  Finding an effective and controllable way to create a sizable energy gap in
graphene-based systems has been a challenging topic of intensive research. We
propose that the hybrid of boron nitride and graphene (h-BNC) at low BN doping
serves as an ideal platform for band-gap engineering and valleytronic
applications. We report a systematic first-principles study of the atomic
configurations and band gap opening for energetically favorable BN patches
embedded in graphene. Based on first-principles calculations, we construct a
tight-binding model to simulate general doping configurations in large
supercells. Unexpectedly, the calculations find a linear dependence of the band
gap on the effective BN concentration at low doping, arising from an induced
effective on-site energy difference at the two C sublattices as they are
substituted by B and N dopants alternately. The significant and tunable band
gap of a few hundred meVs, with preserved topological properties of graphene
and feasible sample preparation in the laboratory, presents great opportunities
to realize valley physics applications in graphene systems at room temperature.
