We study the stability and electronic structure of magic-angle twisted bilayer graphene on the hexagonal boron nitride (TBG/BN). Structural relaxation has been performed for commensurate supercells of the heterostructures with different twist angles ( θ ′ ) and stackings between TBG and BN. We find that the slightly misaligned configuration with θ ′ = 0.54 ∘ and the AA/AA stacking has the globally lowest total energy due to the constructive interference of the moiré interlayer potentials and thus the greatly enhanced relaxation in its 1 × 1 commensurate supercell. Gaps are opened at the Fermi level (E F ) for small supercells with the stackings that enable strong breaking of the C 2 symmetry in the atomic structure of TBG. For large supercells with θ ′ close to those of the 1 × 1 supercells, the broadened flat bands can still be resolved from the spectral functions. The θ ′ = 0.54 ∘ is also identified as a critical angle for the evolution of the electronic structure with θ ′ , at which the energy range of the mini-bands around E F begins to become narrower with increasing θ ′ and their gaps from the dispersive bands become wider. The discovered stablest TBG/BN with a finite θ ′ of about 0.54∘ and its gapped flat bands agree with recent experimental observations.

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Moiré bands separated by the primary and secondary gaps emerge in the superlattices ofmonolayer (MLG) and bilayer graphene (BLG) aligned with the hexagonal boron nitride (BN). We study the tuning of the electronic and transport properties of such moiré superlattices through the periodic electrostatic potentials produced by the one-dimensional (1D) or two-dimensional (2D) patterned gating structure in the devices. The electrostatic potentials in graphene are produced by the spatially varying particle and hole doping due to the local quantum capacitance effect and can be modulated by the voltage (VTG) of the top patterned gating structure and that (VBG) of the uniform bottom gate. For the 1D devices of MLG/BN and BLG/BN, different sets of Fabry-Pérot interference like resistance patterns as a function of VTG and VBG can be observed when the Fermi level is shifted from the charge neutrality point (CNP) to the secondary gaps in MLG/BN and BLG/BN, and the overlapping regions of the patterns exhibit the highest resistance. The electronic states in these various regions of the resistance map show different moiré-band hybridization and spatial distribution. The secondary resistance patterns around the secondary gaps move away from the primary one with increasing twist angle (θ) between graphene and BN and their detailed patterns also depend on the orientation of the 1D potential with respect to that of the superlattice. The 2D periodic potentials can further split the subbands between CNP and the secondary gaps, depending on the commensurability of the moiré superlattice and the 2D potentials, and additional resistance peaks appear as a function of the gate voltages. The calculated resistance map for BLG/BN at θ ∼ 1° is roughly consistent with recent experimental observations.