Correlation Between Micrometer-Scale Ripple Alignment and Atomic-Scale Crystallographic

Correlation between micrometer-scale ripple alignment and atomic-scale crystallographic orientation of monolayer graphene

Jin Sik Choi1+,4, Young Jun Chang2+, Sungjong Woo3+, Young-Woo Son3*, Yeonggu Park1, Mi Jung Lee1, Ik-Su Byun1, Jin-Soo Kim1,4, Choon-Gi Choi4, Aaron Bostwick5, Eli Rotenberg5, Bae Ho Park1*

1Division of Quantum Phases and Devices, Department of Physics, Konkuk University, Seoul 143-701, Korea

2Department of Physics, University of Seoul, Seoul 130-743, Korea

3Korea Institute for Advanced Study, Seoul 130-722, Korea

4Creative Research Center for Graphene Electronics, Electronics and Telecommunications Research Institute (ETRI), Daejeon 305-700, Korea

5Advanced Light Source (ALS), E. O. Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

+These authors contributed equally to this work.

*email: (Y.W.S.) and (B.H.P.)

Supplementary Figure 1

Supplementary Figure 1. Electrode formation. Optical microscopy images of a graphene connected to silver paint, viewed under magnifications of (a)4 ×, (b) 50 ×, and (c) 500 ×. In Fig. 1a, one can see only silver paint as the bright area while graphene sample is barely visible. The enlarged images in Figs. 1b and 1c are obtained from the areas designated by yellow dashed rectangles in Figs. 1a and 1b, respectively.The single-layer graphene (SLG) is extruded from multilayer graphene (MLG), while the MLG has a good contact with the silver paint. By means of these optical microscopy images, with silver paint as a reference for position and direction, we can carry out CTM and ARPES measurements and compare the ripple direction and crystallographic axis of SLG obtained through both methods.

Supplementary Figure 2

Supplementary Figure 2. Ripple direction determination using CTM images, Dependence of (a) TLON and (b) TLAT on the ripple direction.  is the angle (counter-clockwise) of the ripple relative to the forward lateral direction of the cantilever. Figure 2a shows the ripple directions of the domains with bright (black), medium (red), and dark (blue) contrast for the TLON image in Fig. 1(b) in the main article. Figure 2b shows the ripple directions of the domains with bright (blue), medium (black), and dark (red) contrast for the TLAT image in Fig.1(b) in the main article.By comparing the areas with the same color in Figs. 2a and 2b, the ripple direction can be determined within a range of 30° (colored fan shape in Fig. 1(b) in the main article) for each domain indicated by the same color-filled rectangle in the CTM images.1

Supplementary Figure 3

Supplementary Figure 3.Linear energy dispersion near the Fermi level.The energy band cut of the graphene flake from ARPES taken through the K point (a) along and (b) perpendicular to the - direction. The data show linear energy dispersion with Dirac point near the Fermi level, which implies that the sample is barely doped. The dashed arrow corresponds to that in Fig. 2(d) in the main article.

Supplementary Note 1: Theoretical calculations

We estimated the preferred direction and width of ripples in graphene on a SiO2 surface. The stick-spiral model2 was used for calculating the rippling energy of graphene while the Lennard-Jones potential was used for the estimation of the energy of interaction between graphene and the substrate. The energy needed per atom for bending graphene along the zigzag (armchair) direction, as a function of curvature, κ, is given by

,

where α2 = 1.827 × 10−2 eVnm2 , α4,Z = 7.963 × 10−5 eVnm4 , α4,A = 1.516 × 10−4 eV nm4.2 Here, the subscript Z(A) denotes the zigzag (armchair) direction. The difference in energy between zigzag and armchair directions appears in the fourth order term of the curvature. With the surface of a rippled graphene approximated as a sinusoidal shape,, the curvature becomes

,

where A and l are the amplitude and the width of the ripple, respectively. Once graphene adheres to the substrate, ripples can be thought to be caused by the compressive strain originating from the substrate. It is reasonable to assume that such compressive strain induces a curvature in graphene without changing the C−C bond length, as σ bonds are much stiffer than π bonds. Under such conformal deformation, the ratio of A to l is independent of l for a given compressive strain, . For our calculations, is chosen, which corresponds to 10.14% compressive strain. The specific value of is chosen based on our experimental observation. We average along the surface numerically,

,

where . For a given compressive strain, decreases as the ripple width increases since the overall curvature gets smaller with larger ripple width.

On the other hand, the interaction between graphene and the substrate is estimated by a Lennard-Jones potential,

,

where U0, d0 and d are the binding energy, equilibrium distance, and the distance of the graphene from the substrate, respectively. U0 = 0.0736 eVatom-1 = 2.81 eVnm-2and d0 = 0.35 nmare used for the calculation.3 For a rippled graphene, the distance varies as . For large ripples with height comparable with d0, A is added in order to maintain the minimum distance to bed0since the LJ potential is very stiff for dd0.We average along the surface numerically as . For a given compressive strain, increases as the ripple width increases because the average distance between graphene and the substrate gets larger with larger ripple width.

The two curves in Fig. 3 in the main article show the sum of the calculated rippling energy and the graphene-substrate interaction energy, , for zigzag and armchair directions, as a function of the ripple width. It indicates that the total energy minimum occurs at the ripple width of 5.7 nm for both cases, while the ripple along the zigzag direction with the energy-minimum ripple width has a lower energy than that along the armchair direction by 0.16 meVnm-2. The graphene figures in Fig. 3 show relaxed structures, as obtained from ab initio calculations, with the given compressive strain and ripple width at the energy minimum. They confirm that the shapes of the ripple are actually sinusoidal and the deformation is conformal.

Supplementary References

1. Choi, J. S. etal. Facile characterization of ripple domains on exfoliated graphene. Rev. Sci. Instrum. 83, 073905 (2012).

2. Chang, T., Geng, J., Guo, X. Chirality- and size-dependent elastic properties of single-walled carbon nanotubes. App. Phys. Lett. 87, 251929 (2005). Ma, T., Li, B., Chang, T. Chirality- and curvature-dependent bending stiffness of single layer graphene. App. Phys. Lett. 99, 201901 (2011).

3. Koenig, S. P. etal. Ultrastrong adhesion of graphene membranes. Nat. Nanotech. 6, 543-546 (2011).