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.

T. Li, Z. Zhang, Substrate-regulated morphology of graphene, Journal of Physics D: Applied Physics, 43, 075303 (2010).