Non-local phonon thermal conductivity and the ballistic to diffusive crossover
Year: 2016
Type: preprint
Abstract: The local Fourier relation between heat current to temperature gradient, J = -kdT/dr, does not hold on length scales shorter than carrier mean free paths. In insulating crystals, long phonon mean free paths enhance nonlocality of the kernel k(r,r') that relates a steady state heat current J(r) to remote temperature gradients dT(r')/dr'. If the system is spatially homogeneous, k(r,r') = k(r-r'), and in Fourier space, J(q) = -k(q)dT(q)/dr. A local relation has k(q) independent of q, or k(r-r')~delta(r-r'). In nanoscale systems, nonlocality, or equivalently, mixed ballistic/diffusive behavior, complicates heat transfer. Non-local information is starting to be measurable by modern sub-micron imaging methods. This paper derives the formula for k(q) from the Peierls-Boltzmann equation (PBE). There is a given applied thermal power P(k)=ikJ(k), which acts as a source term in the PBE. A new specific form of this source term is presented. Closed form results are obtained in relaxation time approximation (RTA). Different versions of the scattering rate 1/tau_Q are considered, and the Callaway version of the RTA is solved in the nonlocal case. The results are used to analyze a simple example derived from simulations on crystalline GaN. Simplified approximate formulas are presented which assist the analysis. The results provide a trove of new information.
Source: arXiv (Cornell University)
Author: Philip B. Allen
Language: English
Subfield: Materials Chemistry
Field: Materials Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2610486362