First-principles study of graphene edge properties and flake shapes

We perform density-functional-theory calculations to determine the equilibrium shape of graphene flakes as a function of temperature and hydrogen partial pressure. To do this, we first determine the edge orientation dependence of the edge energy and edge stress of graphene nanoribbons. The edge energy is a monotonically increasing function of edge angle; increasing from the armchair orientation to the zigzag orientation. However, reconstruction of the zigzag edge lowers its energy to less than that of the armchair edge in the absence of the hydrogen. The edge stress for all edge orientations is compressive, however, reconstruction of the zigzag edge reduces this edge stress to near zero. Hydrogen adsorption is favorable for all edge orientations at sufficiently high hydrogen partial pressure; dramatically lowering all edge energies and all edge stresses. It also lifts the reconstruction of the zigzag edge. Using these edge energy data within a grand canonical ensemble approach, we determine the equilibrium shape of a graphene flake to be hexagonal with armchair oriented edges at both very small and very large chemical potentials of hydrogen. In the former case, the edges are hydrogen free while in the latter they are hydrogen terminated. At intermediate chemical potentials of hydrogen, graphene flakes show a more complex series of shapes, from zigzag edge terminated (near) hexagons, to dodecagons, to shapes with rounded edges. Zigzag edge reconstruction produces graphene flakes with a sixfold symmetry but with rounded edges. This shape is dominated by near zigzag edges. The compressive edge stresses will lead to edge buckling (out-of-the-plane of the graphene sheet) for all edge orientations, in the absence of hydrogen. Exposing the graphene flake to hydrogen dramatically decreases the magnitude of the compressive edge stresses and reduces the buckling amplitude.