Microsc. Microanal. 20, 1753–1763, 2014 doi:10.1017/S1431927614013105
© MICROSCOPY SOCIETY OF AMERICA 2014
Thickness and Rotational Effects in Simulated HRTEM Images of Graphene on Hexagonal Boron Nitride Avery J. Green* and Alain C. Diebold Optical Physics Group, SUNY College of Nanoscale Science and Engineering, 257 Fuller Road, Albany, NY 12203, USA
Abstract: Recent studies have shown that when graphene is placed on a thin hexagonal boron nitride (h-BN) substrate, unlike when it is placed on a typical SiO2 surface, it can closely approach the ideal carrier mobility observed in suspended graphene samples. This study further examines the epitaxial relationship between graphene and h-BN substrate with high-resolution transmission electron microscopy simulation. Virtual monolayer and multilayer stacks of h-BN were produced with a monolayer of graphene on top, on bottom, and in between h-BN layers, in order to study this interface. Once the simulations were performed, the phase contrast image and Moiré pattern created by this heterostack were analyzed for local and global intensity minima and maxima. In addition, h-BN substrate thickness and rotations between h-BN and graphene were probed and analyzed. The simulated images produced in this work will be used to help understand subsequent transmission electron microscopy images and electron energy-loss studies. Key words: HRTEM, zone axis, power spectrum, Moiré pattern, Dirac point
I NTRODUCTION Since monolayer graphene was identified in 2004, its physics and applications have been quickly growing fields (Novoselov et al., 2004; Geim & Novoselov, 2007; Castro Neto et al., 2009). Its linear energy dispersion and resultant massless Dirac fermion charge carriers (Castro Neto et al., 2009) have drawn the interest of academic researchers and the semiconductor industry. Many steps have been taken toward a viable and efficient method of integration into current field-effect transistor devices, but that goal has not yet been achieved, and there is more preliminary work to be done. It has been shown that the performance of graphene is hindered by the lack of thickness uniformity in typical SiO2 layers (Ishigami et al., 2007; Lui et al., 2009). Thus, finding an insulating substrate to support the structure of graphene that will perpetuate the minimum thickness inconsistency from the oxide sublayer is a key area of research. Hexagonal boron nitride (h-BN) is a promising material for this purpose. The large band gap (5.97 eV) (Watanabe et al., 2004), small lattice mismatch with graphene (~1.7%), atomic flatness, high mechanical strength, high temperature stability, low dielectric constant, and relative lack of dangling bonds and charge traps (Kubota et al., 2007) are all desired characteristics for a graphene channel substrate. Multiple studies have been done on graphene/h-BN stacks, which have included the use of various atomic resolution techniques like atomic force microscopy (AFM) and scanning tunneling microscopy (STM), and have produced results that support h-BN’s prominence as a graphene substrate candidate (Decker et al., 2011; Xue et al., 2011; Yankowitz et al., 2012). However, these studies have been Received January 13, 2014; accepted August 12, 2014 *Corresponding author. [email protected]
limited by surface signal dominance because these techniques lack significant penetration and atomic resolution imaging below the surface of the stack. High-resolution transmission electron microscopy (HRTEM) and the simulations thereof are better designed to observe the atoms below the top layer of graphene or h-BN, and therefore can give more insight into h-BN/graphene interfaces. In this study, HRTEM simulations were done on the [001] zone axis of the materials. We used varying layers of h-BN above and below monolayer graphene, as well as lattice angle mismatches between the graphene and h-BN layers from 0 to 30° in intervals of 5°, in order to observe how these permutations affect the interfaces. The goal was to create a guideline that explains how rotation and thickness affect images in our simulations and helps with transmission electron microscopy (TEM) image interpretation. In order to confirm the validity of these findings, we wanted to compare our simulated images to experimental images from the literature. As we could find no experimental HRTEM or scanning transmission electron microscopy (STEM) images of the stacks simulated in this paper in the literature, we compared our simulated images with experimental images of stacks with similar crystal structures. Namely, these were images of turbostratic bilayer graphene and turbostratic bilayer h-BN, which show some of the same stacking effects observed in our simulated images (Zan et al., 2011). We will use the simulations presented here to correlate heterostack geometry with changes in electron energy-loss data, as described by Nelson (2012).
MATERIALS
AND
METHODS
To create the following images and data, two TEM simulation programs and MATLAB were used. MATLAB was used
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Figure 1. Graphene (a) and hexagonal boron nitride (h-BN) (b) lattices. Both have honeycomb lattices with A and B atomic sites. Graphene’s lattice parameter is 2.46 Å, and h-BN’s lattice parameter is 2.50 Å. Small purple atoms are boron, and large green atoms are nitrogen.
to create crystal simulation data, and to process the images produced by the other software. The simulation programs were MacTempasX and CrystlKitX (Kilaas, 2013). CrystalKitX was used to create virtual crystals to be run through the MacTempasX simulation software. For virtual crystals that could not be created by CrystalKitX, MATLAB was used. These were the crystals in which graphene was sandwiched between h-BN layers. MacTempasX software was used to take virtual crystal data and to create a simulated TEM image of the virtual crystal. MacTempasX implements the multislice algorithm, originally created by Cowley & Moodie (1957), to simulate electron beam propagation through the crystal potential. It does this by iteratively projecting slices of a crystal’s atomic potential onto a plane, operating on the incoming electron wavefunction with the planar atomic potential (also called the phase-grating function), propagating the electron through the slice’s empty space, and repeating until the beam has traveled through the crystal. Once the beam has exited the crystal, it propagates through space until it passes through the objective lens, and then it continues through space until it reaches the image plane. In simulating the propagation and lensing of the beam, MacTempasX makes approximations of nonideal effects such as objective lens defocus, spherical aberration, chromatic aberration, beam divergence, and other nonideal factors. MATLAB was used again to remove edge effects from the virtual crystal images (by simply removing the image edges), pixelate the images to simulate practicable experimental resolution (0.3125 Å/pixel), renormalize the contrast (by linearly scaling the intensity values to a 16-bit integer range such that the minimum and maximum image intensities went to 0 and 216 − 1, respectively), and add in background noise (randomly added by a normal distribution with a standard deviation of 2%). This was necessary because the crystals had finite area, and were thus not completely periodic. This created defective edges, in which intensity reached global minima and maxima, which reduced the contrast in the center of the image. The crystals created were composed of monolayer graphene and at least one layer of h-BN. The monolayer h-BN crystals, like graphene, have a hexagonal
Figure 2. Graphene layered on bilayer hexagonal boron nitride (h-BN), as used in MacTempasX. In this model, one can see the alternating AA stacking of h-BN atoms. Small purple atoms are boron, and large green atoms are nitrogen. The C lattice parameters of graphene and h-BN are 3.37 and 3.33 Å, respectively. The distance between h-BN and graphene was taken as the average of those values.
