Influence of geometry on persistent currents in quantum dots and rings

In a quantum dot the magnetic moment, proportional to the persistent currents, is given by the derivative with respect to the magnetic field of the energy of the system, which for noninteracting electrons is just the sum of the lowest filled one-particle levels. These quantities are calculated for round and square dots, with and without interior holes, using hard wall boundary conditions. The relation is traced between the old results of Landau for a large system and what may be expected in current experiments on small quantum dots of various sizes and shapes. For the square ring geometry at low and moderate magnetic fields we note and discuss the appearance of gaps separating braids of groups of four levels.