Letter pubs.acs.org/NanoLett
Folded Graphene Membranes: Mapping Curvature at the Nanoscale Luca Ortolani,*,† Emiliano Cadelano,‡ Giulio Paolo Veronese,† Cristian Degli Esposti Boschi,† Etienne Snoeck,∥ Luciano Colombo,‡,§ and Vittorio Morandi† †
CNR IMM-Bologna, Via Gobetti, 101, 40129 Bologna, Italy CNR IOM-Cagliari, Cittadella Universitaria, 09042 Monserrato, Italy § Physics Department, University of Cagliari, Cittadella Universitaria, 09042 Monserrato, Italy ∥ CEMES-CNRS, 29 rue Jeanne Marvig, 31055 Toulouse, France ‡
S Supporting Information *
ABSTRACT: While the unique elastic properties of monolayer graphene have been extensively investigated, less knowledge has been developed so far on folded graphene. Nevertheless, it has been recently suggested that fold-induced curvature (without in-plane strain) could possibly affect the local chemical and electron transport properties of graphene, envisaging a material-by-design approach where tailored membranes are used in enhanced nanoresonators or nanoelectromechanical devices. In this work we propose a novel method combining apparent strain analysis from high-resolution transmission electron microscopy (HREM) images and theoretical modeling based on continuum elasticity theory and tight-binding atomistic simulations to map and measure the nanoscale curvature of graphene folds and wrinkles. If enough contrast and resolution in HREM images are obtained, this method can be successfully applied to provide a complete nanoscale geometrical and physical picture of 3D structure of various wrinkle and fold configurations. KEYWORDS: Graphene, folds, wrinkles, curvature, topography, TEM
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the curvature of the lattice.12 Another interesting possibility given by the 2D nature of graphene and its ability to bend and fold is to use these membranes as envelops by wrapping them around other nanostructures.16 In this view, the inner cavity in graphene folds can be easily filled with foreign molecules and atoms, realizing hybrid mesoscale structures similar to hybrid carbon nanotubes,17 as recently experimentally demonstrated.4 The folding graphene, depending on lattice orientations, crystal defects, and possible adsorbed molecules,18−21 is still an open issue, and a deeper understanding of the curvature mechanics in graphene is essential to understand the profound relations between its three-dimensional structure and its properties and to further tune them toward specific technological applications. The mechanism of bending in graphene is indeed at the basis of devices like nanoresonators,22,10 which are one of the most promising technological fallouts for this material. Developing new characterization techniques capable of mapping and measuring the three-dimensional topography in folded graphene with subnanometer resolution is therefore crucial to understand the role of curvature on the physical and chemical properties of this material. Modern aberrationcorrected transmission electron microscopes (TEM) are capable of characterizing the structure of suspended mem-
raphene, the two-dimensional honeycomb lattice of carbon atoms, has a high Young’s modulus1 and it is extremely resilient to in-plane stresses; however, much like a piece of paper it can be easily bent to achieve complex folded structures.2−4 Three-dimensional deformations naturally occur in graphene, as membranes are intrinsically wrinkled5 to achieve thermodynamic stability, and solution-chemistry processes usually produces flakes multiply folded and crumpled.6 Mechanical deformations in graphene, and in particular strain, have been proposed as effective ways to tune the electronic transport behavior of this material. Strain and bending in graphene have been the subject of intense theoretical and experimental studies;7−10 however little is known about the effects of bending without strain. Recent works suggested that the curvature induced by folds, without the presence of in-plane strain, could dramatically change the local chemical reactivity,11,12 the mechanical properties,13 and the charge transport behavior14,15 of graphene membranes. The high degree of curvature in folds induces a strong deformation in the sigma bonds of the honeycomb lattice, and theoretical models suggest that the out-of-plane deformed sigma bonds could transfer charges to the π-orbitals in the convex region, inducing localized dipole moments on the surface.11 This valence charge modification can dramatically enhance molecular adhesion and chemisorption, envisaging the possibility for local and selective functionalization over micrometric regions, or to control the reversible bonding and storage of atoms and molecules, like hydrogen, by controlling © 2012 American Chemical Society
Received: June 26, 2012 Revised: September 10, 2012 Published: September 17, 2012 5207
