Strong gate-tunability of flat bands in bilayer graphene due to moir\'e encapsulation between hBN monolayers

When using hexagonal boron-nitride (hBN) as a substrate for graphene, the resulting moir\'e pattern creates secondary Dirac points. By encapsulating a multilayer graphene within aligned hBN sheets the controlled moir\'e stacking may offer even richer benefits. Using advanced tight-binding simulations on atomistically-relaxed heterostructures, here we show that the gap at the secondary Dirac point can be opened in selected moir\'e-stacking configurations, and is independent of any additional vertical gating of the heterostructure. On the other hand, gating can broadly tune the gap at the principal Dirac point, and may thereby strongly compress the first moir\'e mini-band in width against the moir\'e-induced gap at the secondary Dirac point. We reveal that in hBN-encapsulated bilayer graphene this novel mechanism can lead to isolated bands flatter than 10~meV under moderate gating, hence presenting a convenient pathway towards electronically-controlled strongly-correlated states on demand.