Abstract
The calculus of variations is utilised to study the behaviour of a rippled graphene sheet supported on a metal substrate. We propose a model that is underpinned by two key parameters, the bending rigidity of graphene γ, and the van der Waals interaction strength ξ. Three cases are considered, each of which addresses a specific configuration of a rippled graphene sheet located on a flat substrate. The transitional case assumes that both the graphene sheet length and substrate length are constrained. The substrate constrained case assumes only the substrate has a constrained length. Finally, the graphene constrained case assumes only the length of the graphene sheet is constrained. Numerical results are presented for each case, and the interpretation of these results demonstrates a continuous relationship between the total energy per unit length and the substrate length, that incorporates all three configurations. The present model is in excellent agreement with earlier results of molecular dynamics (MD) simulations in predicting the profiles of graphene ripples.
Highlights
Graphene is a two-dimensional sheet of carbon atoms bonded to each other in a planar hexagonal array
Numerical results are presented for each case, and the interpretation of these results demonstrates a continuous relationship between the total energy per unit length and the substrate length, that incorporates all three configurations
The graphene constrained case can be adapted to address the regime when Lsub > x2g, and here the total energy per unit length Etot remains constant since there are no more interactions that are involved for this section
Summary
Graphene is a two-dimensional sheet of carbon atoms bonded to each other in a planar hexagonal array. Ripples are observed in suspended graphene sheets,[16] as well as graphene on a substrate.[17] Experimental studies nd that the electronic properties of graphene can be affected by the range and height of these ripples.[18,19] Gui et al.[20] use rst-principles calculations to predict the electronic properties of a rippled graphene sheet where a band gap opening is observed in the rippled graphene They evaluate a direct band gap at 0.93 eV which indicates rippled graphene may be a highly tunable semiconductor. Using density functional theory calculations, Wei et al.[10] evaluate the bending rigidity of a single-layer graphene at 1.44 eV They compare this result to earlier obtained values for the bending rigidity of graphene which range from 0.80 to 1.60 eV.
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