Exploring out-of-plane Mechanics of Graphene Membrane by 3D Force Field Spectroscopy on various Nanotubes

Bending rigidity and Gaussian modulus are key parameters to understand flexibilities of two-dimensional (2D) crystalline membrane embedded in three-dimensional (3D) space. These two parameters of graphene membranes are still unclear because of their difficulties to distinguish purely intrinsic characteristics from others due to thermal fluctuations, consequent local strains and so on. Here we present experimental determinations of those properties by analyzing noncontact interaction between the opposing two atoms of probe-tip apex and convexly-curved graphene folding into various nanotubes. We have quantitatively evaluated the relationship between out-of-plane displacement and elasticity of monolayer graphene by 3D force field spectroscopy at low temperature not only on folded nanotubes with well-defined curvatures but also on unfolded one (i.e. graphene nanoribbon) with unknown curvatures. The quantitative evaluations allow us to determine the smaller and locally different curvatures of unfolded monolayer. Our findings to separate the in-plane and out-of-plane contributions allow us to derive the substantially small bending modulus enough to expect the intrinsic characteristics of negligibly small (zero) modulus for the ultimate 2D membrane without any curvature.