Weyl Semimetal Path to Valley Filtering in Graphene

We propose a device in which a sheet of graphene is coupled to a Weyl semimetal, allowing for the physical access to the study of tunneling from two- to three-dimensional massless Dirac fermions. Because of the reconstructed band structure, we find that this device acts as a robust valley filter for electrons in the graphene sheet. We show that, by appropriate alignment, the Weyl semimetal draws away current in one of the two graphene valleys, while allowing current in the other to pass unimpeded. In contrast to other proposed valley filters, the mechanism of our proposed device occurs in the bulk of the graphene sheet, obviating the need for carefully shaped edges or dimensions.

Figure 1

A schematic picture for a graphene-WSM device. The incoming current (region I) is equally populated in the two valleys and the outgoing current (region III) is valley polarized.

Figure 2

Evolution of the projected WSM states in the surface BZ (SBZ). (a) At the nodal energy ($E_F=0$), only the Fermi arcs are present in the SBZ. (b),(c) As the chemical potential is raised, the Fermi arcs get absorbed into the bulk states and eventually disappear in (c). (d) An incommensurate structure of graphene and the WSM shown in momentum space. The green vectors show the three equivalent graphene $\mathbf{K}$ points. For a range of twist angles, the low-energy states in the $\mathbf{K}$ valley of graphene lie in a band of the projected bulk states of the WSM. The graphene states near the ${\mathbf{K}}^{\prime }$ points (red vectors) do not overlap the projected bulk states of the WSM in energy, but instead lie in a band gap.

Figure 3

Graphene and the top surface of our WSM model in a commensurate alignment which we use in our numerical tight-binding calculation. Electrons on the $A$ sublattice of graphene (empty circles) are allowed to hop only to the WSM surface site they overlie, while electrons on the $B$ sublattice (solid black) of graphene can hop to the three WSM surface sites surrounding the given $B$ site.

Figure 4

Results of the tight-binding calculation. (a),(b) The graphene-WSM spectrum is shown near the $\mathbf{K}$ and the ${\mathbf{K}}^{\prime }$ valleys, respectively. The Dirac cone in the $\mathbf{K}$ valley (shown in black) is immersed in the bulk states of the WSM, while the Dirac cone near the ${\mathbf{K}}^{\prime }$ valley is slightly perturbed. (c) An expanded view of the ${\mathbf{K}}^{\prime }$ valley shows an inverted band structure where the colors code the average of $S_z$ operator, with red being spin up and blue spin down. (d) The conductance of the graphene-WSM device based on a simplified model for the states near the ${\mathbf{K}}^{\prime }$ valley is shown in units of $e^2/h$. The conductance shows partial spin polarization as red denotes spin up states, where blue denotes spin down states.