One-electron quantum ring of non-uniform thickness in magnetic field

We consider a model of a quantum ring in the form of a thin layer, whose thickness increases linearly between inner and outer radii. We show that in the structural adiabatic limit, when the quantum ring thickness is much smaller than its lateral dimension, the wave equation for the electron confined in such structure can be completely separated. We use analytical solutions found for this model as the base functions for analyzing the effect of the structural non-homogeneity on the electronic spectrum and the Aharonov–Bohm oscillations of the energy levels, in the framework of the exact diagonalization method we found that the pattern of the electronʹs possible pathways in its displacements generated by the external magnetic field, forms a quasi-one-dimensional region along a guideline marked by a set of highest points of the crater. Therefore, the Aharonov–Bohm oscillations of the energy levels in a crater-shaped quantum dot without non-uniformities are similar to those in 1D quantum ring independently on the crater width. We show that a slight non-uniformity produced by a single valley and single mountain supresses the oscillations of several lower levels due to the localization of the corresponding rotational states close to the mountain. Nevertheless, when the non-uniformity becomes substantial due to the presence of multiple valleys and mountains, the rotational electron motion and the Aharonov–Bohm oscillations generated by the external magnetic field are restored, owing to the electron tunneling through mountains. We consider that our model of crater-shaped structure would be applicable in the analysis of a variety of more complicated problems related to systems of few carriers confined in nanostructures with ring-like geometry, as a starting point in the framework of the diagonalization method.