Low-energy moiré phonons in twisted bilayer van der Waals heterostructures

We develop a low-energy continuum model for phonons in twisted moir\'e bilayers, based on configuration-space approach. In this approach, interatomic force constants are obtained from density functional theory (DFT) calculations of untwisted bilayers with various in-plane shifts. This allows efficient computation phonon properties any small twist angle, while maintaining DFT-level accuracy. Based framework, we show how the modes, including interlayer shearing and layer-breathing vary angle. As angle decreases, frequencies modes reordered atomic displacement fields corresponding to eigenmodes break translational symmetry, developing periodicity length scale. demonstrate capabilities our by calculating three specific structures: bilayer graphene, molybdenum disulfide (MoS$_2$), diselenide-tungsten diselenide (MoSe$_2$-WSe$_2$).