▎ 摘　　要

Recent ab initio calculations without spin [P. Koskinen et al., Phys. Rev. Lett. 101, 115502 (2008)] predict that the zigzag edges of graphene should be reconstructed, albeit with an energy barrier to be overcome. After verifying that spin-polarized calculations give qualitatively the same result, we study the mechanism and the free energy of the reconstruction of the zigzag edges from a periodic hexagon structure (zz) to an alternate pentagon-heptagon structure [zz(57)] using the empirical long-range carbon bond order potential II (LCBOPII) potential. The zz(57) edges are stabilized by an almost triple bond similar to that of the armchair edges, and we propose a way to account for this quantum mechanical effect in the LCBOPII potential. Aside from that, the reconstructed edge is flat as a result of tensile edge stress. The reconstruction occurs spontaneously in molecular dynamics simulations at high temperature, leading to the identification of a reaction coordinate for the reconstruction that allows us to calculate the free-energy barrier by using Monte Carlo simulations and umbrella sampling. At room temperature, we find a free-energy barrier of 0.83 eV for the first transformations of two hexagons to a pentagon-heptagon pair that is higher than the one for a fully reconstructed edge and increasing with temperature.