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Thermal Transport in Crystals as a Kinetic Theory of Relaxons

Andrea Cepellotti and Nicola Marzari
Phys. Rev. X 6, 041013 – Published 17 October 2016; Erratum Phys. Rev. X 10, 049901 (2020)
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Abstract

Thermal conductivity in dielectric crystals is the result of the relaxation of lattice vibrations described by the phonon Boltzmann transport equation. Remarkably, an exact microscopic definition of the heat carriers and their relaxation times is still missing: Phonons, typically regarded as the relevant excitations for thermal transport, cannot be identified as the heat carriers when most scattering events conserve momentum and do not dissipate heat flux. This is the case for two-dimensional or layered materials at room temperature, or three-dimensional crystals at cryogenic temperatures. In this work, we show that the eigenvectors of the scattering matrix in the Boltzmann equation define collective phonon excitations, which are termed here “relaxons”. These excitations have well-defined relaxation times, directly related to heat-flux dissipation, and they provide an exact description of thermal transport as a kinetic theory of the relaxon gas. We show why Matthiessen’s rule is violated, and we construct a procedure for obtaining the mean free paths and relaxation times of the relaxons. These considerations are general and would also apply to other semiclassical transport models, such as the electronic Boltzmann equation. For heat transport, they remain relevant even in conventional crystals like silicon, but they are of the utmost importance in the case of two-dimensional materials, where they can revise, by several orders of magnitude, the relevant time and length scales for thermal transport in the hydrodynamic regime.

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  • Received 25 April 2016

DOI:https://doi.org/10.1103/PhysRevX.6.041013

Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Statistical Physics & ThermodynamicsGeneral PhysicsCondensed Matter, Materials & Applied Physics

Erratum

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Relaxons Heat Up Thermal Transport

Published 17 October 2016

A recasting of the theory that underlies thermal transport in electrical insulators relies on new vibrational modes called relaxons.

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Authors & Affiliations

Andrea Cepellotti and Nicola Marzari*

  • Theory and Simulations of Materials (THEOS), and National Centre for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne, Station 9, 1015 Lausanne, Switzerland

  • *nicola.marzari@epfl.ch

Popular Summary

Microscopic and atomistic heat transport theories have an illustrious history and have evolved from the transport equation that Ludwig Boltzmann first developed in 1872. A modern view of thermal transport was formulated by Rudolph Peierls, who applied the Boltzmann equation to the elementary vibrations of solids, i.e., phonons. Thanks to these accomplishments, it is possible to accurately study thermal transport under a wide variety of conditions. Still, one question has remained unanswered—what are the actual elementary carriers of heat? Textbooks often identify phonons themselves as the heat carriers, but that is just an approximation.

Some inadequacies of the concept of phonons as heat carriers have been studied in the context of ordinary solids in cryogenic conditions, but it turns out that such behavior is dominant even at room temperature when layered or two-dimensional materials are considered. Here, we introduce a new class of excitations, termed relaxons, that are the eigenstates of the scattering matrix in the Boltzmann equation. We show that these relaxons are indeed the elementary carriers of heat. In other words, thermal conductivity can be described with a kinetic theory but where the elementary excitations are relaxons instead of phonons. Each relaxon is, in turn, a collective excitation of phonon populations, with the key property that it has a single well-defined relaxation time as it approaches equilibrium. Therefore, heat flux can be described exactly as a gas of relaxons traveling in a solid, and the time and length scales of these processes can be identified. In a two-dimensional material such as graphene, these scales can be 100 times longer than those of phonons.

Our results, which dramatically change the time scales and length scales of heat transport in two-dimensional and layered materials, pave the way for more advanced models of heat transport.

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Vol. 6, Iss. 4 — October - December 2016

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