Graphene ground states

Abstract

Graphene is locally two-dimensional but not flat. Nanoscale ripples appear in suspended samples and rolling-up often occurs when boundaries are not fixed. We address this variety of graphene geometries by classifying all ground-state deformations of the hexagonal lattice with respect to configurational energies including two-and three-body terms. As a consequence, we prove that all ground-state deformations are either periodic in one direction, as in the case of ripples, or rolled up, as in the case of nanotubes.2010 Mathematics Subject Classification. 70F45, 82D80.

In situ control of graphene ripples and strain in the electron microscope

“…The experimentally observed change in the mean curvature (γ rms ) is directly related to the angle of curvature in the model (θ), and the apparent measured strain ε′ corresponds to ε ¼ ' À ' Ã ð Þ=' Ã , whereas the applied (uniaxial) stretchings are directly equivalent. From the experimental results, it is clear that flattening of graphene continues beyond the point where the material starts to get strained (ε′ > 0), confirming the analytical result indicating that θ * < π, i.e., the optimal configuration for graphene is not flat, 19 at least under these experimental conditions. The experiments further show that the strain in graphene reaches ∼1% for stretching of ca.…”

“…On the theoretical side, the non-flatness in graphene can be addressed by molecular mechanical methods, by analyzing optimal configurations in terms of given interaction energies. This is computationally well known 3,7 and has been recently analytically confirmed 19,20 for a large class of configurational energies. By taking into account two-body interactions over a finite range of neighboring atoms, including at least second neighbors and three-body interactions, it has been shown that optimal configurations are indeed not flat.…”

“…By taking into account two-body interactions over a finite range of neighboring atoms, including at least second neighbors and three-body interactions, it has been shown that optimal configurations are indeed not flat. 19 More precisely, whenever fixed at the boundary, such configurations show ripples with the same orientation having a specific wave-length independent of the size of the sample. 20 These analytical results entail that the whole geometry of the optimal configuration is determined by one of its sections in the direction of the ripple.…”

“…20 These analytical results entail that the whole geometry of the optimal configuration is determined by one of its sections in the direction of the ripple. The 3D optimization problem can hence be efficiently rewritten in terms of a one-dimensional energy minimization problem for the configurational energy E of a chain of locations x i ð Þ 2mnþ1 i¼1 in R 2 , where the locations correspond to single hexagons of the abovementioned section, 19 m stands for the number of waves and 2n + 1 is the number of locations in each wave. One can also assume all pairs of adjacent locations x i and x i + 1 to have the same distance ' ¼ x iþ1 À x i j jand all pairs of consecutive segments x i + 1 − x i and x i − 1 − x i to form the same angle θ.…”

Crystallization in the hexagonal lattice for ionic dimers

“…Recently, a new perspective on this problem was given by De Luca and Friesecke [9] using a discrete Gauss-Bonnet method from discrete differential geometry. Including angular potentials favoring 2π/3 bond-angles, which is typically the case for graphene, crystallization in the hexagonal lattice has been proved by Mainini and Stefanelli [25], see also [14] for a related classification of ground states. In a similar fashion, if π/2 bond-angles are favored, the square lattice can be recovered [24].…”