Growth Mechanism of Hexagonal Shape Graphene Flakes with Zigzag Edges. Johnson, *

Transcription

1 Growth Mechanism of Hexagonal Shape Graphene Flakes with Zigzag Edges Zhengtang Luo, Seungchul Kim, Nicole Kawamoto, Andrew M. Rappe, and A.T. Charlie Johnson, * Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA The Makineni Theoretical Laboratories, Department of Chemistry, University of Pennsylvania, Philadelphia, PA Supplementary Information for Computational Details We modeled the edges of the graphene flake using fragments of the edge bonded to copper slabs which are four layers thick, as shown in Figure S1. Copper (100) surfaces were used in our simulations since the copper foils in our experiments are terminated predominantly in (100) direction. We used two different sizes of supercell depending on the sizes of graphene fragments: the smaller cell is 5 13 (8.10 Å Å of dimension and 97.13º of cell angle), while the larger cell is (11.4 Å 16.2 Å of dimension and º of cell angle). We separated the atoms in adjacent supercells by more than 15 Å in surface normal direction. The sizes of the supercells are large enough to avoid spurious interaction between periodic images in both lateral and vertical directions of the slab. Atomic models for the straight edges and the boundaries between zigzag and armchair edges are presented in Figure S1. All dangling bonds are passivated by hydrogen atoms to mimic the sp 2 type bonds of graphene. Since we did not observe any epitaxial relation between the graphene sheets and the substrates, we averaged over the bonding energies of the all configurations we found instead of using the lowest energy geometry. Density functional theory (DFT) calculations were performed in the framework of plane wave basis set 1 and pseudopotentials, within generalized gradient approximation (GGA-PBE) 2 using Quantum ESPRESSO. 3 Our optimized norm-conserving pseudopotentials 4,5 were generated using OPIUM code. 6 We used Monkhorst-Pack k-grids 7 equivalent to or denser than 7 7 in the 1 1 surface unitcell, and cut off the basis plane waves whose kinetic energy is larger than 50 Ry. S1

2 All atoms were relaxed until its Cartesian force components became smaller than 0.01 ev/å, but two bottom copper layers were fixed to bulk positions. Straight edges of our model stand up on the surface. Since graphene edges must be bent such that flake lies on the substrate, we tested an extremely bent case: the height of opposite side of bonding side was fixed to 3.58 Å, which is predicted as an equilibrium distance between graphene sheet and copper surfaces by Vanin et al. 8 We found that the edge energies of the zigzag and the armchair edges differ by only 0.01 ev/å for fixed heights, and fixed height geometries are about 0.1 ev/å higher in energy than fully relaxed geometries. The energy difference between these two extrema, free relaxation and constrained height relaxation, should be the upper limit of an error in our atomic models. Within this error, the energetics of our models implies that no specific edge is energetically favored. In fact, the error from graphene bending must be much smaller than 0.1 ev/å because the mechanical properties of graphene are isotropic. S2

3 Figure S1 Atomic models for calculations and those schematic drawings of the graphene edges on Cu (100). a & b) θ G/Cu =0º for armchair and zigzag respectively; c & d) θ G/Cu =45º for armchair and zigzag respectively; e & f) adjacent armchair and zigzag edges with angles (θ AC/ZZ ) of 90º and 150º. S3

4 Figure S2. Energy diagrams and atomic models of monomer attachments at (a) the straight edges and (b) the armchair-zigzag boundaries. Only the lowest energy structures of each case are presented among several structures we tested. S4

5 References 1. Ihm, J.; Zunger, A.; Cohen, M. L., Momentum-Space Formalism for the Total Energy of Solids. Journal of Physics C 1979, 12, (21), Perdew, J. P.; Burke, K.; Ernzerhof, M., Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, Giannozzi, P.; Baroni, S.; Nicola Bonini; Matteo Calandra; Roberto Car6; Carlo Cavazzoni; Davide Ceresoli; Guido L Chiarotti; Matteo Cococcioni; Ismaila Dabo11; Andrea Dal Corso; Stefano de Gironcoli; Stefano Fabris; Guido Fratesi; Ralph Gebauer; Uwe Gerstmann; Christos Gougoussis; Anton Kokalj1; Michele Lazzeri; Layla Martin-Samos; Nicola Marzari; Francesco Mauri; RiccardoMazzarello; Stefano Paolini; Alfredo Pasquarello; Lorenzo Paulatto; Carlo Sbraccia1; Sandro Scandolo1; Gabriele Sclauzero; Ari P Seitsonen; Alexander Smogunov; Umari, P.; RenataMWentzcovitch, QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials. J. Phys.: Condens. Matter 2009, 21, Rappe, A. M.; Rabe, K. M.; Kaxiras, E.; Joannopoulos, J. D., Optimized pseudopotentials. Physical Review B 1990, 41, (2), Ramer, N. J.; Rappe, A. M., Designed nonlocal pseudopotentials for enhanced transferability. Physical Review B 1999, 59, (19), Monkhorst, H. J.; Pack, J. D., Special points for Brillouin-zone integrations. Physical Review B 1976, 13, (12), Vanin, M; Mortensen, J. J.; Kelkkanen, A. K.; Garcia-Lastra, J. M.; Thygesen, K. S.; Jacobsen, K. W., Graphene on metals: A van der Waals density functional study, Physical Review B 2010, 81, (8), (R). S5