Within a minimal model, we present analytical expressions for the eigenstates and
eigenvalues of carriers confined in quantum rings in monolayer and bilayer graphene. The
calculations were performed in the context of the continuum model by solving the Dirac
equation for a zero width ring geometry, ie, by freezing out the carrier radial motion. We
include the effect of an external magnetic field and show the appearance of Aharonov-Bohm
oscillations and of a nonzero gap in the spectrum. Our minimal model gives insight on the …

The freezing of the radial motion was included in the calculations by means of the
substitution→ 0 in the Hamiltonians 1 and 16 for single layer and bilayer graphene,
respectively, which resulted in the non-Hermitian Hamiltonians 4 and 18. Solving these non-
Hermitian Hamiltonians resulted in unphysical non-real values for the energy levels around
E= 0. Similar erroneous substitutions were made in eg Ref. 1. It was shown in Ref. 2 that the
radial part of the angular momentum must be taken zero p=− i+ 1