  BN domains are easy to form in the basal plane of graphene due to phase
separation. With first-principles DFT calculations, it is demonstrated
theoretically that the band gap of graphene can be opened effectively around K
(or K') points by doping the small BN domains. It is also found that the random
doping with B or N is possible to open a small gap in the Dirac points, except
for the modulation of Fermi level. The surface charge which belongs to the
$\pi$ states near Dirac points is found to be redistributed locally. The charge
redistribution is contributed to the change of localized potential due to the
doping effects. This may be the reason that the energy states near Dirac points
is just disturbed in the energy region [-0.4eV, 0.4eV] for the doping of BN
domain. Thus, we suggest that the band opening induced by the doped BN domain
is due to the breaking of localized symmetry of the potential. Doping graphene
with BN domains is an effective method to open a band gap with quasi-linear
dissipation near Dirac point.