honeycomb structure. Unlike multilayer graphene, which exhibits Bernal stacking in three dimensions, multilayer h-BN is stacked in an AA pattern such that along the crystal’s C-axis, the atomic sites are alternatively occupied by boron and nitrogen ions, as shown in Figures 1 and 2. The parameters used in the HRTEM simulations presented here are similar to those used in our previous simulation study of graphene stacking, and representative of experimental studies (Meyer et al., 2008; Nelson et al., 2010; Zan et al., 2011; Nelson, 2012). An 80-kV accelerating potential is used because it provides significantly better contrast than higher voltages, which can also damage graphene structure. All images were calculated with a defocus of −147 Å and a defocus spread of 17 Å, a third-order spherical aberration coefficient (CS) of 0 mm, a fifth-order spherical aberration coefficient (C5) of 5 mm, and a convergence angle of 0.15 mrad. Although images were simulated with other defocus and correction settings, it was found that the ideal defocus was minimally dependent on the stack thickness, with ideal values ranging from −156 to −147 Å. As one of the purposes of this study was to determine the impact of stacking orientation on Moiré pattern imaging, the defocus
HRTEM Simulations of Varied h-BN/Graphene Heterostacks
Figure 3. Sinusoidal contrast transfer function of the virtual microscope used to calculate the simulated images. One can see the transferred beams used in the calculations, and their scattering vector magnitudes, as produced by the graphene and boron nitride lattices. Red lines indicate graphene diffraction, and green lines indicate boron nitride diffraction. The first two diffracted beams approach for each lattice the positive dampening limit. The third diffracted beam for each is near null, and thus does not contribute significantly to the simulated high-resolution transmission electron microscopic images.
was kept at the point where Moiré contrast was maximized. Thus, a defocus parameter of −147 Å was used throughout. The sinusoidal portion of the contrast transfer function for the given microscope parameters is shown in Figure 3. Though there are oscillations in the range of interest, the diffracted beams align either to approach the positive dampening limit, or are near null. An objective aperture that transmitted calculated beams with scattering vectors of up to 1 Å − 1 was used throughout. In addition, the MacTempasX program requires the maximum crystal scattering vector to be declared in the multislice diffraction calculation, labeled Gmax. MacTempasX computes phase-grating coefficients out to twice Gmax in order to avoid aliasing in the multislice calculations, as shown in the following equation (Kilaas, 2013): Z ψ 1 ðx; y; z + dzÞ ¼ exp - iσ
z + dz
0
Vðx; y; z Þdz
0
ψðx; y; zÞ
z
qðx; yÞψðx; y; zÞ:
(1)
For all crystals in this study, we used Gmax = 2 Å − 1. Equation (1) shows the propagation of an incoming wavefunction through a crystal with potential V(x,y,z) and thickness dz. In this equation, σ ¼ λ0 πW0 , where λ0 is the incoming electron wavelength in vacuum and W0 the accelerating potential. q(x,y) is defined as the phase-grating function and has an upper limit of 2 Gmax.
RESULTS It is important to note that experimentally producing a single layer of graphene on h-BN is still, despite recent improvements in fabrication, a difficult task. In that light, HRTEM
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simulations were used to study the effect of the number of h-BN layers, the location of graphene within the h-BN layers, and the relative rotation between the graphene and h-BN. Thus, the final crystals created and imaged had 2N h-BN layers, where N ranged from 0 to 3. For each crystal of Z layers, the arrangements of graphene in the crystal were (graphene, h-BN1, h-BN2, … , h-BNZ − 1, h-BNZ), (h-BN1, graphene, h-BN2, … ,h-BNZ − 1, h-BNZ), and so on until the graphene reached the top of the stack. To study the effects of relative rotation (where a stack of two “armchair” or two “zigzag” cells has 0° of relative rotation, and an “armchair” hexagon stacked on a “zigzag” hexagon has 30° of relative rotation), monolayer h-BN was used to accentuate the Moiré pattern changes, as will be explained below. Once each crystal model was created, the image simulated, and the contrast renormalized, a three-dimensional (3D) image was extracted from the data. This was done in order to enhance observation of the superlattice structure, as shown in Figure 4. This 3D creation and manipulation was done in MATLAB. In many 2D images, especially those with four or more h-BN layers, the Moiré pattern is difficult to see. When the image was replotted in three dimensions, where the intensity of the pixels is shown in the third dimension, the Moiré pattern becomes much more apparent. In addition to the 3D image, a fast Fourier transform (FFT) of the renormalized image was calculated using a Hanning window to reduce the image shape bias. This was done in order to confirm the periodicity of the lattices from a different perspective, and to probe the prospective utility of diffraction images. This image processing is shown in Figure 4. Simulated images of graphene on monolayer h-BN at different length scales are shown in Figure 5. These were made in order to evaluate the effect of noise and resolution limits on the images, and to observe the Moiré pattern above and below its periodic size. In these images, one can see the disappearance of the Moiré pattern, the emergence of individual atoms, and the increasing influence of noise as the magnification increases. The middle images have length and width dimensions that are a fourth of those on top. The bottom images have length and width dimensions that are a fourth of those in the middle, or a 16th of those on top. Figures 5b, 5d, and 5f are surfaces designed to fit the Moiré pattern. These data are from maxima in grid areas made by creating a 25 × 25 mesh pattern in the 3D image. The high red areas indicate high intensity values, and the low blue areas indicate low intensity values. One can see that this algorithm breaks down with a sufficiently scaleddown image. In Figure 5f, the physical dimensions are 19.2 × 16.6 Å, and the Moiré pattern is completely gone. Thus, the surface shows atoms instead of the overarching pattern, and ceases to be a useful aid. In the top two, though, the 3D Moiré pattern surface image helps to elucidate the pattern. In graphene/h-BN heterostructures, the Moiré pattern parameters are given by the following equations (Yankowitz et al., 2012): ð1 + δÞa λ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi ; 2ð1 + δÞð1 - cosϕÞ + δ
(2)
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Figure 4. Image processing of a simulated high-resolution transmission electron microscopic (HRTEM) image of graphene on monolayer hexagonal boron nitride (h-BN). The image processing involved taking the original renormalized transmission electron microscopic image (a), and applying a Hanning-windowed fast Fourier transform to obtain diffraction-like power spectra for rotationally dependent studies (d). For h-BN layer number dependence, the image was converted into three dimensional, taking intensity as the third dimension (b). This was then fit with a surface to highlight the Moiré pattern intensity (c).
tanθ ¼
sinϕ : ð1 + δÞ - cosϕ
(3)
In these equations, λ is the Moiré pattern wavelength, which is the minimum distance over which the pattern repeats, and θ the angle of the Moiré pattern with respect to the graphene lattice. The parameters in these equations are a, which is the graphene lattice parameter (2.46 Å), δ, which is the relative lattice size mismatch between graphene and h-BN (0.0163 in this paper), and ϕ, which is the angle that the graphene lattice makes with the h-BN lattice. The angular dependence will be shown in the following paragraphs. One can see from Figures 5b and 5d that there are local intensity maxima that are spaced at a minimum of 101.70 Å away from each other. Throughout the image, there are points where a graphene A site (as shown in Fig. 1) atom
lines up with an h-BN A site atom. Those can be considered the origins of the Moiré pattern unit cell, whose distance from each other defines the Moiré pattern wavelength. There are also points at which a graphene A site lines up with an h-BN B site. When this occurs, the graphene B site aligns with the empty space in the h-BN structure, and the h-BN A site aligns with the empty space in the graphene structure. Likewise, there are points at which a graphene B site lines up with an h-BN A site. The other graphene and h-BN sites then line up with empty space. However, an intensity maximum in the image is still produced by this AB overlay. Thus, there are three local Moiré intensity maxima per pattern unit cell in the HRTEM images calculated in this study. However, as one can see in Figure 5b, one of the three peaks in each Moiré cell is not as large as the other two. This results from the smaller boron not giving as much intensity in the image as
HRTEM Simulations of Varied h-BN/Graphene Heterostacks
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Figure 5. Simulated high-resolution transmission electron microscopy (HRTEM) images and Moiré pattern surfaces of graphene on monolayer hexagonal boron nitride, zooming in iteratively: (a, c, e) show HRTEM images; (b, d, f) show Moiré pattern surfaces. One can see that the more useful figure at high scale is the three-dimensional (3D) surface, whereas at low scale the useful figure is the 2D image. In (f) the red peaks are individual atoms, and the blue minima are voids. Thus, the Moiré pattern is missed entirely.
the larger nitrogen atoms. Thus, when h-BN is a monolayer, the nitrogen–carbon stacked AB Moiré pattern points give higher intensity than the boron–carbon stacked AB pattern points. This effect disappears when h-BN has more than one
layer. In previous STM studies, the Moiré pattern was found to have a honeycomb structure, similar to that of graphene or h-BN (Decker et al., 2011; Xue et al., 2011; Yankowitz et al., 2012). This is likely because of the fact that the surface layers
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Figure 6. a–h: Simulated high-resolution transmission electron microscopy (HRTEM) images and Moiré pattern surfaces of graphene on varying numbers of layers of hexagonal boron nitride (h-BN): (a, b) show graphene on monolayer h-BN; (c, d) show graphene on bilayer h-BN; (e, f) show graphene on tetralayer h-BN; (g, h) show graphene on eight layers of h-BN.
