Multiscale Analysis of Size-Dependent Vibration of Graphene Nanoelectromechanical Resonators

The size-dependent out-of-plane vibrational behavior of graphene-based nanoelectromechanical (NEMS) resonators is investigated using a molecular mechanics (MM) finite element approach. Each carbon-carbon (C-C) bond is modeled as an Euler-Bernoulli beam element, with the bending stiffness derived from the bond-angle potential, yielding an equivalent plate flexural rigidity D = (√3/6) kθ. The natural frequencies of the first four vibration modes are computed for square graphene sheets of increasing size with both zigzag (ZZ) and armchair (AC) chirality configurations under simply supported boundary conditions on all four edges. A chirality-induced frequency deviation δ(L) is defined to quantify the difference between ZZ and AC results, and a threshold size L* is identified as the sheet size at which δ falls below 1%. For mode 1, the threshold is L* = 18.5 nm; the values increase monotonically to 24.5 nm, 28.0 nm, and 31.5 nm for modes 2 through 4, indicating that higher modes require larger sheet dimensions before continuum plate theory becomes reliable. A dimensionless frequency parameter Ω = f_MM/f_CT is introduced to directly compare MM predictions with the Kirchhoff plate theory analytical solution, and the AC frequency ratio Ω = f_MM/f_CT is shown to converge toward unity with increasing sheet size. The present results provide quantitative design guidelines for graphene NEMS resonators and establish the minimum device dimensions for which isotropic continuum models yield accurate dynamic predictions.