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branes with atomic resolution, but the flake is observed projected along the optical axis of the microscope, resulting in a two-dimensional (2D) image, preventing the possibility to retrieve quantitative three-dimensional (3D) information out of a 2D high-resolution TEM (HREM) image. 3D topography and structure determination is conventionally achieved in TEM using transmission electron tomography23 which requires the acquisition of hundreds of images of the same area at different tilt angles to obtain a useful tomogram. However, not only graphene is extremely sensitive to electron radiation damage, even at low beam voltages,24,25 but the spatial resolution achievable in electron tomography is far from sufficient to investigate the 3D morphology of such small distortions. Recently transmission electron diffraction experiments demonstrated that it is possible to measure precisely the amplitude of intrinsic wrinkles over micrometric areas.5,26 Nevertheless, diffraction pattern analysis can provide average information over areas bigger than a few hundreds of nanometers, without any possibility to map nanoscale topography in graphene. In this work we intend to overcome the highlighted limitations of electron tomography and electron diffraction analysis of graphene membranes, proposing a novel methodology to recover 3D information from HREM image analysis. It relies on the intuitive observation that, when looking at a bent crystal from top, as the usual configuration in a TEM, the effect of projection will make the lattice to appear as compressed in sloped regions, as shown in Figure 1. Based on this consideration we use geometric phase analysis (GPA),27 as provided by the STEM Cell software28 to map the apparent compressions observed in the image of the projected lattice, and therefore to reconstruct height variations and local curvatures, with subnanometer spatial resolution. GPA analysis of strain fields in nanomaterials is a well-established experimental technique, and the latest instrumental developments in low-voltage aberration-corrected TEM made it possible to apply it successfully to the study of in-plane strains in graphene membranes29 and carbon nanotubes.30 Nevertheless, we want to highlight that previously GPA has been used to measure in-plane strain fields, while in this work, the strain information is used to recover the 3D structure of the graphene crystal. Folding in graphene naturally occurs in either chemical vapor deposition (CVD) grown membranes or solution processed flakes. In our experiments, we investigated graphene crystals grown by CVD on copper substrates and transferred on a TEM grid. Figure 2A shows a contrast-enhanced HREM image (details in the Supporting Information) of the folded edge of a monolayer membrane, where the border is visible by the (002) graphite lattice fringe. In this case, the crystal is folded, and the two graphenes in the stacked regions are rotated by an angle θ = 21.7°, as can be measured from the fast Fourier transform (FFT) in the inset of Figure 2A. Using GPA the apparent strain tensor in the HREM image is recovered, and Figure 2B and C shows the maps of the Ex and Ey components, where Ex = dUx/ dx and Ey = dUy/dy, Ux (respectively Uy) being the apparent displacements of the atomic columns in x (respectively y) directions (y-axis perpendicular to the folded edge). The graphene lattice is uniformly relaxed in the direction parallel to the folded edge, and the map of Ex in Figure 2B only shows local strain variations below 1% due to noise from the reconstruction. On the other hand the map of the component Ey in Figure 2C reveals areas of large compression running parallel to the edge, where the flake is curved.
Figure 1. Schematics showing the origin for the apparent compressive strains in the TEM image of a curved two-dimensional crystal of graphene. When the crystal is viewed from top, corresponding to the direction of the electron beam in the optic axis of the microscope, regions of the lattice, where the surface is not perpendicular to the beam direction, will appear as compressed.
The fold induces a one-dimensional curvature in the direction perpendicular to the border, and the effect of projection appears in the y-component of the strain tensor. We can use strain information to measure the curvature and topography of the fold. The analysis of the strain profiles acquired over regions indicated by (1) and (2) in Figure 2C shows that in both cases the compression peak is 4%, corresponding to a slope of 16.3°, extending over an area of 2 nm. The height variation in the curved region in the internal part of the flake can be calculated assuming a uniform slope for the whole extension of the strained length. For both areas of the profiles in Figure 2D and E, this analysis reveals steps of 0.6 nm, and it is reasonable to assume the inner curvature radius of the fold to be similar to this value. It is worth noting that close to the fold the curvature of the graphene lattice is expected to increase up to 90°, corresponding to an infinite apparent compression in the imaged lattice. We do not observe this second large dip in the strain profiles of Figure 2D and E, as it is hidden by geometric phase artifacts, arising from phase discontinuity at the interface between the flake and the vacuum. Moreover the spatial resolution achieved in the GPA reconstruction is 0.5 nm, which is the same value of the 5208
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Figure 2. Experimental mapping of the topography of folded graphene edges. All scale bars correspond to 3 nm. (A) HREM image of the graphene folded edge and (inset) FFT of the image, showing the stacking orientation of the two lattices. (B) Strain component Ex = dU/dx in the direction parallel to the border. In the convention adopted, values smaller than 1 indicate apparent compression. No significant compression is visible above the experimental error of 1%. The white rectangle indicates the area used as unstrained reference. (C) Strain component Ey = dU/dy in the direction perpendicular to the edge of the fold. The internal part of the flake shows no significant strain, while parallel to the border we can observe compressed regions. (D, E) Strain profiles acquired, respectively, over the regions (1) and (2) in C.
