Chiral Response of Twisted Bilayer Graphene

We present an effective (minimal) theory for chiral two-dimensional materials. These materials possess an electromagnetic coupling without exhibiting a topological gap. As an example, we study the response of doped twisted bilayers, unveiling unusual phenomena in the zero frequency limit. An in-plane magnetic field induces a huge paramagnetic response at the neutrality point and, upon doping, also gives rise to a substantial longitudinal Hall response. The system also accommodates nontrivial longitudinal plasmonic modes that are associated with a longitudinal magnetic moment, thus endowing them with a chiral character. Finally, we note that the optical activity can be considerably enhanced upon doping and our general approach would enable systematic exploration of 2D material heterostructures with optical activity.

Figure 1

The transverse Drude weight $D_{xy}$ in units of $t/{\hslash }^2$ as a function of the chemical potential for two twist angles ${\theta }_{i=5}=6°$ (left) and ${\theta }_{i=10}=3°$ (right). Also shown are the Drude weights $D_0$ (green) and $D_1$ (blue).

Figure 2

Left: the chiral or Hall conductivity at finite frequencies for several doping levels $\mu /t=0$, 0.1, 0.2 and the two angles $\theta =6°$ (top) and $\theta =3°$ (bottom). Right: the circular dichronism for several doping levels with $v_F/c=300$ and $\epsilon =1$ for the two angles $\theta =6°$ (top) and $\theta =3°$ (bottom). The vertical lines correspond to the two energy scales ${\epsilon }_M=(2\pi /3)\hslash v_F/L_M$ and ${\epsilon }_{vH}\approx {\epsilon }_M-t_{\perp }/3$ with $L_M$ the Moiré-lattice period and $t_{\perp }$ the interlayer hopping amplitude, see also the Supplemental Material [26].

Figure 3

(a) Double-layer structure with two “identical” twisted bilayer graphene samples, but with opposite chiralities (${\chi }_1=-{\chi }_2$), separated by an insulator with dielectric constant $\epsilon$. The current and attached magnetic moment are either parallel or antiparallel. (b) The two plasmon modes hybridize due to electrostatic interaction and form symmetric in-phase (optical) modes and antisymmetric out-of-phase (acoustic) modes. The in-phase mode has purely chargelike character whereas the out-of-phase mode has a purely magneticlike character.