Transport in systems with chaotic dynamics: Lateral superlattices

Abstract

The theoretical investigation of electrons subject to both a periodic electrostatic potential and a magnetic field has a long tradition [1]. On the one hand, this situation can be realized in each crystalline solid. On the other hand, it constitutes a highly nontrivial system due to breaking the rotational symmetry of the magnetic field by the (discrete) translational invariance of the periodic potential. A simple version of the problem is obtained in the tight-binding approximation for the electrons or in perturbation theory for very weak periodic potential. In that case Schrudinger’s equation reduces to Harper’s equation [2], the solution of which has been visualized in the form of the well-known Hofstadter butterfly [3]. The spectrum then critically depends on the number of magnetic flux quanta threading one unit cell of the lattice and becomes a fractal object, a Cantor set, if this number is an irrational one [4]. Experimental studies of such systems promise to provide a confirmation of the theoretical predictions and perhaps new interesting insights into the consequences of the spectral properties on their physical behaviour. However, they turned out to be impossible with conventional crystals because the necessary magnetic fields of several 1000 T for the observation of the peculiarities of the spectrum are still out of the reach of present-day technology.