Edge states of bilayer graphene in the quantum Hall regime

We study the low-energy edge states of bilayer graphene in a strong perpendicular magnetic field. Several possible simple boundary geometries related to zigzag edges are considered. Tight-binding calculations reveal three types of edge-state behaviors: weakly, strongly, and nondispersive edge states. These three behaviors may all be understood within a continuum model, and related by nonlinear transformations to the spectra of quantum Hall edge states in a conventional two-dimensional electron system. In all cases, the edge states closest to zero energy include a holelike edge state of one valley and a particlelike state of the other on the same edge, which may or may not cross, depending on the boundary condition. Edge states with the same spin generically have anticrossings that complicate the spectra, but which may be understood within degenerate perturbation theory. The results demonstrate that the number of edge states crossing the Fermi level in clean, undoped bilayer graphene depends both on boundary conditions and the energies of the bulk states.

Figure 1

Energy bands of a Bernal-stacked graphene bilayer nanoribbon, $N=240$ atoms across in each layer, each with zigzag edges. Perpendicular field is 100 T, interlayer bias $V=0.001t$, and ${\gamma }_1=0.25t$, with $t$ the in-plane hopping amplitude.

Figure 2

Unit cell for Bernal-stacked graphene bilayer nanoribbon, with zigzag edges. Dashed lines indicate bonds on the lower layer, solid lines are bonds on the upper layer.

Figure 3

Detail of tight-binding energy bands of zigzag bilayer ribbon, near left edge of system, for state emerging from Landau levels of the $K$ valley. The unit cell contains 480 atoms, perpendicular field is 100 T, and $V=0.001$ (in units of $t$). Results illustrated for several values of ${\gamma }_1$.

Figure 4

Energy states near zero for graphene ribbon with one layer bearded at each edge. The unit cell contains 474 atoms. Perpendicular field is 100 T, $V=0.001$, and ${\gamma }_1=0.25$ (in units of $t$). Upper inset illustrates edges of the unit cell.

Figure 5

Energy states near zero for graphene ribbon with one layer bearded at each edge. Dimer atoms protrude at edges in this construction. The unit cell contains 478 atoms. Perpendicular field is 100, $V=0.001$, and ${\gamma }_1=0.25$ (in units of $t$). Upper inset illustrates edges of the unit cell.

Figure 6

Energy states near zero for graphene ribbon with one layer bearded at each edge. Parameters are identical to those of Fig. 4, except ${\gamma }_1=0$.