Two different heat-transport mechanisms are discussed in solids: in crystals, heat carriers propagate and scatter like particles, as described by Peierls' formulation of 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 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 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, with case studies of materials for thermal barrier coatings and thermoelectric energy conversion.

Here, we release the crystal structures, interatomic force constants, and computational parameters needed to reproduce the results discussed in the article "Wigner formulation of thermal transport in solids" (arXiv preprint, 2021) by M. Simoncelli, N. Marzari, and F. Mauri.