SUSY design of smooth quantum rings in graphene

TL;DR

This work addresses electrostatic confinement of Dirac fermions in graphene by designing radial potentials that support zero-energy states. It introduces a modified supersymmetric (Darboux) transformation with an asymmetric intertwining relation to generate time-reversal invariant radial deformations of a base potential (such as Coulomb) for which the zero-energy solutions are known analytically. Each concentric ring traps a pair of zero-energy bound states with opposite angular momentum, accompanied by circular probability currents and valley degeneracy inherited from time-reversal symmetry. The resulting ring-decorated Coulomb potentials offer an analytically controllable platform to engineer graphene quantum rings and study phenomena such as atomic collapse, current patterns, and gate-defined ring confinement.

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.

Figures (8)

• Figure 1: The plots of the radial functions and the (non-normalized) probability densities of the two collapse states when $\beta=\frac{5}{2}$ respect to $r/r_0$ are presented from left to write. The constant $r_0$ acts as a natural scale, and as we approach its value to zero, oscillations in the radial functions begin to accumulate near the force center. This behavior is typical of states in the critical regime in general. The probability density decreases slowly as the distance increases, but diverges like $1/r$ when approaching the origin. When considering the regularized problem, one realizes that the particle is most probable to be found at $r=0$.
• Figure 2: The comparison between the original and deformed potentials as a function of $r/r_c$.
• Figure 3: Plots of the normalized probability density (left) and the current vector density times $r_c^2$ (right). The parameter values are $\beta=5/2$ and $m=6$. In this case, the numerical integration determines the normalization constant vale to be $N^2\approx 0.09$.
• Figure 4: Plots for the (not normalized) probability densities of the collapsing states in the case $\beta=5/2$ and $m=6$. For simplicity we take $r_c=1$, while $r_0$ has the same role as in figure \ref{['colapse']}. They corresponds to real combination of the mapping of the collapsing states for the original model.
• Figure 5: Different shapes of the potential $V_{l,m}$ depending on the values of $B$ for $\beta=5/2$, $m=6$ and $l=8$. Left panel: $B=1/2$ (top) and $B=400$ (bottom). Right panel: $B=-1/2$ (top) and $B=-400$ (bottom).