Quantum Hall effect and the different zero energy modes of graphene

TL;DR

This paper analytically investigates the zero energy modes of graphene under an inhomogeneous magnetic field from a long current-carrying wire, revealing equal sublattice probabilities unlike the uniform field case.

Contribution

It introduces an analytical study of zero energy modes in graphene subjected to a magnetic field from a long wire, highlighting differences from uniform magnetic field effects.

Findings

Zero energy solutions exist under the inhomogeneous magnetic field.

Electrons have equal probability on both sublattices A and B.

Contrasts with zero energy modes in uniform magnetic fields.

Abstract

The effect of an inhomogeneous magnetic field which varies inversely as distance on the ground state energy level of graphene is studied. In this work, 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. We 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 single valley Hamiltonian, zero energy solutions in this case reveal that the probability for the electrons to be on the both sublattices, say A and B, are the same.