Antichiral states in twisted graphene multilayers

The advent of topological phases of matter revealed a variety of observed boundary phenomena, such as chiral and helical modes found at the edges of two-dimensional (2D) topological insulators. Antichiral states in 2D semimetals, i.e., copropagating edge modes on opposite edges compensated by a counterpropagating bulk current, are also predicted, but, to date, no realization of such states in a solid-state system has been found. Here, we put forward a procedure to realize antichiral states in twisted van der Waals multilayers, by combining the electronic Dirac-cone spectra of each layer through the combination of the orbital moiré superstructure, an in-plane magnetic field, and interlayer bias voltage. In particular, we demonstrate that a twisted van der Waals heterostructure consisting of graphene/two layers of hexagonal boron nitride $\left[{\left(\mathrm{hBN}\right)}_2\right]/\mathrm{graphene}$ will show antichiral states at in-plane magnetic fields of 8 T, for a rotation angle of $0.2^{\circ }$ between the graphene layers. Our findings engender a controllable procedure to engineer antichiral states in solid-state systems, as well as in quantum engineered metamaterials.

Figure 1

$AB$-stacked bilayer graphene. (a) Real-space lattice of $AB$-stacked bilayer graphene, consisting of two graphene sheets that are shifted with respect to each other, while being aligned only at two points of the unit cell. (b) Corresponding schematic low-energy spectrum in the hexagonal Brillouin zone. The combination of the two graphene sheets results in a quadratic low-energy band structure, appearing at the six corners of the first Brillouin zone.

Figure 2

$AB$-stacked graphene subject to an in-plane magnetic field. Schematic depiction of the low-energy band structure in the first Brillouin zone for (a) a small in-plane magnetic field $B$ and (b) a large $B$ that hybridizes the cones once more. The Lorentz force acting on electrons tunneling between the two graphene layers causes a momentum separation between the lower (red) and upper (blue) layer spectra [arrows in (a)], thus effectively decoupling the layer Dirac-cone spectra from one another. A strong in-plane magnetic field is able to merge Dirac cones by re-combining different valleys, thus selectively producing quadratic spectra at the merged points, while separating the band structures of the layers in other parts of the Brillouin zone.

Figure 3

Band structure of $AB$-stacked bilayer graphene. (a) The spectrum with an in-plane magnetic field that merges Dirac cones from different layers at the $K$ points (quadratic touching) and leaves linear uncoupled Dirac cones at the $\mathrm{\Gamma }$ and $K^{\prime }$ points (marked by blue and red squares); cf. Fig. 2. (b) Since the quadratic touching points are formed by states originating from different layers, an interlayer voltage bias gaps them. Moreover, the bias voltage shifts in energy the Dirac cones of the upper and lower layers relative to one another. (c) This procedure reveals antichiral edge states when considering a finite system of 20 unit cells with zigzag termination, with the color map indicating the position of eigenstates along the finite dimension. (d) The antichiral states propagate in the same direction along the boundary (orange) compensated by a bulk current (black) of opposite direction close to the edge, as can be seen by the spatially resolved group velocity.

Figure 4

Twisted bilayer graphene. (a) Real-space moiré pattern formed by the relative rotation of two graphene layers (red and blue mark atoms in the lower and upper layers, respectively). (b) Corresponding Brillouin zones of the lower (red) and upper (blue) layer rotated by angle $\theta$. The superstructure in real space of length $L_m$ induces a much smaller Brillouin zone in reciprocal space.

Figure 5

Band structure of twisted bilayer graphene in the antichiral regime. (a) In the presence of an in-plane magnetic field that selectively merges Dirac cones from different layers (cf. Figs. 2 and 3), the in-plane magnetic field reduces the number of Dirac cones by creating quadratic touching points in the merged valley. (b) An applied interlayer bias voltage opens a gap at the quadratic touching points and shifts the unpaired Dirac cones from the different layers relative to one another (red and blue squares). (c) Antichiral edge states appear when considering a finite system, with the color map indicating the position of the eigenstates along the finite dimension. It is observed that the states localized on opposite edges propagate in the same direction, as expected from antichiral modes. This regime is achieved in the twisted $\mathrm{graphene}/{\left(\mathrm{hBN}\right)}_2/\mathrm{graphene}$ heterostructure for a rotation angle of $0.2^{\circ }$ and an in-plane magnetic field of 8 T.

Figure 6

Sketch (a) and band structure (b) of a twisted graphene bilayer with a conventional interlayer distance. When two twisted layers of hBN are put in between the graphene layers (c), the effective interlayer coupling becomes weaker (d). By comparing with a structure in which the two layers are artificially separated yielding a reduced interlayer coupling (e) it is observed that the twisted hBN does not create additional perturbations (f).