On the Mechanism Controlling the Relative Orientation of Graphene Bi-Layers
We have measured the relative orientation of rotated graphene bi-layers (RGBs) deposited by chemical vapor deposition and found that there are spontaneously occurring preferred orientations. Measurements were performed using selected area electron diffraction patterns on various regions of the RGBs. These orientations minimize the complexity of the lattice defined by the set of all possible Burgers vectors. By using a precise definition of singularity, we have been able to show that all non-singular preferred orientations are special in the sense that their angular distance ∆θ to the closest singular orientation also complies with the definition of singularity. Our results show that these special interfaces, named secondary singular interfaces, have simpler displacement fields compared to other non-singular RGBs, implying that interfacial dislocations have fewer Burgers vectors to choose from. Since all observed orientations were found to be either singular or secondary singular, we found evidence that RGBs starting out with rotation angles far from singular orientations re-orient themselves into a nearby secondary singular state in order to simplify their strain fields. Secondary singular orientations also account for the spontaneous formation of high Σ interfaces, although the lack of a precise definition of singularity caused them to remain unnoticed.