Gate-induced Dirac cones in multilayer graphenes

We study the electronic structures of $ABA$ (Bernal) -stacked multilayer graphenes in a uniform perpendicular electric field, and we show that the interplay of the trigonal warping and the potential asymmetry gives rise to a number of emergent Dirac cones nearly touching at zero energy. The band velocity and the energy region (typically a few tens of meV) of these gate-induced Dirac cones are tunable with the external electric field. In $ABA$-trilayer graphene, in particular, applying an electric field induces a nontrivial valley Hall state, where the energy gap at the Dirac point is filled by chiral edge modes which propagate in opposite directions between two valleys. In four-layer graphene, in contrast, the valley Hall conductivity is zero and there are no edge modes filling in the gap. A nontrivial valley Hall state generally occurs in asymmetric odd-layer graphenes, and this is closely related to a hidden chiral symmetry which exists only in odd-layer graphenes.

Figure 1

Band structures of $ABA$-trilayer graphene in a perpendicular electric field depicted in a three-dimensional (3D) plot (left panel) and contour plot (right), for the band models (a) only with $v,{\gamma }_1,v_3$ terms and (b) with full parameters. The interlayer potential asymmetry is set to be $\Delta =200$ meV.

Figure 2

Lattice structure of $ABA$-trilayer graphene with tight-binding hopping parameters. The bottom figure is a top view, and the right is a schematic diagram of the lattice structure.

Figure 3

(a) Band structure of gated $ABA$ trilayer with various potential asymmetry $\Delta$. (b) Gate bias dependence of Landau-level energy in $B=0.5$ T. Landau levels in $K_{+}$ and $K_{-}$ valleys are plotted as solid (blue) and dashed (red) lines, respectively. Shaded area represents the charge neutral region, where the number indicates the single-valley Hall conductivity ${\sigma }_{xy}^{K\xi }$ in units of $\xi e^2/h$. (c) Effective mass $m$ and (d) band velocities $v_x,v_y$ for the effective Dirac cones at $p=0,p_{+},p_{-}$, plotted against $\Delta$.

Figure 4

(a) Atomic structure of a zigzag ribbon of $ABA$-trilayer graphene. (b) Band structure around the $K_{+}$ valley of a zigzag ribbon of $ABA$-trilayer graphene only including ${\gamma }_0,{\gamma }_1$, and ${\gamma }_3$, with the asymmetric potential $\Delta =200$ meV. Bulk, left-edge, and right-edge states are plotted with solid lines, red dots, and blue dots, respectively. Numbers indicate the degeneracies of the edge-state bands for a single side (the same for the left and the right edges). (c) Contour plot of the bulk band structure with white dots and labels denoting the position and the chirality of gate-induced Dirac points, respectively. One-dimensional winding number $\gamma \left(k_x\right)$ is indicated between dotted lines penetrating the Dirac points.

Figure 5

Band structure around the $K_{+}$ valley of zigzag ribbon of $ABA$-trilayer graphene with the parameters fully included, in the asymmetric potential (a) $\Delta =200$ meV and (b) $\Delta =400$ meV. Bulk, left-edge, and right-edge states are plotted with solid lines, red dots, and blue dots, respectively.

Figure 6

Band structures of $ABA$-stacked four-layer graphene with the interlayer asymmetric potential $\Delta =100$ meV, depicted in a 3D plot (left panel) and contour plot (right).

Figure 7

Plots similar to Fig. 3 for $ABA$ four-layer graphene. (a) Band structure with various potential asymmetry $\Delta$. (b) Gate bias dependence of Landau-level energy in $B=0.5$ T. Shaded area represents the charge neutral region. (c) Effective mass $m$ and (d) band velocities $v_x,v_y$ for the effective Dirac cones at $p_{+},p_{-}$, plotted against $\Delta$. (e) Band structure similar to Fig. 5 for zigzag ribbon of $ABA$ four layer with $\Delta =200$ meV.