Fast and slow edges in bilayer graphene nanoribbons: Tuning the transition from band to Mott insulator

We show that gated bilayer graphene zigzag ribbons possess a fast and a slow edge, characterized by edge-state velocities that differ due to non-negligible next-nearest-neighbor hopping elements. By applying bosonization and renormalization group methods, we find that the slow edge can acquire a sizable interaction-induced gap, which is tunable via an external gate voltage $V_g$. In contrast to the gate-induced gap in the bulk, the interaction-induced gap depends nonmonotonously on the on-site potential $V$.

Figure 1

(Color online) Upper panel: sketch of a bilayer graphene nanoribbon of lattice constant $a$ and width $W$ (inset: definition of hopping terms). Middle panel: dispersion relation of edge states for a nanoribbon with $W=32\times \sqrt{3}a$ and layer-symmetry breaking potential $V=0.025{\gamma }_0$, focusing on one of the two $K$ points $\left[k_0=2\pi /\left(3a\right)\right]$. (Dashed lines: two innermost bands for $W=40\times \sqrt{3}a$.) Here we use hopping matrix elements ${\gamma }_1={\gamma }_3=0.12{\gamma }_0$, but set ${\gamma }_4=t^{\prime }=0$. The inset shows the transverse charge distribution of two edge states with energies indicated by the markers. Lower panel: the same but including hopping terms ${\gamma }_4=0.05{\gamma }_0$, $t^{\prime }=0.07{\gamma }_0$.

Figure 2

Influence of ${\gamma }_4$ and $t^{\prime }$ on the edge-state velocities (top panel, $v_D=\sqrt{3}{\gamma }_{0}a/2\hslash$) and localization on each isolated edge $\left({\Delta }_h=0\right)$ (bottom panel, expressed in terms of the inverse participation ratio $I_{\text{IPR}}$).

Figure 3

Charge gap ${\Delta }_{\rho }$ as a function of $V$, for selected values of $U$, assuming ${\gamma }_4=t^{\prime }=0$ (upper panel), as well as ${\gamma }_4=0.05{\gamma }_0$, $t^{\prime }=0.07{\gamma }_0$ (middle panel: slow edge; lower panel: fast edge). Note the different scales of the vertical axis.