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 $1.14\text{debye}/\text{Å}$ 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 graphene-based materials, characterized by a magnetic field-dependent resonance in the Hall conductivity.

Figure 1

The atomic structures of bilayer edges from DFT-LDA optimization. (a) and (b) show sideviews (normal projections along the edge direction) of the zigzag and armchair bilayer edges, respectively. (c) and (d) are closeup perspective views of the zigzag and armchair bilayer edges. We call these BLEs zigzag or armchair based on the orientation of ${\text{C}}_6$ hexagon along the edge. As we see in (c), in zigzag BLE a hexagon has two sides perpendicular to the edge direction while in armchair BLE in (d) a hexagon has two sides parallel to the edge. Both BLEs shown lead to $AA$ stacking of the two graphene layers in the flat regions.

Figure 2

(Color online) Extensive dipolar BLEs. (a) Dipole moments of electrically neutral slices of the BLEs. Each half of the unit cell (left or right) is electrically neutral. We subsequently find successive vertical planes, between which electrical neutrality is satisfied. The local dipole moment density (averaged over the width of the unit cell along the longitudinal direction), $\Delta P$ $\left(\text{debye}/\text{Å}\right)$, is calculated for each of the neutral slices. Valence electron density of (b) armchair and (c) zigzag bilayer edges, respectively. The values plotted are the integration along the longitudinal (out-of-plane) direction in a unit cell. Scale bar units: $\text{e}/{\text{Å}}^2$. (d) A simple triple-arc model describing the charge distribution in the BLE. Blue and red lines represent the continuous distribution of negative and positive charges, respectively. Overall charge neutrality is maintained in this model.

Figure 3

(Color online) Tight-binding formulation of the electronic structure of graphene and BLE structures. (a) In a single layer graphene, there are two sublattices, designated A (green) and B (yellow). In each of the sublattices, all carbon atoms are related by walks prescribed by the translational symmetry of the underlying hexagonal lattice. (b) The model described in our VP1 (see text). The vertical dotted lines represent the interlayer hopping with $AA$ stacking. (c) The model described by our VP2 (see text). The bent honeycomb net of continuous bilayer edge is shown by black lines. (d) The dispersion near the Dirac point calculated from the tight-binding formulation according to our VP1. (e) The DFT-LDA band structure calculated for an extended graphene bilayer with $AA$ stacking. (f) A schematic illustration of the quantum Hall effect in $AA$-stacked bilayer, including the DOS of Landau levels (with an arbitrary width) and expected Hall conductivity, for an applied magnetic field with flux density of 21 T (along the $z$ direction). The degeneracy factor, $g$, accounts for intervalley and spin symmetries. (g) Landau levels between $-t^{\prime }/2$ and $t^{\prime }/2\text{eV}$ as functions of $B$ up to $n=10$, given by ${\text{LL}}_n=\pm t^{\prime }/2\mp 0.0362\sqrt{nB}$ (in eV if $B$ is in teslas). The levels belonging to the lower double cone are labeled.

Figure 4

(Color online) Electronic structure of finite-sized BLE ribbons. (a) The level discretization due to finite size of armchair and zigzag BLE ribbons. The green hexagons are the 1BZ of an extended graphene bilayer. Dashed lines are the loci in the 2D $\mathbf{k}$ space along which the cuts of the 2D band structures are made. (b) The DFT DOS near the Fermi level of infinite graphene bilayer with $AA$ stacking and of the armchair and zigzag BLE nanoribbons, corresponding to our VP1 and VP2, respectively. $A_{\text{atom}}\equiv 3\sqrt{3}{b}^2/4$ is the area per carbon atom on a monolayer graphene, where $b$ is the C-C bond length.