Helical network model for twisted bilayer graphene

In the presence of a finite interlayer displacement field, bilayer graphene has an energy gap that is dependent on stacking and largest for the stable AB and BA stacking arrangements. When the relative orientations between layers are twisted through a small angle to form a moiré pattern, the local stacking arrangement changes slowly. We show that for nonzero displacement fields the low-energy physics of twisted bilayers is captured by a phenomenological helical network model that describes electrons localized on domain walls separating regions with approximate AB and BA stacking. The network band structure is gapless and has of a series of two-dimensional bands with Dirac band-touching points and a density of states that is periodic in energy with one zero and one divergence per period.

Figure 1

Helical model band structure over half of the rhombic Brillouin zone (BZ) defined in Fig. 3. The bands in the other half of the BZ can be obtained by the reflection. The model's band energies ${\epsilon }_{\mathbf{q}}^{n0}$ are given by Eq. (11) and depend on a single controlling parameter $\alpha$ which was set to $\alpha =1.1$ in this illustration. The bands touch at Dirac points located at high-symmetry $K$, $K^{\prime }$, and $\mathrm{\Gamma }$ points.

Figure 2

Spatial distribution of the the gap parameters in Eq. (4): (a) the gap minimum ${\delta }_{-}$; (b) the angle $\theta \left(\mathbf{r}\right)$ which specifies the direction in momentum space at which minima are achieved; (c) the gap maximum ${\delta }_{+}$. The dashed lines highlight the network of domain walls that separate regions in which the hybridization is dominated by $T_{\mathrm{AB}}$ from regions in which it is dominated by $T_{\mathrm{BA}}$.

Figure 3

(a) Elementary cell of the network. The wave-function amplitudes, links 1, 2, and 3, are denoted by ${\psi }_{ij}=\left\{{\psi }_{ij}^1,{\psi }_{ij}^2,{\psi }_{ij}^3\right\}$. (b) Node with three incoming and three outgoing channels characterized by the scattering matrix $\widehat{S}$. (c) First Brillouin zone of the network in hexagonal and rhombohedral representations.

Figure 4

The energy dependence of the density of states $\nu \left(\epsilon \right)$ per valley, spin and per Dirac point in the ring. It has three dips and three maxima separated from each other by $\mathrm{\Delta }{\epsilon }_{\mathrm{D}}={\epsilon }_{\mathrm{L}}^{\left|\right|}/3$. The primer correspond to Dirac points, while the latter to saddle points of the $\mathrm{moir}\acute{\mathrm{e}}$ pattern band structure presented in Fig. 1. The corresponding scale for the density of states is ${\nu }_{\mathrm{L}}=\sqrt{3}\pi /{\epsilon }_{\mathrm{L}}^{\left|\right|}{L}^2$.