Composite Fermions with a Warped Fermi Contour

Via measurements of commensurability features near the Landau filling factor $\nu =1/2$, we probe the shape of the Fermi contour for hole-flux composite fermions confined to a wide GaAs quantum well. The data reveal that the composite fermions are strongly influenced by the characteristics of the Landau level in which they are formed. In particular, their Fermi contour is warped when their Landau level originates from a hole band with significant warping.

Figure 1

(a) Sample schematics. The electron-beam resist grating covering the top surface of each Hall bar arm is shown as blue stripes. (b) Dispersions for the two lowest, spin-split subbands along [110] for a wide GaAs QW, calculated self-consistently for $B=0$. (c) Calculated Fermi contours, exhibiting significant warping. (d) Magnetoresistance trace from the $[\overline{1}10]$ arm of a sample with period $a=150\mathrm{nm}$. The trace for the [110] arm (not shown) is similar. Inset: Pronounced COs are seen after subtracting the background resistance using a second-order polynomial fit. Observed resistance minima do not match the $B_{\perp }$ positions expected for a spin-degenerate, circular, Fermi contour (red tick marks) but lie in between the black solid and dotted tick marks that denote the expected COs positions for the majority and minority spin Fermi contours, respectively. All expected positions are calculated according to $2R_c/a=i-1/4$ (see text).

Figure 2

Magnetoresistance trace from the $[\overline{1}10]$ Hall bar for a GaAs QW of width $W=35\mathrm{nm}$. The two prominent minima near $\nu =1/2$ are signatures of commensurability of CF cyclotron orbit diameter with the period ($a=200\mathrm{nm}$) of the density modulation. Inset: Enlarged trace near $\nu =1/2$ shows that the positions of the minima are measurably farther from $B_{\perp }^{*}=0$ than expected for a circular Fermi contour of fully spin-polarized CFs (marked by vertical red tick marks). For comparison, we also include data (dotted blue trace) from a narrower 2DHS ($W=17.5\mathrm{nm}$) [8, 9, 11]. In this case, similar to their 2D hole counterparts near zero magnetic field, CFs also show no warping in their Fermi contour, as illustrated by commensurability minima near $\nu =1/2$, which agree well with the red tick marks that are based on a circular Fermi contour.

Figure 3

(a),(b) Evolution of the magnetoresistance in the vicinity of $\nu =1/2$ measured along [110] and $[\overline{1}10]$, respectively. Traces are shifted vertically for clarity and tilt angle $\theta$ is given for each trace. The vertical red lines mark the expected positions of the CF commensurability resistance minima if the CF cyclotron orbit were circular. (c) Crossing between the LLs with LH and HH character as a function of $\theta$. Note that the lowest LL, in which $\nu =1/2$ CFs are formed, changes from “LH” to “HH” as $\theta$ increases.

Figure 4

(a),(b) Measured values of the CF Fermi wave vectors $k_F^{*}$ along $[\overline{1}10]$ and [110], normalized to $k_{F,\text{cir}}^{*}$, as a function of $B_{||}$ for both positive and negative $B_{\perp }^{*}$. Filled squares (black) and filled circles (red) are from measurements in two different systems. (c) Relative anisotropy of the CF Fermi contour is plotted as the ratio of $k_F^{*}$ along $[\overline{1}10]$ and [110]; data for $B_{\perp }^{*}>0$ were used. (d) The geometric mean of the measured $k_F^{*}$ along $[\overline{1}10]$ and [110], divided by $k_{F,\text{cir}}^{*}$, as a measure of how much the Fermi contour deviates from an ellipse. The yellow region signifies the LL crossing.