HRTEM Simulations of Varied h-BN/Graphene Heterostacks Table 1. Normalized Moiré Pattern Intensity is a Ratio of the Intensity of the Surface Fit to the Moiré Pattern to the Total Intensity of the Image. Relative Moiré pattern intensity is a ratio of the normalized Moiré intensity to the normalized Moiré intensity given by monolayer h-BN. h-BN Layers Normalized Moiré Intensity Relative Moiré Intensity 1 2 4 8
0.3797 0.3077 0.2367 0.1734
1 0.81038 0.62339 0.45668
of the sample dominate the intensity of STM images. Thus, a honeycomb lattice is found regardless of h-BN thickness. If we were to remove the smaller minima created by boron– carbon AB alignment, we would also find a honeycomb Moiré pattern. The next subject to be studied is the effect of multiple layers of h-BN with graphene, and the orientation of the layers. From Figure 6, one can clearly see in the surface images that as the number of layers of h-BN increases, the prominence of the Moiré pattern decreases. In the monolayer h-BN case, the Moiré pattern intensity (measured as maximum intensity minus the minimum superlattice surface energy) is 38% of the total intensity in the image (216 − 1 in the 16-bit TIFF produced). When there are eight layers of h-BN, the normalized Moiré pattern intensity is 17%. These numbers are outlined in Table 1. Not surprisingly, adding more h-BN layers shifts the image from having a Moiré quality to representing a pure h-BN lattice. The h-BN layers dominate the intensity of the image. To further analyze the effect of number of h-BN layers on Moiré pattern intensity, the data were plotted and
Figure 7. Plot of Moiré pattern intensities as a function of number of hexagonal boron nitride (h-BN) layers, along with power function fit lines. R2 fit parameter is 0.99 for both lines. Using this trend, we predict that normalized Moiré intensity will dip below 10% after 42 h-BN layers.
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Table 2. Studied Values of Moiré Pattern Wavelength and Rotation, as Given by Equations (2) and (3). Relative Layer Rotation (°) 0 5 10 15 20 25 30
Moiré Pattern Wavelength (Å)
Moiré Pattern Rotation (°)
153.75 27.953 14.166 9.482 7.133 5.725 4.789
0 77.035 79.734 78.994 77.381 75.417 73.276
fit to power function curves in Figure 7. These curves are given by the formula, I(N) = ANB, where A and B are fit parameters, I(N) the normalized Moiré pattern intensity, and N the number of h-BN layers. For both curves, we found that B = − 0.362 ± 0.023B. These curves gave us adjusted R2 (coefficient of determination) values of 0.99, indicating a good fit. Thus, given constant imaging conditions that are optimized for graphene on monolayer h-BN, one can expect that it would take 42 layers of h-BN for the Moiré pattern intensity to dip below 10% of the total intensity of the image. Another finding that came from varying the layers of h-BN and the graphene around those layers was that the images produced appear to be independent of graphene location in the crystal, that is, the images with graphene on top of, below, and in between h-BN were visually indistinguishable, even with the aid of the analytical methods used in this study. We expected them to be similar, as the low sample thickness and low Z of the atoms in the simulations should keep diffraction events at 1 or 0. The raw numerical data sets were not identical, but slightly shifted owing to edge effects, which made a direct comparison of the intensities very difficult. Thus, we were unable to complete a robust direct comparison of these differently oriented layered structures. Though we believe that stacking effects merit further investigation, based on our results, we expect these different heterostructures to be very difficult to distinguish experimentally. The last factor to be studied was rotational mismatching between the graphene and h-BN layers. As mentioned before, rotation mismatching is defined as the angle between graphene and h-BN lattices, where two overlaid “armchair” cells or “zigzag” cells have 0° of relative rotation, and an “armchair” overlaid with a “zigzag” cell has 30° of relative rotation. In this study, monolayer graphene and monolayer h-BN were used to make bilayer stacks in all trials, in order to maximize the contrast from the graphene layer and Moiré pattern. The angles studied were 0 through 30°, in intervals of 5°. We stopped at 30° because the structures are symmetric around this point, and any further rotations would be redundant. The equations that determine the Moiré pattern were shown previously, as equations (2) and (3), and the parameters that these equations produced are outlined in Table 2. The simulated TEM images (Fig. 8) clearly show the
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decreasing size and changing rotation of the Moiré pattern that occurs as the interlayer rotation increases. The individual graphene and h-BN layers are nearly unobservable; only the Moiré pattern can be seen. The interlayer atomic rotation, however, is very apparent in the power spectra, which resulted from taking the intensity of our Hanning-windowed FFTs. One can see that the h-BN spots (which have a slightly smaller radius in inverse space) in the power spectrum shown in Figure 8b remain in place for rotations from 0 to 30°. By contrasting these stable h-BN spots with the rotating graphene spots that appear above 0° of interlayer rotation, and tracking them through the rotational change, one can see the full rotation from 0 to 30°. The Moiré wavelength can be seen relatively easily at and below 15° of rotation in the TEM image. However, for rotations >15°, one cannot readily assess the wavelength or rotation of the Moiré pattern, nor see the angle between the layers. Thus, one can quickly ascertain that experimental images of bilayer heterostacks, which do not show obvious periodic superlattice contrast, are from samples with more than 15° of interlayer rotation. Fortunately, the FFTs give valuable information for any rotation, making them more useful for determining the relative orientation of graphene/h-BN bilayers. One way to ascertain the rotational information from the TEM image, which can be used at or below 20°, is to look at the number of hexagons in each Moiré pattern unit cell. One can see that at 20°, there are many cells with three hexagons, and some with one hexagon. Likewise, at 15° there are cells with four or seven hexagons, at 10° there are cells with close to 19 hexagons, and at smaller rotations, there are many more hexagons per cell. Despite the patterns that may be seen in the simulated TEM image itself, rotations are much more easily assessed in the power spectrum. To verify our work, we compared our simulated images with experimental STEM images of bilayer turbostratic graphene and bilayer turbostratic h-BN obtained by Zan et al. (2011). In their images (Figs. 9, 10), one can see their raw image data collected in the STEM (a), their filtered data (b), and their images produced by simulation (c). Taking the layer rotations from their FFTs (8° for graphene, 7° for h-BN, FFT images in the insets of Figs. 9, 10), we predict that their Moiré wavelengths should be 17.6 Å for turbostratic graphene and 14.2 Å for turbostratic h-BN, using equations (2) and (3). One can see from the scale bar in Figure 10 that this is a reasonable prediction. We can accurately determine the Moiré wavelength from the interlayer rotation, and vice versa, because of the one-to-one correspondence of the Moiré pattern wavelength and interlayer rotation. We note that our simulated images are similar in appearance to the data taken by Zan et al. Figure 8. a–n: Simulated high-resolution transmission electron microscopy images and power spectra of graphene on monolayer hexagonal boron nitride with varying degrees of relative rotation: (a, b) have 0° of relative rotation; (c, d) have 5° of relative rotation; (e, f) have 10° of relative rotation; (g, h) have 15° of relative rotation; (i, j) have 20° of relative rotation; (k, l) have 25° of relative rotation; (m, n) have 30° of relative rotation.
