Helical networks in twisted bilayer graphene under interlayer bias

A twisted graphene bilayer exhibits a triangular Moiré pattern in the local stacking, that smoothly alternates between the three basic types $A{A}^{\prime }$, $A{B}^{\prime }$, and $B{A}^{\prime }$. Under an interlayer bias $U$, the latter two types develop a spectral gap, characterizsed by opposite valley Chern numbers. We show that for large enough Moiré periods and bias, these regions become depleted electronically, and topologically protected helical modes appear at their boundaries. This gives rise to a delocalized topological network of the Chalker-Coddington type, composed of valley current vortices. This network can be tailored by controlled deposition of valley-mixing adsorbates, which block transmission in selected links, thus opening the possibility of custom topological nanoelectronics.

Figure 1

(a) Band structure and (b) DOS for a $\theta =4^{\circ }$ twisted bilayer with and without interlayer bias $U$. At such angles, a bias merely shifts the two Dirac cones, creating a flat DOS plateau for $\left|\epsilon \right|<\left|U\right|/2$ [to the left of the dashed line in (b)].

Figure 2

(a) Density of states of a $L_M=290a_0$ ($\theta \approx 0.2^{\circ }$) twisted bilayer with increasing bias $U$. The region above the dotted line denotes the helical network regime. (b)–(f) Periodic spatial profile of electron probability density for several characteristic eigenstates in the spectrum [arrows in (a)], all chosen at one of the $K$ points. Black color corresponds to zero density, and white to maximum. Panel (b) shows a fully developed helical network (Chalker-Coddington) state, with arrows indicating the orientation of the valley currents. Panels (c)–(e) show various resonant states, all localized away from $A{B}^{\prime }$/$B{A}^{\prime }$ regions by the finite bias. Panel (f) shows a $U=0$ delocalized state.

Figure 3

(a) Spatial density of a helical network state similar to that of Fig. 2b, computed within a tight-binding model of a biased twisted bilayer. (b), (c) The same state in the presence of isolated atomic impurities adsorbed onto the top layer at positions marked by the red dots. Intervalley scattering on the defects suppresses the transmission through the corresponding link.