Structural analysis of multilayer graphene via atomic moir\'e interferometry

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Structural analysis of multilayer graphene via atomic moiré interferometry

David L. Miller, Kevin D. Kubista, Gregory M. Rutter, Ming Ruan, Walt A. de Heer, Phillip N. First, and Joseph A. Stroscio

Phys. Rev. B 81, 125427 – Published 24 March 2010

Abstract

Rotational misalignment of two stacked honeycomb lattices produces a moiré pattern that is observable in scanning tunneling microscopy as a small modulation of the apparent surface height. This is known from experiments on highly oriented pyrolytic graphite. Here, we observe the combined effect of three-layer moiré patterns in multilayer graphene grown on $SiC (000 \bar{1})$ . Small-angle rotations between the first and third layer are shown to produce a “double-moiré” pattern, resulting from the interference of moiré patterns from the first three layers. These patterns are strongly affected by relative lattice strain between the layers. We model the moiré patterns as a beat-period of the mismatched reciprocal lattice vectors and show how these patterns can be used to determine the relative strain between lattices, in analogy to strain measurement by optical moiré interferometry.

Received 15 January 2010

DOI:https://doi.org/10.1103/PhysRevB.81.125427

Authors & Affiliations

David L. Miller¹, Kevin D. Kubista¹, Gregory M. Rutter², Ming Ruan¹, Walt A. de Heer¹, Phillip N. First¹, and Joseph A. Stroscio²

¹School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
²Center for Nanoscale Science and Technology, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA

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Vol. 81, Iss. 12 — 15 March 2010

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Figure 1
(Color online) STM topographs of two moiré patterns on multilayer epitaxial graphene (sample bias $V_{S} = 0.5 V$ , tunnel current $I = 100 pA$ ). (a) Two similarly sized moiré patterns interfering, resulting in a large superlattice unit cell. Layers 1 (surface layer) and 2, and layers 2 and 3 have comparable rotation angles in opposite directions (shown schematically in d). (b) High resolution image of a region in (a). (c) FT of (a). (d) Schematic of layer orientations based on the moiré pattern shown in (a) and (b). (e) STM image of a different moiré superlattice. (f) High resolution image from within the same area. (g) FT of (e). Spots on the dotted circle result from the interference of two moiré patterns (three graphene layers). The top-layer atomic reciprocal lattice from a FT of (f) determines which spots correspond to the moiré formed from layers 1 and 2 (see Eq. (1)). (h) schematic of layer orientations based on the moiré pattern in (e).Reuse & Permissions
Figure 2
Multibias imaging on the moiré pattern in Fig. 1a. Each image is $5 nm \times 5 nm$ , and was taken at the same spatial location and tunnel current $(100 pA)$ . In (a), only the moiré pattern of the top two layers is imaged. (b) and (c) show the transition as the tip moves closer to the surface. In (d), the tip has pushed in far enough to only image the lower layer moiré pattern. We believe that by $V_{S} = - 0.05 V$ , the tip is actually in contact with the surface.Reuse & Permissions
Figure 3
(Color online) Strain induced between two stacked hexagonal (real space) lattices in the case where one lattice (black/dashed line) is (a) stretched but not rotated and (b) stretched and rotated with respect to the undistorted lattice (blue/solid line). (c) displays equal strain for both lattices. For rotated lattices under relative strain, the angular change in the lattice vectors affect the interference pattern differently for each direction. As a result, the moiré pattern observed in STM will appear stretched or distorted. Significantly higher strain is required for similarly sized distortions if the lattices are equally strained along the same direction. At the low moiré angles $(θ)$ , the lattice vectors (when equally strained) will obtain similarly sized angular distortions $φ_{i}$ which will nullify each other.Reuse & Permissions
Figure 4
(Color online) Curves display the length distortion between two vectors of the moiré cell as a function of the moiré rotation angle $θ$ . The amount of distortion for a small strain rapidly increases with decreasing moiré rotation angles (i.e., when the rotation angle is comparable to the angular distortion, $θ \sim φ$ ).Reuse & Permissions
Figure 5
(Color online) (a) STM image demonstrating the effect of strain on the moiré superlattice ( $V_{S} = 0.35 V$ , $I = 100 pA$ ). (b) FT of the image in (a). Five sets of moiré spots can be identified, requiring interactions from at least 4 layers. Strain may be caused by subsurface low-angle grain boundaries [marked approximately with green dotted lines in (a)] or by nearby “pleats.” (c) Large-area AFM image showing pleats; i.e., folds in the stacked graphene layers. Pleats tend to run along the SiC step or perpendicular to them. (d) Atomically resolved STM image on the top of a pleat. (e) Schematic of a pleat in epitaxial graphene. Mismatches in the thermal contraction after growth can induce such buckling of the lattice.Reuse & Permissions

Physical Review B

covering condensed matter and materials physics

Structural analysis of multilayer graphene via atomic moiré interferometry

David L. Miller, Kevin D. Kubista, Gregory M. Rutter, Ming Ruan, Walt A. de Heer, Phillip N. First, and Joseph A. Stroscio

Phys. Rev. B 81, 125427 – Published 24 March 2010

Abstract

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Physical Review B

covering condensed matter and materials physics

Structural analysis of multilayer graphene via atomic moiré interferometry

David L. Miller, Kevin D. Kubista, Gregory M. Rutter, Ming Ruan, Walt A. de Heer, Phillip N. First, and Joseph A. Stroscio

Phys. Rev. B 81, 125427 – Published 24 March 2010

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