Fundamental relation between longitudinal and transverse conductivities in the quantumHall system

Abstract

We investigate the relation between the diagonal (σxx) andoff-diagonal (σxy) components of the conductivity tensor in the quantum Hall system. We calculate theconductivity components for a short-range impurity potential using the linear responsetheory, employing an approximation that simply replaces the self-energy by a constantvalue with τ the scattering time. The approximation is equivalent to assuming that the broadening of aLandau level due to disorder is represented by a Lorentzian with the width . Analytic formulae are obtained for bothσxx andσxy withinthe framework of this simple approximation at low temperatures. By examining the leading terms inσxx andσxy, we find a proportionalrelation between dσxy/dB and Bσxx2. The relation, after slight modification to account for the long-range nature of the impuritypotential, is shown to be in quantitative agreement with experimental results obtained inthe GaAs/AlGaAs two-dimensional electron system at the low magnetic field regime wherespin splitting is negligibly small.