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

Abstract

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 $\ensuremath{\theta}$ 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 ${0}^{\ensuremath{\circ}}<\ensuremath{\theta}<{60}^{\ensuremath{\circ}}$ (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 $\ensuremath{\theta}$ 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 $\ensuremath{\theta}$ of the formation energy parallels that of the out-of-plane distortions, with a secondary minimum at $\ensuremath{\theta}=32.{2}^{\ensuremath{\circ}}$ where the grain boundary is made of a flat zigzag array of only 5 and 7 rings. For $\ensuremath{\theta}>32.{2}^{\ensuremath{\circ}}$, other nonhexagonal rings are also possible. We discuss the importance of these findings for the interpretation of recent experimental results.