Magnetic-field dependence of low-lying spectra in bilayer graphene-based magnetic dots and rings

The low-lying energy spectra of bilayer graphene in a perpendicular magnetic field B ( r ) z ^ were obtained by numerical diagonalization of the Hamiltonian. We assumed that B(r) takes on the shape of a circular dot or a ring. Under such a nonuniform field, the lowest-energy Landau levels, with N − = 0 , 1 , remain invariant with a zero eigenvalue. For other Landau levels, complicated level-splitting and level-crossings take place when the effective radius of the dot or ring increases.