Evolution of the bilayer $\nu =1$ quantum Hall state under charge imbalance

We use high-mobility bilayer hole systems with negligible tunneling to examine how the bilayer $\nu =1$ quantum Hall state evolves as charge is transferred from one layer to the other at constant total density. We map bilayer $\nu =1$ state stability versus imbalance for five total densities spanning the range from strongly interlayer coherent to incoherent. We observe competition between single-layer correlations and interlayer coherence. Most significantly, we find that bilayer systems that are incoherent at balance can develop spontaneous interlayer coherence with imbalance, in agreement with recent theoretical predictions.

Figure 1

Longitudinal ${\rho }_{\mathrm{x}\mathrm{x}}$ and Hall ${\rho }_{\mathrm{x}\mathrm{y}}$ resistivity traces obtained at balance $(n_{\mathrm{t}}=n_{\mathrm{b}})$ for $d∕l_B=1.36(n_{\mathrm{T}}=9.7\times {10}^{10}{\mathrm{cm}}^{-2})$. The inset shows two equivalent methods of determining the width of the $\nu =1$ QH state based on ${\rho }_{\mathrm{x}\mathrm{y}}$ (dashed) and ${\rho }_{\mathrm{x}\mathrm{x}}$ (solid).

Figure 2

Normalized width $\mathrm{\Delta }B∕B$ of the $\nu =1$ ${\rho }_{\mathrm{x}\mathrm{x}}$ minima gate imbalance $N_z$ for five $d∕l_B$. The arrows indicate the composite $1∕3+2∕3$ states (see text) and the three solid and one open circles correspond to those shown in the simplified phase diagram (Fig. 3). Consecutive traces are offset by $\mathrm{\Delta }B∕B=0.04$ for clarity.

Figure 3

Simplified ground-state phase diagram for this bilayer $\nu =1$ system. Solid and dashed lines mark the boundary between SILC and non-SILC states with and without intralayer correlations, respectively. Dotted line indicates the transition between bilayer and single-layer charge distributions. The symbols correspond to the data points highlighted in Fig. 2.

Figure 4

Grey-scale plot of ${\rho }_{\mathrm{x}\mathrm{x}}$ ($z$ axis) $B$ and $N_{\mathrm{z}}$. Dark regions correspond to QH states. The four diagonal dotted lines denote single-layer fractional QH states and the vertical white dotted line follows the bilayer $\nu =1$ QH state.