Experimental investigation of the mobility/quality dichotomy in ultra-high quality AlGaAs/GaAs two-dimensional electron gases

We detailed the relationship between mobility lifetime ($\tau _{\mathrm{tr}})$, and quantum scattering lifetime ($\tau_{\mathrm{q}})$ measured near zero magnetic field and the excitation gap ($\Delta _{\mathrm{5/2}})$ at $\nu =$5/2 in ultra-high quality AlGaAs/GaAs two-dimensional electron gases. While the lack of correlation between $\tau _{\mathrm{tr}}$ and $\Delta_{\mathrm{5/2\thinspace }}$has been noted previously, we demonstrate that $\tau_{\mathrm{q}}$ is also a poor predictor of $\Delta_{\mathrm{5/2.\thinspace }}$The impact of small density inhomogeneities on determination of $\tau_{\mathrm{q}}$ in the limit of very large $\tau_{\mathrm{q}}$ and low temperatures is discussed. We define, analyze and discuss the utility of a different metric $\rho_{\mathrm{cf}}$, the so-called composite fermion resistivity, as a high temperature (T$=$0.3K) predictor of $\Delta_{\mathrm{5/2.}}$