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 ( r, t) = P0 exp(i k · r − iωt), 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 ( r, t) = P (x) from isolated evenly spaced segments with period L in x. The ratio L/ min is varied from 6 to ∞, where min is the minimum mean free path. The Debye phonons are assumed to scatter anharmonically with mean free paths varying as min(qD /q) 2 where qD is the Debye wavevector. The results illustrate the expected local (diffusive) response for min L, and a diffusive to ballistic crossover as 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 occurs in RTA. arXiv:1803.10757v2 [cond-mat.mes-hall]

Generalized Fourier's law for nondiffusive thermal transport: Theory and experiment

“…The first part represents a convolution between the temperature gradient along the ξ direction and a space-and time-dependent thermal conductivity κ μξ (ξ ). As reported previously, this convolution indicates the nonlocality of the thermal conductivity [13,43,46]. However, a second term exists that is determined by the inhomogeneous term originating from the boundary conditions and source terms.…”

“…dξ , is nonlocal, meaning the contribution at a given point is determined by convolving the heat source function with an exponential decay function with a decay length of μx . While the nonlocality of thermal conductivity was identified by earlier works on phonon transport [6,[11][12][13]20,39,43,54], the contribution from the inhomogeneous term has been neglected. Recently, Allen and Perebeinos [43] considered the effects of external heating and derived a thermal susceptibility based on the PBE that links external heat generation to temperature response and a thermal conductivity that links temperature response to heat flux.…”

“…Nondiffusive responses observed in experiments [4,6,10,[35][36][37] have been explained using the phonon-hydrodynamic heat equation [38], a truncated Levy formalism [39], a two-channel model in which lowand high-frequency phonons are described by the PBE and heat equation [40], and a McKelvey-Shockley flux method [41]. Methods based on solving the PBE under the relaxation time approximation (RTA), where each phonon mode relaxes towards thermal equilibrium at a characteristic relaxation rate, have been developed to investigate nonlocal transport in an infinite domain [13,[42][43][44], a finite one-dimensional slab [45,46], and experimental configurations such as transient grating [44,47] and thermoreflectance experiments [39,[48][49][50]. An efficient Monte Carlo scheme has been used to solve the PBE under the RTA in complicated geometries involving multiple boundaries [51][52][53].…”

Heat transport in semiconductor crystals: Beyond the local-linear approximation

“…Among the different studies, the one of non-diffusive and quasiballistic phonon transport in SC and dielectric crystals including the transition between the ballistic and diffusive regimes has attracted a growing interest in the last decade. 10,12,[14][15][16][17][57][58][59][60][61][62][63]65,[68][69][70][71] In fact, some recent works have shown that the average MFP of phonons in SC crystals can be up to two orders of magnitude longer than the prediction of the kinetic theory at room temperature. 5,6 This has raised the fundamental question about the real contribution of low frequency phonons to the effective thermal conductivity of SC crystals that has been measured by several photothermal methods, mainly time domain thermoreflectance (TDTR), 10,16,21,29,30,59,60,62,70 frequency domain thermoreflectance (FDTR), 22,25 and transient thermal grating (TTG) 17,20,31,35,41,45 techniques.…”

Non-local thermal transport modeling using the thermal distributor

“…Fourier's law requires generalization. Many theoretical studies of submicron thermal transport have been conducted in recent decades [4,[7][8][9][10][11]. The problem is that there is no rigorous thermodynamic definition of local temperature.…”