Effect of symmetry on the electronic properties of arbitrarily shaped quantum rings in a magnetic field

The energy states and persistent current oscillations considered here are formulated in a one-dimensional model for a quantum ring with a nearly circular arbitrary contour. Using conformal mapping, the curvature of the ring is introduced into the Schrodinger equation via the Lamé coefficient of the conformal mapping. Asymptotic methods were employed in order to derive analytical solutions. Energy levels and current oscillations were studied for three cases of the possible non-circular QR symmetries: a QR with two axes of symmetry (ellipse), a QR with one axis of symmetry and also on QRs with no axes of symmetry, i.e. asymmetric shapes. We obtain explicit expressions for the periodic energy and the energy gaps opened at the half-integer and integer values of the flux. The spectrum behavior is found to be dependent on the order of symmetry. In particular, small asymmetries of the QR cause some perceptible chaos in the width of the gaps.