Isolated and hybrid bilayer graphene rings

Using the continuum model, we investigate the electronic properties of two types of bilayer graphene ring (BLGR) geometries: (i) an isolated BLGR and (ii) a monolayer graphene (MLG) ring put on top of an infinite graphene sheet. Solving the Dirac-Weyl equation in the presence of a perpendicular magnetic field and applying infinite-mass boundary condition at the ring boundaries, we obtain analytical results for the energy levels and corresponding wave spinors for both structures. Our results show that the presence of interface boundary in a hybrid BLGR modifies drastically the energy levels as compared to that of an isolated BLGR. The intervalley symmetry $ E^K_e (m) = -E^{K'}_h(m) $ between the electron (e) and hole (h) states ($ m $ being the angular momentum quantum number) is found for the energy spectrum of an isolated BLGR, while it is broken for the hybrid BLGR. Our analytical results agree with those obtained within the tight-binding model (TBM). Sizeable and magnetically tunable band gap, in contrast to that in the zero-width BLGR model, is predicted for both ring structures, which experimentally can be measured in, e.g., magnetotransport measurements.