Three-electron spectra and generalized Hund's rule in nanorings

Energy spectra and the spin order in a three-electron nanoring with an impurity are studied. It is found that the structure of energy levels is determined by the size and shape of the nanoring, and that the doublet ground state can be modified into the quadruplet one by changing the size and shape of the ring or adding an impurity. For a fixed ratio of width to radius, i.e., a fixed ring shape, the critical width W-c of the spin transition of three electrons is obtained. The size and shape effect on the spin transition and the occupation of single electron states are clearly shown, and they lead to the generalized Hund's rule, i.e., the W-c rule that can be useful in understanding the occupation and the Fermi liquid-Wigner crystal crossover in many-electron nanostructures.