Topological flat bands in tetralayer graphene on boron nitride moire superlattices*

We show that rhombohedral four-layer graphene (4LG) nearly aligned with a hexagonal boron nitride (hBN) substrate has nearly flat low energy bands with generally non-zero valley Chern numbers. These bands are isolated even in the absence of a perpendicular electric field thanks to the opening of a primary bandgap at charge neutrality and secondary gaps near the moire Brillouin zone corners. The bandwidths are controllable through an electric field and they can become as narrow as ∼5meV when the interlayer potential differences between top and bottom layers amount up to |Δ|≈40meV. The local density of states (LDOS) analysis shows that the nearly flat band states wave functions are associated to the non-dimer low energy sublattice sites at the top or bottom layer graphene and their degree of localization is strongly gate tunable. Similar behaviors are seen in nLG/BN for n=5−8 where generally the valley Chern number of the first valence band of nLG/BN is equal to the number of graphene layers, C^ν=±1 =±n.