  On the basis of first-principles calculations, we report that a novel
interfacial atomic structure occurs between graphene and the surface of silicon
carbide, destroying the Dirac point of graphene and opening a substantial
energy gap there. In the calculated atomic structures, a quasi-periodic
$6\times 6$ domain pattern emerges out of a larger commensurate
$6\sqrt{3}\times6\sqrt{3}R30^\circ$ periodic interfacial reconstruction,
resolving a long standing experimental controversy on the periodicity of the
interfacial superstructures. Our theoretical energy spectrum shows a gap and
midgap states at the Dirac point of graphene, which are in excellent agreement
with the recently-observed anomalous angle-resolved photoemission spectra.
Beyond solving unexplained issues of epitaxial graphene, our atomistic study
may provide a way to engineer the energy gaps of graphene on substrates.
