Suppression of chaotic dynamics and localization of two-dimensional electrons by a weak magnetic field

We study a two-dimensional motion of a charged particle in a weak random potential and a perpendicular magnetic field. The correlation length of the potential is assumed to be much larger than the de Broglie wavelength. Under such conditions, the motion on not too large length scales is described by classical equations of motion. We show that the phase-space averaged diffusion coefficient is given by the Drude-Lorentz formula only at magnetic fields $B$ smaller than certain value $B_c.$ At larger fields, the chaotic motion is suppressed and the diffusion coefficient becomes exponentially small. In addition, we calculate the quantum-mechanical localization length as a function of $B$ at the minima of ${\sigma }_{\mathrm{xx}}.$ At $B<\mathrm{}{B}_c$ it is exponentially large but decreases with increasing $B$. At $B>\mathrm{}{B}_c,$ this decrease becomes very rapid and the localization length ceases to be exponentially large at a field $B_{*},$ which is only slightly larger than $B_c.$ Implications for the crossover from the Shubnikov–de Haas oscillations to the quantum Hall effect are discussed.