HRTEM Simulations of Varied h-BN/Graphene Heterostacks
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Figure 9. Images of turbostratic graphene taken by Zan et al. (2011): (a) is a raw scanning transmission electron microscopy (STEM) image; (b) is a filtered STEM image; and (c) is a simulation image. The rotated layers give rise to a rotated, constricted Moiré pattern, as shown in our graphene/hexagonal boron nitride simulation images.
Figure 10. Images of turbostratic hexagonal boron nitride (h-BN) taken by Zan et al. (2011): (a) is a raw scanning transmission electron microscopy (STEM) image; (b) is a filtered STEM image; and (c) is a simulation image. The rotated layers give rise to a rotated, constricted Moiré pattern, as shown in our graphene/h-BN simulation images.
(2011), thus we expect that our simulations will accurately correspond to future data taken via TEM.
D ISCUSSION Although devices fabricated using graphene on multilayer h-BN have superior device performance when compared with graphene—SiO2/Si (Dean et al., 2010), there are various phenomena arising from layer stacking orientations that are not well understood (Yankowitz et al., 2012; Eckmann et al., 2013; Yang et al., 2013). This includes the appearance of superlattice Dirac points, and their resultant local resistivity maxima (Yang et al., 2013), and change in dielectric function, as well as other possible unexplored effects. Thus, the ability to use HRTEM and STEM to determine the stacking orientation of graphene on h-BN is very useful for interpreting electrical and optical measurements of these stacks.
The superlattice Dirac point locations are determined by the Moiré pattern unit cell, which is determined by the rotation of the layers. Thus, using the rotational information from an FFT, the location of charge neutral points (CNPs) and local resistivity maxima can be predicted via the following equation (Yankowitz et al., 2012): 2πhvF ESDP ¼ pffiffiffi ; 3λ
(4)
where ESDP is the superlattice Dirac point energy, ħ the Planck’s constant, vF the Fermi velocity of graphene (≅1 × 106 m/s), and λ the superlattice wavelength as given by equations (2) and (3). There may be additional factors that affect these new CNPs in graphene/h-BN stacks, such as substrate thickness. The dielectric function of graphene has been shown to be highly dependent on the substrate (Nelson, 2012). As the CNP emergence infers a change in the band structure of
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graphene owing to rotation, it is reasonable to assume that the dielectric function of graphene is affected by the rotation of layers as well. In order to study these and other effects, we must first identify and characterize the heterostructures. Atomic resolution TEM should give good information about the stack, which can then be correlated to electronic changes. However, in order to utilize any correlations, we must be able to analyze our images. These simulated images, and comparison thereof to real data, provide a guide to analyzing TEM images of graphene on h-BN that can aid in studies into electronic structure.
CONCLUSIONS Virtual h-BN and graphene have been created and used to produce simulated HRTEM images of heterolayered stacks of these promising materials, with the help of CrystalKitX, MacTempasX, and MATLAB software programs. The validity of this study was verified by comparing these simulations with published TEM images of turbostratic bilayer h-BN and turbostratic bilayer graphene (Zan et al., 2011). Power spectra and Moiré pattern contrast images were extracted from simulated images, thereby highlighting methods that can be used to analyze future TEM data. We found that, by using HRTEM, one can observe three contrast maxima per Moiré unit cell, differing from that seen by surface-specific methods like STM and AFM (Decker et al., 2011; Xue et al., 2011; Yankowitz et al., 2012). We were able to map the Moiré pattern contrast dependence on the number of h-BN layers in the stack, and use the extracted FFTs to facilitate observation of interlayer rotations between 0 and 30°. As stated, owing to the low thickness and atomic weight of the stack, there was no effect observed after moving the position of graphene between, above, or below a stack of h-BN layers. Regardless, we believe these differently layered stacks merit further investigation. Despite these findings, there are inherent limitations to simulation-based studies. These simulations had the benefit of an ideal microscope and sample set-up. There was no vibration or astigmatism to deal with, there were no local or nonlocal sample defects, and despite pixelating our images and adding noise, our calculated data had ideal resolution. This study will be used to identify and characterize graphene/h-BN stacks in future TEM investigations, so that we may correlate geometric layer orientations and epitaxial effects to future electrical, optoelectronic, and metrological measurements.
ACKNOWLEDGMENTS The authors would like to thank Dr. Florence Nelson for help with simulations, Dr. Recep Zan for permission to use his figures, and Dr. Juan Carlos Idrobo of ORNL for helpful discussions on practical microscope resolution limits. The authors would also like to thank the SRC INDEX program for funding.
REFERENCES CASTRO NETO, A.H., GUINEA, F., PERES, N.M.R., NOVOSELOV, K.S. & GEIM, A.K. (2009). The electronic properties of graphene. Rev Mod Phys 81, 109–162. COWLEY, J.M. & MOODIE, A.F. (1957). The scattering of electrons by atoms and crystals. I. A new theoretical approach. Acta Cryst 10, 609–619. DEAN, C.R., YOUNG, A.F., MERIC, I., LEE, C., WANG, L., SORGENFREI, S., WATANABE, K., TANIGUCHI, T., KIM, P., SHEPARD, K.L. & HONE, J. (2010). Boron nitride substrates for high-quality graphene electronics. Nat Nanotech 5, 722–726. DECKER, R., WANG, Y., BRAR, V.W., REGAN, W., TSAI, H., WU, Q., GANNETT, W., ZETTL, A. & CROMMIE, M.F. (2011). Local electric properties of graphene on a BN substrate via scanning tunneling microscopy. Nano Lett 11, 2291–2295. ECKMANN, A., PARK, J., YANG, H., ELIAS, D., MAYOROV, A.S., YU, G., JALIL, R., NOVOSELOV, K.S., GORBACHEV, R.V., LAZZERI, M., GEIM, A.K. & CASIRAGHI, C. (2013). Raman fingerprint of aligned graphene/h-BN superlattices. Nano Lett 13, 5242–5246. GEIM, A.K. & NOVOSELOV, K.S. (2007). The rise of graphene. Nat Mat 6, 183–191. ISHIGAMI, M., CHEN, J.H., CULLEN, W.G., FUHRER, M.S. & WILLIAMS, E.D. (2007). Atomic structure of graphene on SiO2. Nano Lett 7, 1643–1648. KILAAS, R. (2013). CrystalKitX computer program. Version 1.9.81. Berkeley, CA: Total Resolution. Accompanied by 1 manual. KILAAS, R. (2013). MacTempasX computer program. Version 2.3.40. Berkeley, CA: Total Resolution. Accompanied by 1 manual. KUBOTA, Y., WATANABE, K., TSUDA, O. & TANIGUCHI, T. (2007). Deep ultraviolet light-emitting hexagonal boron nitride synthesized at atmospheric pressure. Science 317, 932–934. LUI, C.H., LIU, L., MAK, K.F., FLYNN, G.W. & HEINZ, T.F. (2009). Ultraflat graphene. Nature 462, 339–341. MEYER, J.C., KISIELOWSKI, C., ERNI, R., ROSSELL, M.D., CROMMIE, M.F. & ZETTL, A. (2008). Direct imaging of lattice atoms and topological defects in graphene membranes. Nano Lett 8, 3582–3586. NELSON, F. (2012). Study of the dielectric function of graphene from spectroscopic ellipsometry and electron energy loss spectroscopy. Doctoral dissertation. Retrieved from ProQuest Dissertations and Theses (Accession Order No. AAT 3550002). NELSON, F., DIEBOLD, A.C. & HULL, R. (2010). Simulation study of aberration-corrected high-resolution transmission electron microscopy imaging of few-layer-graphene stacking. Microsc Microanal 16, 194–199. NOVOSELOV, K.S., GEIM, A.K., MOROZOV, S.V., JIANG, D., ZHANG, Y., DUBONOS, S.V., GRIGORIEVA, I.V. & FIRSOV, A.A. (2004). Electric field effect in atomically thin carbon films. Science 306, 666–669. WATANABE, K., TANIGUCHI, T. & KANDA, H. (2004). Direct-bandgap properties and evidence for ultraviolet lasing of hexagonal boron nitride single crystal. Nat Mat 3, 404–409. XUE, J., SANCHEZ-YAMAGISHI, J., BULMASH, D., JACQUOD, P., DESHPANDE, A., WATANABE, K., TANIGUCHI, T., JARILLO-HERRERO, P. & LEROY, B.J. (2011). Scanning tunneling microscopy and spectroscopy of ultra-flat graphene on hexagonal boron nitride. Nat Mat 10, 282–285.
