The morphology of doubly-clamped graphene nanoribbons

In figure 1 we show the length-dependent changes in morphology for free-standing nanoribbons with an initial width w_i of 150 nm. For small aspect ratios the nanoribbons remain flat. However, when we increase the length, the edge of the ribbon folds over and realizes a bilayer edge, as shown in figure 1(b). In this region we also observe significant uni-axial crumpling in the middle of the ribbons. Finally, for ribbons with a length above 1 µm, full scrolls are formed.

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To study this process systematically, we vary w_i between 100 and 200 nm, and the length between 320 and 1100 nm. We use the minimum width w_m of the ribbons as an indicator for structural changes. To assess w_m , we vertically sum over a 30 nm high region centered at the smallest width of the ribbon and take the full width at half height of the resulting trace.

Figure 2 illustrates the impact of the ribbons' length L on the morphology for different values of w_i . For short ribbons, w_m decreases linearly with increasing L. This corresponds to the transition from straight over crumpled ribbons to complete scrolls, as shown in figure 1. The slope s is relatively independent of w_i , suggesting that the process depends only on L. For w_i  = 100 nm and $L\gt 600$ nm, we observe a second linear regime, where the slope decreases from $s = (-147\pm 4)\times 10^{-3}$ to $s = (-17\pm 3)$ $\times\, 10^{-3}$ (figure 2(c)). In this region at least part of the ribbon is scrolled, realizing its minimum diameter. While further increase in L may lead to longer scrolls, it has no significant effect on w_m .

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In the experiments we observe a mixture of folding and scrolling. To investigate the dependence of the ribbons' morphology on their geometry, we perform molecular dynamics simulations (see section 5). We consider ribbons with w_i  = 37 and 49 nm and induce the formation of folded bilayer edges as well as scrolls. The covalent carbon-carbon bonds are described with the Adaptive Intermolecular Reactive Empirical Bond Order (AIREBO) potential with an added Moire potential to describe the van der Waals interaction [32, 33]. In all cases w_m decreases linearly with L as soon as the van der Waals energy overcompensates the strain (supplementary information (available online at stacks.iop.org/2DM/8/025035/mmedia)). For folded ribbons, the slopes amount to $s = (-86\pm 2)$ $\times\, 10^{-3}$ (w_i  = 37 nm) and $s = (-93\pm 2)\times 10^{-3}$ (w_i  = 49 nm). Interestingly, scrolling minimizes w_m more effectively ($s = (-154\pm 5)\times 10^{-3}$ for w_i  = 49 nm) than folding. Due to computational expenses, the simulations were only carried out for one orientation (and thus edge structure), and provide therefore no statistical variance. The stated error result from the uncertainty of the linear fit. In all cases, for structures with the same number of atoms, the folded ones were energetically favored over the scrolled ones.

From figure 2(d) we can see that the experimental results are between the two limiting cases of a pure scroll and a pure fold. Furthermore, the value for a scroll (−154 ± 4) × 10⁻³ is in very good agreement with the values of the ribbons with an initial width of 100 and 200 nm.

Changes in the morphology of the nanoribbons inevitably lead to strain, which sometimes is even the driving force for these changes. This is studied in an aberration-corrected scanning transmission electron microscope (Nion UltraSTEM 100) by recording convergent electron beam diffraction maps (see section 5). The method combines a series of diffraction patterns with a spatial resolution of 2 nm to reconstruct the forces acting on the ribbon. The strain is deduced from deformations of the lattice geometry as determined from the eccentricity and the orientation of an ellipse fitted to the diffraction pattern at each location. Results for a scroll are shown in figure 3, the analysis of a ribbon is shown in the supplementary information. The first row shows high-resolution annular dark field (HAADF) images, the other rows display the measured strain and the rotation of the fitted ellipse. The labels on the smaller images identify the sample position for each 4D map.

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The first clear observation is that the nanoribbons are largely amorphized by ion beam milling. This is illustrated in the leftmost strain map (1) in figure 3: reliable results over large regions are mostly obtained for the left half of the region although the membrane extends further to the right. For the ribbon itself the strain maps are very noisy—amorphized areas provide unreliable results. However, each map contains also areas where the strain can be measured, as seen from the locally less noisy data. The width of the amorphous region extends dozens of nanometers away from the cuts. In these areas the tail of the focused ion beam has reached a line dose sufficient to amorphize the material, but not high enough to cut [34]. In spite of the amorphization, the ribbons are stable over months as observed in molecular interference experiments [35], for which the impact of ion beam milling on the diffraction pattern has been discussed before [36, 37]. From maps recorded in the pristine region we see that the membrane is mainly elongated along the scroll axis (ε_yy ) and compressed perpendicular to it (ε_xx ). We point out, that the strain measurement for the two-dimensional graphene has an inherent uncertainty, since there is no guaranteed reference value from an aligned bulk crystal structure. From the strain maps we estimate the expansive and compressive strain to remain below 4%. However, these values should be treated with some caution.

