Several experiments had revealed the presence of grain boundaries in graphene that might change its electronic and elastic properties. Here, a general theory for the structure of [00•1] tilt grain boundaries in graphene was presented which was based upon coincidence site lattice theory. It was shown that the latter theory uniquely classifies the grain boundaries in terms of the misorientation angle θ and periodicity d using two grain-boundary indices (m,n), similar to the nanotube indices. The structure and formation energy of a large set of grain boundaries generated by the coincidence site lattice theory for 0∘<θ<60∘ (up to 15608 atoms) were optimized by a hierarchical methodology and validated by density functional calculations. It was found that low-energy grain boundaries in graphene can be identified as dislocation arrays. The dislocations form hillocks like those observed by scanning tunneling microscopy in graphene grown on Ir(111) for small θ that flatten out at larger misorientation angles. It was found that, in contrast to three-dimensional materials, the strain created by the grain boundary could be released via out-of-plane distortions that led to an effective attractive interaction between dislocation cores. Therefore, the dependence upon θ of the formation energy paralleled that of the out-of-plane distortions, with a secondary minimum at θ=32.2∘ where the grain boundary was made of a flat zig-zag array of only 5 and 7 rings. For θ>32.2∘, other non-hexagonal rings were also possible.

Theory and Hierarchical Calculations of the Structure and Energetics of [0001] Tilt Grain Boundaries in Graphene. J.M.Carlsson, L.M.Ghiringhelli, A.Fasolino: Physical Review B, 2011, 84[16], 165423