HRTEM Simulations of Varied h-BN/Graphene Heterostacks YANG, W., CHEN, G., SHI, Z., LIU, C., ZHANG, L., XIE, G., CHENG, M., WANG, D., YANG, R., SHI, D., WATANABE, K., TANIGUCHI, T., YAO, Y., ZHANG, Y. & ZHANG, G. (2013). Epitaxial growth of singledomain graphene on hexagonal boron nitride. Nat Mat 12, 792–797. YANKOWITZ, M., XUE, J., CORMODE, D., SANCHEZ-YAMAGISHI, J.D., WATANABE, K., TANIGUCHI, T., JARILLO-HERRERO, P., JACQUOD, P. & LEROY, B.J. (2012). Emergence of superlattice Dirac points in graphene on hexagonal boron nitride. Nat Phys 8, 382–386.
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© MICROSCOPY SOCIETY OF AMERICA 2014
Thickness and Rotational Effects in Simulated HRTEM Images of Graphene on Hexagonal Boron Nitride Avery J. Green* and Alain C. Diebold Optical Physics Group, SUNY College of Nanoscale Science and Engineering, 257 Fuller Road, Albany, NY 12203, USA
Abstract: Recent studies have shown that when graphene is placed on a thin hexagonal boron nitride (h-BN) substrate, unlike when it is placed on a typical SiO2 surface, it can closely approach the ideal carrier mobility observed in suspended graphene samples. This study further examines the epitaxial relationship between graphene and h-BN substrate with high-resolution transmission electron microscopy simulation. Virtual monolayer and multilayer stacks of h-BN were produced with a monolayer of graphene on top, on bottom, and in between h-BN layers, in order to study this interface. Once the simulations were performed, the phase contrast image and Moiré pattern created by this heterostack were analyzed for local and global intensity minima and maxima. In addition, h-BN substrate thickness and rotations between h-BN and graphene were probed and analyzed. The simulated images produced in this work will be used to help understand subsequent transmission electron microscopy images and electron energy-loss studies. Key words: HRTEM, zone axis, power spectrum, Moiré pattern, Dirac point
I NTRODUCTION Since monolayer graphene was identified in 2004, its physics and applications have been quickly growing fields (Novoselov et al., 2004; Geim & Novoselov, 2007; Castro Neto et al., 2009). Its linear energy dispersion and resultant massless Dirac fermion charge carriers (Castro Neto et al., 2009) have drawn the interest of academic researchers and the semiconductor industry. Many steps have been taken toward a viable and efficient method of integration into current field-effect transistor devices, but that goal has not yet been achieved, and there is more preliminary work to be done. It has been shown that the performance of graphene is hindered by the lack of thickness uniformity in typical SiO2 layers (Ishigami et al., 2007; Lui et al., 2009). Thus, finding an insulating substrate to support the structure of graphene that will perpetuate the minimum thickness inconsistency from the oxide sublayer is a key area of research. Hexagonal boron nitride (h-BN) is a promising material for this purpose. The large band gap (5.97 eV) (Watanabe et al., 2004), small lattice mismatch with graphene (~1.7%), atomic flatness, high mechanical strength, high temperature stability, low dielectric constant, and relative lack of dangling bonds and charge traps (Kubota et al., 2007) are all desired characteristics for a graphene channel substrate. Multiple studies have been done on graphene/h-BN stacks, which have included the use of various atomic resolution techniques like atomic force microscopy (AFM) and scanning tunneling microscopy (STM), and have produced results that support h-BN’s prominence as a graphene substrate candidate (Decker et al., 2011; Xue et al., 2011; Yankowitz et al., 2012). However, these studies have been Received January 13, 2014; accepted August 12, 2014 *Corresponding author. [email protected]
limited by surface signal dominance because these techniques lack significant penetration and atomic resolution imaging below the surface of the stack. High-resolution transmission electron microscopy (HRTEM) and the simulations thereof are better designed to observe the atoms below the top layer of graphene or h-BN, and therefore can give more insight into h-BN/graphene interfaces. In this study, HRTEM simulations were done on the [001] zone axis of the materials. We used varying layers of h-BN above and below monolayer graphene, as well as lattice angle mismatches between the graphene and h-BN layers from 0 to 30° in intervals of 5°, in order to observe how these permutations affect the interfaces. The goal was to create a guideline that explains how rotation and thickness affect images in our simulations and helps with transmission electron microscopy (TEM) image interpretation. In order to confirm the validity of these findings, we wanted to compare our simulated images to experimental images from the literature. As we could find no experimental HRTEM or scanning transmission electron microscopy (STEM) images of the stacks simulated in this paper in the literature, we compared our simulated images with experimental images of stacks with similar crystal structures. Namely, these were images of turbostratic bilayer graphene and turbostratic bilayer h-BN, which show some of the same stacking effects observed in our simulated images (Zan et al., 2011). We will use the simulations presented here to correlate heterostack geometry with changes in electron energy-loss data, as described by Nelson (2012).
