Edge state transmission, duality relation and its implication to measurements

TL;DR

This paper investigates the duality in the Chalker-Coddington model, deriving relations for edge state transmission and exploring their implications for measurements, revealing limitations of duality in explaining experimental results.

Contribution

It derives a duality relation for edge state transmission in a specific symmetric geometry and analyzes its implications for conductance measurements.

Findings

Duality relation holds only for symmetric Hall geometry.

Resistances are not always direct measures of resistivity.

Duality alone cannot fully explain experimental observations.

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 $\sigma_{xx}$ and $\sigma_{xy}$ in the diffusive limit. The edge state scattering problem is reduced to solving the diffusion equation 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 necessarily measures of the resistivity and $\rho_{xx}=L/W R/T h/e^2$ ($R=1-T$) holds only when $\rho_{xy}$ 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.