Electronic transport and the localization length in the quantum Hall effect

Abstract

We report on recent experimental results from transport measurements with large Hall bars made of high- mobility ${\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}/\mathrm{A}\mathrm{l}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{As}\mathrm{}$ heterostructures. Thermally activated conductivities and hopping transport were investigated in the integer quantum Hall regime. The predominant transport processes in two dimensions are discussed. The implications of transport regime on prefactor universality and on the relation between ${\ensuremath{\rho}}_{\mathrm{xx}}$ and ${\ensuremath{\rho}}_{\mathrm{xy}}$ are studied. Particularly in the Landau-level tails, a strictly linear dependence $\ensuremath{\delta}{\ensuremath{\rho}}_{\mathrm{xy}}({\ensuremath{\rho}}_{\mathrm{xx}})$ was found, with pronounced asymmetries with respect to the plateau center. At low temperatures, Ohmic (temperature-dependent) as well as non-Ohmic (current-dependent) transport were investigated and analyzed on the basis of variable-range hopping theory. The non-Ohmic regime could successfully be described by an effective electron temperature model. The results from either the Ohmic transport or from a comparison of Ohmic and non-Ohmic data allowed us to determine the localization length $\ensuremath{\xi}$ in two different ways. The observed divergence of $\ensuremath{\xi}(\ensuremath{\nu})$ with the filling factor $\ensuremath{\nu}$ approaching a Landau-level center, is in qualitative agreement with scaling theories of electron localization. The absolute values of $\ensuremath{\xi}$ far from the ${\ensuremath{\rho}}_{\mathrm{xx}}$ peaks are compared with theoretical predictions. On one hand, discrepancies between the $\ensuremath{\xi}$ results obtained from the two experimental methods are attributed to an inhomogeneous electric-field distribution. Extrapolation yields an effective width of dominant potential drop of about $100$ $\ensuremath{\mu}$m. On the other hand, our analysis suggests a divergence of the dielectric function ${\ensuremath{\epsilon}}_{r}\ensuremath{\propto}{\ensuremath{\xi}}^{\ensuremath{\beta}}$ with $\ensuremath{\beta}\ensuremath{\simeq}1$.