An enhanced Fourier law derivable from the Boltzmann transport equation and a sample application in determining the mean-free path of nondiffusive phonon modes

Ramu, Ashok T.; Ma, Yanbao

doi:10.1063/1.4894087

Cited by 21 publications

(17 citation statements)

References 24 publications

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“…The quasiparticle distribution N Q is driven by external heating at a rate P ( r, t). This driving is described by a term in the PBE which has only recently been discussed [15][16][17][18] . The heating P causes the temperature to have new spatial variation T = T 0 + ∆T ( r, t).…”

Section: Introductionmentioning

confidence: 99%

Temperature in a Peierls-Boltzmann treatment of nonlocal phonon heat transport

Allen

Perebeinos

2018

Phys. Rev. B

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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]

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Section: Introductionmentioning

confidence: 99%

Temperature in a Peierls-Boltzmann treatment of nonlocal phonon heat transport

Allen

Perebeinos

2018

Phys. Rev. B

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“…This simulation is on the basis of the Fourier law for better accessibility, but more sophisticated simulation models on the basis of the Boltzmann transport equation will give more precise evaluation of thermal conductivity in such semi-ballistic phonon transport systems. 20,21 The effect of the geometry of the NWs and 2D PnC systems on thermal conductivity was also investigated, from the viewpoint of nanoscale phonon transport. Figure 2 shows thermal conductivities for NWs and 2D PnC nanostructures with a variety of widths and neck sizes.…”

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confidence: 99%

Thermal phonon transport in silicon nanowires and two-dimensional phononic crystal nanostructures

Nomura

Nakagawa

Kage

et al. 2015

Applied Physics Letters

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Thermal phonon transport in silicon nanowires (Si NWs) and two-dimensional phononic crystal (2D PnC) nanostructures was investigated by measuring thermal conductivity using a micrometer-scale time-domain thermoreflectance. The impact of nanopatterning on thermal conductivity strongly depends on the geometry, specularity parameter, and thermal phonon mean free path (MFP) distribution. Thermal conductivities for 2D PnC nanostructures were found to be much lower than that for NWs with similar characteristic length and surface-to-volume ratio due to stronger phonon back scattering. In single-crystalline Si, PnC patterning has a stronger impact at 4 K than at room temperature due to a higher specularity parameter and a longer thermal phonon MFP. Nanowire patterning has a stronger impact in polycrystalline Si, where thermal phonon MFP distribution is biased longer by grain boundary scattering.

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“…A McKelveyShockley flux method was adapted to describe phonon transport [20]. Analyses within the framework of the BTE have been recently reported in a number of studies [3,[21][22][23]. Most of the works simplified the BTE either by assuming a gray model or by asymptotic expansion of the spatial part of the phonon distribution.…”

Section: Introductionmentioning

confidence: 99%

Heat dissipation in the quasiballistic regime studied using the Boltzmann equation in the spatial frequency domain

Hua

Minnich

2018

Phys. Rev. B

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Quasiballistic heat conduction, in which some phonons propagate ballistically over a thermal gradient, has recently become of intense interest. Most works report that the thermal resistance associated with nanoscale heat sources is far larger than predicted by Fourier's law; however, recent experiments show that in certain cases the difference is negligible despite the heaters being far smaller than phonon mean-free paths. In this work, we examine how thermal resistance depends on the heater geometry using analytical solutions of the Boltzmann equation. We show that the spatial frequencies of the heater pattern play the key role in setting the thermal resistance rather than any single geometric parameter, and that for many geometries the thermal resistance in the quasiballistic regime is no different than the Fourier prediction. We also demonstrate that the spectral distribution of the heat source also plays a major role in the resulting transport, unlike in the diffusion regime. Our work provides an intuitive link between the heater geometry, spectral heating distribution, and the effective thermal resistance in the quasiballistic regime, a finding that could impact strategies for thermal management in electronics and other applications.

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An enhanced Fourier law derivable from the Boltzmann transport equation and a sample application in determining the mean-free path of nondiffusive phonon modes

Cited by 21 publications

References 24 publications

Temperature in a Peierls-Boltzmann treatment of nonlocal phonon heat transport

Temperature in a Peierls-Boltzmann treatment of nonlocal phonon heat transport

Thermal phonon transport in silicon nanowires and two-dimensional phononic crystal nanostructures

Heat dissipation in the quasiballistic regime studied using the Boltzmann equation in the spatial frequency domain

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