We analyze the spectrum of a neutral donor located inside or outside of a finite-barrier toroidal-shaped nanoring whose radius is much larger than the height. We derive a one-dimensional wave equation which describes the low-lying donor levels corresponding to the slow electron motion along the ring, by using the adiabatic approximation. Numerical solution of this equation has been obtained by using the trigonometric sweep method. The dependence of the energy spectrum on the donor position, radius and height ring has been studied.

The ground state energies of off-axis negatively charged donors in axially symmetrical quantum dots, with different shapes but in all cases with a small height-to-base radius aspect ratio are calculated in adiabatic approximation by using the Hylleraas-type trial function. The dependencies of the neutral and negative donor binding energies and their ratios on the base radius in the pyramid, lens and disk are calculated and compared with previously obtained results for the spherical quantum dot. We also present the contour plots of the binding energies of the neutral and negative donors with different positions along a vertical cross section in the middle of the quantum dots.

A problem of two electrons spatially separated in vertically coupled one-dimensional rings is solved exactly by using the numerical trigonometric sweep method. The change of the level-ordering and the crossover of the curves of the energy levels as a functions of the rings radii, the separation between rings and the magnetic field, applied along the axis, are found and discussed. As the distance between rings tends to zero our results are in an excellent agreement with those obtained previously for the single two-electron one-dimensional ring.