Figure 1

A schematic of the sample, the aluminum SET and the circuitry for the compressibility measurement. The SET consists of an $80\times 500$-nm${}^2$ aluminum bar connected by two electrodes.Reuse & Permissions

Figure 2

(a) Color rendition of the longitudinal resistance $R_{xx}$ as a function of magnetic field $B$ and backgate voltage $V_{BG}$. On the bottom axis also the total density $n$ has been plotted. Total filling factors ${\nu }_{tot}$ are denoted in white numbers. They are determined from the height of the QH plateaus in the $R_{xy}$ traces. The QH state at ${\nu }_{tot}=3/5+2/3$ corresponds to ${\nu }_{upper}=3/5$ and ${\nu }_{lower}=2/3$. Dotted (dashed) lines mark the $B$-field positions where the upper (lower) layer is expected to condense in a quantum Hall state. Lines are plotted for ${\nu }_{upper,lower}=1/3$, 2/3, 1, 4/3, 2, and 3. They were theoretically calculated with two fitting parameters. Solid lines highlight the region where $\nu =1$. Gray color marks the same $\nu =1$ region but at 850 mK with the arrow indicates the bending of the ${\nu }_{upper}=1$ QH state. (b) Schematic drawings of the conduction band profile at different backgate voltages. The Fermi level is drawn below the subband to emphasize that the chemical potential is negative for a low-density 2DES. In order to fill the lower quantum well, one has to align the lowest subband of the lower quantum well with the Fermi level. This alignment requires a backgate voltage of $V_0$. (c) Hall ($R_{xy}$) and longitudinal ($R_{xx}$) resistances as a function of magnetic field for $V_{BG}=$0.6 V at different temperatures. Two minima can be observed in the $R_{xx}$ traces in the region where $\nu =1$. These minima are attributed to the single layer $\nu =1$ and the correlated ${\nu }_{tot}=1$ QH states, respectively. (d) Comparison between the Hall and longitudinal resistance traces taken at $V_{BG}=0.30$ V, where only the upper layer is populated, and at $V_{BG}=1.04$ V, where the two layers are balanced.Reuse & Permissions

Figure 3

(a) $\eta$ vs the averaged density $n$. The blue traces were obtained using the same SET at $B=0$ T (top panel) and $B=2.3$ T (bottom panel), respectively. The red and gray traces were obtained from the other two SETs at $B=2.3$ T. The solid black lines are from the theoretical calculations. The dashed black curve illustrates the smoothed background for the data represented by the red curve. This background was later subtracted from the original data to determine $\Delta \mu$ at ${\nu }_{tot}=1$. (b) $\eta$ plotted as a function of the magnetic field and the density. The dotted black lines separate QH states identified from transport.Reuse & Permissions

Figure 4

$\Delta \mu$ and $E_g$ at ${\nu }_{tot}=1$. Data points denoted by SET1, SET2, and SET3 are derived from the compressibility data of three different SETs on the same sample. The black squares represent $E_g$ determined from thermal activation. Lines are guides to the eye.Reuse & Permissions

Figure 5

The local inverse compressibility at ${\nu }_{tot}=1$ plotted as a function of the average density and the magnetic field. A smoothed background has been subtracted to highlight the discrete charging lines (black).Reuse & Permissions

Figure 6

Densities as a function of the backgate voltage calculated for $B=0$ T. The inset shows the conduction band profile of the bilayer system. We took the growth parameters $l_{FG}=450$ nm and $d=28.6$ nm. $l_{BG}$ is experimentally determined to be 1780 nm from the linear fitting of the total density as a function of $V_{BG}$ (we used a relative dielectric constant of $12.25$ for Al${}_{0.23}$Ga${}_{0.77}$As).Reuse & Permissions