FIG. 2: The AFM and PM zigzag 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. oscillation magnitude of edge energy decays quickly with the increasing width and converges to ∼ 1.0 eV/ ˚ A, which agrees well with the previous first-principles values[8]. In contrast, the oscillation magnitude of edge stress decays much slower with a mean value of ∼ -1.45 eV/ ˚ A(using negative sign as convention for compressive stress). The much larger oscillation in edge stress than in edge energy is possibly caused by the fact that edge stress equals to the derivative of edge energy with respect to strain, so that stress is much more sensitive to the width-dependent quantum confinement effect. There is also a slight revival effect in the stress oscillations at ∼ 40 ˚ A width, whose origin is not clear and needs further study. Figure 2 shows the calculated edge energy and edge stress of GNR zigzag edges as a function of ribbon width ranging from ∼ 5.0 to ∼ 85 ˚ A. In this case, both edge energy and edge stress show very weak width dependence and converges quickly, again consistent with their corresponding electronic-structure behavior[15–18]. However, the zigzag edge is known to have an antiferromagnetic (AFM) ground state[17]. The AFM edge energy is calculated to be ∼ 1.2 eV/ ˚ A, about 0.2 eV/ ˚ A lower than the paramagnetic (PM) edge energy[8, 9, 19]. Thus, we have calculated the edge stress of both AFM and PM states for comparison. We found that spin polarization have a sizable effect reducing the compressive stress from ∼ -0.7 eV/ ˚ A in the PM edge to ∼ -0.5 eV/ ˚ A in the AFM edge. Our first-principles stress calculations confirm qualitatively the recent empirical-potential results[10] that both edges are under compressive stress. However, there are also some 3