Atomic-Site-Specific Analysis on Out-of-Plane Elasticity of Convexly Curved Graphene and Its Relationship to sp2 to sp3 Re-Hybridization

The geometry of two-dimensional crystalline membranes is of interest given its unique synergistic interplay with their mechanical, chemical, and electronic properties. For one-atom-thick graphene, these properties can be substantially modified by bending at the nanometer scale. So far variations of the electronic properties of graphene under compressing and stretching deformations have been exclusively investigated by local-probe techniques. Here we report that the interatomic attractive force introduced by atomic force microscopy triggers single-atom displacement and consequently enables us to determine out-of-plane elasticities of convexly curved graphene including its atomic-site-specific variation. We have quantitatively evaluated the relationship between the out-of-plane displacement and elasticity of convexly curved graphene by three-dimensional force field spectroscopy on a side-wall of a hollow tube with a well-defined curvature. The substantially small intrinsic modulus that complies with continuum mechanics has been found to increase significantly at atomically specific locations, where sp(2) to sp(3) re-hybridization would certainly take place.