MATERIALS
AND
METHODS
To create the following images and data, two TEM simulation programs and MATLAB were used. MATLAB was used
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Figure 1. Graphene (a) and hexagonal boron nitride (h-BN) (b) lattices. Both have honeycomb lattices with A and B atomic sites. Graphene’s lattice parameter is 2.46 Å, and h-BN’s lattice parameter is 2.50 Å. Small purple atoms are boron, and large green atoms are nitrogen.
to create crystal simulation data, and to process the images produced by the other software. The simulation programs were MacTempasX and CrystlKitX (Kilaas, 2013). CrystalKitX was used to create virtual crystals to be run through the MacTempasX simulation software. For virtual crystals that could not be created by CrystalKitX, MATLAB was used. These were the crystals in which graphene was sandwiched between h-BN layers. MacTempasX software was used to take virtual crystal data and to create a simulated TEM image of the virtual crystal. MacTempasX implements the multislice algorithm, originally created by Cowley & Moodie (1957), to simulate electron beam propagation through the crystal potential. It does this by iteratively projecting slices of a crystal’s atomic potential onto a plane, operating on the incoming electron wavefunction with the planar atomic potential (also called the phase-grating function), propagating the electron through the slice’s empty space, and repeating until the beam has traveled through the crystal. Once the beam has exited the crystal, it propagates through space until it passes through the objective lens, and then it continues through space until it reaches the image plane. In simulating the propagation and lensing of the beam, MacTempasX makes approximations of nonideal effects such as objective lens defocus, spherical aberration, chromatic aberration, beam divergence, and other nonideal factors. MATLAB was used again to remove edge effects from the virtual crystal images (by simply removing the image edges), pixelate the images to simulate practicable experimental resolution (0.3125 Å/pixel), renormalize the contrast (by linearly scaling the intensity values to a 16-bit integer range such that the minimum and maximum image intensities went to 0 and 216 − 1, respectively), and add in background noise (randomly added by a normal distribution with a standard deviation of 2%). This was necessary because the crystals had finite area, and were thus not completely periodic. This created defective edges, in which intensity reached global minima and maxima, which reduced the contrast in the center of the image. The crystals created were composed of monolayer graphene and at least one layer of h-BN. The monolayer h-BN crystals, like graphene, have a hexagonal
Figure 2. Graphene layered on bilayer hexagonal boron nitride (h-BN), as used in MacTempasX. In this model, one can see the alternating AA stacking of h-BN atoms. Small purple atoms are boron, and large green atoms are nitrogen. The C lattice parameters of graphene and h-BN are 3.37 and 3.33 Å, respectively. The distance between h-BN and graphene was taken as the average of those values.
honeycomb structure. Unlike multilayer graphene, which exhibits Bernal stacking in three dimensions, multilayer h-BN is stacked in an AA pattern such that along the crystal’s C-axis, the atomic sites are alternatively occupied by boron and nitrogen ions, as shown in Figures 1 and 2. The parameters used in the HRTEM simulations presented here are similar to those used in our previous simulation study of graphene stacking, and representative of experimental studies (Meyer et al., 2008; Nelson et al., 2010; Zan et al., 2011; Nelson, 2012). An 80-kV accelerating potential is used because it provides significantly better contrast than higher voltages, which can also damage graphene structure. All images were calculated with a defocus of −147 Å and a defocus spread of 17 Å, a third-order spherical aberration coefficient (CS) of 0 mm, a fifth-order spherical aberration coefficient (C5) of 5 mm, and a convergence angle of 0.15 mrad. Although images were simulated with other defocus and correction settings, it was found that the ideal defocus was minimally dependent on the stack thickness, with ideal values ranging from −156 to −147 Å. As one of the purposes of this study was to determine the impact of stacking orientation on Moiré pattern imaging, the defocus
HRTEM Simulations of Varied h-BN/Graphene Heterostacks
Figure 3. Sinusoidal contrast transfer function of the virtual microscope used to calculate the simulated images. One can see the transferred beams used in the calculations, and their scattering vector magnitudes, as produced by the graphene and boron nitride lattices. Red lines indicate graphene diffraction, and green lines indicate boron nitride diffraction. The first two diffracted beams approach for each lattice the positive dampening limit. The third diffracted beam for each is near null, and thus does not contribute significantly to the simulated high-resolution transmission electron microscopic images.
was kept at the point where Moiré contrast was maximized. Thus, a defocus parameter of −147 Å was used throughout. The sinusoidal portion of the contrast transfer function for the given microscope parameters is shown in Figure 3. Though there are oscillations in the range of interest, the diffracted beams align either to approach the positive dampening limit, or are near null. An objective aperture that transmitted calculated beams with scattering vectors of up to 1 Å − 1 was used throughout. In addition, the MacTempasX program requires the maximum crystal scattering vector to be declared in the multislice diffraction calculation, labeled Gmax. MacTempasX computes phase-grating coefficients out to twice Gmax in order to avoid aliasing in the multislice calculations, as shown in the following equation (Kilaas, 2013): Z ψ 1 ðx; y; z + dzÞ ¼ exp - iσ
z + dz
0
Vðx; y; z Þdz
0
ψðx; y; zÞ
z
qðx; yÞψðx; y; zÞ:
(1)
For all crystals in this study, we used Gmax = 2 Å − 1. Equation (1) shows the propagation of an incoming wavefunction through a crystal with potential V(x,y,z) and thickness dz. In this equation, σ ¼ λ0 πW0 , where λ0 is the incoming electron wavelength in vacuum and W0 the accelerating potential. q(x,y) is defined as the phase-grating function and has an upper limit of 2 Gmax.
RESULTS It is important to note that experimentally producing a single layer of graphene on h-BN is still, despite recent improvements in fabrication, a difficult task. In that light, HRTEM
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simulations were used to study the effect of the number of h-BN layers, the location of graphene within the h-BN layers, and the relative rotation between the graphene and h-BN. Thus, the final crystals created and imaged had 2N h-BN layers, where N ranged from 0 to 3. For each crystal of Z layers, the arrangements of graphene in the crystal were (graphene, h-BN1, h-BN2, … , h-BNZ − 1, h-BNZ), (h-BN1, graphene, h-BN2, … ,h-BNZ − 1, h-BNZ), and so on until the graphene reached the top of the stack. To study the effects of relative rotation (where a stack of two “armchair” or two “zigzag” cells has 0° of relative rotation, and an “armchair” hexagon stacked on a “zigzag” hexagon has 30° of relative rotation), monolayer h-BN was used to accentuate the Moiré pattern changes, as will be explained below. Once each crystal model was created, the image simulated, and the contrast renormalized, a three-dimensional (3D) image was extracted from the data. This was done in order to enhance observation of the superlattice structure, as shown in Figure 4. This 3D creation and manipulation was done in MATLAB. In many 2D images, especially those with four or more h-BN layers, the Moiré pattern is difficult to see. When the image was replotted in three dimensions, where the intensity of the pixels is shown in the third dimension, the Moiré pattern becomes much more apparent. In addition to the 3D image, a fast Fourier transform (FFT) of the renormalized image was calculated using a Hanning window to reduce the image shape bias. This was done in order to confirm the periodicity of the lattices from a different perspective, and to probe the prospective utility of diffraction images. This image processing is shown in Figure 4. Simulated images of graphene on monolayer h-BN at different length scales are shown in Figure 5. These were made in order to evaluate the effect of noise and resolution limits on the images, and to observe the Moiré pattern above and below its periodic size. In these images, one can see the disappearance of the Moiré pattern, the emergence of individual atoms, and the increasing influence of noise as the magnification increases. The middle images have length and width dimensions that are a fourth of those on top. The bottom images have length and width dimensions that are a fourth of those in the middle, or a 16th of those on top. Figures 5b, 5d, and 5f are surfaces designed to fit the Moiré pattern. These data are from maxima in grid areas made by creating a 25 × 25 mesh pattern in the 3D image. The high red areas indicate high intensity values, and the low blue areas indicate low intensity values. One can see that this algorithm breaks down with a sufficiently scaleddown image. In Figure 5f, the physical dimensions are 19.2 × 16.6 Å, and the Moiré pattern is completely gone. Thus, the surface shows atoms instead of the overarching pattern, and ceases to be a useful aid. In the top two, though, the 3D Moiré pattern surface image helps to elucidate the pattern. In graphene/h-BN heterostructures, the Moiré pattern parameters are given by the following equations (Yankowitz et al., 2012): ð1 + δÞa λ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi ; 2ð1 + δÞð1 - cosϕÞ + δ
(2)
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Figure 4. Image processing of a simulated high-resolution transmission electron microscopic (HRTEM) image of graphene on monolayer hexagonal boron nitride (h-BN). The image processing involved taking the original renormalized transmission electron microscopic image (a), and applying a Hanning-windowed fast Fourier transform to obtain diffraction-like power spectra for rotationally dependent studies (d). For h-BN layer number dependence, the image was converted into three dimensional, taking intensity as the third dimension (b). This was then fit with a surface to highlight the Moiré pattern intensity (c).