Simulated strain maps for both scrolled and folded edges are shown in figure 4. Here the strain is computed from the average bond length. The spatial maps show a quarter of the final structure and are created by averaging the strain over an area of 1 × 1 nm² and interpolating on the edges. For both geometries the strain remains below 1% and is mostly located at the folded/scrolled edge and the flat region close to the ribbon; the hexagons in the remaining ribbon are not elongated by much. On the contrary, the structures are compressed slightly in the middle of the ribbon. The results are thus in general agreement with the experimental data, indicating strain mainly along the scroll axis. Since the computational model uses average bond lengths, where as the experimental map is estimated from a converged beam electron diffraction pattern, the values are not directly comparable but rather provide complementary information.

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Single-layer graphene suspended over an array of holes with a diameter of 3 µm is patterned using a focused beam of gallium ions at a kinetic energy of 35 keV (Raith Ionline FIB). The gaps are produced by using a current of 6.3 pA and a line dose of 2.3 nC cm⁻¹. Each opening is made by milling four single lines separated by 20 nm. The initial width w_i of the ribbons is set to 100, 150, or 200 nm and the length is varied between 320 and 1100 nm.

Low-resolution characterization of the nanoribbons is done with a tabletop transmission electron microscope (TEM, Delong Instruments LVEM5). From the micrographs, we determine the length of each ribbon, the initial width close to the edge of the cut, and its minimum width w_m . The length of the ribbons is determined from a vertical line profile for each cut. To assess w_i and w_m , we vertically sum over 30 nm high regions. These are located either 15 nm away from the top of the cut (w_i ) or at the position of minimum diameter (w_m ). The respective width corresponds to the distance between the points of half height. Ribbons that are strongly contaminated are excluded from the data set and in total about 100 ribbons contribute to each data point in figures 2(a–c).

High-resolution annular dark field images and nanodiffraction maps are recorded in an aberration-corrected scanning transmission electron microscope (Nion UltraSTEM 100) at an acceleration voltage of 60 kV. To assess the strain in the patterned regions, we measure the distortion of the diffraction pattern of graphene at each location. We use a convergent electron beam with an estimated size of 2 nm, which is scanned with a step size of ca. 2 nm to record a four-dimensional data set covering each of the measured ribbons (several partially overlapping images are recorded for each). Due to the partial amorphization of the ribbons via the tails of the focused ion beam, some parts of the ribbons do not display clear diffraction spots, but diffuse diffraction rings, which are excluded from the strain analysis. The microscopy images were recorded with the high angle annular dark field (HAADF) detector with an annular range of 80–300 mrad.

In the MD-simulations, we consider graphene nanoribbons with a width of 37 and 49 nm and vary their length between 103 and 251 nm. To assess the strain in the flat regions close to the ribbons, each sheet is capped at both ends by a 22 nm long buffer region, where out-of-plane movements are prohibited. In total these sheets contain between 323 000 and 565 000 carbon atoms. Unfortunately, the large system sizes and the number of required systems prevented us from extending the simulations to the experimental sizes.

The edge morphology leads to different edge forces and energies, which in turn affect the response to strain and stress [40]. To mimic the experiment, where the cuts are not aligned with the high symmetry lattice directions, the graphene lattice vectors are rotated by 15^∘ with respect to the cut of the ribbon. We use the Adaptive Intermolecular Reactive Empirical Bond Order (AIREBO) with a Morse potential for the intermolecular interaction [32, 33] as the empirical manybody potential for the covalent and van der Waals (vdW) interactions, respectively. The potential is implemented in the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) code [49, 50].

To induce folding in the simulations, we increase the temperature of the system and simultaneously apply an out-of-plane force to the edges in the middle of the nanoribbons. This way the membrane is brought in contact with itself (in the cut-off range of the vdW potential) and bilayer edges are created. During the simulations the geometry of the hexagons in the buffer regions remains fixed. The maximum applied force is 0.2 eV Å⁻¹ and the whole process runs for 2.75 fs. In case of scrolling, the direction of applied force is changed accordingly during the dynamical simulation to stimulate the creation of a scrolled structure. After the modified structure is created, the periodicity of the model is restored, the applied fixes to the buffer regions are released, and the nanoribbons are allowed to relax until the maximum forces are less than 5 × 10⁻⁴ eV Å⁻¹.