Structure and electronic properties of moiré periodic and quasiperiodic crystals

Stacking two atomic crystals with a twist between their crystal axes produces tunable moiré potentials that can radically alter the electronic band structure[1,2]. An even richer variety of structures and electronic properties emerges when three non-identical atomic crystals are superposed with a twist. The resulting double moiré potentials create a new class of tunable quasiperiodic structures that alter the symmetry and spatial distribution of the electronic wavefunctions. By using scanning-tunneling-microscopy (STM) and spectroscopy (STS) to study twisted bilayer graphene on hexagonal Boron-Nitride (hBN), we unvailed a phase diagram spanned by the lattice vectors of the two overlaid moiré structures (graphene-on-graphene and graphene-on-hBN), which includes both commensurate periodic and incommensurate quasiperiodic crystals. The 1:1 commensurate crystal, which theoretically should exist at only one point on this phase-diagram, is observed over a wide range, demonstrating the presence of an unexpected self-alignment mechanism. The quasiperiodic crystals include quasicrystals featuring a Bravais-forbidden dodecagonal symmetry, and intercrystals which are also quasiperiodic but lack forbidden symmetries. STS maps of the quasiperiodic crystals reveal a strongly energy dependent Bragg peak structure generated by scattering of the electronic wave functions off the quasiperiodic potential. Furthermore, we find that despite the broken translational symmetry, the quasiperiodic crystals feature flat bands and correlation induced gaps that mimic the electronic properties of magic angle twisted graphene.

[1] G. Li, A. Luican, J. M. B. Lopes dos Santos, A. H. Castro Neto, A. Reina, J. Kong, and E. Y. Andrei, Nature Physics 6, 109 (2010).

[2] E. Y. Andrei and A. H. MacDonald, Nature Materials 19, 1265 (2020).