Charge imbalance and bilayer two-dimensional electron systems at ${\nu }_T=1$

We use interlayer tunneling to study bilayer two-dimensional electron systems at ${\nu }_T=1$ over a wide range of charge-density imbalance $\Delta \nu ={\nu }_1-{\nu }_2$ between the two layers. We find that the strongly enhanced tunneling associated with the coherent excitonic ${\nu }_T=1$ phase at small layer separation can survive at least up to an imbalance of $\Delta \nu =0.5$, i.e., $\left({\nu }_1,{\nu }_2\right)=\left(3/4,1/4\right)$. Phase transitions between the excitonic ${\nu }_T=1$ state and bilayer states which lack significant interlayer correlations can be induced in three different ways: by increasing the effective interlayer spacing $d/\ell$, the temperature $T$, or the charge imbalance $\Delta \nu$. We observe that close to the phase boundary the coherent ${\nu }_T=1$ phase can be absent at $\Delta \nu =0$, present at intermediate $\Delta \nu$, and then absent again at large $\Delta \nu$, thus indicating an intricate phase competition between it and incoherent quasi-independent layer states. At zero imbalance, the critical $d/\ell$ shifts linearly with temperature, while at $\Delta \nu =1/3$ the critical $d/\ell$ is only weakly dependent on $T$. At $\Delta \nu =1/3$ we report on an observation of a direct phase transition between the coherent excitonic ${\nu }_T=1$ bilayer integer quantum Hall phase and the pair of single-layer fractional quantized Hall states at ${\nu }_1=2/3$ and ${\nu }_2=1/3$.

Figure 1

(Color online) (a) Longitudinal magnetoresistance trace in the top 2D gas at low magnetic field $B$ for zero applied top and bottom gate voltages $V_{\text{tg}}=V_{\text{bg}}=0$ and $T=100\text{mK}$. A density of $n_1=5.4\times {10}^{10}{\text{cm}}^{-2}$ is extracted from the Shubnikov-de Haas oscillations. (b) Hall resistance traces for the bottom 2D gas at $T=700\text{mK}$. The carrier density is extracted from the Hall slope. The upper solid (black) trace is acquired with $V_{\text{tg}}=-450.3\text{mV}$ and $V_{\text{bg}}=-3.64\text{V}$. Under these conditions $n_2=4.5\pm 0.15\times {10}^{10}{\text{cm}}^{-2}$, while the top 2DES is fully depleted, $n_1=0$. The lower solid (red) trace is acquired with $V_{\text{tg}}=-353.7\text{mV}$ and $V_{\text{bg}}=-3.33\text{V}$. With these gate voltages $n_1>0$. Below $B\approx 1\text{T}$ the two traces overlap and hence yield essentially identical values for the lower-layer density. Above $B=1\text{T}$ the traces begin to separate. The dashed line is a linear fit to the lower Hall resistance trace at high fields giving $n_2=5.3\pm 0.15\times {10}^{10}{\text{cm}}^{-2}$. (c) Filling factor imbalance $\Delta {\nu }_{\text{high}}$ deduced from high $B$-field Hall measurements versus $\Delta {\nu }_{\text{low}}$, the imbalance deduced from low-field Shubnikov-de Haas oscillations under the same gating conditions for three different $d/\ell$’s at ${\nu }_T=1$. The dashed line indicates $\Delta {\nu }_{\text{high}}=\Delta {\nu }_{\text{low}}$. The solid line is the result of the simulation of the charge-transfer effect described in the text and Appendix.

Figure 2

(Color online) (a) Tunneling conductance spectra $dI/dV$ vs $V$ at ${\nu }_T=1$, $T=55\text{mK}$, and effective layer spacing $d/\ell =1.56$ for various charge imbalances $\Delta \nu ={\nu }_1-{\nu }_2$. (b) Temperature dependence of $G\left(0\right)$, the height of the tunneling resonance at ${\nu }_T=1$, for various $\Delta \nu$. The solid lines are guides for the eyes. The arrows associated with the $\Delta \nu =0.28$, 0.40 and 0.50 data traces indicate the crossover temperature $T^{\ast }$ defined in the text.

Figure 3

Tunneling conductance spectra $dI/dV$ vs $V$ at ${\nu }_T=1$ and $T=85\text{mK}$ vs imbalance $\Delta \nu$ for (a) $d/\ell =1.65$ and (b) 1.80. In each panel the middle trace corresponds to $\Delta \nu =0$ while the remaining traces are offset by an amount proportional to $\Delta \nu$ for clarity. The interlayer voltage range is from $-150$ to $+150\mu \text{V}$ in all cases.

Figure 4

Tunneling conductance spectra $dI/dV$ vs $V$ at ${\nu }_T=1$ and $d/\ell =1.71$ vs imbalance $\Delta \nu$ for (a) $T=55$ and (b) 125 mK. In each panel the middle trace corresponds to $\Delta \nu =0$ while the remaining traces are offset by an amount proportional to $\Delta \nu$ for clarity. The interlayer voltage range is from $-150$ to $+150\mu \text{V}$ in all cases.

Figure 5

Tunneling conductance at zero bias $G\left(0\right)$ at ${\nu }_T=1$ vs $\Delta \nu$ for various $d/\ell$. (a) $T=85$, (b) 55, and (c) 200 mK. The solid lines are guides for the eyes.

Figure 6

(Color online) (a) Phase boundary separating the interlayer coherent ${\nu }_T=1$ phase at small $d/\ell$ from the incoherent states at large $d/\ell$ vs $\Delta \nu$ at $T=55$, 85, 125, and 200 mK. The data points represent the position in $d/\ell$ of the smallest measured tunneling resonance. Arrows denote $\Delta \nu =\pm 1/3$. (b) Phase boundary in the $\left(d/\ell \right)\text{−}T$ plane extracted from panel (a) for fixed $\Delta \nu =0$, 0.16, and 1/3. The lines are guides for the eyes.

Figure 7

(Color online) Observation of the transition from the interlayer coherent ${\nu }_T=1$ phase to two quasi-independent fractional quantized Hall states. (a) Tunneling conductance spectra at ${\nu }_T=1$ and $T=85\text{mK}$ at fixed imbalance $\Delta \nu =+1/3$ (i.e., ${\nu }_1=2/3$ and ${\nu }_2=1/3$). Zero-bias peak collapses rapidly as $d/\ell$ increases and interlayer coherence is lost. (b) Hall resistance ${\rho }_{xy,\text{top}}$ in the top 2DES vs inverse total filling fraction $1/{\nu }_T$ at $d/\ell =1.62$, 1.71, 1.80, 1.84, and 2.10. For each trace the imbalance at ${\nu }_T=1$ is $\Delta \nu =+1/3$. A clear transition from a Hall plateau at $h/e^2$ to one at $3h/2e^2$ is observed at ${\nu }_T=1$.

Figure 8

(Color online) Tunneling peak height vs imbalance at $T=85\text{mK}$ and $d/\ell =1.56$. The dashed line is prediction of pseudospin moment projection model. The solid line is Gaussian fit to the data.