tanθ ¼
sinϕ : ð1 + δÞ - cosϕ
(3)
In these equations, λ is the Moiré pattern wavelength, which is the minimum distance over which the pattern repeats, and θ the angle of the Moiré pattern with respect to the graphene lattice. The parameters in these equations are a, which is the graphene lattice parameter (2.46 Å), δ, which is the relative lattice size mismatch between graphene and h-BN (0.0163 in this paper), and ϕ, which is the angle that the graphene lattice makes with the h-BN lattice. The angular dependence will be shown in the following paragraphs. One can see from Figures 5b and 5d that there are local intensity maxima that are spaced at a minimum of 101.70 Å away from each other. Throughout the image, there are points where a graphene A site (as shown in Fig. 1) atom
lines up with an h-BN A site atom. Those can be considered the origins of the Moiré pattern unit cell, whose distance from each other defines the Moiré pattern wavelength. There are also points at which a graphene A site lines up with an h-BN B site. When this occurs, the graphene B site aligns with the empty space in the h-BN structure, and the h-BN A site aligns with the empty space in the graphene structure. Likewise, there are points at which a graphene B site lines up with an h-BN A site. The other graphene and h-BN sites then line up with empty space. However, an intensity maximum in the image is still produced by this AB overlay. Thus, there are three local Moiré intensity maxima per pattern unit cell in the HRTEM images calculated in this study. However, as one can see in Figure 5b, one of the three peaks in each Moiré cell is not as large as the other two. This results from the smaller boron not giving as much intensity in the image as
HRTEM Simulations of Varied h-BN/Graphene Heterostacks
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Figure 5. Simulated high-resolution transmission electron microscopy (HRTEM) images and Moiré pattern surfaces of graphene on monolayer hexagonal boron nitride, zooming in iteratively: (a, c, e) show HRTEM images; (b, d, f) show Moiré pattern surfaces. One can see that the more useful figure at high scale is the three-dimensional (3D) surface, whereas at low scale the useful figure is the 2D image. In (f) the red peaks are individual atoms, and the blue minima are voids. Thus, the Moiré pattern is missed entirely.
the larger nitrogen atoms. Thus, when h-BN is a monolayer, the nitrogen–carbon stacked AB Moiré pattern points give higher intensity than the boron–carbon stacked AB pattern points. This effect disappears when h-BN has more than one
layer. In previous STM studies, the Moiré pattern was found to have a honeycomb structure, similar to that of graphene or h-BN (Decker et al., 2011; Xue et al., 2011; Yankowitz et al., 2012). This is likely because of the fact that the surface layers
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Figure 6. a–h: Simulated high-resolution transmission electron microscopy (HRTEM) images and Moiré pattern surfaces of graphene on varying numbers of layers of hexagonal boron nitride (h-BN): (a, b) show graphene on monolayer h-BN; (c, d) show graphene on bilayer h-BN; (e, f) show graphene on tetralayer h-BN; (g, h) show graphene on eight layers of h-BN.
HRTEM Simulations of Varied h-BN/Graphene Heterostacks Table 1. Normalized Moiré Pattern Intensity is a Ratio of the Intensity of the Surface Fit to the Moiré Pattern to the Total Intensity of the Image. Relative Moiré pattern intensity is a ratio of the normalized Moiré intensity to the normalized Moiré intensity given by monolayer h-BN. h-BN Layers Normalized Moiré Intensity Relative Moiré Intensity 1 2 4 8
0.3797 0.3077 0.2367 0.1734
1 0.81038 0.62339 0.45668
of the sample dominate the intensity of STM images. Thus, a honeycomb lattice is found regardless of h-BN thickness. If we were to remove the smaller minima created by boron– carbon AB alignment, we would also find a honeycomb Moiré pattern. The next subject to be studied is the effect of multiple layers of h-BN with graphene, and the orientation of the layers. From Figure 6, one can clearly see in the surface images that as the number of layers of h-BN increases, the prominence of the Moiré pattern decreases. In the monolayer h-BN case, the Moiré pattern intensity (measured as maximum intensity minus the minimum superlattice surface energy) is 38% of the total intensity in the image (216 − 1 in the 16-bit TIFF produced). When there are eight layers of h-BN, the normalized Moiré pattern intensity is 17%. These numbers are outlined in Table 1. Not surprisingly, adding more h-BN layers shifts the image from having a Moiré quality to representing a pure h-BN lattice. The h-BN layers dominate the intensity of the image. To further analyze the effect of number of h-BN layers on Moiré pattern intensity, the data were plotted and
Figure 7. Plot of Moiré pattern intensities as a function of number of hexagonal boron nitride (h-BN) layers, along with power function fit lines. R2 fit parameter is 0.99 for both lines. Using this trend, we predict that normalized Moiré intensity will dip below 10% after 42 h-BN layers.
