Wigner Formulation of Thermal Transport in Solids

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. Recently, it has been shown that these two conduction mechanisms emerge as limiting cases from a unified transport equation, which describes on an equal footing solids ranging from crystals to glasses; moreover, in materials with intermediate characteristics the two conduction mechanisms coexist, and it is crucial to account for both. 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.