Moir\\'e Periodic and Quasiperiodic Crystals in Heterostructures of\n Twisted Bilayer Graphene and Hexagonal Boron Nitride

Stacking two atomic crystals with a twist between their crystal axes produces\nmoir\\'e potentials that modify the electronic properties. Here we show that\ndouble moir\\'e potentials generated by superposing three atomic crystals create\na new class of tunable quasiperiodic structures that alter the symmetry and\nspatial distribution of the electronic wavefunctions. By using scanning\ntunneling microscopy and spectroscopy to study twisted bilayer graphene on\nhexagonal boron nitride (hBN), we unveil a moir\\'e phase diagram defined by the\nlattice constants of the two moir\\'e lattices (graphene-on-graphene and\ngraphene-on-hBN), comprising both commensurate periodic and incommensurate\nquasiperiodic crystals. Remarkably, the 1:1 commensurate crystal, which should\ntheoretically exist at only one point on this phase diagram, is observed over a\nwide range, demonstrating an unexpected self-alignment mechanism. The\nincommensurate crystals include quasicrystals, which are quasiperiodic and\nfeature a Bravais-forbidden dodecagonal symmetry, and intercrystals, which are\nalso quasiperiodic but lack forbidden symmetries. This rich variety of tunable\ndouble moir\\'e structures offers a synthetic platform for exploring the unique\nelectronic properties of quasiperiodic crystals, which are rarely found in\nnature.\n