Indexing moiré patterns of metal-supported graphene and related systems: strategies and pitfalls

Abstract

We report on strategies for characterizing hexagonal coincidence phases by analyzing the involved spatial moiré beating frequencies of the pattern. We derive general properties of the moiré regarding its symmetry and construct the spatial beating frequency ${\vec{K}}_{\text{moir{\'e}}}$ as the difference between two reciprocal lattice vectors ${\vec{k}}_{i}$ of the two coinciding lattices. Considering reciprocal lattice vectors ${\vec{k}}_{{i}}$ , 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 n th-order moiré pattern. We predict how many beating frequencies occur in n th-order moirés 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 moiré 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 $(6\sqrt{3}\times 6\sqrt{3}){R}_{{30}^{^\circ }}$ moiré of graphene on SiC(0001). We also show how to apply our model for the characterization of hexagonal moiré phases, found for transition metal-supported graphene and related systems. We explicitly 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. We also point out the close relation of spatial beating frequencies in a moiré and multiple scattering in electron diffraction data and show how this fact can be explicitly used to extract high-precision data.

Keywords

• moiré
• graphene
• 2-dimensional hexagonal systems
• low-energy electron diffraction
• x-ray diffraction
• scanning tunneling microscopy