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Table 2. Studied Values of Moiré Pattern Wavelength and Rotation, as Given by Equations (2) and (3). Relative Layer Rotation (°) 0 5 10 15 20 25 30
Moiré Pattern Wavelength (Å)
Moiré Pattern Rotation (°)
153.75 27.953 14.166 9.482 7.133 5.725 4.789
0 77.035 79.734 78.994 77.381 75.417 73.276
fit to power function curves in Figure 7. These curves are given by the formula, I(N) = ANB, where A and B are fit parameters, I(N) the normalized Moiré pattern intensity, and N the number of h-BN layers. For both curves, we found that B = − 0.362 ± 0.023B. These curves gave us adjusted R2 (coefficient of determination) values of 0.99, indicating a good fit. Thus, given constant imaging conditions that are optimized for graphene on monolayer h-BN, one can expect that it would take 42 layers of h-BN for the Moiré pattern intensity to dip below 10% of the total intensity of the image. Another finding that came from varying the layers of h-BN and the graphene around those layers was that the images produced appear to be independent of graphene location in the crystal, that is, the images with graphene on top of, below, and in between h-BN were visually indistinguishable, even with the aid of the analytical methods used in this study. We expected them to be similar, as the low sample thickness and low Z of the atoms in the simulations should keep diffraction events at 1 or 0. The raw numerical data sets were not identical, but slightly shifted owing to edge effects, which made a direct comparison of the intensities very difficult. Thus, we were unable to complete a robust direct comparison of these differently oriented layered structures. Though we believe that stacking effects merit further investigation, based on our results, we expect these different heterostructures to be very difficult to distinguish experimentally. The last factor to be studied was rotational mismatching between the graphene and h-BN layers. As mentioned before, rotation mismatching is defined as the angle between graphene and h-BN lattices, where two overlaid “armchair” cells or “zigzag” cells have 0° of relative rotation, and an “armchair” overlaid with a “zigzag” cell has 30° of relative rotation. In this study, monolayer graphene and monolayer h-BN were used to make bilayer stacks in all trials, in order to maximize the contrast from the graphene layer and Moiré pattern. The angles studied were 0 through 30°, in intervals of 5°. We stopped at 30° because the structures are symmetric around this point, and any further rotations would be redundant. The equations that determine the Moiré pattern were shown previously, as equations (2) and (3), and the parameters that these equations produced are outlined in Table 2. The simulated TEM images (Fig. 8) clearly show the
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decreasing size and changing rotation of the Moiré pattern that occurs as the interlayer rotation increases. The individual graphene and h-BN layers are nearly unobservable; only the Moiré pattern can be seen. The interlayer atomic rotation, however, is very apparent in the power spectra, which resulted from taking the intensity of our Hanning-windowed FFTs. One can see that the h-BN spots (which have a slightly smaller radius in inverse space) in the power spectrum shown in Figure 8b remain in place for rotations from 0 to 30°. By contrasting these stable h-BN spots with the rotating graphene spots that appear above 0° of interlayer rotation, and tracking them through the rotational change, one can see the full rotation from 0 to 30°. The Moiré wavelength can be seen relatively easily at and below 15° of rotation in the TEM image. However, for rotations >15°, one cannot readily assess the wavelength or rotation of the Moiré pattern, nor see the angle between the layers. Thus, one can quickly ascertain that experimental images of bilayer heterostacks, which do not show obvious periodic superlattice contrast, are from samples with more than 15° of interlayer rotation. Fortunately, the FFTs give valuable information for any rotation, making them more useful for determining the relative orientation of graphene/h-BN bilayers. One way to ascertain the rotational information from the TEM image, which can be used at or below 20°, is to look at the number of hexagons in each Moiré pattern unit cell. One can see that at 20°, there are many cells with three hexagons, and some with one hexagon. Likewise, at 15° there are cells with four or seven hexagons, at 10° there are cells with close to 19 hexagons, and at smaller rotations, there are many more hexagons per cell. Despite the patterns that may be seen in the simulated TEM image itself, rotations are much more easily assessed in the power spectrum. To verify our work, we compared our simulated images with experimental STEM images of bilayer turbostratic graphene and bilayer turbostratic h-BN obtained by Zan et al. (2011). In their images (Figs. 9, 10), one can see their raw image data collected in the STEM (a), their filtered data (b), and their images produced by simulation (c). Taking the layer rotations from their FFTs (8° for graphene, 7° for h-BN, FFT images in the insets of Figs. 9, 10), we predict that their Moiré wavelengths should be 17.6 Å for turbostratic graphene and 14.2 Å for turbostratic h-BN, using equations (2) and (3). One can see from the scale bar in Figure 10 that this is a reasonable prediction. We can accurately determine the Moiré wavelength from the interlayer rotation, and vice versa, because of the one-to-one correspondence of the Moiré pattern wavelength and interlayer rotation. We note that our simulated images are similar in appearance to the data taken by Zan et al. Figure 8. a–n: Simulated high-resolution transmission electron microscopy images and power spectra of graphene on monolayer hexagonal boron nitride with varying degrees of relative rotation: (a, b) have 0° of relative rotation; (c, d) have 5° of relative rotation; (e, f) have 10° of relative rotation; (g, h) have 15° of relative rotation; (i, j) have 20° of relative rotation; (k, l) have 25° of relative rotation; (m, n) have 30° of relative rotation.
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Figure 9. Images of turbostratic graphene taken by Zan et al. (2011): (a) is a raw scanning transmission electron microscopy (STEM) image; (b) is a filtered STEM image; and (c) is a simulation image. The rotated layers give rise to a rotated, constricted Moiré pattern, as shown in our graphene/hexagonal boron nitride simulation images.
Figure 10. Images of turbostratic hexagonal boron nitride (h-BN) taken by Zan et al. (2011): (a) is a raw scanning transmission electron microscopy (STEM) image; (b) is a filtered STEM image; and (c) is a simulation image. The rotated layers give rise to a rotated, constricted Moiré pattern, as shown in our graphene/h-BN simulation images.
(2011), thus we expect that our simulations will accurately correspond to future data taken via TEM.
D ISCUSSION Although devices fabricated using graphene on multilayer h-BN have superior device performance when compared with graphene—SiO2/Si (Dean et al., 2010), there are various phenomena arising from layer stacking orientations that are not well understood (Yankowitz et al., 2012; Eckmann et al., 2013; Yang et al., 2013). This includes the appearance of superlattice Dirac points, and their resultant local resistivity maxima (Yang et al., 2013), and change in dielectric function, as well as other possible unexplored effects. Thus, the ability to use HRTEM and STEM to determine the stacking orientation of graphene on h-BN is very useful for interpreting electrical and optical measurements of these stacks.
The superlattice Dirac point locations are determined by the Moiré pattern unit cell, which is determined by the rotation of the layers. Thus, using the rotational information from an FFT, the location of charge neutral points (CNPs) and local resistivity maxima can be predicted via the following equation (Yankowitz et al., 2012): 2πhvF ESDP ¼ pffiffiffi ; 3λ
(4)
where ESDP is the superlattice Dirac point energy, ħ the Planck’s constant, vF the Fermi velocity of graphene (≅1 × 106 m/s), and λ the superlattice wavelength as given by equations (2) and (3). There may be additional factors that affect these new CNPs in graphene/h-BN stacks, such as substrate thickness. The dielectric function of graphene has been shown to be highly dependent on the substrate (Nelson, 2012). As the CNP emergence infers a change in the band structure of
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graphene owing to rotation, it is reasonable to assume that the dielectric function of graphene is affected by the rotation of layers as well. In order to study these and other effects, we must first identify and characterize the heterostructures. Atomic resolution TEM should give good information about the stack, which can then be correlated to electronic changes. However, in order to utilize any correlations, we must be able to analyze our images. These simulated images, and comparison thereof to real data, provide a guide to analyzing TEM images of graphene on h-BN that can aid in studies into electronic structure.
CONCLUSIONS Virtual h-BN and graphene have been created and used to produce simulated HRTEM images of heterolayered stacks of these promising materials, with the help of CrystalKitX, MacTempasX, and MATLAB software programs. The validity of this study was verified by comparing these simulations with published TEM images of turbostratic bilayer h-BN and turbostratic bilayer graphene (Zan et al., 2011). Power spectra and Moiré pattern contrast images were extracted from simulated images, thereby highlighting methods that can be used to analyze future TEM data. We found that, by using HRTEM, one can observe three contrast maxima per Moiré unit cell, differing from that seen by surface-specific methods like STM and AFM (Decker et al., 2011; Xue et al., 2011; Yankowitz et al., 2012). We were able to map the Moiré pattern contrast dependence on the number of h-BN layers in the stack, and use the extracted FFTs to facilitate observation of interlayer rotations between 0 and 30°. As stated, owing to the low thickness and atomic weight of the stack, there was no effect observed after moving the position of graphene between, above, or below a stack of h-BN layers. Regardless, we believe these differently layered stacks merit further investigation. Despite these findings, there are inherent limitations to simulation-based studies. These simulations had the benefit of an ideal microscope and sample set-up. There was no vibration or astigmatism to deal with, there were no local or nonlocal sample defects, and despite pixelating our images and adding noise, our calculated data had ideal resolution. This study will be used to identify and characterize graphene/h-BN stacks in future TEM investigations, so that we may correlate geometric layer orientations and epitaxial effects to future electrical, optoelectronic, and metrological measurements.
ACKNOWLEDGMENTS The authors would like to thank Dr. Florence Nelson for help with simulations, Dr. Recep Zan for permission to use his figures, and Dr. Juan Carlos Idrobo of ORNL for helpful discussions on practical microscope resolution limits. The authors would also like to thank the SRC INDEX program for funding.
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