In-plane breathing and shear modes in low-dimensional nanostructures

We use continuum elasticity theory to revise scaling laws for radial breathing-like and shear-like vibration modes in quasi-2D nanostructures including finite-width nanoribbons and finite-size thin circular discs. Such modes can be observed spectroscopically in corresponding nanostructures of graphene and phosphorene and can be determined numerically by atomistic ab initio density functional theory and classical force-field calculation. The revised scaling laws differ from previously used expressions, some of which display an unphysical asymptotic behavior. Apart from model assumptions describing the effect of edge termination, the continuum scaling laws have no adjustable parameters and display correct asymptotic behavior. These scaling laws yield excellent agreement with experimental and numerical results for vibration frequencies in both isotropic and anisotropic structures as well as useful expressions for the frequency dependence on structure size and edge termination.