Analysis of nonlocal phonon thermal conductivity simulations showing the ballistic to diffusive crossover

Abstract

Simulations (e.g. Zhou et al., Phys. Rev. B 79, 115201 (2009)) show nonlocal effects of the ballistic/diffusive crossover. The local temperature has nonlinear spatial variation not contained in the local Fourier law $\vec{j}(\vec{r})=-\kappa \vec{\nabla }T(\vec{r})$. The heat current $\vec{j}(\vec{r})$ depends not just on the local temperature gradient $\vec{\nabla }T(\vec{r})$, but also on temperatures at points ${\vec{r}}^{\prime }$ within phonon mean free paths, which can be micrometers long. This paper uses the Peierls-Boltzmann transport theory in non-local form to analyze the spatial variation $\Delta T(\vec{r})$. The relaxation-time approximation (RTA) is used because full solution is very challenging. Improved methods of extrapolation to obtain the bulk thermal conductivity $\kappa$ are proposed. Callaway invented an approximate method of correcting RTA for the $\vec{q}$ (phonon wavevector or crystal momentum) conservation of N (normal as opposed to Umklapp) anharmonic collisions This method is generalized to the non-local case where $\kappa (\vec{k})$ depends on wavevector of the current $\vec{j}(\vec{k})$ and temperature gradient $i\vec{k}\Delta T(\vec{k})$.

10 pages and 5 figures. This new version replaces the versions from April 2017 and March 2018. The source term is included in the Boltzmann equation, for the heat driving the phonon distribution. This method agrees with other recent work, and gives a better fit to the simulation of kappa(k). Recent changes are a clearer presentation thanks to help from an anonymous referee

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Cited by in corpus (5)

• Generalization of Fourier's law into viscous heat equations
• Experimental demonstration of a generalized Fourier's Law for non-diffusive thermal transport
• Non-local linear-response functions for thermal transport computed with equilibrium molecular-dynamics simulation
• Non-local thermal transport modeling using the thermal distributor
• Thermal transport in crystals: from the quantum Dyson equation to mesoscopic phonon hydrodynamics