SUSY design of smooth quantum rings in graphene

TL;DR

This paper introduces a supersymmetric design method for creating zero-energy graphene quantum rings with electrostatic confinement, enabling analytical solutions and novel current flow properties.

Contribution

It develops a modified supersymmetric transformation technique to design zero-energy states in graphene with concentric electrostatic rings, including analytical solutions and current flow analysis.

Findings

Analytical zero-energy bound states in graphene rings.

Circular probability currents flow in opposite directions for different angular momenta.

Constructed a Coulomb potential with zero energy collapse and bound states.

Abstract

We develop a suitable technique to design zero-energy graphene models with radial electrostatic potentials capable of achieving electrostatic confinement. Using the Gaussian law for electrostatics, we derive the charge density associated with these potentials that correspond to concentric electrostatic rings. The technique is based on a modified supersymmetric transformation that allows to design time-reversal invariant interaction terms and to find the corresponding zero-energy bound states in analytical form. Consequently, solutions with the same probability density but different angular momentum are characterized by circular probability currents flowing in opposite directions. The energies of the systems defined in two Dirac valleys (one-valley) have a fourfold (twofold) degeneracy. As an example of the technique, we construct a ring-decorated Coulomb potential that exhibits zero energy collapse and bound states together.