Impact of Heterostructure Design on Transport Properties in the Second Landau Level of in-situ Back-Gated Two-Dimensional Electron Gases

We report on transport in the second Landau level in \emph{in-situ} back-gated two-dimensional electron gases in GaAs/Al$_x$Ga$_{1-x}$As quantum wells. Minimization of gate leakage is the primary heterostructure design consideration. Leakage currents resulting in dissipation as small as $\sim$ 10 pW can cause noticeable heating of the electrons at 10 mK, limiting the formation of novel correlated states. We show that when the heterostructure design is properly optimized, gate voltages as large as 4V can be applied with negligible gate leakage, allowing the density to be tuned over a large range from depletion to over 4 $\times$ 10$^{11}$ cm$^{-2}$. As a result, the strength of the $\nu = 5/2$ state can be continuously tuned from onset at n $\sim 1.2 \times 10^{11}$ cm$^{-2}$ to a maximum $\Delta_{5/2} = 625$ mK at n = $3.35 \times 10^{11}$ cm$^{-2}$. An unusual evolution of the reentrant integer quantum Hall states as a function of density is also reported. These devices can be expected to be useful in experiments aimed at proving the existence of non-Abelian phases useful for topological quantum computation.