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 $R_{ xx} \propto B \frac{ dR_{ xy}}{ dB}$ where $R_{ xy}$ is the Hall resistance, and $B$ is the magnetic field.
Categories: cond-mat