Figure 3. Three-dimensional rendering of the TB simulated graphene folds. (A) Armchair fold. (B) Zigzag fold.
geometrical parameters using zero-temperature atomistic relaxation simulations adopting a TB semiempirical scheme31 plus a van der Waals interaction.32 If the central region, where the layers keep parallel, is large enough, as shown in Figure 3, any further constrain is not needed. The atomic coordinates calculated so far allow us to simulate HREM images. It is important to notice that in two-dimensional out-of-plane deformations it is impossible to achieve bending without introducing strain,9 and there is always interplay between real bond-length variations and the apparent strains due to the effect of projection of a bent structure. However, atomistic simulations performed on folded monolayer structures show that bond-length variations in the folded regions are less than 0.1% in the direction perpendicular to the edge.9 This indicates that graphene stiffness ensures that changes in the interatomic distances are small compared to the effect of projection on the measured strain in the image. We used the modeled atomic structures previously described to simulate the HREM images with optical parameters similar to the experimental conditions. Details about the HREM image simulation conditions are available in the Supporting Information. Figure 4A shows the HREM image obtained from the atomic positions of the folded zigzag edge, rendered three-dimensionally in Figure 4B. Using GPA the strain tensor in the image is recovered, and Figure 4C shows the maps of Ex
estimated fold curvature radius, making impossible to map such a large and rapid variation of the crystal slope. To validate the experimental results we modeled the 3D atomic structure of folded graphene membranes in different geometries and lattice configurations by using a combination of continuum elasticity theory and atomistic tight-binding (TB) simulations. From the continuum elasticity theory point of view, the equilibrium shape of a folded graphene can be predicted by solving the corresponding Euler−Poisson problem (details in the Supporting Information), providing specific geometrical boundary conditions, as the length of the bended ribbon (L), the attack angle (φ0) between the folded region and the flat one, and the distance (a) between the two parallel layers (see Figure 3). The starting value for the attack angle φ0 is fixed by continuity reasons, while the length of the folded region results from the competition between the energy stored by bending process (i.e., by the elastic moduli of graphene), which tends to open the structure, and the attractive van der Waals potential. To be more precise, in our case the generic cylindrical configuration involves only the mean curvature on the surface; therefore the elastic moduli depends only by the bending rigidity. We started from a configuration made by an hexagonal configuration mapped on the predicted shape with the desired chirality; then we obtained the correct values for the 5209
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Figure 4. Modeling of a graphene zigzag folded edge and curvature mapping. (A) Details of the HREM image simulations of the modeled atomic structures and (B) three-dimensional rendering of the structure shown in A. (C) Strain tensor components calculated by GPA along the directions parallel (Ex) and perpendicular (Ey) to the folded border. (D) Ex strain profile (blue line) compared to the discrete strain values (red dots) calculated from the modeled atomic structure.
