Geometric and Electronic Structure of Graphene Bilayer Edges

We present a computational investigation of free-standing graphene bilayer edge (BLE) structures, aka “fractional nanotubes.” We demonstrate that these curved carbon nanostructures possess a number of interesting properties, electronic in origin. The BLEs, quite atypical of elemental carbon, have large permanent electric dipoles of 0.87 and 1.14 debye/Å for zigzag and armchair inclinations, respectively. An unusual, weak AA interlayer coupling leads to a twinned double-cone dispersion of the electronic states near the Dirac points. This entails a type of quantum Hall behavior markedly different from what has been observed in graphenebased materials, characterized by a magnetic field-dependent resonance in the Hall conductivity. Disciplines Engineering | Materials Science and Engineering Comments Suggested Citation: Feng, J., Qi, L., Huang, J.Y. and Li, J. (2009). "Geometric and electronic structure of graphene bilayer edges." Physical Review B. 80, 165407. © 2009 The American Physical Society http://dx.doi.org/10.1103/PhysRevB.80.165407 This journal article is available at ScholarlyCommons: http://repository.upenn.edu/mse_papers/192 Geometric and electronic structure of graphene bilayer edges Ji Feng,1 Liang Qi,1 Jian Yu Huang,2 and Ju Li1,* 1Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA 2Center for Integrated Nanotechnologies, Sandia National Laboratories, Albuquerque, New Mexico 87185, USA Received 2 September 2009; published 9 October 2009 We present a computational investigation of free-standing graphene bilayer edge BLE structures, aka “fractional nanotubes.” We demonstrate that these curved carbon nanostructures possess a number of interesting properties, electronic in origin. The BLEs, quite atypical of elemental carbon, have large permanent electric dipoles of 0.87 and 1.14 debye /Å for zigzag and armchair inclinations, respectively. An unusual, weak AA interlayer coupling leads to a twinned double-cone dispersion of the electronic states near the Dirac points. This entails a type of quantum Hall behavior markedly different from what has been observed in graphenebased materials, characterized by a magnetic field-dependent resonance in the Hall conductivity. DOI: 10.1103/PhysRevB.80.165407 PACS number s : 73.22. f, 73.43. f, 81.05.Uw, 81.05.Zx Free-standing graphene monolayers GMLs Ref. 1 have attracted tremendous interest owing to a variety of exotic electronic properties, including in particular integer quantum Hall effects QHEs , as a consenquence of a Berry phase of .1–5 An ideal infinite GML’s pzpz energy spectrum is characterized by being gapless, with a cone-shaped dispersion around the Fermi level, at the vertices K and K , often referred to as Dirac points for graphene of the first Brillouin zone 1BZ . This linear dispersion leads to a Fermi velocity of vF 106 m /s and a vanishingly small effective mass for low-energy excitations. Graphene bilayers with AB stacking have also been prepared. Graphene and AB-stacked graphene bilayer show two types quantum Hall conductivity stairs.6 In the AB bilayer, there is a 2 Berry phase and zero-level anomaly in QHE conductivity6,7 because of a weak interlayer interaction.6,8 The zero-level anomaly is not seen in the GMLs. A novel form of graphene nanostructure was recently discovered,9 which can be viewed as two flat graphene layers continuously connected by a curved bilayer edge BLE, see Fig. 1 . Geometrically, a BLE can be considered a “fractional nanotube.”10,11 Detailed transmission electron microscopy TEM revealed that in these BLE structures the flat bilayer regions are forced into AA stacking,10 a geometry that is not usually seen. Lammert et al.12 have studied theoretically the electronic structure of “squashed nanotubes,” where the electronic structure is sensitive to the interlayer coupling. Here we investigate the geometric and electronic structures of graphene BLEs using a combination of density-functional theory13 DFT and tight-binding TB methods. We show that these BLE structures have extensive permanent electric dipoles. Our analysis of the electronic structure also indicates that the BLE structures will show a type of QHE, characterized by magnetic field-dependent anomalous resonance between two separate QHE sequences as the doping level is continuously varied, which is markedly different from QHE previously observed in graphene.6 In situ TEM observations of Joule-heated few-layer graphene reveal that graphene monolayer edges MLEs are atomically rough and not strongly faceted crystallographically.11,14 In contrast, when two MLEs react to form a more stable BLE, the newly formed BLE tends to be atomically sharp and strongly faceted into zigzag and armchair inclinations.11,14 This suggests that in the Wulff plot15 of edge energy versus inclination angle, zigzag and armchair inclinations are strongly favored in the BLE Wulff plot, whereas it is not the case in the MLE Wulff plot. Consequently, we propose that there is a geometrical reason for this strong preference of zigzag and armchair inclined BLEs. Unlike rolling carbon nanotubes which involves 360° rotation of graphene, rolling graphene into a BLE requires 180° rotation only see Fig. 1 . Generally speaking the lattice orientation of the top and bottom graphene layers can differ, like what happens when one folds a piece of ruled writing-pad paper along an arbitrary crease line. Yet, there is an orientation constraint, enforced through the common preparation