Transport waves as crystal excitations

Cepellotti, Andrea; Marzari, Nicola

doi:10.1103/physrevmaterials.1.045406

Cited by 28 publications

(34 citation statements)

References 35 publications

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“…Further developments of the LBTE combined with state-of-the-art first-principles simulations have also recently predicted the existence of hydrodynamic phenomena at non-cryogenic temperatures ( > ∼ 100 K) in graphene [30,39,40], in other 2D materials [30], in carbon nanotubes [41] and in graphite [42]. These theoretical suggestions have now been confirmed by the experimental evidence of second sound in graphite [43].…”

Section: Introductionmentioning

confidence: 78%

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Generalization of Fourier’s Law into Viscous Heat Equations

2020

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Heat conduction in dielectric crystals originates from the propagation of atomic vibrational waves, whose microscopic dynamics is well described by linearized or generalized phonon Boltzmann transport. Recently, it was shown that the thermal conductivity can be resolved exactly and in a closed form as a sum over relaxons, i.e. the collective phonon excitations that are eigenvectors of Boltzmann equation's scattering matrix [Cepellotti and Marzari, Phys. Rev. X 6, 041013 (2016)]. Relaxons have a well-defined parity and only odd relaxons contribute to the thermal conductivity. Here, we show that the complementary set of even relaxons determines another quantity -the thermal viscosity -that enters into the description of heat transport in the hydrodynamic regime, where dissipation of crystal momentum by Umklapp scattering phases out. We also show how the thermal viscosity and conductivity parametrize two novel viscous heat equations -two coupled equations for the local temperature and drift velocity fields -which represent the thermal counterpart of the Navier-Stokes equations of hydrodynamics in the linear, laminar regime. These viscous heat equations are derived from a coarse-graining of the linearized Boltzmann transport equation for phonons, and encompass both limits of Fourier's law or second sound for strong or weak Umklapp dissipation, respectively. Last, we introduce the Fourier deviation number, a dimensionless parameter that quantifies the steady-state deviations from Fourier's law due to hydrodynamic effects. We showcase these findings in a test case of a complex-shaped device made of silicon or diamond. This formulation generalizes rigorously Fourier's heat equation, and extends the reach of microscopic computational techniques to characterize the fundamental parameters governing heat conduction. * michele.simoncelli@epfl.ch ics. Sussmann and Thellung [12], starting from the linearized BTE (LBTE) in absence of momentumdissipating (Umklapp) phonon-phonon scattering events, derived mesoscopic equations in terms of temperature and phonon drift velocity, the thermal counterpart of pressure and fluid velocity in liquids. Further advances came from Gurzhi [13,14] and Guyer & Krumhansl [15,16] who, including the effect of weak crystal momentum dissipation, obtained equations for damped second sound and Poiseuille heat flow. Among early works, we also mention the discussions on phonon hydrodynamics using approaches different from the LBTE of Refs. [17,18]. While correctly capturing the qualitative features of phonon hydrodynamics, all the theoretical investigations mentioned above are heuristic, e.g. they assume simplified phonon dispersion relations (either power-law [13,14] or linear isotropic [12,15,16]), or neglect momentum dissipation. A more rigorous and general formulation -albeit valid only in the hydrodynamic regime of weak Umklapp scattering -was introduced by Hardy, who extended the discussion of second sound [19] and, together with Albers, of Poiseuille flow in terms of mesoscopic transport equations...

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Section: Introductionmentioning

confidence: 78%

“…In this section we show that drifting second sound [16,19,40,43], i.e. thermal transport in terms of a temperature damped wave, is described by the viscous heat equations.…”

Section: Appendix F: Second Soundmentioning

confidence: 98%

Generalization of Fourier’s Law into Viscous Heat Equations

2020

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“…The Callaway model has been widely employed to investigate many of the characteristic phenomena in hydrodynamic transport such as phonon Poiseuille flow [12,35], second sound [28,29,37], phonon Knudsen minimum [12,30] and phonon viscous flow [38]. Second sound refers to the propogation of heat in a phonon gas, in analogy to the propogation of ordinary sound waves in solids [39,40]. Second sound is of particular interet [40] in thermal transport since it is the most direct demonstration that heat can travel as waves, in contrast to the diffusion process underlying the Fourier heat conduction law.…”

Section: Second Sound Propagation Lengthmentioning

confidence: 99%

Umklapp scattering is not necessarily resistive

et al. 2018

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Since Peierls' pioneering work, it is generally accepted that phonon-phonon scattering processes consist of momentum-conserving normal scatterings and momentum-destroying Umklapp scatterings, and that the latter always induce thermal resistance. We show in this work that Umklapp scatterings are not necessarily resistive -no thermal resistance is induced if the projected momentum is conserved in the direction of heat flow. This distinction is especially important in anisotropic materials such as graphite and black phosphorous. By introducing a direction-dependent definition of normal and Umklapp scattering, we can model thermal transport in anisotropic materials using the Callaway model accurately. This accuracy is physically rooted in the improved description of mode-specific phonon dynamics. With the new definition, we predict that second sound might persist over much longer distance than previously expected.

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“…and Λ is a Lagrange multiplier, adjusted so that the distribution n * Q contains all the crystal momentum. That means Q qN Q = Q qn * Q (43) or, since the equilibrium distribution n Q has no net crystal momentum,…”

Section: Appendix C: Nonlocal Callaway Modelmentioning

confidence: 99%

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

Allen

2018

Phys. Rev. B

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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 j( r) = −κ ∇T ( r). The heat current j( r) depends not just on the local temperature gradient ∇T ( r), but also on temperatures at points r within phonon mean free paths, which can be micrometers long. This paper uses the Peierls-Boltzmann transport theory in nonlocal form to analyze the spatial variation ∆T ( r). The relaxation-time approximation (RTA) is used because full solution is very challenging. Improved methods of extrapolation to obtain the bulk thermal conductivity κ are proposed. Callaway invented an approximate method of correcting RTA for the 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 κ( k) depends on wavevector of the current j( k) and temperature gradient i k∆T ( k). GaN simulated T vs. x (Zhou et al.)2J J J

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Transport waves as crystal excitations

Cited by 28 publications

References 35 publications

Generalization of Fourier’s Law into Viscous Heat Equations

Generalization of Fourier’s Law into Viscous Heat Equations

Umklapp scattering is not necessarily resistive

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

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