Chaotic transport and fractal spectra in lateral superlattices

Abstract

We show that chaos and nonlinear resonances are clearly reflected in the magnetotransport of lateral surface superlattices and thereby explain a series of magnetoresistance peaks observed in antidot arrays on semiconductor heterojunctions. For small magnetic fields we find the counterintuitive result that electrons move in the opposite direction to the free-electron E*B drift when subject to a two-dimensional periodic potential. We show that this phenomenon arises from chaotic channelling trajectories, and by a subtle mechanism leads to a negative value of the Hall resistivity for small magnetic fields. For a quantum mechanical description of Bloch electrons in magnetic fields Harper's equation has been studied extensively; this is integrable in the classical limit and thus fails for lateral surface superlattices where chaotic trajectories prevail near the classical limit. We therefore derive a new model, which is exact under the most general conditions, study the influence of classical chaos on the fractal spectrum known as Hofstadter's butterfly, and make predictions on its observability in lateral surface superlattices.