Magnetic moment of a two‐dimensional degenerate electron gas in mesoscopic samples

The orbital magnetism of two‐dimensional electrons in mesoscopic samples is studied in models where the interaction between electrons is neglected. Various geometries are considered as there are disc, plaquette, bracelet with hard wall confinement and also a confinement with a parabolic potential. We calculate the average magnetic moment which means an average with respect to size fluctuations and de Haas‐van Alphen oscillations which arise in the case of a sharp Fermi cutoff. We see three distinct ranges in the magnetic field: (i) small field region where perturbation theory applies; (ii) moderate fields where edge currents play a prominent role; and (iii) the high field range with a Landau type susceptibility. In a quasiclassical picture, the electronic orbits are not qualitatively changed by a magnetic field in (i); skipping orbits are important in (ii); and in (iii), the cyclotron radius is smaller than the sample size. As a rule, we find an enhancement of the magnetic response which increases with k F L, that is, with sample size divided by the Fermi wave length. Also, we have found out that the quasiclassical approximation fails in the calculation of the magnetic properties; on the other hand, we have seen no essential differences between the canonical ensemble (fixed particle number) and the grand canonical ensemble (chemical potential given). In the case of plaquettes, in particular for samples in the form of squares, we have found agreement with experimental results by Lévy, Reich, Pfeiffer and West.