Pinning of a two-dimensional membrane on top of a patterned substrate: The case of graphene

We study the pinning of a two-dimensional membrane to a patterned substrate within elastic theory both in the bending rigidity and in the strain-dominated regimes. We find that both the in-plane strains and the bending rigidity can lead to depinning. We show from energetic arguments that the system experiences a first-order phase transition between the attached configuration to a partially detached one when the relevant parameters of the substrate are varied, and we construct a qualitative phase diagram. Our results are confirmed through analytical solutions for some simple geometries of the substrate's profile. We apply our model to the case of graphene on top of a SiO₂ substrate and show that typical orders of magnitude for corrugations imply graphene will be partially detached from the substrate.