5 Distortions can also be made from equally stretching both lat- tices along one direction as in Fig. 3c; however the distortions of the resulting moir´e patterns are generally much smaller than for relative strains (on the order of several percent strain for a comparable length distortion). The reason is that for small moir´e rotation angles, the lattice vectors will obtain similarly sized angular distortions for each lattice vector pair, i , thereby yielding a relative angular distortion of ’$_{i}$ GLYPH<0> ’ 0 i GLYPH<25> 0. In this case, we can use the original moir´e equations to approximate the length distortion simply by replacing the lattice spacing a in equation (1) with its distorted values, a$_{i}$ = (1 + GLYPH<15>$_{i}$ ) a . Given a low enough angle rotation, the relative strain e GLYPH<11> ects are indeed observed. Low angle moir´e patterns are even more likely to occur when multiple patterns can be simultaneously observed. Figure 5a shows a fairly large moir´e pattern with an average unit cell length of D = 10 : 5 nm GLYPH<6> 0 : 9 nm, correspond- ing to a rotation angle GLYPH<18> = 1 : 36 GLYPH<6> 0 : 11 $^{GLYPH<14>}$; however, distortion from relative strain causes the actual size of the moir´e pattern to range from 9 : 50 nm to 11 : 7 nm, depending on which axis is Multilayer epitaxial graphene is known to have car- rier mean free times comparable to suspended monolayer graphene, 21,32,33 and measurements to date indicate that its electronic structure is e GLYPH<11> ectively that of independent graphene monolayers. 15,34 However, the physical structure of MEG is complex. Using the methods developed here, surface STM imaging can be used to obtain layer alignments and local strain from at least 3 graphene layers in the multilayer stack. These microscopic measurements will be useful for decipher- ing the formation of rotational domains and for mapping inho- mogeneous strain fields, each of which could a GLYPH<11> ect the quan- tum states of this material—especially under the influence of electric and magnetic fields. We thank C. Berger, M. Sprinkle, N. Sharma, S. Blanken- ship, A. Band, and F. Hess for their technical contributions to this work. Funding from NSF (DMR-0804908), the Semicon- ductor Research Corporation Nanoelectronics Research Ini- tiative (NRI-INDEX), and the W. M. Keck Foundation are gratefully acknowledged. Graphene production facilities were developed under NSF grant ECCS-0521041. measured. 26 This corresponds to a moir´e length distortion of 23% ( D$_{max}$ = D$_{min}$ = 11 : 7 = 9 : 50 = 1 : 23); thus we estimate the parameter " GLYPH<25> 0 : 37% for the relative strain between the two lattices producing the pattern. There are a few mechanisms by which a graphene sheet can undergo strain. In Fig. 5a, low angle grain boundaries are vis- ible. The moir´e pattern bends with the boundary on the right. In this case, the subsurface rippling of the lattice is likely caus- ing a slight relative strain. Another type of strain possible in graphene grown on SiC(000 ¯ 1) is a bending of the graphene sheet caused by unequal thermal contractions between the SiC and graphene sheets after growth. 30 In order to relieve stress, the lattice buckles (see Fig. 5e). The image in Fig. 5c was taken by atomic force microscopy, and shows these “pleats” of the graphene sheet. They are typically 5 nm to 10 nm in height, and are visible in Fig. 5c as white lines running par- allel and transverse to the SiC steps. Despite bending, the graphene lattice stays in tact over a pleat, and the graphene lattice can extend for several microns before a pleat occurs. The STM image in Fig. 5d was taken on top of a graphene pleat. The lattice is continuous, but it is clearly contorted as a result of the lattice buckling. The strain reilef looks much like the strain relief in nanotubes. 31 Flat areas near these pleats could show strain e GLYPH<11> ects. Immediately around pleats, slight layer delamination is possible which may not even produce a recognizable moir´e pattern to be observed. V. CONCLUSIONS FIG. 5: (a) An image of how strain can a GLYPH<11> ect a moir´e pattern. (b) displays the FT of the image in (a). As many as 5 sets of moir´e spots can be identified, requiring interactions from at least 4 layers. Strain can be caused by low angle grain boundaries seen in (a) or “pleats”. The AFM image in (c) shows these structures. The pleats are folds in the graphene lattice and tend to run along the SiC step or at angles roughly perpendicular to them. The STM image in (d) shows atomic imaging on top of a pleat. (e) Schematic of a pleat in epitaxial graphene. Mismatches in the thermal contraction after growth can induce buckling in the lattice. Acknowledgments