Moiré Valleytronics: Realizing Dense Arrays of Topological Helical Channels

We propose a general and robust platform, the moiré valleytronics, to realize high-density arrays of 1D topological helical channels in real materials at room temperature. We demonstrate the idea using a long-period 1D moiré pattern of graphene on $h\mathrm{BN}$ by first-principles calculation. Through calculating the Berry curvature and topological charge of the electronic structure associated with various local graphene/$h\mathrm{BN}$ stackings in the moiré pattern, it is revealed that the helical channel arrays originate intrinsically from the periodic modulation of the local topological orders by the moiré pattern. For a freestanding wavelike moiré pattern, two groups of helical channel arrays are spatially separated out of plane, validating the structural robustness of the moiré topology. The generality and experimental feasibility of moiré valleytronics are demonstrated by investigating a broad range of moiré systems.

Figure 1

(a) The graphene and $h\mathrm{BN}$ lattices, strained along the zigzag direction to match while left free along the armchair direction (insets) to form the 1D moiré pattern. $a$ and $b$ are the lattice constants of primitive graphene, with the value of 2.4595 and 4.26 Å, respectively. (b) The 1D moiré pattern periodic along the armchair direction formed by three different high-symmetry local stacking configurations indicated as $\mathrm{N}\text{−}h$, $\mathrm{B}\text{−}h$, and B-N stacking.

Figure 2

(a) Band structure of the 1DMP. The linear bands near the Fermi level (energy zero)—red and blue lines—are the two valley helical channels, and the gray lines are bulk bands. The valley points $K$ and $K^{\prime }$ are at $k_x=1/3$ and $2/3$, respectively. (b) Wave functions $|{\psi }_y{|}^2$ along the armchair ($y$) direction of the two helical channels with the same color scheme; vertical dashed lines indicate central positions of the three high-symmetry local stacking configurations. The left (right) panels are for $k_1=0.3329$ ($k_2=0.3338$), located at the left (right) of the $K$ valley. (c) Illustration of valley helical channels and valley current. Upper: Solid (empty) circles denote carriers with positive (negative) group velocity, i.e., right movers (left movers). Solid (dashed) lines are helical channels with valley index $K$ or $K^{\prime }$, respectively. Lower: The red (blue) color is for the channels located at edge 1 (2); purple and green arrows indicate valley currents. (d) Topological valley currents flow at the moiré edges.

Figure 3

Topology analysis of the lattice-matched $\mathrm{Gr}/h\mathrm{BN}$ bilayer. (a) Three high-symmetry stacking configurations in the moiré structure, where $\alpha$ and $\beta$ are two subsites of the graphene lattice. (b) Corresponding $k$-resolved Berry curvature $\mathrm{\Omega }(k)$ of the linear conducting band. (c) Valley-dependent topological phase diagram parametrized by $r_0$, which is the lattice displacement between Gr and $h\mathrm{BN}$ layers, and $\widetilde{N_3}$ is the calculated topological charge of the $K$ valley.

Figure 4

(a) Relaxed freestanding $\mathrm{Gr}/h\mathrm{BN}$ bilayer with out-of-plane ($z$-direction) corrugation of about 13 Å. (b) Left: Band structure; right: spatial distribution of modular squared wave functions of the two helical channels, where the plotted isosurface (pink and light blue areas) is $4\times {10}^{-9}\mathrm{a}.\mathrm{u}.$ (c) Topological valley current in the freestanding moiré structure.