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

TL;DR

This paper investigates how a weak magnetic field influences the chaotic behavior and localization of two-dimensional electrons in a random potential, revealing a suppression of chaos and a sharp decrease in localization length beyond a critical field.

Contribution

It provides a detailed analysis of the transition from classical to quantum behavior in 2D electron systems under weak magnetic fields, including the suppression of chaos and localization length behavior.

Findings

Diffusion coefficient follows Drude-Lorentz formula at low B

Chaotic motion is suppressed at B > B_c

Localization length decreases sharply beyond B_c

Abstract

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 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 $B$ in the minima of $\sigma_{xx}$. At $B < B_c$ it is exponentially large but decreases with increasing $B$. At $B > B_c$, the localization length drops precipitously, and ceases to be exponentially large at a field $B_\ast$, which is only slightly above $B_c$. Implications for the crossover from the Shubnikov-de Haas oscillations to the quantum Hall effect are discussed.