in the vacuum chambers of either a Cryomech pulse-tube cryocooler, $T_{min}$=2.7K, or a Lakeshore cryogenic probe-station, $T_{min}$=8K, equipped with a 2T superconducting magnet. A small ($\sim1\mu$A) constant current is applied between S and D (Fig.\[figure:device\]a) by using a two-channel Agilent B2912A source-measure unit (SMU). The longitudinal and transverse voltage drops are measured with an Agilent 34420 low-noise nanovoltmeter. Thermoelectrical and other offset voltages are eliminated by measuring each resistance value and inverting the source current in each measurement[@DagheroPRL2012]. Gate biases are applied between the corresponding G and D with the same two-channel SMU (liquid gate) or a Keithley 2410 SMU (back gate). Samples are allowed to degas in vacuum ($<10^{-5}$mbar) at room $T$ for at least$\sim1$h before measurements, in order to remove residual water traces in the electrolyte.

![image](Figure_transport_1.eps){width="80.00000%"}

We first characterize the T dependence of the sheet resistance, $R_s$, under the effect of the liquid top gate. We apply the liquid gate voltage, $V_{LG}$, at 240K, where the electrolyte is still liquid, and under high-vacuum ($<10^{-5}$mbar) to minimize unwanted electrochemical interactions and extend the stability window of the ionic liquid[@UenoReview2014]. After $V_{LG}$ is applied, we allow the ion dynamics to settle for$\sim$10min before cooling to a base T=2.7K.

Fig.\[figure:transport\]a 