Moiré minibands in graphene heterostructures with almost commensurate √3×√3 hexagonal crystals

We present a phenomenological theory of the low-energy moiré minibands of Dirac electrons in graphene placed on an almost commensurate hexagonal underlay with a unit cell approximately three times larger than that of graphene. A slight incommensurability results in a periodically modulated intervalley scattering for electrons in graphene. In contrast to the perfectly commensurate Kekulé distortion of graphene, such superlattice perturbation leaves the zero-energy Dirac cones intact, but is able to open a band gap at the edge of the first moiré subbands, asymmetrically in the conduction and valence bands.