Magnetic breakdown in twisted bilayer graphene

We consider magnetic breakdown in twisted bilayer graphene where electrons may hop between semiclassical $k$-space trajectories in different layers. These trajectories within a doubled Brillouin zone (BZ) constitute a network in which an $S$ matrix at each saddle point (SP) is used to model interlayer tunneling. Although semiclassical orbits at a given energy all have the same shape, we find that the doubled BZ supports two distinct quantization conditions above the SP, versus only one below it, yielding a more intricate spectrum. Possible experimental signatures are discussed.

Figure 1

Semiclassical orbits in the doubled BZ. Solid (red) trajectories are in the top layer, and dashed (blue) are on the bottom. Circular orbits correspond to energies below the saddle point, and starlike (purple) orbits are above. Saddle points are labeled by $M_a$, $M_b$, $M_c$. Cross symbols represent the Dirac points.

Figure 2

Energy spectrum near the SP energy for the interlayer hopping $\alpha =0.05$. Here the reference of energy corresponds to zero of $E^{\prime }\equiv E-v_F{k}_{\theta }/2$. Energy units are $E_0=v_F{k}_{\theta }$, magnetic field units are $B_0=hc{k}_{\theta }^2/e$.