Cones, Pringles, and Grain Boundary Landscapes in Graphene Topology

A polycrystalline graphene consists of perfect domains tilted at angle a to each other and separated by the grain boundaries (GB). These nearly one-dimensional regions consist in turn of elementary topological defects, 5-pentagons and 7-heptagons, often paired up into 5-7 dislocations. Energy G(alpha) of GB computed for all range 0 <= alpha <= pi/3, shows a slightly asymmetric behavior, reaching similar to 5 eWrim in the middle, where the 5's and 7's qualitatively reorganize in transition from nearly armchair to zigzag interfaces. Analysis shows that two-dimensional (20) nature permits the off-plane relaxation, unavailable in three-dimensional (3D) materials, qualitatively reducing the energy of defects on one hand while forming stable 3D landscapes on the other. Interestingly, while the GB display small off-plane elevation, the random distributions of 5's and 7's create roughness that scales inversely with defect concentration, h similar to n(-1/2).