by LG cannot be altered for $T\lesssim 220$K, as the ions are locked when the electrolyte is frozen. The kink that appears early in the metallic state, on the other hand, extends across a small range of $n_{2d}$ ($1\lesssim n_{2d} \lesssim 2\cdot10^{13}$cm$^{-2}$), and is ideally suited to be explored continuously by exploiting the dual-gate configuration.

We thus bias our samples in the low-density range of the metallic state ($n_{2d}\sim7\cdot 10^{12}$cm$^{-2}$) by applying V$_{LG}=0.9$V, and cool the system to 2.7K. We then apply $V_{BG}$ and fine-tune $n_{2d}$ across the kink. We constantly monitor $I_{BG}$ to avoid dielectric breakdown. Fig.\[figure:transport\]c plots $\sigma_{2d}$ of a representative device subject to multiple $V_{BG}$ sweeps, as $n_{2d}$ is tuned across the kink. This reproduces well the behavior observed for low $V_{LG}$ ($1\lesssim n_{2d} \lesssim 2\cdot10^{13}$cm$^{-2}$). The hysteresis between increasing and decreasing $V_{BG}$ is minimal. This kink is suppressed by increasing T, similar to LG gating.

$V_{BG}$ provides us an independent tool to estimate $n_{2d}$: If $V_{LG}$ is small enough ($V_{LG}\lesssim 1$V) so that conduction in the channel can be switched off by sufficiently large negative $V_{BG}$ ($V_{BG}\lesssim -25$V), we can write $n_{2d}=C_{ox}/e \cdot (V_{BG}-V_{th})$. Here, $C_{ox}=\epsilon_{ox}/d_{ox}$ is the back gate oxide specific capacitance, $e=1.602\cdot10^{-19}$C is the 