Experimental studies of low-field Landau quantization in two-dimensional GaAs electron systems

We studied effects of low-field Landau quantization in two-dimensional systems by applying a magnetic field $B$ perpendicular to GaAs electron systems. With increasing $B$, Shubnikov-de Haas (SdH) oscillations appear in the longitudinal resistivity \textit{$\rho $}$_{xx}$ before the appearance of the quantum Hall effect (QHE). Universal properties based on the modular symmetry become invalid at low $B$, where we should consider SdH theory, low-field localization, and quantum diffusion model. The crossover from SdH oscillations to the QHE is studied by sweeping $B$ and changing the temperature $T$. By investigating the peak values of the longitudinal resistivity $\rho _{xx}^{pk}$, it is shown that we shall consider the refinement to the theory for the low-field Landau quantization.