2 FIG. 1: Comparison of two moir´e patterns in MEG showing multiple layer e GLYPH<11> ects. (a) An image of two similar sized moir´e patterns interfering, resulting in a large unit cell pattern. Layers 1 (Surface Layer) and 2, and 2 and 3 have comparable rotation angles in opposite directions (shown in d). (b) zoomed in view of (a). (c) FT of (a). (d) schematic of layer orientation based on the moir´e pattern in (a). (e) A di GLYPH<11> erent superstructure that indicates low rotation angles between layers 2 and 3. The spots in the FT relative to the reciprocal lattice vectors were used in this case to determine which spots correspond to the rotation between layer 1 and 2. (f) zoomed in view of (e). (g) FT of (e). (h) schematic of layer orientation based on the moir´e pattern in (e). patterns are often seen during imaging (Figure 1). Unlike HOPG, Bernal stacking does not frequently occur; thus, sub- surface layers will also have moir´e alignment, which can be observed. Various publications have reported an attenuation factor (AF) associated with overlayer coverage of a moir´e pat- tern in HOPG. 8,9,24 A fit of this data yields AF$_{n}$ = e 0 : 81 $^{n}$, where n is the number of overlayers. 10 In view of this, a second / third layer moir´e pattern would be visible ( AF$_{1}$ = 2 : 25) if present but the visibility of patterns from deeper layers decays expo- nentially. The length of a superlattice cell can be evaluated as the wavelength of an interference pattern resulting from the mis- aligned reciprocal lattice vectors. The di GLYPH<11> erence between a reciprocal lattice vector k$_{i}$ and a reciprocal lattice vector ro- tated by angle GLYPH<18> , k 0 i = ˆ R ( GLYPH<18> ) k$_{i}$ , is used to determine the length ( D ) and orientation ( GLYPH<30> ) of the pattern. Since we are interested in long wavelength moir´e patterns (rotation angles, GLYPH<18> GLYPH<20> 5 $^{GLYPH<14>}$), the only relevant vectors of the interference pattern will be GLYPH<1> k = k$_{i}$ GLYPH<0> k 0 j for i = j where i is any of six reciprocal lattice vectors. For unstrained lattices, only one pair of vectors k and k 0 needs to be considered because of symmetry. Thus, for two rotated lattices of undistorted length, we obtain the equation for the size of the moir´e unit cell length 5,25 and orientation with respect to the atomic lattice, D = 2 GLYPH<25> j k GLYPH<0> k 0 j = a 2 sin ( GLYPH<18>= 2) ; (1) GLYPH<30> = GLYPH<25> 6 GLYPH<0> GLYPH<18>= 2 : (2) Figure 1a shows an example of two interfering moir´e patterns (the collective interference of three graphene layers), obtained by STM measurements. On first glance, one might think that the moir´e pattern constitutes a significant modulation in the tip height. In fact, the graphene lattice is extremely smooth, such that any small changes in tip height become the foremost feature. Height modulations due to moir´e patterns are typi- cally only 0 : 05 nm peak-to-peak. From Fig. 1a and Fig. 1b, two periodicities can be seen. The large periodicity has a length of D GLYPH<25> 26 : 5 nm while the smaller periodicity has a length of D GLYPH<25> 3 nm, still much larger than the atomic lat- tice spacing. Because more than a single moir´e pattern is ob- served, we can conclude that this is a combined e GLYPH<11> ect of at least three layers (in another area of the sample, 3 coexisting moir´e patterns were found, implying the participation of at least 4 graphene layers). From the discrete fast Fourier trans- form (FT) in Fig. 1c, we find that two sets of spots are present in reciprocal space. These spots correspond to moir´e lengths of D$_{1}$ = 2 : 95 GLYPH<6> 0 : 05 nm and D$_{2}$ = 3 : 35 GLYPH<6> 0 : 09 nm. 26 Using equations (1) and (2) and a = 0 : 246 nm as the graphene lattice constant, we deduce the rotation angles needed to produce a moir´e of size D$_{1}$ and D$_{2}$ as GLYPH<18>$_{1}$ = 4 : 78 GLYPH<6> 0 : 07 GLYPH<14> and GLYPH<18>$_{2}$ = 4 : 21 GLYPH<6> 0 : 11 $^{GLYPH<14>}$. Because GLYPH<1> GLYPH<30> = GLYPH<30>$_{1}$ GLYPH<0> GLYPH<30>$_{2}$ GLYPH<25> 0 $^{GLYPH<14>}$, as seen in the Fourier transform, we propose the orientation of the top three layers in Fig. 1d. The first and third layers (starting from the surface and counting into the bulk) must be nearly aligned in order to obtain the observed GLYPH<1> GLYPH<30> in the FT. The measured value is GLYPH<1> GLYPH<30> = 0 : 61 GLYPH<6> 0 : 23 $^{GLYPH<14>}$, slightly higher than the expected value of GLYPH<1> GLYPH<30> = 0 : 29 GLYPH<14> from the proposed three