Edge state transmission, duality relation and its implication to measurements

Abstract

The duality in the Chalker-Coddington network model is examined. We are able to write down a duality relation for the edge state transmission coefficient, but only for a specific symmetric Hall geometry. Looking for broader implication of the duality, we calculate the transmission coefficient $T$ in terms of the conductivity $σ_{xx}$ and $σ_{xy}$ in the diffusive limit. The edge state scattering problem is reduced to solving the diffusion equation with two boundary conditions $(\partial_y-(σ_{xy})/(σ_{xx})\partial_x)ϕ=0$ and $[\partial_x+(σ_{xy}-σ_{xy}^{lead})/(σ_{xx}) \partial_y]ϕ=0$. We find that the resistances in the geometry considered are not necessarily measures of the resistivity and $ρ_{xx}=L/W R/T h/e^2$ ($R=1-T$) holds only when $ρ_{xy}$ is quantized. We conclude that duality alone is not sufficient to explain the experimental findings of Shahar et al and that Landauer-Buttiker argument does not render the additional condition, contrary to previous expectation.