Localization of Dirac-like excitations in graphene in the presence of smooth inhomogeneous magnetic fields

TL;DR

This paper investigates how inhomogeneous magnetic fields influence Dirac excitations in graphene, providing exact solutions for certain field profiles and analyzing localized zero-energy states.

Contribution

It introduces new magnetic field profiles affecting graphene excitations and relates their spectra to known quantum systems, offering exact and approximate solutions.

Findings

Exact spectrum for inverse-radial magnetic field in graphene.

Localized zero-energy states in a realistic fading magnetic field.

Spectral relations to two-dimensional hydrogen and Coulomb problems.

Abstract

The present article discusses magnetic confinement of the Dirac excitations in graphene in presence of inhomogeneous magnetic fields. In the first case a magnetic field directed along the z axis whose magnitude is proportional to $1/r$ is chosen. In the next case we choose a more realistic magnetic field which does not blow up at the origin and gradually fades away from the origin. The magnetic fields chosen do not have any finite/infinite discontinuity for finite values of the radial coordinate. The novelty of the two magnetic fields is related to the equations which are used to find the excited spectra of the excitations. It turns out that the bound state solutions of the two-dimensional hydrogen atom problem are related to the spectra of graphene excitations in presence of the $1/r$ (inverse-radial) magnetic field. For the other magnetic field profile one can use the knowledge of the bound state spectrum of a two-dimensional cut-off Coulomb potential to dictate the excitation spectra of the states of graphene. The spectrum of the graphene excitations in presence of the inverse-radial magnetic field can be exactly solved while the other case cannot be. In the later case we give the localized solutions of the zero-energy states in graphene.