Quantum Hall Effect and Different Zero-Energy Modes of Graphene

The effect of an inhomogeneous magnetic field, which varies inversely with distance on the ground state energy level of graphene, is studied. We analytically show that graphene under the influence of a magnetic field arising from a straight long current-carrying wire (proportional to the magnetic field from carbon nanotubes and nanowires) exhibits zero-energy solutions and find that contrary to the case of a uniform magnetic field for which the zero-energy modes show the localization of electrons entirely on just one sublattice corresponding to a single valley Hamiltonian, zero-energy solutions in this case reveal that the probabilities for the electrons to be on both sublattices, say A and B, are the same.