Breakdown of continuum mechanics for nanometre-wavelength rippling of graphene

Transcription

1 SUPPLEMENTARY INFORMATION DOI: /NPHYS2389 Breakdown of continuum mechanics for nanometre-wavelength rippling of graphene 1. Growth conditions and surface morphology of graphene on Cu(111) surface Our graphene samples have been grown on single crystalline Cu (111) substrates by thermal chemical vapor deposition of carbon from methane. The Cu (111) substrate was both mechanically and electrochemically polished before heating it up in a furnace to 990 C and annealing for 30 minutes in hydrogen atmosphere (1sccm). This process is supposed to remove the copper oxide capping layer forming under ambient conditions. After the annealing step, a high purity methane/hydrogen gas mixture was introduced into the furnace for 2 minutes, during which step the graphene overlayer was grown. Then the temperature of the sample was lowered to room temperature at a cooling rate of 50 C/min. Afterwards the sample was kept under ambient conditions, the graphene cover layer protecting the underlying Cu substrate from oxidation 1. The surface of the as synthesized sample typically displayed steps and terraces ( 10nm high steps, and 100 nm wide terraces) but also atomically flat areas of a few square microns. Surface areas containing nanotrenches (and adatom clusters) shown in Fig. S1 were first accidentally found among the large flat areas. Afterwards several such areas could be identified upon purposeful searching. Fig. S1. STM image of a flat, graphene-covered, reconstructed Cu(111) area displaying adatom clusters and trench-like vacancy islands. 1 Chen, S. et al. Oxidation resistance of graphene-coated Cu and Cu/Ni alloy. ACS nano 5, (2011). NATURE PHYSICS 1

2 It is well-known that at the graphene synthesis temperatures ( 1000 C) the surface morphology of the Cu substrate is substantially reconstructed through step formation and migration as well as a massive diffusion of the surface Cu atoms. Similar surface reconstructions consisting of adatom clusters (pads) and vacancy islands have been previously reported on Cu and Ag surfaces 2 as well as on Cu(100) single crystals during the graphene growth 3. However, the formation of the typically observed nanotrench-like features with well-defined width ( 5nm) has not been reported previously. This indicates that the atomic configuration of the Cu surface together with the presence of graphene might determine the formation of such structures. It was previously shown that the presence of different atoms (e.g. Nitrogen) might substantially influence the surface reconstructions of Cu 4. Furthermore, the fact that adatom clusters have a tendency of decorating graphene grain boundaries (Fig. S2) indicates that the reconstruction of the Cu substrate continues also after the graphene capping layer is grown. Fig. S2 STM image of a 300 nm x300nm reconstructed Cu (111)/graphene surface revealing a graphene grain boundary triple junction with the rectangular adatom clusters showing some preference towards decorating the graphene GB. The grain boundaries of graphene either act as nucleation sites for the growth of the adatom clusters or block their migration. Either way the graphene (grain boundary) has to be already present during the reconstruction of the Cu surface. 2 Pai, W.W., et al. Evolution of two-dimensional wormlike nanoclusters on metal surfaces. Phys. Rev. Lett. 86, (2001) 3 Rasool, H.I., Song, E.B., Mecklenburg, M., Regan, B.C., Wang, K.L., Weiller, B.H. & Gimzewski, J.K. Atomic-scale characterization of graphene on copper (100) single crystals. JACS 133, (2011) 4 Driver SM, Woodruff DP, Nitrogen-induced pseudo-(100) reconstruction of the Cu(111) surface identified by STM. Surf. Sci. 422, 1-8 (1999)

3 2. Suspended graphene over nanotrenches To validate the interpretation of the experimentally observed structural rippling of graphene nanomembranes it is important to make sure that the graphene sheet is indeed suspended over the trenches. The often observed sagging of the rippled graphene membrane along the width of the trenches already indicates the suspended nature of these graphene regions. However, as complementary experimental methods such as Atomic Force Microscopy do not have the necessary resolution to test the mechanical properties of rippled graphene over the trenches, we employed the STM tip to mechanically interact with the sample. This can be achieved by setting small tip-sample distances (10mV, 3nA), that imply mechanical tip-sample interactions. This way, and by choosing the slow scanning direction parallel to the trench edges we were able, to push the graphene membrane into the trench, with the help of the STM tip, as evidenced in Fig S3. Comparing the two STM images provides direct evidence that originally the rippled graphene area is suspended over the trench. Fig. S3. Left: 3D STM image of rippled graphene membrane over a nanotrench, acquired under typical atomic resolution imaging conditions (10 mv, 1 na). Right: STM image of the same trench acquired with (10 mv, 3 na) and slow scanning direction parallel to the trench edge, revealing the graphene pushed into the trench, due to the strong mechanical interaction with the STM tip during the scanning.

