  The electric-field effect on the electronic and magnetic properties of
triangular and hexagonal graphene quantum rings with zigzag edge termination is
investigated by means of the single-band tight-binding Hamiltonian and the
mean-field Hubbard model. It is shown how the electron and spin states in the
nanoring structures can be manipulated by applying an electric field. We find
different spin-depolarization behaviors with variation of electric field
strength due to the dependence of spin densities on the shapes and edges of
this kind of nanorings. In the case of triangular quantum rings, the
magnetization on the inner and outer edges can be selectively tuned and the
spin states depolarize gradually as the field strength is increased, while in
the case of hexagonal nanorings, the transverse electric field reduces the
magnetic moments on both inner and outer edges symmetrically and rapidly.
