Competition between a Fractional Quantum Hall Liquid and Bubble and Wigner Crystal Phases in the Third Landau Level

Magnetotransport measurements were performed in an ultrahigh mobility $\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}/\mathrm{A}\mathrm{l}\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}$ quantum well of density $\sim 3.0\times {10}^{11}{\mathrm{c}\mathrm{m}}^{-2}$. The temperature dependence of the magnetoresistance $R_{xx}$ was studied in detail in the vicinity of $\nu =\frac{9}{2}$. In particular, we discovered new minima in $R_{xx}$ at a filling factor $\nu \simeq 4\frac{1}{5}$ and $4\frac{4}{5}$, but only at intermediate temperatures $80\lesssim T\lesssim 120\mathrm{m}\mathrm{K}$. We interpret these as evidence for a fractional quantum Hall liquid forming in the $N=2$ Landau level and competing with bubble and Wigner crystal phases favored at lower temperatures. Our data suggest that a magnetically driven insulator-insulator quantum phase transition occurs between the bubble and Wigner crystal phases at $T=0$.

Figure 1

Longitudinal resistance $R_{xx}$ as a function of magnetic field measured with an excitation current of 4 nA. In panel (a), $R_{xx}$ is shown over the whole field range measured between 0 and 16 T, with the principal filling factor denoted with arrows. Panels (b) and (c) show the same data, but rescaled such as to show the smaller features. New developing minima in panel (c) are labeled with their tentative fractional assignment, $\frac{12}{7}$, $\frac{13}{8}$, and $\frac{15}{11}$. The RIQHE state near $\nu =\frac{9}{2}$ is labeled with the letter $B$.

Figure 2

Magnetotransport $R_{xx}$ at various temperatures. The upper (lower) panel shows the data taken below (above) $T\lesssim 65\mathrm{m}\mathrm{K}$. The half-integer filling factors are indicated with arrows. The position of the $\nu =4\frac{1}{5}$ and $\nu =4\frac{4}{5}$ fillings are indicated in the lower panel, together with the bubble states in the upper panel, labeled with the letter $B$. The horizontal arrows indicate the magnetic field range spanned by the $N=2$, three Landau levels. The upper inset shows an Arrhenius plot of the temperature dependence of the resistance maximum labeled “peak1”. The inset in the lower panel shows an enlarged version around $\nu =4\frac{1}{5}$ at $T\sim 22\mathrm{m}\mathrm{K}$, with I1 and I2 denoting the two insulating phase.