4 3. Microscopic modeling of rippled graphene In our microscopic simulations aimed at regaining the experimentally observed subnanometer rippling of graphene, we used the non-orthogonal, density functional theory based tight-binding approximation 5 for the force model, as implemented in the code Trocadero 6. The method permits simulations on large enough systems (600 C atoms), yet has proven to be accurate for describing the inplane linear elasticity 7 and complex static rippling modes 8,9 developed in carbon systems. A supercell, like the one shown in Fig. 3a of the manuscript, placed under periodic boundary conditions along the z direction models the portion of the graphene membrane suspended over the trench of width L aligned along the z direction. Calculations were performed using 10 point k-sampling, as the employed repeating domain is large enough along the axial direction to ensure good electronic energy convergence. To capture the substrate pining at the trench edges, the two outermost lines of atoms spaced L apart, were kept fixed during simulations. Biaxial compressive strain, typically applied in small increments, was imposed on these two fixed edges along the direction parallel (axial strain) and perpendicular (longitudinal strain) with the trench. To bring the system closer to the (meta-) stable state of interest, the conjugate gradient relaxations were started with the compressed graphene morphology modulated by out-of-plane wavelike deformations with parameters close to those experimentally found. The handsoff relaxation process, leading to subnanometer rippling, was considered to be complete when maximum forces on atoms were less than 10-4 atomic units and the total energy reduction was less than 10-7 atomic units. The rippling amplitude of 0.5 Å reported in Fig. 3(b) was measured in the middle of the 5% biaxially compressed ribbon, where the amplitude was found to be the largest. Further systematic simulations confirmed that the identified wavelike rippling pattern is robust: (i) Repeating the relaxation process with a supercell doubled in size (1200 C atoms) led to an identical wavelength. (ii) The removal of the longitudinal strain (i.e., only 5% axial compression was applied) led to the same wavelike pattern characterized by an amplitude of 0.35 Å. Further increasing the axial strain to 7%, led to a rippling amplitude of 0.5 Å. In both cases, the C-C bond lengths were compressed, confirming the invalidity of the inextensible deformation hypothesis. Finally, (iii) the subnanometer rippling mode with nearly identical wavelengths and amplitudes was regained under similar strain conditions using the alternative second-generation force model of Brenner 10. This force model is known to describe well the bending rigidity of graphene, of particular importance here. 5 Porezag, D., Frauenheim, T., Köhler, Th., Seifert, G. & Kaschner, R. Construction of tight-binding-like potentials on the basis of density-functional theory: Application to carbon. Phys. Rev. B 51, (1995). 6 Rurali, R., Hernandez, E. Trocadero: a multiple-algorithm multiple-model atomistic simulation program. Comp. Mat. Sci. 28, (2003). 7 Zhang D.B., Dumitrica T. Elasticity of ideal single-walled carbon nanotubes via symmetry-adapted tight-binding objective modeling. Appl. Phys. Lett. 93, (2008). 8 Zhang D. -B., James, R.D., Dumitrica T. Electromechanical Characterization of Carbon Nanotubes in Torsion via Symmetry- Adapted Tight-Binding Objective Molecular Dynamics. Phys. Rev. B 80, (2009). 9 Zhang D.B., Dumitrica T. Effective Strain in Helical Rippled Carbon Nanotubes: A Unifying Concept for Understanding Electromechanical Response ACS Nano. 4, (2010). 10 Brenner, D.W., et al. A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons. J. Phys.: Condens. Matter 14, (2002)

5 4. Oblique graphene nanoripples Over some of the investigated trenches superposed on the 0.7 nm wavelength ripples with their crests always perpendicular to the trench edges an additional oblique oscillation has been revealed with a larger wavelength as exemplified in Fig. S4. These oblique ripples typically enclose an angle of about with the trench edges, their wavelength is about 6±1nm, and display a modulation in the range of Å. Fig. S4. STM image revealing the presence of oblique ripples marked by the arrows and characterized by λ 6 nm and A 0.4Å. The periodic features between the trenches are due to Moiré-patterns. Such oblique ripples have been predicted to occur when the strain has a shear component, due to the parallel displacement of the trench edges 11,12. Thermal strain usually induces an isotropic (biaxial) strain, which does not lead to the appearance of oblique rippling. Consequently their observation indicates that due to the local morphology of the sample additional strain components can arise (e.g. in the vicinity of adatom clusters). 11 Duan WH, Gong K, Wang Q. Controlling the formation of wrinkles in a single layer graphene sheet subjected to in-plane shear. Carbon 49, (2011) 12 Wang, Z.; & Devel, M.; Periodic ripples in suspended graphene. Phys. Rev. B 83, (2011)

