Electronic energy levels of nanorings with impurities and Aharonov–Bohm effects

By modeling impurities along a nanoring as general potential forms the Schrödinger equation for ballistic electrons is shown to separate in cylindrical coordinates. We find an analytical eigenvalue equation for N delta-function-barrier impurities in the presence of magnetic flux. Previous calculations of the electronic states of a one-dimensional (1D) and two-dimensional (2D) nanoring for only one or two impurities modeled by equal square barriers is explicitly extended to three and four different or equal impurities modeled as delta-barrier, square-barrier, or delta-well potential forms. This is shown to be generalizable to any number N. Effects on the energy spectra due to magnetic flux and different kinds and numbers of impurities are compared in 1D and 2D nanorings.