Non-Universal Behavior of Finite Quantum Hall Systems as a Result of Weak Macroscopic Inhomogeneities

We show that, at low temperatures, macroscopic inhomogeneities of the electron density in the interior of a finite sample cause a reduction in the measured conductivity peak heights $\sigma_{xx}^{\rm max}$ compared to the universal values previously predicted for infinite homogeneous samples. This effect is expected to occur for the conductivity peaks measured in standard experimental geometries such as the Hall bar and the Corbino disc. At the lowest temperatures, the decrease in $\sigma_{xx}^{\rm max}(T)$ is found to saturate at values proportional to the difference between the adjacent plateaus in $\sigma_{xy}$, with a prefactor which depends on the particular realization of disorder in the sample. We argue that this provides a possible explanation of the ``non-universal scaling'' of $\sigma_{xx}^{\rm max}$ observed in a number of experiments. We also predict an enhancement of the ``non-local'' resistance due to the macroscopic inhomogeneities. We argue that, in the Hall bar with a sharp edge, the enhanced ``non-local'' resistance and the size corrections to the ``local'' resistance $R_{xx}$ are directly related. Using this relation, we suggest a method by which the finite-size corrections may be eliminated from $R_{xx}$ and $R_{xy}$ in this case.