Moiré scaling of the sliding force in twisted bilayer graphene

The weak interlayer binding in two-dimensional layered materials such as graphite gives rise to distinguished low-friction properties if the atomic lattices at the interface are rotated with respect to one another. The lack of crystal symmetry leads to poorly understood correlations and cancelations of the interlayer atomic forces. Here we report on a powerful tiling method based on the moir\'e superstructure which allows us to study the intricate interplay of the interlayer forces in a systematic manner. Based on numerical simulation data for a circular graphene flake on an infinite graphene substrate, it is shown that the sliding force is dominated by a rim area consisting of incomplete moir\'e tiles. This rim force, which scales with the number of atoms in a moir\'e tile and as the radius to the power of 0.5, is minimal whenever the sliding structure can be approximated by a hexagon composed of an integer number ${N}_{t}$ of moir\'e tiles. Intriguingly, the corresponding area force scales as ${N}_{t}$ to the power of 0.25, i.e., it increases with size, whereas it has been often argued that interlayer forces should add up to a zero value for large twisted systems. However, at specific twist angles the moir\'e structure is commensurate with the graphene lattice, leading to a perfect force correlation in the moir\'e tiles. Correspondingly, the area force becomes dominant and scales as ${N}_{t}$, i.e., as the radius to the power of 2.