Determination of the Bending Rigidity of Graphene via Electrostatic Actuation of Buckled Membranes

Abstract

Classical continuum mechanics is used extensively to predict the properties of nanoscale materials such as graphene. The bending rigidity, κ, is an important parameter that is used, for example, to predict the performance of graphene nanoelectromechanical devices and also ripple formation. Despite its importance, there is a large spread in the theoretical predictions of κ for few-layer graphene. We have used the snap-through behavior of convex buckled graphene membranes under the application of electrostatic pressure to determine experimentally values of κ for double-layer graphene membranes. We demonstrate how to prepare convex-buckled suspended graphene ribbons and fully clamped suspended membranes and show how the determination of the curvature of the membranes and the critical snap-through voltage, using AFM, allows us to extract κ. The bending rigidity of bilayer graphene membranes under ambient conditions was determined to be 35.5−15.0 +20.0 eV. Monolayers are shown to have significantly lower κ than bilayers.

Graphene as a Prototypical Model for Two-Dimensional Continuous Mechanics

“…For bilayer graphene, G can be multiplied by a factor of two, σ gr-SiO 2 can be reduced to 0.3 J/m 2 , as reported in Section 4. Then, the observed value θ = 20 • requires D = 30 eV-quite close to the value of 35 eV measured for the bending rigidity of bilayer graphene [39]. In view of this result, mechanical tearing is likely to bend the atomic layers by keeping them coupled, by opposition with self-folding where the apparent bending modulus was found to be much smaller (Section 4).…”

“…In case the atomic planes slip over each other, continuum theory predicts that D for multilayer graphene is proportional to the number of planes, which has been confirmed by some atomistic simulations [37,38]. On the experimental side, atomic force microscopy (AFM) measurement of the deformation of convex-buckled suspended graphene ribbons yielded D = 35.5 eV for bilayer graphene, with relative uncertainty of 50% [39]. The last row of Table 1 corresponds to this parameter; h has been doubled because the inter-sheet distance must be measured from the plane at mid-thickness.…”

A general procedure for thermomechanical calibration of nano/micro-mechanical resonators

“…Circular membranes, used in a number of experiments [65][66][67], are another interesting nano/micro-mechanical resonator that have an analytically solvable mode shape.…”

Apparent stiffening of a graphene nanomembrane with initial curvature

“…(7) and (8) imply that for a positive κ 0 , the initial shape of the membrane is concave, in accord with AFM images reported in Refs. 4,12,14,20,21,24. It follows from the first equation in Eq.…”