Complex quantum dynamics in the integer quantum Hall regime.

Abstract

The scattering matrix for potential scattering of planar electrons in a perpendicular magnetic field is required to understand normal and Hall conductivity off of the quantum Hall plateaus, at nonzero temperature and frequency, and fluctuations and nonlinear transport. Little, however, was previously known about it. We have developed a method to calculate amplitudes and phases for impurity scattering in the quantum Hall regime, by classically evolving appropriate Wigner distribution functions and projecting them onto quantum states. To check the reliability of this approximate calculation, we have also developed a numerical method that solves exactly the quantum scattering problem for an infinitely long strip lattice. We consider mainly nonadiabatic scattering, in which the potential variation is rapid or moderate. Quantum scattering is shown to exhibit sharp resonances, even from repulsive potentials. The results of the two approaches are compared and the relation between quantum and complex classical scattering is made apparent. The existence of classical periodic orbits is demonstrated for general potentials, and the relation between classical periodic orbits and quantum resonances and phase shifts is analyzed. The possibility that breakdown in the quantum Hall effect is due to inter-Landau-level tunneling caused by rapidly varying impurity potentials is investigated. The prediction for the critical Hall field improves on previous explanations by about two orders of magnitude and is in reasonable agreement with experimental results.