Observation of geometry-dependent conductivity in two-dimensional electron systems

Abstract

We report electrical conductivity $\ensuremath{\sigma}$ measurements on a range of two-dimensional electron gases (2DEGs) of varying linear extent. Intriguingly, at low temperatures ($T$) and low carrier density (${n}_{\mathrm{s}}$) we find the behavior to be consistent with $\ensuremath{\sigma}\ensuremath{\sim}{L}^{\ensuremath{\alpha}}$, where $L$ is the length of the 2DEG along the direction of transport. Importantly, such scale-dependent behavior is precisely in accordance with the scaling hypothesis of localization [Abrahams et al., Phys. Rev. Lett. 42, 673 (1979)] which dictates that in systems where the electronic wave function $\ensuremath{\xi}$ is localized, $\ensuremath{\sigma}$ is not a material-specific parameter but depends on the system dimensions. From our data we are able to construct the ``$\ensuremath{\beta}$ function'' $\ensuremath{\equiv}(h/{e}^{2})dln\ensuremath{\sigma}/dlnL$ and show this to be strongly consistent with theoretically predicted limiting values. These results suggest, remarkably, that the electrons in the studied 2DEGs preserve phase coherence over lengths $\ensuremath{\sim}10\phantom{\rule{0.28em}{0ex}}\ensuremath{\mu}\mathrm{m}$. This suggests the utility of the 2DEGs studied towards applications in quantum information as well as towards fundamental investigations into many-body localized phases.