One-Parameter Scaling of the Localization Length in High Magnetic Fields

Abstract

A disordered two-dimensional system in the presence of a strong perpendicular magnetic field is of considerable theoretical interest. Most theoretical tools fail in dealing with this situation. The only quantity accessible to exact calculations is the density of states which consists of a series of disorder broadened Landau bands [1]. For other physical quantities analytical treatments have to rely on resummations of diverging series [2] or lead to models that seem to be too complicated to solve [3]. Only in the limit of a long-range correlated random potential are semi-classical arguments applicable and analytical results can be obtained [4, 5, 6]. In such a situation numerical calculations might help to understand the properties of the system under consideration. Besides the density of states the most easily calculated quantity is the localization length ζ. If ζ is finite then the DC conductivity σ _ii (= 0) vanishes and the system is an insulator. Furthermore, the localization length acts as a scaling parameter that governs the critical behaviour of physical quantities near the center of the Landau bands [7]. This behaviour can be obtained by investigating the exponential decay length in a finite system as a function of the size of the system. It can further be argued that at finite temperatures this system size dependence is replaced by a dependence on a temperature dependent phase coherence length. As a consequence the critical behaviour of the localization length shows up in the temperature dependence of physical quantities, like the conductivity, when the Fermi energy lies near the center of a Landau band.