Localized States at Zigzag Edges of Bilayer Graphene

We report the existence of zero-energy surface states localized at zigzag edges of bilayer graphene. Working within the tight-binding approximation we derive the analytic solution for the wave functions of these peculiar surface states. It is shown that zero-energy edge states in bilayer graphene can be divided into two families: (i) states living only on a single plane, equivalent to surface states in monolayer graphene and (ii) states with a finite amplitude over the two layers, with an enhanced penetration into the bulk. The bulk and surface (edge) electronic structure of bilayer graphene nanoribbons is also studied, both in the absence and in the presence of a bias voltage between planes.

Figure 1

Charge density for the two families of edge states in bilayer graphene: (a) monolayer [Eq. (11)]; (b) bilayer [Eq. (12)].

Figure 2

Bilayer graphene ribbon with zigzag edges.

Figure 3

Charge density for bilayer edge states at (a) $ka/2\pi =0.36$ and (b) $ka/2\pi =0.34$.

Figure 4

(a) Energy spectrum for a bilayer ribbon with zigzag edges: $N=400$, $t_{\perp }=0.2t$. (b) Zoom of (a).

Figure 5

Energy spectrum for a bilayer ribbon with zigzag edges, $N=400$, $t_{\perp }=0.2t$: (a) $V=t_{\perp }/10$, (b) $V=t_{\perp }/2$, (c) $V=t_{\perp }$. Dashed lines are from Eq. (15).