Electronic eigenstates in quantum rings: Asymptotics and numerics

The electronic eigenstates of quantum rings of constant width and arbitrary shapes are studied in the two-dimensional infinite hard-wall potential approximation. An asymptotic method is developed to evaluate the eigenenergies and eigenfunctions under the assumption that the ratio of ring half-width to a typical radius of curvature is small, and this provides a significant improvement over a more conventional zero-curvature approximation. A direct numerical scheme based on spectral methods is also developed and this is free of the small curvature limitation. To illustrate the versatility and accuracy of our general formulas we also treat a specific illustrative case, a pseudoelliptic annulus, and the two methods are compared. The asymptotic model is demonstrated to be very accurate while being orders of magnitude faster than the direct numerics. The effects of varying ring curvature on spectral and transport properties of quantum rings are studied. In particular, the existence and structure of eigenstates localized at the regions of maximal curvature is investigated.