We report electrical conductivity $\sigma$ 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 $\sigma \sim L^{\alpha }$, 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 $\xi$ is localized, $\sigma$ is not a material-specific parameter but depends on the system dimensions. From our data we are able to construct the “$\beta$ function” $\equiv (h/e^2)dln\sigma /dlnL$ 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 $\sim 10\mu \mathrm{m}$. This suggests the utility of the 2DEGs studied towards applications in quantum information as well as towards fundamental investigations into many-body localized phases.

• Received 31 March 2015
• Revised 28 October 2015

DOI:https://doi.org/10.1103/PhysRevB.92.235427