Fractional Quantum Hall Effect and Wigner Crystal of Interacting Composite Fermions

In two-dimensional electron systems confined to GaAs quantum wells, as a function of either tilting the sample in a magnetic field or increasing density, we observe multiple spin-polarization transitions of the fractional quantum Hall states at filling factors $\nu =4/5$ and $5/7$. The number of observed transitions provides evidence that these are fractional quantum Hall states of interacting two-flux composite fermions. Moreover, the fact that the reentrant integer quantum Hall effect near $\nu =4/5$ always develops following the transition to full spin polarization of the $\nu =4/5$ fractional quantum Hall state links the reentrant phase to a pinned ferromagnetic Wigner crystal of composite fermions.

Figure 1

Schematic figures illustrating two different explanations of the RIQHS near $\nu =4/5$. (a) The hole Wigner crystal picture. The electrons at $\nu \sim 4/5$ are equivalent to holes at ${\nu }^h\sim 1/5$. These holes condense into a liquid phase when the short-range interaction dominates (left), and into a crystal phase when the long-range interaction dominates in thicker 2DESs (right). The gray background represents electrons, and white circles represent the holes. (b) The ${}^2\mathrm{CF}$ Wigner crystal picture. The electrons at $\nu \sim 4/5$ are equivalent to ${}^2\mathrm{CFs}$ at $|{\nu }^{\mathrm{CF}}|\sim 4/3$. There is one fully filled, spin-up $\mathrm{\Lambda }$-level (the gray background), and the rest of the ${}^2\mathrm{CFs}$ can either be spin-down (red) and form a liquid phase when the Zeeman energy ($E_Z$) is small (left panel), or be spin-up (blue) at large $E_Z$ and form a ferromagnetic crystal phase of ${}^2\mathrm{CFs}$ (right panel).

Figure 2

Longitudinal ($R_{xx}$) and Hall ($R_{xy}$) magnetoresistance traces for 2D electrons confined to a 65-nm-wide GaAs QW near $\nu =3/4$ as a function of (a) increasing density, and (b) tilting the sample in the magnetic field. The density $n$ (in units of ${10}^{11}{\mathrm{cm}}^{-2}$) or tilting angle $\theta$ for each trace is indicated, and traces are shifted vertically for clarity. The top ($B_{\perp }$) axis in (a) is valid only for the $n=1.00\times {10}^{11}{\mathrm{cm}}^{-2}$ trace. Note also that the angular-dependent traces in (b) were taken in a dilution refrigerator with a slightly warmer base temperature ($T\simeq 30$ mK) compared to the one used to take traces in (a) ($T\simeq 25\mathrm{mK}$). (c) Energy gap of the $\nu =4/5$ FQHS as a function of $n$. (d) Arrehnius plot of $R_{xx}$ vs $1/T$ at $n=0.93$ and $1.41\times {10}^{11}{\mathrm{cm}}^{-2}$.

Figure 3

(a) $R_{xx}$ measured in a 60-nm-wide QW near $\nu =3/4$, at a fixed density $n=0.44\times {10}^{11}{\mathrm{cm}}^{-2}$ and different tilting angles $\theta$. (b) $R_{xx}$ for a 65-nm-wide QW near $\nu =5/4$, at $\theta =0°$ and different densities $n=0.86$ to $1.94\times {10}^{11}{\mathrm{cm}}^{-2}$. (c),(d) Schematic plots showing multiple configurations of the $\nu =4/5$ and $6/5$, and $5/7$ and $9/7$ FQHSs with different spin-polarizations.

Figure 4

Summary of the spin-polarization energy in units of the Coulomb energy, $E_Z/V_C$, at different filling factors ($\nu$). Dotted lines are guides to the eye. All data points were measured in a symmetric 60-nm-QW. The transitions at $\nu =2/3$, $3/5$, and $4/7$ were measured in perpendicular magnetic field by changing $n$, and the rest at a fixed density $n=0.44$ by changing $\theta$. For each filling, only the (last) transition into a fully spin-polarized configuration is shown.