Gate-controlled conductance through bilayer graphene ribbons

We study the conductance of a biased bilayer graphene flake with monolayer nanoribbon contacts. We find that the transmission through the bilayer ribbon strongly depends on the applied bias between the two layers and on the relative position of the monolayer contacts. Besides the opening of an energy gap on the metallic bilayer, the bias allows to tuning the electronic density on the bilayer flake, making possible the control of the electronic transmission by an external parameter.

Figure 1

Schematic band structure of a biased graphene bilayer with and without interlayer coupling ${\gamma }_1$. (a) Biased bands without interlayer hopping. A positive (negative) bias $V/2$ is applied to the bottom (top) layer, producing a rigid shift of the corresponding linear dispersion relation (red dotted lines for the bottom layer, blue dashed lines for the top one). (b) When ${\gamma }_1$ is switched on, gaps open at the band crossings, yielding the well-known Mexican hat shape and split-off bands of the bulk bilayer band structure. This picture allows identifying the top (T)/bottom (B) character of the different branches of the bilayer dispersion relation.

Figure 2

Schematic band structure of the metallic bilayer armchair nanoribbon with monolayer contacts. Upper and lower panels show the bottom-bottom (1$\to$1) and bottom-top (1$\to$2) configurations. Left and right dispersions correspond to the metallic monolayer ribbons acting as electrodes, and the central dispersion corresponds to the biased bilayer ribbon. The applied gate bias between bottom and top layers is $V$. For widths for which armchair monolayer ribbons are metallic, the band structure of the bilayer is not affected by the confinement. In the band structure of the bilayer we indicate the three relevant gaps, $V$, ${\Delta }_1$, and ${\Delta }_2$.

Figure 3

Same as Fig. 2 but for zigzag terminated nanoribbons. For zigzag monolayer nanoribbons, states in different valleys transport charge in opposite directions. In the biased bilayer ribbon, central part, edge localized states with energies $\pm V/2$ appear, connecting the Dirac points. At the Dirac points of the bilayer ribbons, topological surface states exist in the gap. In wide ribbons these surface states close the energy gap. In narrow bilayer ribbons states in different edges interact and open a gap $\Delta$ in the spectrum.

Figure 4

Transmission as a function of energy and applied voltage for metallic armchair nanoribbons in the 1$\to$1 and 1$\to$2 configuration within the one-mode energy range. The value of the interlayer hopping parameter is ${\gamma }_1=0.1t$. The length of the bilayer region is $L=35\sqrt{3}a$. The values of the gap edges $\pm {\Delta }_1/2$, $\pm {\Delta }_2/2$, and $\pm V/2$ are plotted with white lines; see Fig. 2.

Figure 5

Transmission as a function of energy and applied voltage for zigzag graphene nanoribbons in the 1$\to$1 and 1$\to$2 configurations. The interlayer hopping parameter is ${\gamma }_1=0.1t$. The number of atoms across the nanoribbon is 16, corresponding to a width $W=11a/\sqrt{3}$. The bilayer region length is $L=60a$. The values of the gap edges $\pm {\Delta }_1/2$ are plotted in white lines, see Fig. 2.

Figure 6

Conductance in units of $2e^2/h$ as a function of energy and applied voltage for an armchair-terminated nanoribbon system in the 1$\to$1 configuration. The interlayer hopping parameter is ${\gamma }_1=0.1t$. The number of dimers across the nanoribbon is 23, corresponding to a width $W=11a$. The bilayer region length $L=35\sqrt{3}a$.

Figure 7

Conductance in units of $2e^2/h$ as a function of energy and applied voltage for a semiconducting armchair-terminated nanoribbon system in the 1$\to$1 configuration. The interlayer hopping parameter is ${\gamma }_1=0.1t$. The number of dimers across the nanoribbon is 19, corresponding to a width $W=9a$. The bilayer region length $L=35\sqrt{3}a$.