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Using a method of integration over commuting and anticommuting variables, we study the motion of noninteracting electrons in a system with short-range disorder and in a strong magnetic field. An explicit calculation shows that the static conductivities cr, and o.~v anish if the Fermi energy is situated in the lower tail of the lowest Landau level. At the same time, the Hall resistivity p~r emains finite, although the longitudinal resistivity diverges. These results are valid both for two-dimensional (2D) and for 3D systems. We compare our theoretical findings with experiments on magnetic-field-induced metal-insulator transitions in 3D systems. Furthermore, the result for the Hali conductivity of 2D systems is generalized to higher localization regions in order to study the deformation of the Hall plateaus in microwave experiments on GaAs-Al Gal "As heterostructures.

We comment on recently observed oscillations of the Hall resistivity around the classical value with particular emphasis on the coexistence of localized and extended states in the case of a spinsplit lowest Landau level. We prove the existence of localization at the bottom of the 0t level, thus explaining the observed plateaulike feature of the Hall resistivity.Although the Shubnikov-de Haas (SdH) oscillations of the magnetoresistance are by now well understood, no convincing theoretical explanation for the oscillatory deviations of the Hall resistivity from the classical behavior has been presented so far. In recent experiments' the Hall resistivity p"~w as investigated in narrow-gap, bulk semiconductors such as InAs, InSb, and Hg~-"Cd,Te with particular emphasis on the magnetic-field region where the last SdH minimum of the longitudinal resistivity p" is observed. The interesting feature was a distinct decrease of the slope of the p"~v ersus 8 graph in the same region. Murzin' was the first to suggest a relation to the quantum Hall eA'ect (QHE) in two-dimensional (2D) systems and conjectured that electron-electron interaction might account for an additional phasea Hao insulatorin three dimensions. Mani tried to reproduce the essential experimental features in a simple model by assuming the existence of localized states at the bottom of each 3D Landau level (LL). His model suggested real plateaus at T=O coinciding with the SdH minima. In Refs. 1 and 2, however, only one plateaulike structure was observed together with the last SdH minimum. From the theoretical point of view three questions arise. (i) Does p, , exhibit plateaus at T=O or is only the slope decreasing? (ii) Can the existence of localized states be proven? How can the problem of coexistence of localized and extended stateswhich was put forward in Ref. 3be solved in the presence of disorder?(iii) Does the magnetic-field range in which the last SdH minimum occurs play a distinguished role concerning the plateaulike feature of p", , ? Finally, we have also to comment on the observation of a Hall resistivity being smaller than the classical value for magnetic fields below the last SdH oscillation as well as before approaching the freezing-out Mott-insulator (MI) transition at high fields.In the following we want to illustrate that the decreasing slope of p, -,, can be understood within a model of 3D disordered systems studied previously, provided spinsplitting is taken into account. Let us first recall that the g factor of InSb decreases from about 50 to 35 for a doping concentration varying from n =10' to 10' cm (cf. , e.g. , Ref. 5). With the eA'ective mass m=0. 014m, this yields a spin splitting of one third of the LL separation so that at least for the lowest Landau level (LLL) the spin splitting is well resolved. Consequently, in the vicinity of the last observed SdH minimum the Fermi energy crosses the bottom of' the 0) level. Since in the absence of magnetic impurities the one-particle states of the lowest spin-up and spin-down LL cannot mix, all...

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