Moiré patterns form when two periodic structures overlay one another with a slight mismatch, such as a difference in lattice spacing or small twist angle. In two-dimensional (2D) materials, stacking two single layers with these mismatches creates a moiré pattern, introducing a new effective structure with a much larger periodicity than that of the original atomic lattices. This phenomenon is widely studied owing to its crucial role in describing the physical properties of 2D systems. A prominent example is twisted bilayer graphene (TBG). In TBG, when the twist angle between the two graphene are around 1°, the moiré pattern significantly slows down the highly mobile Dirac electrons. This results in strongly correlated phenomena, such as superconductivity.

In addition to electronic effects, moiré patterns also influence lattice vibrations. When two layers are stacked, atoms undergo structural relaxation, forming domains at the moiré scale, thus altering the mechanical excitations—phonons—of the system. In particular, the interlayer antisymmetric modes, where the atoms in each layer move in opposite directions, transform into macroscopic collective vibrations of the effective moiré domain structure. These changes in phonon behavior raise questions about their impact on thermal transport.

Recently, we theoretically investigated the thermal conductivity due to in-plane phonons in various twisted bilayer systems. To efficiently capture the huge number of atoms within a single moiré unit cell, we utilized the continuum model, in which distortion within the layers are smooth in the atomic scale. In the absence of moiré-induced interlayer coupling, the phonon modes behave as in single layers, where they move at constant velocities. In the actual systems, however, the moiré pattern significantly slows down the interlayer antisymmetric phonons with energies from few meV to tens of meV. This velocity reduction leads to lower thermal conductivity at temperatures where these phonons dominate (such as a few K to ~100 K in TBG), as slower phonon transport heat less efficiently. On the contrary, at extremely low temperatures where only the few lowest phonon bands contribute to thermal transport, the reduced velocity of the lowest moiré phonon bands boosts thermal conductivity because the linear dispersive nature makes thermal conductivity inversely proportional to their velocity.

In typical 2D materials with only in-plane acoustic phonons, thermal conductivity increases quadratically with temperature. However, moiré systems deviate from this pattern because of the non-monotonic changes over temperature introduced by the altered phonon spectrum. This results to a temperature dependence unique to moiré systems. We expect that this study contributes to the development of new moiré material properties and offers the possibility to observe the previously elusive moiré effects on phonons. In the future, it is interesting to examine the role of phonons in various phenomena in moiré systems.

(Written by L. P. A. Krisna on behalf of all authors.)

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