Explanation for the Resistivity Law in Quantum Hall System

Abstract

We consider a 2D electron system in a strong magnetic field, where the local Hall resistivity ${\rho }_{xy}(\vec{r})$ is a function of position and ${\rho }_{xx}(\vec{r})$ is small compared to ${\rho }_{xy}$. Particularly if the correlations fall off slowly with distance, or if fluctuations exist on several length scales, one finds that the macroscopic longitudinal resistivity $R_{xx}$ is only weakly dependent on ${\rho }_{xx}$ and is approximately proportional to the magnitude of fluctuations in ${\rho }_{xy}$. This may provide an explanation of the empirical law $R_{xx}\propto B\frac{d{R}_{xy}}{dB}$ where $R_{xy}$ is the Hall resistance, and $B$ is the magnetic field.

11 pages (REVTeX 3.0). Revised Version. Complete postscript file for this paper is available on the World Wide Web at http://cmtw.harvard.edu/~simon/ ; Preprint number HU-CMT-94S03

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