Theory and hierarchical calculations of the structure and energetics of [0001] tilt grain boundaries in graphene

Several experiments have revealed the presence of grain boundaries in graphene that may change its electronic and elastic properties. Here, we present a general theory for the structure of [0001] tilt grain boundaries in graphene based on the coincidence site lattice (CSL) theory. We show that the CSL theory uniquely classifies the grain boundaries in terms of the misorientation angle theta 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 CSL theory for 0 degrees < theta < 60 degrees (up to 15 608 atoms) were optimized by a hierarchical methodology and validated by density functional calculations. We find 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 theta that flatten out at larger misorientation angles. We find that, in contrast to three-dimensional materials, the strain created by the grain boundary can be released via out-of-plane distortions that lead to an effective attractive interaction between dislocation cores. Therefore, the dependence on theta of the formation energy parallels that of the out-of-plane distortions, with a secondary minimum at theta = 32.2 degrees where the grain boundary is made of a flat zigzag array of only 5 and 7 rings. For theta > 32.2 degrees, other nonhexagonal rings are also possible. We discuss the importance of these findings for the interpretation of recent experimental results.