▎ 摘　　要

Graphene has extremely high mobility with unique linear band dispersions at the Fermi level, referred to as the Dirac cones, but the absence of the energy gap limits its application for switching devices. To open an energy gap, theoretical studies so far have introduced certain perturbations to graphene in the real or momentum space and checked whether they open a gap at the Dirac point. Here, as a reverse approach, we directly enforce energy splittings at the Dirac point with perturbations in the minimal Hilbert space at the Dirac point and then characterize the perturbations in the real space to obtain perturbed band structures throughout the Brillouin zone. Our approach provides refined descriptions of the sublattice symmetry breaking and the intervalley scattering, distinguishing clearly the sublattice symmetry breaking without intervalley scattering, the sublattice mixing without intervalley scattering, the intervalley scattering within each sublattice, and the intersublattice intervalley scattering. For fully gapped cases, the effective mass is obtained as a function of the energy gap. Our present method can be applied to band-gap engineering of graphene-like hexagonal layered materials, in general.