Quantum diffusion, fractal spectra, and chaos in semiconductor microstructures

Abstract

Abstract We review how unbounded quantum mechanical diffusion is related to multifractal properties of the spectrum, its level statistics, and to the algebraic decay of correlations. This new field could be called “quantum chaology” of fractal spectra and should be contrasted with the dynamical localization of the kicked rotator and other previously studied quantum systems. These fascinating properties are found in systems described by a quasiperiodic Schrodinger equation, e.g. the Fibonacci chain for quasicrystals and the Harper model, a single-band description of Bloch electrons in magnetic fields. The semiclassical limit of Bloch electrons in magnetic fields is realized in recent experiments on lateral surface superlattices. There we show that classical chaos and nonlinear resonances are clearly reflected in the magnetotransport and thereby explain a series of magnetoresistance peaks observed in antidot arrays on semiconductor heterojunctions. We also find the counterintuitive result that electrons move in opposite direction to the free electron E × B - drift when subject to a two-dimensional periodic potential. This phenomenon arises from chaotic channeling trajectories and by a subtle mechanism leads to a negative value of the Hall resistivity for small magnetic fields.