Fractal energy gaps and topological invariants in hBN/Graphene/hBN double moiré systems

Abstract

We calculate the electronic structure in quasiperiodic double-moiré systems of graphene sandwiched by hexagonal boron nitride, and identify the topological invariants of energy gaps. We find that the electronic spectrum contains a number of minigaps, and they exhibit a recursive fractal structure similar to the Hofstadter butterfly when plotted against the twist angle. Each of the energy gaps can be characterized by a set of integers, which are associated with an area in the momentum space. The corresponding area is geometrically interpreted as a quasi Brillouin zone, which is a polygon enclosed by multiple Bragg planes of the composite periods and can be uniquely specified by the plain wave projection in the weak potential limit.

17 pages, 11 figures

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Cited by in corpus (18)

• Superconductivity and strong interactions in a tunable moiré quasiperiodic crystal
• Multi-scale lattice relaxation in general twisted trilayer graphenes
• 3/2 magic-angle quantization rule of flat bands in twisted bilayer graphene and relationship with the Quantum Hall effect
• Why the first magic-angle is different from others in twisted graphene bilayers: interlayer currents, kinetic and confinement energy and wavefunction localization
• Moiré phonons in graphene/hexagonal boron nitride moiré superlattice
• Extended magic phase in twisted graphene multilayers
• Higher-order Bragg gaps in the electronic band structure of bilayer graphene renormalized by recursive supermoiré potential
• Electric field-tunable layer polarization in graphene/boron nitride twisted quadrilayer superlattices
• Topological gap labeling with the third Chern numbers in three-dimensional quasicrystals
• Characterization of a graphene-hBN superlattice field effect transistor
• Strong gate-tunability of flat bands in bilayer graphene due to moiré encapsulation between hBN monolayers
• Ferromagnetically ordered states in the Hubbard model on the $H_{00}$ hexagonal golden-mean tiling
• Hexagonal and trigonal quasiperiodic tilings
• Moiré fractals in twisted graphene layers
• Gap labeling theorem for multilayer thin film heterostructures
• Energy Spectrum Theory of Incommensurate Systems
• Electric field tunable bands in doubly aligned bilayer graphene hBN moire superlattice
• One-dimensional moiré engineering in zigzag graphene nanoribbons on hBN