Topological flat bands in rhombohedral tetralayer and multilayer graphene on hexagonal boron nitride moire superlattices

We show that rhombohedral four-layer graphene (4LG) nearly aligned with a hexagonal boron nitride (hBN) substrate often develops nearly flat isolated low-energy bands with nonzero valley Chern numbers. The bandwidths of the isolated flat bands are controllable through an electric field and twist angle, becoming as narrow as similar to 10 meV for interlayer potential differences between top and bottom layers of |Delta| approximate to 10-15 meV and theta similar to 0.5 degrees at the graphene and boron nitride interface. The local density of states analysis shows that the nearly flat band states are associated to the nondimer low-energy sublattice sites at the top or bottom graphene layers and their degree of localization in the moire superlattice is strongly gate tunable, exhibiting at times large delocalization despite the narrow bandwidth. We verified that the first valence band's valley Chern numbers C-V1(nu= +/- n) = +/- n, proportional to layer number for nLG/BN systems up to n = 8 rhombohedral multilayers.