Online Determination of Graphene Lattice Orientation Through Lateral Forces

Rapid progress in graphene engineering has called for a simple and effective method to determine the lattice orientation on graphene before tailoring graphene to the desired edge structures and shapes. In this work, a wavelet transform-based frequency identification method is developed to distinguish the lattice orientation of graphene. The lattice orientation is determined through the different distribution of the frequency power spectrum just from a single scan line. This method is proven both theoretically and experimentally to be useful and controllable. The results at the atomic scale show that the frequencies vary with the lattice orientation of graphene. Thus, an adjusted angle to the desired lattice orientation (zigzag or armchair) can easily be calculated based on the frequency obtained from the single scan line. Ultimately, these results will play a critical role in wafer-size graphene engineering and in the manufacturing of graphene-based nanodevices. Electronic supplementary material The online version of this article (doi:10.1186/s11671-016-1553-z) contains supplementary material, which is available to authorized users.

(a) shows the graphene sample for atomic imaging. The black rectangle indicates the scanning site. The thicknesses of graphene are around 2.367nm, 2.455nm and 2.226nm, respectively, as shown in Fig. S1(a-1), (b-1) and (c-1).   1.0nm

Theoretical Modeling for atomic friction behavior
A two-dimensional Tomlinson model is used to study the atomic friction behavior on graphene. In this model, the AFM tip is handled as a point mass which is coupled to a support M ('AFM cantilever') by springs and the support M moves along thex direction with a constant velocity v M . The interaction between the tip and the surface is described by a periodic potential ) , ( t t y x V . All energy dissipation is considered by a simple velocity-dependent damping term  .
The motion of the tip in the interaction potential is given by [1]  The interaction potential of graphene used here is the same as the one of graphite [1] : where 246 . 0  a nm. The lattice structure can be rotated by a simple coordinate transformation, whereas the [100] direction (zigzag orientation) is defined as lattice angle 0°.
The equation (1) is solved using a 4th order Runge-Kutta method with the following initial conditions. , , , To obtain a numericalsolutiontoequation (1), suitable parameters must to be found. However, the exact values of these parameterscannot be obtained from experiments directly and these parameters have to be estimated. All the numerical calculations presented here are obtained using the set of parameters in Ref. [1]: typically used for AFM experiments), kg 10 8 (with these effective masses, the system has a common resonance frequency of  FIG. S2. Simulation frequency power spectrums of different lattice orientations.  =0° is defined as the zigzag orientation.  =30° indicate armchair orientation.  is the spatial frequency ratio. Figure S2 shows the simulation spatial frequency power spectrums of friction force signals at different lattice angles. The simulation friction force signals were obtained from two-dimensional Tomlinson model depicted above. The lattice angle β=0°

Simulation spatial frequency power spectrums of different lattice orientations
is defined as the zigzag orientation, thusβ=30°indicate armchair orientations.  is the spatial frequency ratio of two main peaks of the spectrums. The results show that the frequency power spectrum distributions vary with different lattice orientations. There are three types of distributions: one peak, two peaks and three peaks. The zigzag orientation has one-peak distribution; the armchair orientation has double-peak distribution. The angles in between with these two orientations have three-peak distribution, but the positions of the peaks are shifted with different orientations. Therefore, the frequency ratio varies with lattice orientations.

Wavelet transforms
Wavelet transforms include continuous wavelet transform (CWT) and discrete wavelet transform (DWT). The CWT of a signal x(t) is defined as [2] :

The effect of scanning parameters on the frequency ratio
FIG. S4.  at different scan rate for the same lattice orientation (lattice angle:14°). (a-1),(b-1) "Atomic resolution" AFM lateral force images (filtered by FFT) with lattice structures. The blue line shows 0° direction (zigzag direction) and the yellow line shows the scan direction. (a-2), (b-2) spatial frequency power spectrum.  is the spatial frequency ratio of the scan line L=58.