Localization Investigations in the Regime of the Quantum Hall Effect

Abstract

Already in the original work of K. v. Klitzing [1] the occurence of the plateaus in the Halleffect has been attributed to the existence of localized and extended states in the Landau levels [2], but the electronic properties of a two-dimensional electron gas (2DEG) in high magnetic fields has yet to be fully understood. Especially, it was not possible to describe transport through localized and extended states for realistic samples under realistic conditions. This lack of a full understanding originates partly from the fact that the theoretical description of localization in high magnetic fields is inherently very diffcult. A localization length ξ has been introduced as a measure of the spatial extent (or correlation) of the electronic wavefunction. Some numerical results for the variation of the localization length within one Landau-level were published already many years ago [3]. Also the assumption that the localization length diverges with a power law in approaching a critical energy E c is quite old (see e.g. [4]): $$ \xi (E)\alpha |E -{E_c}{|^{ -v}}$$ (1) Here v is the critical exponent of the localization length. Nevertheless, the determination of this critical exponent and the calculation of transport properties have been of great interest for many theory groups even during the last years [5]–[14]. On the experimental side the results of different groups are not very consistent [15]–[21], which might be due to the fact that the disorder leading to the localization of electronic states can be of very different, usually unknow origin in different samples.