Wigner formulation of thermal transport in solids

TL;DR

This paper presents a unified Wigner transport equation that describes both particle-like and wave-like heat conduction mechanisms in solids, providing a comprehensive framework for understanding thermal transport across different material types.

Contribution

It introduces a Wigner phase-space formulation of quantum mechanics to unify and extend existing models of thermal transport in crystals and glasses, addressing complex materials with mixed conduction mechanisms.

Findings

The Wigner formulation yields a size-consistent, invariant conductivity.

It explains the crossover from particle-like to wave-like heat conduction.

The approach overcomes limitations of the Peierls-Boltzmann model for ultralow thermal conductivity materials.

Abstract

Two different heat-transport mechanisms are discussed in solids: in crystals, heat carriers propagate and scatter particle-like as described by Peierls' formulation of the Boltzmann transport equation for phonon wavepackets. In glasses, instead, carriers behave wave-like, diffusing via a Zener-like tunneling between quasi-degenerate 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 to describe also complex crystals where particle-like and wave-like 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 non-unique choice of the crystal's unit cell and is size-consistent. We rationalize the conditions determining the crossover from particle-like to wave-like 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 wave-like capability to interfere and tunnel. Finally, we show that the present approach overcomes the failures of the Peierls-Boltzmann formulation for materials with ultralow or glass-like thermal conductivity.