Twisted bilayer graphene as a phononic metamaterial

The twist angle within stacked van der Waals materials represents a novel degree of freedom to tune electronic properties. In bilayer graphene, varying the twist angle hybridizes the Dirac cones from each layer, resulting in flat bands that localize charge and induce unconventional superconductivity. Recently, graphene-like Dirac cones were observed in the phononic band structure of a metamaterial consisting of a honeycomb lattice of steel pillars in air. However, varying the twist angle has not been explored in metamaterials, due to the difficulty in coupling two macroscopic layers. Here we develop a method to couple layered phononic metamaterials using intermediary membranes, and we numerically demonstrate a classical system with flat phononic bands analogous to the electronic structure at magic angle twisted bilayer graphene. Our results provide a more tangible route to comprehending the behavior of quantum materials and may yield applications in super-resolution imaging.