Collimation and splitting of valley electron diffraction in graphene

We reported the collimation and splitting effects of the diffraction of valley electrons in graphene. When the incident energy increases from the neutral point, the diffraction tends to be collimated for one valley and split for the other valley. The difference in the diffraction between valleys results in valley-dependent transport. We investigated the left-right conductance of a four-terminal graphene device. The conductance ratio between the two valleys was derived to be $1-(8/3)E$, where $E$ is the incident energy in units of the atom-atom hopping. The ratio is independent of the device dimensions and reflects the intrinsic properties of the electronic structure of graphene.

Figure 1

(a) Lattice of the lead-bulk graphene system. (b) Sketch of the four-terminal device.

Figure 2

(a) Contour plot of the conduction band of graphene with the inset showing the relation between the diffraction ray and the energy contour. (b) Orientation of diffraction rays for valley $K$ and valley $K^{\prime }$.

Figure 3

The distribution of local density of states calculated (i) from Eq. (10) and (ii) from Eq. (11). The electron beams of valley $K$ or $K^{\prime }$ are injected from the left lead and diffracted in the semi-infinite region. The lead width is set to be 30 units (20 lateral atoms).

Figure 4

(a) Dispersion structure of the zigzag-edged graphene ribbon with 20 lateral atoms. (b) and (c) Conductance for the two valleys as a function of energy. The dimensions of the center region are $W=30$ (20 atoms in the $y$ direction) and $L=15\sqrt{3}$ (30 atoms in the $x$ direction). (d) Conductance of valley $K$ as a function of energy for different choices of $L$ and fixed $W=30$.

Figure 5

(a) Conductance of the two valleys and (b) conductance ratio between them as functions of energy for $W=150$ and $L=10W$. (c) Conductance ratio between two valleys as a function of $\nu =L/W$ with $W=150$.