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 as the difference between two reciprocal lattice vectors of the two coinciding lattices. Considering reciprocal lattice vectors , 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 moiré pattern. We predict how many beating frequencies occur in nth-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 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.
Highlights
Moiré patterns are generally observed if two lattices are stacked on top of each other, while having either a different lattice constant or a different rotational orientation or both at the same time
The problem has been known in epitaxy and surface science for a long time [1,2,3,4], but the topic regained interest after transition metal (TM) catalyzed chemical vapor deposition was identified as a promising growth protocol for graphene (g) [5,6,7,8]
Calculating commensurability plots for first- and second-order moirés using augmented cells is a very convenient way to derive unit cells without having to include higher-order lattice frequencies. This is discussed for the case of the (6 3 ́ 6 3 )R30 moiré with the unit cell notation (12, 6)TM/(13, 0)g, which is reported for graphene on SiC(0001) [24]
Summary
Yat-Sen University, Guangzhou 510275, People’s Republic of China 4 Author to whom any correspondence should be addressed. We derive general properties of the moiré regarding its symmetry and constru ct the spatial beating frequency Kmoiré as the difference between two reciprocal lattice vectors ki of the two coinciding lattices. Considering reciprocal lattice vectors ki, 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 moiré pattern. 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 3 ́ 6 3 )R30 moiré of graphene on SiC(0001). We 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
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