Phase Diagram of Interacting Composite Fermions in the Bilayer $\nu =2/3$ Quantum Hall Effect

We study the phase diagram of composite fermions (CFs) in the presence of spin and pseudospin degrees of freedom in the bilayer $\nu =2/3$ quantum Hall (QH) state. Activation studies elucidate the existence of three different QH states with two different types of hysteresis in the magnetotransport. While a noninteracting CF model provides a qualitative account of the phase diagram, the observed renormalization of tunneling gap and a non-QH state at high densities are not explained in the noninteracting CF model, and are suggested to be manifestations of interactions between CFs.

Figure 1

Lower-lying energy levels of noninteracting CFs schematically shown as a function of the magnetic field. While pseudo-Zeeman energy ${\Delta }_{\mathrm{B}\mathrm{A}\mathrm{B}}$ is constant, the Zeeman ${\Delta }_Z\propto B_{\mathrm{t}\mathrm{o}\mathrm{t}}$ and CF-cyclotron ${\Delta }_{\mathrm{c}\mathrm{y}}\propto \sqrt{B_{\perp }}$ energies depend on the magnetic field. At $\nu =2/3$, the lowest two levels are filled (thick curves), and there are three possible QH ground states. Here, SU-PP means the spin-unpolarized pseudospin-polarized state. ($0,b,\uparrow$) represents the $N_{\mathrm{C}\mathrm{F}}=0$ Landau level in the bonding state with up-spin.

Figure 2

$R_{xx}$ at the balanced-density point ($\sigma =0$) and the monolayer point ($\sigma =1$) for several total densities, plotted as a function of $1/\nu =e{B}_{\perp }/h{n}_t$. Each trace is vertically offset by $5\mathrm{k}\Omega$. Hysteresis between the upward (solid trace) and downward (dashed trace) field sweeps is seen at $\sigma =1$.

Figure 3

Phase diagram for $\nu =2/3$. The horizontal and vertical axes are the normalized density difference between two layers, $\sigma$, and the total electron density, $n_t$, respectively. The black and white areas represent QH and non-QH states, respectively. Type-G hysteresis (Fig. 2) was observed in the hatched area. The small white square in black region indicates the occurrence of a different type of hysteresis (Fig. 5).

Figure 4

The activation energy $\Delta$ as a function of $B_{\mathrm{t}\mathrm{o}\mathrm{t}}$ at $\nu =2/3$ in the four QH areas. $\Delta$ is determined from the temperature dependence, $R_{xx}\propto \exp (-\Delta /2T)$ [13]. On the top axis, we show the tilt angle $\theta$. The leftmost point corresponds to $\theta =0$ where $B_{\mathrm{t}\mathrm{o}\mathrm{t}}=B_{\perp }$. The slope of the lines included in I, III, and IV correspond to $\pm g^{*}{\mu }_B{B}_{\mathrm{t}\mathrm{o}\mathrm{t}}$. In the insets, we showed the ground state of the QH states (thick line) and excitation levels (thin line).

Figure 5

The evolution of $R_{xx}$ from areas II to III. At $\sigma =0.5$ and 0.6, type-E hysteresis is seen.