bending modulus for this chirality.33 In addition, it is worth noticing that, in the experimental measurements, we have to take into account the defects induced by the electron beam (100 kV) on the graphene lattice.25 The curvature radius of the internal part of the fold also depends on the interlayer coupling and defects can significantly change the interaction between the stacked layers, and they can also modify the bending modulus. In this perspective, the experimental values for the slope of the curved region and the calculation of the height variation can be considered in a satisfactory agreement with the simulated data. For the sake of completeness, a theoretical model capable to fully reproduce the experimental system should take into account both the elasticity and the interlayer interactions induced by chirality, as well as the presence of defects. At present, the addition of chirality and defects into the atomistic tight-binding model is prohibitive in terms of computing time. Nevertheless, in future experiments we plan to extend the model of the elastic properties of the membranes to chiral and defected structures, using multiscale semiclassical algorithms to overcome these difficulties. The results show that the analysis of the apparent strains in the HREM images of graphene membranes can provide threedimensional topographic information with subnanometer height and spatial resolution, in excellent agreement with predictions by atomistic tight-binding simulations. On this basis, the proposed experimental methodology can be used to characterize the structure of freely suspended graphene membranes in the TEM, combining direct information on the atomic structure through diffraction and HREM imaging, with the ability of mapping and measuring the actual three-
and Ey components. Figure 4D shows the strain profile along the y-direction compared to the discrete strain values (red dots) calculated using the displacements in the projected positions from the modeled atomic structure. Even in the simulated HREM image, the recovered strain map has not sufficient lateral resolution to recover the rapid curvature and the corresponding apparent compression close to the edge, but it does map perfectly the internal curvature of the fold. The value of the strain is directly interpretable as a value of the local slope between the beam and the surface of the flake. From the strain map of Figure 4C, we measure a maximum compression of 11%, corresponding to a maximum lattice slope of 27° over a length of 0.7 nm, with an estimated height variation of 0.32 nm. In the atomistic model of the zigzag edge the maximum slope is 28.4°, extending for 0.7 nm, corresponding to a height variation of 0.24 nm. We performed a similar analysis on the modeled structure of an armchair edge, finding again an excellent agreement between the reconstruction and the actual 3D structure (see the Supporting Information), thus showing that the geometry reconstructed from the apparent strain maps faithfully reproduces the atomic structure of the folded edge. It is worth noticing that the simulated structures of zigzag and armchair folds show smaller curvature radii and height variations with respect to the experimental ones shown in Figure 2. The whole fold of the modeled flakes occurs over lengths of less than 1.5 nm, with smaller curvature radiuses with respect to the experimental ones. Besides the limited experimental precision arising from working with low contrast HREM images, the higher value for the curvature radius and the sloped area could be an effect of the possibly increased 5210
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(5) Meyer, J. C.; et al. The structure of suspended graphene sheets. Nature 2007, 446, 60−63. (6) Zhang, J.; et al. Free Folding of Suspended Graphene Sheets by Random Mechanical Stimulation. Phys. Rev. Lett. 2010, 104, 166805. (7) Topsakal, M.; Bagci, V. M. K.; Ciraci, S. Current-voltage (I-V) characteristics of armchair graphene nanoribbons under uniaxial strain. Phys. Rev. B 2010, 81, 205437. (8) Cadelano, E.; Palla, P.; Giordano, S.; Colombo, L. Nonlinear Elasticity of Monolayer Graphene. Phys. Rev. Lett. 2009, 102, 235502. (9) Cadelano, E.; Giordano, S.; Colombo, L. Interplay between bending and stretching in carbon nanoribbons. Phys. Rev. B 2010, 81, 144105. (10) Poetschke, M.; Rocha, C. G.; Foa Torres, L. E. F.; Roche, S.; Cuniberti, G. Modeling graphene-based nanoelectromechanical devices. Phys. Rev. B 2010, 81, 193404. (11) Feng, J.; Qi, L.; Huang, J.; Li, J. Geometric and electronic structure of graphene bilayer edges. Phys. Rev. B 2009, 80, 165407. (12) Tozzini, V.; Pellegrini, V. Reversible hydrogen storage by controlled buckling of graphene layers. J. Phys. Chem. C 2011, 115, 25523−25528. (13) Zheng, Y.; Wei, N.; Fan, Z.; Xu, L.; Huang, Z. Mechanical properties of grafold: a demonstration of strengthened graphene. Nanotechnology 2011, 22, 405701. (14) Prada, E.; San-Jose, P.; Brey, L. Zero Landau Level in Folded Graphene Nanoribbons. Phys. Rev. Lett. 2010, 105, 106802. (15) Zhu, W. Structure and Electronic Transport in Graphene Wrinkles. Nano Lett. 2012, 12, 3431−3436. (16) Yuk, J. M.; et al. Graphene Veils and Sandwiches. Nano Lett. 2011, 11, 3290−3294. (17) Monthioux, M.; Flahaut, E. Meta- and hybrid-CNTs: A clue for the future development of carbon nanotubes. Mater. Sci. Eng., C 2007, 27, 1096−1101. (18) Pang, A. L. J.; Sorkin, V.; Zhang, Y.