Wigner formulation of thermal transport in solids

Abstract

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 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 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.

Many-body Green's function approach to lattice thermal transport

“…In the last few years, the technological interest in increasing the efficiency of thermoelectric energyconversion devices [1][2][3], or in optimizing thermal shields and thermal barrier coatings [4][5][6][7][8], has stimulated an intense research on materials with ultralow thermal conductivity. So-called complex crystals, defined as materials with a phonon spectrum featuring interband spacings smaller than the linewidths [9,10], are promising candidates for these applications, since their thermal conductivity is very low (typically < ∼ 1 W m•K around room temperature). More precisely, the thermal properties of complex crystals can be regarded as intermediate between those of simple crystals, where the interband spacings between phonon branches are much larger than their linewidths [9,10], and those of glasses, where vibrational eigenstates are quasi-degenerate.…”

“…So-called complex crystals, defined as materials with a phonon spectrum featuring interband spacings smaller than the linewidths [9,10], are promising candidates for these applications, since their thermal conductivity is very low (typically < ∼ 1 W m•K around room temperature). More precisely, the thermal properties of complex crystals can be regarded as intermediate between those of simple crystals, where the interband spacings between phonon branches are much larger than their linewidths [9,10], and those of glasses, where vibrational eigenstates are quasi-degenerate. More specifically, while in simple crystals the thermal conductivity κ(T ) follows the universal [11] Peierls-Boltzmann κ(T )∼T −1 decay for T θ D (where θ D is the Debye temperature), in complex crystals κ(T ) has a much milder asymptotic decay, which resembles the saturating trend typical of glasses [12][13][14][15].…”

“…On the other hand, some of us followed a different approach, where the heat flux is directly computed from a generalization of the BTE named Wigner transport equation (WTE) [9], further detailed in Ref. [10]. This formulation exploits the Wigner transformation of the density matrix, that naturally suggests the introduction of bosonic operators which describe localized vibrational excitations.…”

“…Even though in Refs. [9,10] the corresponding quantum operator for the heat current is not explicitly given, from the form of the matrix elements of phonon velocities presented in [9] one can already infer a different structure for it with respect to the one given by Hardy. A second apparent source of discrepancies between recent [19,47] and previous [25,26] works based on the Green-Kubo formalism concerns the implementation of interaction effects for phonons.…”

Single-channel or multi-channel thermal transport? Effect of higher-order anharmonic corrections on the predicted phonon thermal transport properties of semiconductors

“…using the Wallace/smooth representation of the dynamical matrix (Eqn. 2) [23]. Further details on these methods are available in Refs.…”