Edge state transmission, duality relation and its implication to measurements

摘要

The duality in the Chalker-Coddington network model is examined. We are ableto write down a duality relation for the edge state transmission coefficient,but only for a specific symmetric Hall geometry. Looking for broaderimplication of the duality, we calculate the transmission coefficient $T$ interms of the conductivity $\sigma_{xx}$ and $\sigma_{xy}$ in the diffusivelimit. The edge state scattering problem is reduced to solving the diffusionequation with two boundary conditions$(\partial_y-(\sigma_{xy})/(\sigma_{xx})\partial_x)\phi=0$ and$[\partial_x+(\sigma_{xy}-\sigma_{xy}^{lead})/(\sigma_{xx}) \partial_y]\phi=0$.We find that the resistances in the geometry considered are not necessarilymeasures of the resistivity and $\rho_{xx}=L/W R/T h/e^2$ ($R=1-T$) holds onlywhen $\rho_{xy}$ is quantized. We conclude that duality alone is not sufficientto explain the experimental findings of Shahar et al and that Landauer-Buttikerargument does not render the additional condition, contrary to previousexpectation.