Moiré^2 : self alignment, topology and quasi-crystals

Using scanning-tunneling-microscopy and spectroscopy on heterostructures of twisted bilayer graphene and hexagonal Boron-Nitride, we show that the ensuing super-moiré structures display a rich landscape of moiré-crystals and quasicrystals. We reveal a phase-diagram comprised of commensurate moiré-crystals embedded in swaths of moiré quasicrystals. The 1:1 commensurate moiré crystal, which is expected to display an anomalous quantum Hall effect, should be practically impossible to realize experimentally. Surprisingly we find that commensurate moiré crystals exist over a wide range of parameters, providing evidence of a relaxation mechanism favoring self-alignment. The remainder of the phase-diagram is comprised of moiré quasicrystals with emergent rotational symmetries of the electronic wave function.  These structures afford a new platform for exploring the unique electronic-properties of the, rarely found in nature, quasicrystal structures.