Boron Nitride and Graphene Heterostructures Modeled by Quantum Graphs

We use the theory of periodic quantum graphs to model stackings of hexagonal materials with two and three layers, such as hexagonal boron nitride (hBN) and graphene, with pairs of parameters \(\delta _N\) and \(\delta _B\) associated with the types of distinct atoms located at their vertices. We analyze equal bilayers in AA and \(AA'\) stackings, as well as heterostructures and “sandwiches” of graphene between layers of hBN. In each of these configurations, we use the Schrödinger operator on the edges, with their respective boundary conditions, and introduce a weak interaction parameter \(t_0\) between the connections of different layers. We analytically study the dispersion relations obtained in these models with respect to the existence of conic and parabolic touches and confirm, in a rigorous way, known results in the physics literature: hBN bilayers do not have Dirac cones, but in the AA stacking, we identify parabolic touches. In mixed bilayers, the inclusion of an hBN layer over graphene can induce a gap, the amplitude of which depends on \(t_0\) and \(\delta _N\). In the “sandwiches,” we find that graphene between two layers of hBN reduces the width of the spectral gap, while in the graphene-hBN-graphene configuration, some of the graphene cone persists, but gaps open in other Dirac cones.