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  • Open Access

Wigner Formulation of Thermal Transport in Solids

Michele Simoncelli1,*,†, Nicola Marzari1, and Francesco Mauri2

  • 1Theory and Simulation of Materials (THEOS) and National Centre for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne, Lausanne CH-1015, Switzerland
  • 2Dipartimento di Fisica, Università di Roma La Sapienza, Piazzale Aldo Moro 5, I-00185 Roma, Italy

  • *ms2855@cam.ac.uk
  • †Present address: Theory of Condensed Matter Group of the Cavendish Laboratory and Gonville & Caius College, University of Cambridge CB3 0HE, Cambridge, United Kingdom.
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Phys. Rev. X 12, 041011 – Published 31 October, 2022

DOI: https://doi.org/10.1103/PhysRevX.12.041011

Abstract

Two different heat-transport mechanisms are discussed in solids. In crystals, heat carriers propagate and scatter particlelike as described by Peierls’s formulation of the Boltzmann transport equation for phonon wave packets. In glasses, instead, carriers behave wavelike, diffusing via a Zener-like tunneling between quasidegenerate vibrational eigenstates, as described by the Allen-Feldman equation. Recently, it has been shown that these two conduction mechanisms emerge from a Wigner transport equation, which unifies and extends the Peierls-Boltzmann and Allen-Feldman formulations, allowing one to describe also complex crystals where particlelike and wavelike conduction mechanisms coexist. Here, we discuss the theoretical foundations of such transport equation as is derived from the Wigner phase-space formulation of quantum mechanics, elucidating how the interplay between disorder, anharmonicity, and the quantum Bose-Einstein statistics of atomic vibrations determines thermal conductivity. This Wigner formulation argues for a preferential phase convention for the dynamical matrix in the reciprocal Bloch representation and related off-diagonal velocity operator’s elements; such convention is the only one yielding a conductivity which is invariant with respect to the nonunique choice of the crystal’s unit cell and is size consistent. We rationalize the conditions determining the crossover from particlelike to wavelike heat conduction, showing that phonons below the Ioffe-Regel limit (i.e., with a mean free path shorter than the interatomic spacing) contribute to heat transport due to their wavelike capability to interfere and tunnel. Finally, we show that the present approach overcomes the failures of the Peierls-Boltzmann formulation for crystals with ultralow or glasslike thermal conductivity, with case studies of materials for thermal barrier coatings and thermoelectric energy conversion.

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Corrections

21 November, 2022

Correction: The third sentence of the abstract was partially duplicated owing to a processing error and has been set right.

Popular Summary

Two different microscopic mechanisms for heat transport are known in solids. In crystals, heat carriers behave akin to particles in a gas, whereas in glasses, heat transfer bears analogies to the physics of waves. This paper elucidates how quantum wave-particle duality emerges in thermal transport, discussing how particlelike and wavelike heat-transport mechanisms can emerge and coexist, and providing a quantitative criterion to assess their relative strength and the crossover between the regimes where one or the other dominates.

In 1929, Peierls formulated the phonon Boltzmann transport equation to explain heat conduction in crystalline solids, discussing how quantized atomic vibrations mediated heat transport. This formulation successfully explained the temperature-conductivity curve observed in good thermal conductors with crystalline structure but could not describe glasses or low thermal conductors. In 1989, Allen and Feldman made a key step forward on the description of thermal transport in glasses, envisioning that in glasses, atomic vibrations with very similar energies can interfere constructively, enabling a wavelike tunneling mechanism through which heat can propagate.

In this work, we discuss the theoretical foundations of a “Wigner” heat-transport equation (named after the Wigner formulation of quantum mechanics used to derive it) that naturally encompasses the coexistence of particlelike and wavelike conduction mechanisms, unifying and extending the Peierls and Allen-Feldman formulations. We discuss the conditions determining the relative strength of particlelike and wavelike conduction mechanisms, showing that in the intermediate case of complex crystals with ultralow thermal conductivity these can be equally relevant.

Our findings pave the way for the theory-driven optimization of thermal barriers and thermoelectrics, since in these materials it is crucial to account for heat’s particle-wave duality to correctly describe the thermal conductivity.

Article Text

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  119. We stress that this definition of phonon lifetime is general and always valid, but its implications on the thermal conductivity depend on the regime of thermal transport considered. In the kinetic regime of thermal transport umklapp processes dominate, thus the SMA approximation is valid [11, 84, 120], and the phonon lifetime enters directly in the populations’ conductivity [Eq. (49)]. Instead, in the hydrodynamic regime of thermal transport where normal processes dominate, the phonon lifetime can still be defined but it no longer determines directly the populations conductivity, since this latter is determined by the lifetime of collective excitations of phonon wave packets (relaxons [13, 14]) and the relaxon lifetime is related in a nontrivial way to the phonon lifetime [13].

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Phys. Rev. X 12, 041011– Published 31 October, 2022
  • Received 14 December 2021
  • Revised 1 May 2022
  • Accepted 30 June 2022
  • Corrected 21 November 2022
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