QUANTUM DIFFUSION, FRACTAL SPECTRA, AND CHAOS IN SEMICONDUCTOR MICROSTRUCTURES

We review how unbounded quantum mechanical diffusion is related to mul tifractal properties of the spectrum, its level statistics, and to the algebraic decay of correlations. This new field could be called ''qua ntum chaology'' of fractal spectra and should be contrasted with the d ynamical localization of the kicked rotator and other previously studi ed 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 descriptio n of Bloch electrons in magnetic fields. The semiclassical Limit of Bl och 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 antid ot arrays on semiconductor heterojunctions. We also find the counterin tuitive result that electrons move in opposite direction to the free e lectron E x B-drift when subject to a two-dimensional periodic potenti al. 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.