Moiré periodic and quasiperiodic crystals in heterostructures of twisted bilayer graphene on hexagonal boron nitride.

Stacking two atomic crystals with a twist between their crystal axes produces moiré potentials that modify the electronic properties. Here we show that double-moiré potentials generated by superposing three atomic crystals create a unique class of tunable quasiperiodic structures that alter the symmetry and spatial distribution of the electronic wavefunctions. By using scanning tunnelling microscopy and scanning tunnelling spectroscopy to study twisted bilayer graphene on hexagonal boron nitride, we unveil a moiré phase diagram defined by the lattice constants of the two moiré lattices (graphene-on-graphene and graphene-on-hexagonal boron nitride), comprising both commensurate periodic and incommensurate quasiperiodic crystals. Remarkably, the 1:1 commensurate crystals, which should theoretically exist at only one point on this phase diagram, are observed over a wide range, demonstrating an unexpected self-alignment mechanism. The incommensurate crystals include quasicrystals, which are quasiperiodic and feature a Bravais-forbidden dodecagonal symmetry, and intercrystals, which are also quasiperiodic but lack forbidden symmetries. This rich variety of tunable double-moiré structures offers a synthetic platform for exploring the unique electronic properties of quasiperiodic crystals, which are rarely found in nature.