Energy spectrum of on-axis negatively charged donor in toroidal-shaped ring

Low-lying states of an on-axis negatively charged donor in a finite barrier height toroidal-shaped quantum ring with cross-section of small diameter are analyzed. Following the well-known adiabatic procedure, we have deduced a one-dimensional wave equation with periodic conditions, which describe all low-lying levels related to the two-electrons rotation around the axis under attraction to the donor and repulsion between them given by effective potentials that depend on the ring radius, thickness and distance of the donor from the symmetry plane.