Temperature in a Peierls-Boltzmann Treatment of Nonlocal Phonon Heat Transport

Abstract

In nonmagnetic insulators, phonons are the carriers of heat. If heat enters in a region and temperature is measured at a point within phonon mean free paths of the heated region, ballistic propagation causes a nonlocal relation between local temperature and heat insertion. This paper focusses on the solution of the exact Peierls-Boltzmann equation (PBE), the relaxation time approximation (RTA), and the definition of local temperature needed in both cases. The concept of a non-local "thermal susceptibility" (analogous to charge susceptibility) is defined. A formal solution is obtained for heating with a single Fourier component $P(\vec{r},t)=P_0\exp (i\vec{k}\cdot \vec{r}-i\omega t)$, where $P$ is the local rate of heating). The results are illustrated by Debye model calculations in RTA for a three-dimensional periodic system where heat is added and removed with $P(\vec{r},t)=P(x)$ from isolated evenly spaced segments with period $L$ in $x$. The ratio $L/{\ell }_{\min }$ is varied from 6 to $\infty$, where ${\ell }_{\min }$ is the minimum mean free path. The Debye phonons are assumed to scatter anharmonically with mean free paths varying as ${\ell }_{\min }(q_D/q{)}^2$ where $q_D$ is the Debye wavevector. The results illustrate the expected local (diffusive) response for ${\ell }_{\min }\ll L$, and a diffusive to ballistic crossover as ${\ell }_{\min }$ increases toward the scale $L$. The results also illustrate the confusing problem of temperature definition. This confusion is not present in the exact treatment but is fundamental in RTA.

12 pages including 7 figures. The newer version posted in August 2018 has a modified discussion and additional references

References in corpus (9)

• Distribution of phonon lifetime in Brillouin zone
• almaBTE: a solver of the space-time dependent Boltzmann transport equation for phonons in structured materials
• Towards More Accurate Molecular Dynamics Calculation of Thermal Conductivity. Case Study: GaN Bulk Crystals
• Superdiffusive heat conduction in semiconductor alloys -- I. Theoretical foundations
• The Big World of Nanothermodynamics
• Gauge invariance of thermal transport coefficients
• Heat dissipation in the quasiballistic regime studied using Boltzmann equation in the spatial frequency domain
• Transport Waves as Crystal Excitations
• Nonlocality in microscale heat conduction

Cited by in corpus (10)

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• Experimental demonstration of a generalized Fourier's Law for non-diffusive thermal transport
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• Analysis of nonlocal phonon thermal conductivity simulations showing the ballistic to diffusive crossover
• Analytical Green's Function of Multidimensional Boltzmann Transport Equation for Modeling Hydrodynamic Second Sound
• Analysis of ballistic transport and resonance in the $\alpha$-Fermi-Pasta-Ulam-Tsingou model
• Imaging heat transport in suspended diamond nanostructures with integrated spin defect thermometers
• Non-local thermal transport modeling using the thermal distributor
• Phonon Boltzmann equation non-local in space and time: the partial failure of the generalized Fourier law
• Thermal transport in crystals: from the quantum Dyson equation to mesoscopic phonon hydrodynamics