Chapter 5: Graphene and interlayer interaction
5.3 Direct growth of mm-size twisted bilayer graphene by
5.3.3 Simulations for the Moiré pattern and FFT of
reciprocal lattice points are still recognizable. We will treat 8 Moiré superlattices side to side as the minimum requirement to observe a clear Moiré reciprocal lattice in the FFT image. The STM image in Figure 5.23 (c) happens to have 8 Moiré superlattices from side to side, and therefore we can see a clear Moiré reciprocal lattice.
To achieve atomic resolution, 4 pixels per atom is the minimum requirement.
Consequently, we set 50 pm/pixel as the minimum requirement to resolve atoms. For a twist angle near the magic angle, the size of the Moiré superlattices is about 12 to 15 nm.
Therefore, a (100 nm × 100 nm) topographic map is required to observe clear a Moiré reciprocal lattice while (2000 × 2000) pixels are necessary to obtain atomic resolution and, therefore, the outer hexagon associated with graphene in FFTs. STM images with a large number of pixels generally take a long time to acquire, and thermal shifts during the long scan could distort the resulting image. Therefore, taking one larger area and another smaller area scan of topography and combine their FFTs may be a good compromise to provide good estimates of the twist angle.
Figure 5.26: Simulation of twist bilayer graphene. The first row shows a comparison between real STM topography and a simulated topography at 10.32° and AB stacking (0°). The second row explains the deviation of simulations from the real image when the bottom lattice has holes.
Simulations can help us optimize the choice of scan parameters. From Figure 5.23, we can see that the Moiré pattern simulation used in Section 5.2 gives us an accurate measurement of the Moiré superlattices in topography and FFT. However, closer inspection of the topography of simulation is not perfectly the same as the STM image (Figure 5.26, first row).
This problem arises from how my code deals with a hole in the middle of the hexagonal lattice: When the hole of a hexagon is on top of the hole of another hexagon at the bottom, our simulation function creates an artificially deeper hole. This is because, for an STM tip scanning over the surface of graphene, the effective hole to the STM tip is only a little lower than the nearby atoms because of the finite size of the atom on the STM tip. Generally speaking, the simulated pattern for atoms or lattices is still correct with height, although in the case of holes associated with the top layer, they would appear more in-depth than the real situation.
On the other hand, if the atoms at the bottom can fill the void of the top lattice, the simulated topography is accurate. For example, AB stacking bilayer graphene in Figure 5.26 is nearly the same as what we observe from STM topography. The simulation of graphene on a copper substrate is also accurate because of no holes in the copper substrate as well as the bigger size of copper atoms.
Deeper holes usually introduce more reciprocal lattice points in the FFT, but they will not remove correct features from the lattice points. As we have seen in Section 5.2, the FFTs of our simulation always provide more points than the FFTs of real STM images. Hence, we need to be careful in order to recognize the correct reciprocal lattice points in the FFTs from simulations.
Simulation for Moiré pattern and FFT of twist bilayer graphene at the magic angle In Figure 5.27, simulations of the Moiré pattern and its FFT for the magic angle are done for different sizes, (100 nm × 100 nm) and (30 nm × 30 nm), and with different pixels, (256
× 256), (512 × 512) and (2048 × 2048), from side to side. The (100 nm × 100 nm)
topography includes around 8 superlattices. The Moiré pattern is clear despite of the limited resolution. Therefore, the inner hexagon in FFT images is clear for all resolution.
The outer hexagon is still complete for (512 × 512) and (2048 × 2048) pixels. However, for (256 × 256) pixels, the outer hexagon disappears. If we look at the topography for (256
× 256) pixels and (100 nm × 100 nm), stripe features appear due to poor resolution.
Quantitatively, for the 100-nm/256-pixel topography, each pixel spans 0.4 nm, which is larger than the size of a graphene atom. Therefore, it is not surprising that the outer hexagon disappears in the FFT.
For the (30 nm × 30 nm) topography, it spans about 3 superlattices from side to side.
Surprisingly, even at (256 × 256) pixels, both the inner and outer hexagons can already be resolved. However, the Bragg spots of the inner hexagon are much larger than those in the FFT of the (100 nm × 100 nm) topography. On the other hand, the outer hexagon is much clearer due to higher resolution.
From the simulation, we know that Bragg spots of Moiré superlattices can be resolved in FFTs even without real-space atomic resolution. Therefore, properly calibrated STM topography can provide the size of Moiré superlattices either in topography or in FFT space, thereby giving us a good estimate of the twist angle. Additionally, simulations can provide a good idea about what the shape of the Moiré pattern should look like, thereby helping us identify from STM measurements the real Moiré patterns associated with the sample and excluding the substrate features.
Since the STM calibration conditions can vary easily with the environment, temperature, time, and various other factors, the topography of Moiré patterns with atomic resolutions can provide self-calibration and help us estimate the twist angle more accurately. From our simulations, we have found that topography with a size of 4 superlattice constants and with atomic resolution will be sufficient to reveal a complete outer hexagon and an inner hexagon in the FFT. Therefore, simulations help set a lower limit on the spatial size and the number of pixels, thereby enabling more efficient STM data acquisition.
Figure 5.27: Simulation of the Moiré pattern and its FFT near the magic-angle 𝜃𝜃 = 1.1° for different sizes of areas and resolutions. For topography, the top image has (2048 × 2048) pixels, while the bottom image has (256 × 256) pixels. For FFT, the top image is around the first Brillouin zone, and the bottom image is a zoom-in of the top image.