-W.; Srolovitz, D. J. Selfassembly of free-standing graphene nano-ribbons. Phys. Lett. A 2012, 376, 973−977. (19) Qi, L.; Huang, J. Y.; Feng, J.; Li, J. In situ observations of the nucleation and growth of atomically sharp graphene bilayer edges. Carbon 2010, 48, 2354−2360. (20) Patra, N.; Wang, B.; Král, P. Nanodroplet Activated and Guided Folding of Graphene Nanostructures. Nano Lett. 2009, 9, 3766−3771. (21) Catheline, A.; et al. Solutions of Fully Exfoliated Individual Graphene Flakes in Low Boiling Points Solvents. Soft Matter 2012, 8, 7882−7887. (22) Bunch, J. S. NEMS Putting a damper on nanoresonators. Nat. Nanotechnol. 2011, 6, 331−332. (23) Midgley, P. A.; Dunin-Borkowski, R. E. Electron tomography and holography in materials science. Nat. Mater. 2009, 8, 271−280. (24) Warner, J. H.; Ruemmeli, M. H.; Bachmatiuk, A.; Buechner, B. Examining the stability of folded graphene edges against electron beam induced sputtering with atomic resolution. Nanotechnol. 2010, 21, 325702. (25) Girit, C.; Meyer, J.; Erni, R.; Rossell, M. Graphene at the Edge: Stability and Dynamics. Science 2009, 323, 1705−1708. (26) Kirilenko, D.; Dideykin, A.; Van Tendeloo, G. Measuring the corrugation amplitude of suspended and supported graphene. Phys. Rev. B 2011, 84, 235417. (27) Hytch, M.; Snoeck, E.; Kilaas, R. Quantitative measurement of displacement and strain fields from HREM micrographs. Ultramicroscopy 1998, 74, 131−146. (28) D’Addato, S.; et al. Structure and stability of nickel/nickel oxide core-shell nanoparticles. J. Phys.: Condens. Matter 2011, 23, 175003. (29) Warner, J. H.; et al. Dislocation-Driven Deformations in Graphene. Science 2012, 337, 209−212. (30) Warner, J. H.; et al. Resolving strain in carbon nanotubes at the atomic level. Nat. Mater. 2011, 10, 958−962. (31) Xu, Y.; et al. Electronic transport in monolayer graphene with extreme physical deformation: ab initio density functional calculation. Nanotechnology 2011, 22, 365202.
dimensional morphology of the membranes. In this work the combination of apparent strain analysis and theoretical modeling has been used to characterize folded edges of monolayer graphene, but provided that enough contrast and resolution in the HREM image are obtained, it can be successfully applied to investigate the 3D structure of various wrinkle and fold configurations.4 The proposed methodology can apply to graphene membranes of different thickness, and it can be extended to other two-dimensional crystal like BN or MoS2 membranes, as well as to hybrid multilayer thin-films composed by these materials. To completely extend the method to a wider range of materials and experimental conditions, two main limitations should be overcome. From the computational point of view, to fully reproduce the experimental systems, the described model should be improved, taking into account both elasticity and interlayer interactions induced by chirality and lattice defects, and we plan to extend it using multiscale semiclassical algorithms. On the other hand, from the experimental point of view, the main limitation of the proposed methodology relies in the low contrast of the lattice fringes in the HREM images. This problem limits the final spatial resolution and the signal-to-noise ratio in reconstructed strain maps. Nevertheless, we strongly believe that precision and resolution of this structural and topographical mapping can be greatly improved using new-generation low-voltage and aberration-corrected conventional and scanning transmission electron microscopes, opening new and unrevealed possibilities to investigate mechanical properties at the nanoscale in twodimensional materials.
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ASSOCIATED CONTENT
S Supporting Information *
Additional information and figures. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail address:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS L.O. wants to thank Vincenzo Grillo for the assistance with the HREM image simulations and GPA analysis. This work was partially supported by the project “FlexSolar” within the framework of the Italian “Industry 2015” program. E.C. and L.C. acknowledge financial support from the Regional Government of Sardinia under project “Ricerca di Base” titled “Modellizzazione Multiscala della Meccanica dei Materiali Complessi” (RAS-M4C).
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