Area dependence of interlayer tunneling in strongly correlated bilayer two-dimensional electron systems at ${\nu }_T=1$

The area and perimeter dependences of the Josephson-like interlayer tunneling signature of the coherent ${\nu }_T=1$ quantum Hall phase in bilayer two-dimensional electron systems is examined. Electrostatic top gates of various sizes and shapes are used to locally define distinct ${\nu }_T=1$ regions in the same sample. Near the phase boundary with the incoherent ${\nu }_T=1$ state at large layer separation, our results demonstrate that the tunneling conductance in the coherent phase is closely proportional to the total area of the tunneling region. This implies that tunneling at ${\nu }_T=1$ is a bulk phenomenon in this regime.

Figure 1

(Color online) (a) Schematic diagram of sample A. White rectangle represents $200\text{−}\mu \text{m}$-wide mesa containing the bilayer 2DES; ohmic contacts (not shown) to the individual 2DES layers are attached near both ends of the mesa. Gray rectangle under the mesa is the global back gate; hatched rectangles crossing above the mesa are the various top gates under which tunneling is established. (b) Zero voltage tunneling conductance $G\left(0\right)$ versus top gate bias taken at $B=0$ and $T=60\text{mK}$ and with an applied back gate bias of $-15.18\text{V}$. Peaks correspond—in order of decreasing height—to the 100, 50, 20, and $10\mu \text{m}$ top gates. (c) Background subtracted peak heights from (b) versus top gate length. The line fitted to this data has an $x$ intercept of $3\pm 2\mu \text{m}$.

Figure 2

(Color online) Tunneling vs gate length at ${\nu }_T=1$ in sample A at $d/\ell =1.81$. Gate layout as shown in Fig. 1. (a) Tunneling conductance resonances $dI/dV$ vs $V$ at $T=85\text{mK}$. Gate lengths 100, 50, 20, and $10\mu \text{m}$; tallest to shortest. (b) Peak heights plotted vs top gate length; open circles 60 mK, solid circles 85 mK, and squares 125 mK. Straight lines are linear least-squares fits.

Figure 3

(Color online) (a) Schematic diagram of the mesa and gates in the active region of sample B. (b) Zero voltage tunneling conductance $G\left(0\right)$ versus top gate bias taken at $B=0$ and with an applied back gate bias of $-17.08\text{V}$. (c) Tunneling conductance resonances $dI/dV$ vs $V$ at ${\nu }_T=1$ with $d/\ell =1.64$ and $T=61\text{mK}$. In (b) and (c) the solid trace (black) corresponds to the $100\mu \text{m}$ square tunneling region and dotted trace (red) to the $200\times 50\mu {\text{m}}^2$ rectangular region. The red trace in (c) has been multiplied by 1.086, as described in the text.