Abstract
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 s p 2 to s p 3 re-hybridization would certainly take place.
Highlights
There is a long-standing discussion [1,2,3] about how the properties of conformal elastic two-dimensional (2D) crystalline membranes [4,5] are affected by geometry [6,7]
The first ultimate 2D crystalline membrane is graphene, a monolayer of carbon atoms arranged in a honeycomb lattice, capable of being isolated from a layered crystal, graphite, known as one of the most inert materials [4,12,13]
While the continuum mechanics would still be available at the hollow site of the honeycomb lattice, out-of-plane elasticity would be significantly enhanced at the individual carbon atom site, where the carbon atom more favorably moves out of plane toward the probe-tip apex against the bending rigidity
Summary
There is a long-standing discussion [1,2,3] about how the properties of conformal elastic two-dimensional (2D) crystalline membranes [4,5] are affected by geometry [6,7]. We report that the attractive force field introduced by the local probe technique triggers out-of-plane relaxation of the individual carbon atoms (resulting in the local acumination of ripples) of convexly curved graphene and enables us to determine its out-of-plane elastic properties including its atomistic variation. Three-dimensional force-field mapping reveals an atomically specific local enhancement of out-of-plane elasticity beyond the substantially small elastic moduli resulting from the flexible elastic 2D membrane of one-atom-thick graphene. Our findings suggest that the tension applied by attractive forces in AFM would provide insight into the atomically specific properties of graphene mechanics and provide experimental evidence of local-curvature-induced sp to sp re-hybridization, which results in a local enhancement of chemical reactivity
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