Geometric resonance of four-flux composite fermions

Two-dimensional interacting electrons exposed to strong perpendicular magnetic fields generate emergent, exotic quasiparticles phenomenologically distinct from electrons. Specifically, electrons bind with an even number of flux quanta, and transform into composite fermions (CFs). Besides providing an intuitive explanation for the fractional quantum Hall states, CFs also possess Fermi-liquid-like properties, including a well-defined Fermi sea, at and near even-denominator Landau-level filling factors such as $\nu =1/2$ or $1/4$. Here, we directly probe the Fermi sea of the rarely studied four-flux CFs near $\nu =1/4$ via geometric resonance experiments. The data reveal some unique characteristics. Unlike in the case of two-flux CFs, the magnetic field positions of the geometric resonance resistance minima for $\nu <1/4$ and $\nu >1/4$ are symmetric with respect to the position of $\nu =1/4$. However, when an in-plane magnetic field is applied, the minima positions become asymmetric, implying a mysterious asymmetry in the CF Fermi sea anisotropy for $\nu <1/4$ and $\nu >1/4$. This asymmetry, which is in stark contrast to the two-flux CFs, suggests that the four-flux CFs on the two sides of $\nu =1/4$ have very different effective masses, possibly because of the proximity of the Wigner crystal formation at small $\nu$.

Figure 1

GR features for ${}^4\mathrm{CFs}$ near $\nu =1/4$. (a) Lateral surface superlattice of period $a$, inducing a periodic density perturbation in the 2DES. When the ${}^4\mathrm{CFs}$' cyclotron orbit becomes commensurate with the period of the perturbation, the $i=1$ GR occurs. (b) Magnetoresistance trace revealing GR features near $\nu =1/4$ and $\nu =1/2$. Inset: The L-shaped Hall bar along [110] and $\left[\overline{1}10\right]$ directions used for the measurements. (c) Magnetoresistance near $\nu =1/4$ demonstrating the $i=1{}^4\mathrm{CF}$ GR features, resistance minima flanking $\nu =1/4$. Black solid and orange dashed lines mark the expected positions for the $i=1$ GR for fully spin-polarized ${}^4\mathrm{CFs}$ with circular Fermi contour assuming $k_F^{*}=\sqrt{4\pi n}$ and $k_F^{*}=\sqrt{4\pi n}\times \sqrt{B/B_{\nu =1/4}}$, respectively. The extra minimum near $B_{\perp }=29.75$ T stems from the $i=2$ GR.

Figure 2

Tilt evolution of the ${}^4\mathrm{CF}$ GR features near $\nu =1/4$ along (a) [110] and (b) $\left[\overline{1}10\right]$ directions. The insets show the orientation of the Hall bars, and the ${}^4\mathrm{CF}$ cyclotron orbit for the $i=1$ GR. Magnetoresistance traces are vertically offset for clarity; the tilt angle $\theta$ is given for each trace. The expected positions for the $i=1{}^4\mathrm{CF}$ GRs are marked with vertical dotted lines assuming that $k_F^{*}=\sqrt{4\pi n}$. In both panels, the scale for the applied external field $B_{\perp }$ is shown on the bottom while the top scale is the effective magnetic field $B_{\perp }^{*}$ experienced by the ${}^4\mathrm{CFs}$.

Figure 3

(a) Normalized ${}^4\mathrm{CF}$ Fermi wave vectors $k_F^{*}$ from the positions of $B_{\perp }^{*}$ for the primary ${}^4\mathrm{CF}$ GR minima along the [110] and $\left[\overline{1}10\right]$ directions. Open and filled symbols represent the data for $\nu <1/4$ and $\nu >1/4$, respectively. The typical error bar for the data points is of the order of $3\%$. (b) Anisotropy of the ${}^4\mathrm{CF}$ Fermi sea for $\nu <1/4$ (open symbols) and $\nu >1/4$ (filled symbols) deduced from dividing the (interpolated) measured values of $k_F^{*}$ along $\left[\overline{1}10\right]$ by those along [110]. Orange lines correspond to the theoretical estimate of the anisotropy using Eq. (1) assuming $m_{\parallel }=2.5,1.9,1.4$, and 1.0 (see text). Inset: Geometric mean of the measured values of $k_F^{*}$ (${\tilde{k}}_F^{*}=\sqrt{k_F^{*}\left[110\right]\times k_F^{*}\left[\overline{1}10\right]}$) along the two directions normalized to $k_{F0}^{*}$ for $\nu <1/4$ and $\nu >1/4$ denoted by solid and dashed lines, respectively. Up to the highest $B_{\parallel }, {\tilde{k}}_F^{*}/k_F^{*}\simeq 1$ to within $5\%$, implying that the measured Fermi seas are nearly elliptical.

Figure 4

Comparison between the evolution with $B_{\parallel }$ of the calculated Fermi contour of electrons [(a)–(d)] and measured Fermi contour of ${}^4\mathrm{CFs}$ near $\nu =1/4$ [(e)–(h)]. For simplicity, in (a)–(d) only the majority-spin contour is shown. In (e)–(h), solid and dotted contours denote the ${}^4\mathrm{CF}$ Fermi contours for $\nu <1/4$ and $\nu >1/4$, respectively. Even though the electron Fermi sea completely splits at large $B_{\parallel }$, the ${}^4\mathrm{CF}$ Fermi sea near $\nu =1/4$ remains intact.