FIG. 1: The armchair edge stresses and edge energies of graphene nanoribbons as a function of ribbon width. Inset: schematics of the nanoribbon; the rectangle marks one unit cell (supercell) of the ribbon. [13]. The electron-ion interaction was described by the projector augmented wave method, and the energy cutoff was set to 500 eV. The structures were fully optimized using the conju- gate gradient algorithm until the residual atomic forces to be smaller than 10 meV / ˚ A. The supercell method with periodic boundary conditions was adopted to model the graphene nanoribbons (GNR), with a vacuum layer larger than 15 ˚ A to guarantee that there was no interaction between the GNR images in the neighboring cells. The edge energy is calculated as E$_{edge}$ = ( E$_{ribbon}$ − E$_{atom}$ ) / 2 L , where E$_{ribbon}$ is the total energy of the graphene nanoribbon in the supercell, E$_{atom}$ is the energy per atom in a perfect graphene without edge, and L is the length of edge. The edge stress is calculated as σ$_{edge}$ = σ$_{xx}$/ 2 L , where σ$_{xx}$ is the diagonal component of supercell stress tensor in the x -direction (defined along the edge), which is calculated using the Nielsen-Martin algorithm [14]. All other components of stress tensor vanish. We also note that DFT is suitable for calculating ground-state properties of lattice energies and stresses, to which the non-local many-body effects are not expected to be important. Figure 1 shows the calculated edge energy and edge stress of GNR armchair edges as a function of ribbon width ranging from ∼ 3.5 to ∼ 48 ˚ A. One notices that both edge energy and edge stress oscillate with the increasing width having a period of 3 but out of phase with each other. The oscillations are originated from the quantum confinement effect, as also manifested in the similar oscillations of electron band structures[15–18]. The 2