  The recently observed unconventional ferroelectricity in AB bilayer graphene
sandwiched by hexagonal Boron Nitride (hBN) presents a new platform to
manipulate correlated phases in multilayered van der Waals heterostructures. We
present a low-energy continuum model for AB bilayer graphene encapsulated by
the top and bottom layers of either hBN or graphene, with two independent twist
angles. For the graphene/hBN heterostructures, we show that twist angle
asymmetry leads to a layer polarization of the valence and conduction bands. We
also show that an out-of-plane displacement field not only tunes the layer
polarization but also flattens the low-energy bands. We extend the model to
show that the electronic structures of quadrilayer graphene heterostructure
consisting of AB bilayer graphene encapsulated by the top and bottom graphene
layers can similarly be tuned by an external electric field.
