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

Abstract

Abstract. The orbital magnetism of two-dimensional electrons in mesoseopic samples is studied in models where the interaction between electrons is neglected. Various geometries are considered as there are disc, plaquettc, 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 ~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 LCvy, Reich, Pfeiffer and West.

Comment on “Weak‐field orbital magnetism of mesoscopic samples”

“…Averaging over m eliminates the oscillatory part and the Landau result c 9 c L is obtained Ð in severe contradiction to the result c jc L j k F b of [1]. Similarly, the result of [1] for a disc is also in disagreement with the results derived in [3,7]. This list of disagreement can be continued.…”

“…(4.18), as well as from the opening statement in Sec. 5 of [1], the reader can infer that the result holds in the limit a 3 I. More recently [9], Ovchinnikov insisted, however, that the result is valid only for finite a, albeit much larger than b.…”

“…[6]), and of the linear response theory in general. Therefore, contrary to what is stated in [1], it is quite straightforward to obtain the Landau susceptibility by perturbation theory: one just has to work at finite temperatures, k B T ) G. This was explained e.g. in [3] where the condition for the applicability of linear response theory was discussed in detail.…”

“…1. In a recent publication in this journal Ovchinnikov, Lehle and Schmid [1] studied the orbital magnetisation in 2d mesoscopic samples of different geometries. Some of their results are in disagreement with those obtained previously by the present authors [2,3].…”

“…We would like to point out that the difference between ``large aº and ``infinite aº can be of importance only at extremely small temperatures such that k B T `" hv F aa G a . Presumably, the authors of [1] have in mind temperatures larger than G a ak B (otherwise c would exhibit a lot of ``fine structureº on an energy scale of order G a ). Thus, one would expect some connection between the results of Ovchinnikov et al and those for an infinite strip.…”