6 5. Spatially resolved scanning tunneling spectroscopy on nanorippled graphene We have performed spatially resolved tunneling spectroscopy measurements to explore the electronic properties of the nanorippled graphene membranes over the trenches and compare with the surrounding supported (flat) areas. As the tunneling spectroscopy measurements have been performed under ambient conditions, to achieve reproducible results (reduce the noise of di/dv curves), each I-V characteristic is in fact an average of 20 individual curves taken at the same position. This way the fine structure of the curves is smeared out but the remaining robust features become highly reproducible. Besides the local density of states (LDOS) maps discussed also in the main text we have also investigated the spatial distribution of their minimum position. Since our tunneling spectra show no signature of gaplike features the conductivity minima is expected to correlate with the position of the Dirac point 13. On the other hand, the position of the Dirac point relative to the Fermi energy (zero bias) is indicative of the local doping. a Topography LDOS Doping b c d e f Fig. S5 Topographic image (a) and the corresponding color map of the spatially resolved LDOS (b) and differential tunneling conductivity minima positions (c). The latter is expected to correlate with the local doping of the sheet. Individual di/dv characteristics on flat-supported (d) low curvature - suspended (e) and high curvature - suspended graphene regions (f). 13 Teague, M.L.; Lai, A.P.; Velasco, J.; Hughes, C.R.; Beyer, A.D.; Bockrath, M.W.; Lau, C.N.; & Yeh, N.C. Evidence for strain induced local conductance modulations in single-layer grapheneon Si O 2 Nano Lett. 9, (2009)

7 A substantial difference is apparent on the conductance minima map above the trenches and over the flat supported graphene area (Fig S5 c). The rippled region is more p-doped, indicative of a charge transfer between the rippled and the flat graphene lattice. The individual di/dv characteristics shown in Fig S5 d - f, are rather U shaped, instead of the V shape usually expected for graphene, which is a known signature of strain 13. Although the minima are relatively shallow a clear shift in their positions for the rippled areas as compared to the flat graphene surface is obvious form comparing individual di/dv curves. The experimentally observed trend of (more) p-doped ripples is in full agreement with the theoretical prediction that the charge neutrality point of graphene will be offset in the presence of curvature in a way that the curved regions become locally hole doped 14. The same p-shift of the Dirac point has been predicted for graphene nanoribbons with 1D periodic corrugations 15. The above results clearly indicate that systematic differences occur in the electronic structure of the flat and nanorippled graphene regions. The major challenge in the detailed interpretation of these findings is that at least three factors can contribute to their origin. The nanoscale rippling introduces a locally varying potential, which certainly contributes to the observed effect given the experimentally found spatial modulation of the LDOS on the structural ripples. However, additional factors can complicate the picture: the graphene sheet over the trenches is also suspended, in contrast to the flat regions where it is in direct contact with the Cu substrate. Consequently, the substrate proximity might also play a role. Moreover, the flat graphene region is significantly compressed (up to a few %), while the rippled region is (at least partially) relaxed. Compressive strain is known to be able to modify the local tunneling conductivity of the graphene sheets as well as induce charging effects 13,16. Separating the effects of the above mentioned factors is a challenging task and requires a more detailed analysis preferentially at low temperatures to reveal the fine structure of the tunneling spectra. However, cooling down the sample to low temperatures induces even more strain, which leads to the crumpling of graphene all over the Cu substrate. In this case, most probably, the elastic energy due to compression of the graphene sheet becomes larger than the binding (pinning) of the graphene to the Cu substrate and the graphene crumples all over the Cu (111) surface area, which makes any subsequent measurement highly complicated. 14 Kim, E. & Castro-Neto, A.H. Graphene as an electronic membrane. EPL 84, (2008) 15 Isacsson, A.; Jonsson, L.M.; Kinaret, J.M. & Jonson M. Electronic superlattices in corrugated graphene. Phys. Rev. B 77, (2008) 16 Yeh, N.C.; et al. Strain-induced pseudo-magnetic fields and charging effects on CVD-grown graphene. Surf. Sci. 605, (2011)

8 6. Width dependence of the observed nanorippling mode Studying the dependence of the rippling wavelength and amplitude as a function of suspended membrane width L would be very useful in exploring the nanomechanical characteristics of graphene. Unfortunately, the nature of the Cu (111) surface reconstructions - under the given synthesis conditions - results in the formation of nanotrenches with a quite monodisperse width of about 5 nm (+/- 0.5 nm); as also can be seen in Fig 1a of the main text. Therefore, all of the more than 30 suspended membranes investigated by us were about 5 nm wide. Accordingly, they always displayed ripples characterized by λ = 0.7 ± 0.1 nm, A = 0.5 ± 0.2 Å. Since the formation mechanism of the nanotrench-like surface reconstructions is not yet fully understood it is very challenging to control their width. Nevertheless, on one occasion we were able to find and investigate also a 10 nm wide nanotrench covered by graphene. Fig S6. STM image displaying the characteristic rippling of the suspended graphene membrane over a 10 nm wide nanotrench in Cu (111) The observed graphene ripples, over the 10 nm wide trench are characterized by λ = 0.8 ± 0.1 nm, and A = 0.7 ± 0.2 Å. It is apparent that the increase in λ is substantially lower than predicted by the continuum theory, which indicates the scaling law λ ~L 1/2 (see eq. 1). Consequently, upon doubling the trench width the rippling wavelength measured on 5 nm wide trenches (0.7 nm) should increase to 1 nm for the 10 nm wide trench, while we measured only 0.8 nm. We note that the measured increase is very close to the experimental error. However, the observed difference between the continuum predicted (1 nm) and measured (0.8 nm) values is above the experimental error. It is not possible to establish a width dependence based on two measurement points, but this finding is already indicative that also the predictions concerning the width dependence of the continuum model will most probably fail.