Quenching of the Hall Effect

Abstract

We argue that for ballistic transport through a narrow conductor (of width $W$) a threshold magnetic field exists below which the Hall resistance vanishes. The field is of order $\left(\frac{h}{e}\right)k_{\mathrm{F}}^{}{}_{}{}^{-1\mathrm{}}{W}^{-3\mathrm{}}$, and is reached when the transverse wavelength of quantum edge states becomes comparable to the width. This is offered as a mechanism for the quenching of the Hall effect discovered experimentally in a narrow two-dimensional electron-gas wire by Roukes et al.