Phase diagrams for the stability of the $\nu =\frac{1}{2}$ fractional quantum Hall effect in electron systems confined to symmetric, wide GaAs quantum wells

We report an experimental investigation of the fractional quantum Hall effect (FQHE) at the even-denominator Landau-level filling factor $\nu =1/2$ in very-high-quality wide GaAs quantum wells and at very high magnetic fields up to 45 T. The quasi-two-dimensional electron systems we study are confined to GaAs quantum wells with widths $W$ ranging from 41 to 96 nm and have variable densities in the range of $\simeq$$4\times {10}^{11}$ to $\simeq$$4\times {10}^{10}$ cm${}^{-2}$. We present several experimental phase diagrams for the stability of the $\nu =1/2$ FQHE in these quantum wells. In general, for a given $W$, the 1/2 FQHE is stable in a limited range of intermediate densities where it has a bilayerlike charge distribution; it makes a transition to a compressible phase at low densities and to an insulating phase at high densities. The densities at which the $\nu =1/2$ FQHE is stable are larger for narrower quantum wells. Moreover, even a slight charge distribution asymmetry destabilizes the $\nu =1/2$ FQHE and turns the electron system into a compressible state. We also present a plot of the symmetric-to-antisymmetric subband separation (${\Delta }_{\mathrm{SAS}}$), which characterizes the interlayer tunneling, vs density for various $W$. This plot reveals that ${\Delta }_{\mathrm{SAS}}$ at the boundary between the compressible and FQHE phases increases linearly with density for all the samples. There is no theoretical explanation for such a simple dependence. Finally, we summarize the experimental data in a diagram that takes into account the relative strengths of the interlayer and intralayer Coulomb interactions and ${\Delta }_{\mathrm{SAS}}$. We conclude that consistent with the conclusions of some of the previous studies, the $\nu =1/2$ FQHE observed in wide GaAs quantum wells with symmetric charge distribution is stabilized by a delicate balance between the interlayer and intralayer interactions and is very likely described by a two-component (${\Psi }_{331}$) state.

Figure 1

Longitudinal ($R_{xx}$) and Hall ($R_{xy}$) resistance data for a 41-nm-wide QW at a density of $4.34\times {10}^{11}$ cm${}^{-2}$. The inset shows the electron distribution (blue curve) and potential energy (black curve) calculated self-consistently at zero magnetic field. The “interlayer distance” $d$ and full width at half maximum $\lambda$ of each “layer” are also indicated.

Figure 2

$R_{xx}$ vs inverse filling factor for a 48-nm-wide QW for different densities. All traces were taken for symmetric total charge distributions (balanced QW). Except for the lowest density, the traces are offset vertically for clarity. In each trace, $R_{xx}\simeq$ 0 at $\nu =2/3$. Note that for the $n=3.69\times {10}^{11}$ cm${}^{-2}$ (dark blue) trace, $R_{xx}$ at $\nu =1/2$ essentially vanishes. For the $n=3.42\times {10}^{11}$ cm${}^{-2}$ (red) trace, in the top section of the figure we also show its corresponding $R_{xy}$ trace (in red), illustrating a well-developed $R_{xy}$ plateau quantized at $2h/e^2$; the top axis is the perpendicular magnetic field for these two red traces. The inset shows the self-consistently calculated charge distributions for $n=2.80$ (black) and 3.26 $\times {10}^{11}$ cm${}^{-2}$ (green).

Figure 3

$R_{xx}$ vs inverse filling factor for a 45-nm-wide QW for different densities while the total charge distribution was kept symmetric. The upper two traces are shifted vertically for clarity. The top axis is the perpendicular magnetic field for the $n=3.66\times {10}^{11}$ cm${}^{-2}$ trace. The inset shows the calculated charge distributions for $n=3.39$ (black) and 3.66 $\times {10}^{11}$ cm${}^{-2}$ (red).

Figure 4

Evolution of the $\nu =1/2$ FQHE in the 45-nm-wide GaAs quantum well with total density of $n=3.53\times {10}^{11}$cm${}^{-2}$ as the charge distribution is made asymmetric via applying front- and back-gate voltage biases. Inset: Self-consistently calculated electron distributions for $\delta n=0$ and $\delta n/n=0.034$.

Figure 5

Experimentally measured subband separation energy (${\Delta }_{\mathrm{SAS}}$) as a function of density ($n$) for electron systems with symmetric charge distributions confined to wide GaAs QWs. The accuracy of the measured ${\Delta }_{\mathrm{SAS}}$ is about $\pm$2%. The well widths are given (in units of nm) next to each set of data points, and are determined from fitting each set to our self-consistent (Hartree) calculations, examples of which are shown by thin red curves for the 48-nm-wide QW (see text). Data points in black are from Refs. 23, 24, 25, 26 and those in green from Ref. 30. Filled symbols represent the presence of a $\nu =$ 1/2 FQHE and open symbols represent its absence; the size of each filled symbol for the 45-, 48-, 56-, and 77-nm-wide QWs provides an estimate for the strength of the observed $\nu =1/2$ FQHE. The boundary between the FQHE and compressible states appears to be a $straight$ line (dashed line) over the entire range of QW widths and densities in our study.

Figure 6

The well width ($W$) vs density ($n$) phase diagram for the state of the electron system at $\nu =1/2$ in symmetric, wide GaAs QWs. The symbols have the same meaning as in Fig. 5.

Figure 7

The $d/l_B$ vs $\alpha ={\Delta }_{\mathrm{SAS}}/(e{}^2/4\pi \epsilon l_B)$ phase diagram for the observation of the $\nu =1/2$ FQHE in symmetric, wide GaAs QWs. The well widths $W$ are given in units of nm. The upper-left inset shows a typical electron charge distribution calculated self-consistently; the “interlayer distance” $d$ and full width at half maximum $\lambda$ of each layer are also indicated. The upper-right inset is the $d/l_B$ vs $\lambda /l_B$ phase diagram; for clarity, only data points for well widths $W=45$, 56, 77, and 96 nm are shown.