Tuning commensurability in twisted van der Waals bilayers.

Moiré superlattices based on van der Waals bilayers 1-4 created at small twist angles lead to a long wavelength pattern with approximate translational symmetry. At large twist angles (θ t ), moiré patterns are, in general, incommensurate except for a few discrete angles. Here we show that large-angle twisted bilayers offer distinctly different platforms. More specifically, by using twisted tungsten diselenide bilayers, we create the incommensurate dodecagon quasicrystals at θ t  = 30° and the commensurate moiré crystals at θ t  = 21.8° and 38.2°. Valley-resolved scanning tunnelling spectroscopy shows disparate behaviours between moiré crystals (with translational symmetry) and quasicrystals (with broken translational symmetry). In particular, the K valley shows rich electronic structures exemplified by the formation of mini-gaps near the valence band maximum. These discoveries demonstrate that bilayers with large twist angles offer a design platform to explore moiré physics beyond those formed with small twist angles.