Magnetic-field and spin-orbit interaction in restricted geometries: Solvable models

Properties of a two-dimensional degenerate gas of noninteracting electrons, subject to a perpendicular magnetic field, are investigated by studying two solvable models. These models amount to restricted geometries in the plane. The first reveals three types of oscillations (as a function of the applied field) of thermodynamic quantities: de Haasvan Alphen (dHvA) oscillations, Aharonov-Bohm (AB) oscillations, and oscillations related to correlations among energy levels associated with different Landau levels. In the second model the latter correlations are absent, and the third kind of oscillations is manifested as nonperiodic fluctuations of the thermodynamic quantities at hand. Accounting for spin-orbit (s.o.) interaction, it is found that the dHvA oscillations split into a sum of spin-up and spin-down branches. The expressions for the AB oscillations acquire universal s.o. reduction factors. Introducing a modification of the second model (which is treated using a WKB approach), an enhanced s.o. coupling, and consequently a larger s.o. effect, is obtained. Both the phenomena of quantum Hall effect and longitudinal conductance quantization are demonstrated in the two models, with AB oscillations superimposed on the conductance plateaus.