  We use ab initio density functional calculations to determine the interaction
of a graphene monolayer with the Si(111) surface. We found that graphene forms
strong bonds to the bare substrate and accommodates the 12% lattice mismatch by
forming a wavy structure consisting of free-standing conductive ridges that are
connected by ribbon-shaped regions of graphene, which bond covalently to the
substrate. We perform quantum transport calculations for different geometries
to study changes in the transport properties of graphene introduced by the wavy
structure and bonding to the Si substrate. Our results suggest that wavy
graphene combines high mobility along the ridges with efficient carrier
injection into Si in the contact regions.
