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Zeller, Patrick; Ma, Xinzhou; Günther, Sebastian (2017): Indexing moire patterns of metal-supported graphene and related systems: strategies and pitfalls. In: New Journal of Physics, Vol. 19, 13015
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Abstract

We report on strategies for. characterizing hexagonal coincidence phases by analyzing the involved spatial moire beating frequencies of the pattern. Wederive general properties of the moire regarding its symmetry and construct the spatial beating frequency (K) over right arrow (moire) as the difference between two reciprocal lattice vectors. (k) over right arrow (i) of the two coinciding lattices. Considering reciprocal lattice vectors. (k) over right arrow (i), with lengths of up to n times the respective (1, 0) beams of the two lattices, readily increases the number. of beating frequencies of the nth-order moire pattern. Wepredict how many beating frequencies occur in nth-order moires and show that for one hexagonal lattice rotating above another the involved beating frequencies follow circular trajectories in reciprocal-space. The radius and lateral displacement of such circles are defined by the order n and the ratio x of the two lattice constants. The question of. whether the moire pattern is commensurate or not is addressed by using our derived concept of commensurability plots. When searching potential commensurate phases we introduce a method, which we call cell augmentation, and which avoids the need to. consider high-order beating frequencies as discussed using the reported (6 root 3 x 6 root 3) R-30 degrees. moire of graphene on SiC(0001). We also show how to apply our model for the characterization of hexagonal moire phases, found for transition metal-supported graphene and related systems. Weexplicitly treat surface x-ray diffraction, scanning tunneling microscopy-and low-energy electron diffraction data to extract the unit cell of commensurate phases or to find evidence for incommensurability. For each data type, analysis strategies are outlined and avoidable pitfalls are discussed. Wealso point out the close relation of spatial beating frequencies in a moire and multiple scattering in electron diffraction data and show how this fact can be explicitly used to extract high-precision data.