  We delineate a general theoretical framework to determine the
substrate-regulated graphene morphology through energy minimization. We then
apply such a framework to study the graphene morphology on a substrate with
periodic surface grooves. Depending on the substrate surface roughness and the
graphene-substrate interfacial bonding energy, the equilibrium morphology of
graphene ranges from 1) closely conforming to the substrate, to 2) remaining
flat on the substrate. Interestingly, in certain cases, the graphene morphology
snaps between the above two limiting states. Our quantitative results envision
a promising strategy to precisely control the graphene morphology over large
areas. The rich features of the substrate-regulated graphene morphology (e.g.,
the snap-through instability) can potentially lead to new design concepts of
functional graphene device components.
