Magnetoresistance and dephasing in a two-dimensional electron gas at intermediate conductances

Abstract

We study, both theoretically and experimentally, the negative magnetoresistance (MR) of a two-dimensional (2D) electron gas in a weak transverse magnetic field $B$. The analysis is carried out in a wide range of zero-$B$ conductances $g$ (measured in units of ${e}^{2}∕h$), including the range of intermediate conductances $g\ensuremath{\sim}1$. Interpretation of the experimental results obtained for a 2D electron gas in $\mathrm{Ga}\mathrm{As}∕{\mathrm{In}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ single quantum well structures is based on a theory that takes into account terms of higher orders in $1∕g$. We show that the standard weak localization (WL) theory is adequate for $g\ensuremath{\gtrsim}5$. Calculating the corrections of second order in $1∕g$ to the MR, stemming from both the interference contribution and the mutual effect of WL and Coulomb interaction, we expand the range of a quantitative agreement between the theory and experiment down to significantly lower conductances $g\ensuremath{\sim}1$. We demonstrate that at intermediate conductances the negative MR is described by the standard WL ``digamma-functions'' expression, but with a reduced prefactor $\ensuremath{\alpha}$. We also show that at not very high $g$ the second-loop corrections dominate over the contribution of the interaction in the Cooper channel, and therefore appear to be the main source of the lowering of the prefactor $\ensuremath{\alpha}\ensuremath{\simeq}1\ensuremath{-}2∕\ensuremath{\pi}g$. The fitting of the MR allows us to measure the true value of the phase breaking time within a wide conductance range $g\ensuremath{\gtrsim}1$. We further analyze the regime of a ``weak insulator,'' when the zero-$B$ conductance is low $g(B=0)<1$ due to the localization at low temperature, whereas the Drude conductance is high ${g}_{0}⪢1$, so that a weak magnetic field delocalizes electronic states. In this regime, while the MR still can be fitted by the digamma-functions formula, the experimentally obtained value of the dephasing rate has nothing to do with the true one. The corresponding fitting parameter in the low-$T$ limit is determined by the localization length and may therefore saturate at $T\ensuremath{\rightarrow}0$, even though the true dephasing rate vanishes.