Magnetic quantum dots and rings in two dimensions

TL;DR

This paper explores how inhomogeneous magnetic fields in two-dimensional electron systems can create bound states in magnetic quantum dots, revealing new solvable models and mechanisms for localization and spin separation.

Contribution

It introduces new quasi-exactly solvable models for magnetic quantum dots with inhomogeneous fields and demonstrates how magnetic field modulation can control electron localization and spin states.

Findings

Bound states exist in slowly-decaying magnetic fields.

New solvable models involve confluent and biconfluent Heun polynomials.

Magnetic field modulation can induce localization-delocalization transitions.

Abstract

We consider the motion of electrons confined to a two dimensional plane with an externally applied perpendicular inhomogeneous magnetic field, both with and without a Coulomb potential. We find that as long as the magnetic field is slowly-decaying, bound states in magnetic quantum dots are indeed possible. Several example cases of such magnetic quantum dots are considered in which one can find the eigenvalues and eigenfunctions in closed form, including two hitherto unknown quasi-exactly solvable models treated with confluent and biconfluent Heun polynomials. It is shown how a modulation of the strength of the magnetic field can exclude magnetic vortex-like states, rotating with a certain angular momenta and possessing a definite spin orientation, from forming. This indicates one may induce localization-delocalization transitions and suggests a mechanism for spin-separation.