  We present a density functional study of graphene adhesion on a realistic
SiO$_2$ surface taking into account van der Waals (vdW) interactions. The
SiO$_2$ substrate is modeled at the local scale by using two main types of
surface defects, typical for amorphous silica: the oxygen dangling bond and
three-coordinated silicon. The results show that the nature of adhesion between
graphene and its substrate is qualitatively dependent on the surface defect
type. In particular, the interaction between graphene and silicon-terminated
SiO$_2$ originates exclusively from the vdW interaction, whereas the
oxygen-terminated surface provides additional ionic contribution to the binding
arising from interfacial charge transfer ($p$-type doping of graphene). Strong
doping contrast for the different surface terminations provides a mechanism for
the charge inhomogeneity of graphene on amorphous SiO$_2$ observed in
experiments. We found that independent of the considered surface morphologies,
the typical electronic structure of graphene in the vicinity of the Dirac point
remains unaltered in contact with the SiO$_2$ substrate, which points to the
absence of the covalent interactions between graphene and amorphous silica. The
case of hydrogen-passivated SiO$_2$ surfaces is also examined. In this
situation, the binding with graphene is practically independent of the type of
surface defects and arises, as expected, from the vdW interactions. Finally,
the interface distances obtained are shown to be in good agreement with recent
experimental studies.
