  Graphene monolayers supported on oxide substrates have been demonstrated with
superior charge mobility and thermal transport for potential device
applications. Morphological corrugation can strongly influence the transport
properties of the supported graphene. In this paper, we theoretically analyze
the morphological stability of a graphene monolayer on an oxide substrate,
subject to van der Waals interactions and in-plane mismatch strains. First, we
define the equilibrium separation and the interfacial adhesion energy as the
two key parameters that characterize the van der Waals interaction between a
flat monolayer and a flat substrate surface. By a perturbation analysis, a
critical compressive mismatch strain is predicted, beyond which the graphene
monolayer undergoes strain-induced instability, forming corrugations with
increasing amplitude and decreasing wavelength on a perfectly flat surface.
When the substrate surface is not perfectly flat, the morphology of graphene
depends on both the amplitude and the wavelength of the surface corrugation. A
transition from conformal (corrugated) to non-conformal (flat) morphology is
predicted. The effects of substrate surface corrugation on the equilibrium mean
thickness of the supported graphene and the interfacial adhesion energy are
analyzed. Furthermore, by considering both the substrate surface corrugation
and the mismatch strain, it is found that, while a tensile mismatch strain
reduces the corrugation amplitude of graphene, a corrugated substrate surface
promotes strain-induced instability under a compressive strain. These
theoretical results suggest possible means to control the morphology of
graphene monolayer on oxide substrates by surface patterning and strain
engineering.
