive heat flow and alter the thermal properties of materials. Simple theoretical models that accurately capture non-diffusive transport physics are valuable for experimental analysis, technology design, and providing physical insight. In this work, we utilize and extend the McKelvey-Shockley (McK-S) flux method, a simple and accurate framework, to investigate ballistic effects in transient phonon transport submitted to a spatially sinusoidal heating profile, simulating a transient thermal grating. We begin by extending a previous McK-S formulation to include inelastic scattering, then obtain an analytical solution in the single phonon energy case (gray approximation), and after show how this approach can readily support a full phonon dispersion and mean-free-path distribution. The results agree with experimental data and compare very well to solutions of the phonon Boltzmann transport equation in the diffusive and weakly quasi-ballistic transport regimes. We discuss the role of ballistic and non-equilibrium physics, and show that inelastic scattering is key to retrieving the heat equation solution in the diffusive limit. Overall the McK-S flux method, which takes the form of a diffusion-like equation, proves to be a simple and accurate framework that is applicable from the ballistic to diffusive transport regime.

Phonon transport across a Si–Ge interface: The role of inelastic bulk scattering

“…The McK-S describes the spatial and temporal evolution of the forward-and reverse-moving phonon fluxes (along the transport direction, taken here as x). Instead of dealing with directed fluxes, it is possible to rewrite the governing equations directly in terms of temperature and heat current [38]. Temperature, in steady-state, is given by…”

“…The expression for T 0 (x) is obtained by imposing conservation of energy (see Ref. [38]). In the form given by Eq.…”

“…Separate BCs are adopted for the Si-Ge junction (discussed below). The effective, energy-integrated temperature and heat current are obtained using [38] T…”

“…This approach was shown to compare well to the more rigorous and computationally demanding BTE, particularly when solved using a full phonon dispersion [34][35][36][37][38].…”

Ab initio based investigation of thermal transport in superlattices using the Boltzmann equation: Assessing the role of phonon coherence

“…For instance, several works have reported a modified Fourier's Law to approximately capture the size effects in the superlattice, but these works cannot explicitly account for modal transmission and reflection at an interface. [41][42][43] Hua and Minnich presented a method to solve the BTE in a 2-layer system using a cosine series expansion, making the computation efficient enough to eliminate the gray assumption. 44 However, their work must be adapted to describe periodic superlattices and include modal transmission and reflection at the interfaces.…”