Current flow in biased bilayer graphene: Role of sublattices

We investigate here how the current flows over a bilayer graphene in the presence of an external electric field perpendicularly applied (biased bilayer). Charge density polarization between layers in these systems is known to create a layer pseudospin, which can be manipulated by the electric field. Our results show that current does not necessarily flow over regions of the system with higher charge density. Charge can be predominantly concentrated over one layer, while current flows over the other layer. We find that this phenomenon occurs when the charge density becomes highly concentrated over only one of the sublattices, as the electric field breaks layer and sublattice symmetries for a Bernal-stacked bilayer. For bilayer nanoribbons, the situation is even more complex, with a competition between edge and bulk effects for the definition of the current flow. We show that, in spite of not flowing trough the layer where charge is polarized to, the current in these systems also defines a controllable layer pseudospin.

Figure 1

(a) Schematic representation of a BLG nanoribbon, with zigzag edges and width $W$, between left (L) and right (R) semi-infinite contacts. There is an electrostatic potential difference of 2 V between the two layers. Different sublattices, A and B, are represented in different colors. (b) Detail of the sublattices A and B in top and bottom layers, indicating the nearest hoppings: ${\gamma }_0$ in-plane and ${\gamma }_1$ coupling the dimer sites interlayer.

Figure 2

The left column shows results for a bulk biased BLG, while the right column are for BLG nanoribbon with zigzag edges (for both cases a potential difference $V=0.07$ eV and a width of $N=300$ atoms is considered). (a) and (b) Band structures. (c) and (d) Zoom into the band structure regions marked by the dashed lines in (a) and (b). $\Delta$ is the minimum separation between the flat and the dispersive edge state bands for the zigzag ribbon. (e) and (f) Percentage of the charge density in each layer. (g) and (h) Current density on each layer. (i) and (j) Total current density on the BLG. (k) and (l) The contribution of each sublattice to the charge density.

Figure 3

Spatial distribution of charge densities (left) and current densities (right) over each layer of the BLG. Results for a bulk system (periodical boundary conditions) are shown in (a) and (b) for $N=300$ and $E/V=1.003$. Results for a BLG nanoribbon with zigzag edges and $N=300$ are shown in (c) and (d) for $E/V=0.0965$; and in (e) and (f) for $E/V=1.003$ (e) and (f). The charge densities are schematically represented here for a narrower nanoribbon, where the radius of each circle is proportional to the amplitude of charge density and different colors stand for different sublattices. The current densities are evaluated at different sites using Eq. (2).

Figure 4

(a) and (b) Percentage of charge density on the top and bottom layers as a function of the bias $V$ applied between the layers, calculated for $E/V=1$. (c) and (d) Total current density on top and bottom layers as a function of bias V for $E/V=1$. Several widths of zigzag biased BLG are shown: from $N=20$ to 640. (e) Band structure for a BLG nanoribbon with zigzag edges, $V=0.07$ eV, for two different sizes: $N=80$ and 160.

Figure 5

Dashed lines are for nondisordered systems, while solid lines show the effects of the inclusion of on-site correlated disorder ($W/{\gamma }_0=0.5$) in the biased BLG ($V=0.07$ eV). Systems considered here have $80\times 40$ sites in each layer. The left column shows results for a bulk, while the right column is for BLG nanoribbon with zigzag edges. (a) and (b) Percentage of the charge density in each layer. (c) and (d) Current density on each layer.

Figure 6

Analysis with the inclusion of next-nearest neighbor hoppings in the model for the biased BLG. The left column is for a bulk, while the right column is for BLG nanoribbon with zigzag edges (for both cases $V=0.07$ eV and a width of $N=300$ atoms is considered). (a) and (b) Band structures. (c) and (d) Zoom into the band structure regions marked by the dashed lines in (a) and (b). (e) and (f) Percentage of the charge density in each layer. (g) and (h) Current density on each layer.