Figure 1

(a) Calculated 2D electron Fermi contours for a 30-nm-wide GaAs quantum well when $B_{\parallel }$ is applied along the $\left[110\right]$ direction. The majority- (minority-) spin contour is given by solid (dotted) lines. (b) Schematic of the L-shaped Hall bar. The arms of the Hall bar, oriented along the $\left[110\right]$ and $\left[\overline{1}10\right]$ directions, are covered with stripes of negative electron-beam resist. Part of the Hall bar is intentionally left unpatterned, and its magnetoresistance ($R_{\text{ref}}$) is used to measure Shubnikov-de Haas oscillations. (c) The geometry of the Hall bar is designed to use the commensurability of the ballistic cyclotron orbits with the period of the potential modulation induced by the stripes to probe the size of the Fermi wave vector along the $\left[110\right]$ and $\left[\overline{1}10\right]$ directions directly. Note that the real-space orbits are rotated by ${90}^{\circ }$ with respect to the Fermi contours.[15]Reuse & Permissions

Figure 2

Low-field magnetoresistance measured in the [110] direction at $B_{\parallel }=0$ showing pronounced COs. Vertical lines mark the positions of the expected COs resistance minima according to $2R_C/a=i-1/4$. Shubnikov-de Haas oscillations are visible on top of the COs above 0.2 T. The large number of COs minima (up to $\simeq 17$) attests to the very high quality of the sample and the periodic modulation. Inset: An example SEM image of the 200-nm-period grating of negative electron-beam resist.Reuse & Permissions

Figure 3

(a), (b) Magnetoresistance data for the patterned sections of the L-shaped Hall bar in the $\left[110\right]$ and $\left[\overline{1}10\right]$ directions at different values of $B_{\parallel }$. (c), (d) Normalized Fourier transform spectra of the COs data shown in (a) and (b), respectively. The anticipated $B_{\parallel }=0$ COs frequency, based on a spin-degenerate, circular Fermi contour, is marked with dashed lines. The low-frequency parts of the spectra (below the vertical dotted lines) are severely affected by the Hamming window used in the Fourier analysis and are shown here suppressed by a factor of 100.Reuse & Permissions

Figure 4

Summary of the Fermi wave vector ($k_F$) peak deduced from the positions of the COs’ FT spectra for the two Hall bar arms. Filled circles represent $k_F\perp B_{\parallel }$ and open circles the $k_F\parallel B_{\parallel }$ experimental data. Values of $k_F$ in the two directions calculated using $V_{xc}=0$ and $V_{xc}\ne 0$ are given by red and blue lines, respectively. The calculated $k_F$ corresponding to the majority-spin species are plotted using solid lines, and the minority-spin species using dotted lines.Reuse & Permissions

Figure 5

(a) Shubnikov-de Haas oscillations measured in the unpatterned (reference) region of the Hall bar as $B_{\parallel }$ increases. (b) Fourier transform spectra of the SdH oscillations as a function of $B_{\parallel }$. The dashed line shows the expected position of the spin-unresolved FT peak. The signal in the region to the left of the vertical dotted line is shown suppressed. (c) Summary of the SdH FT peak positions normalized to $f_{\text{SdH}}^{\circ }$, the frequency at $B_{\parallel }=0$. Open squares represent the measured frequencies. The frequencies predicted by the calculations with $V_{xc}=0$ and $V_{xc}\ne 0$ are shown using red and blue lines, respectively. For each calculation, the solid lines represent the majority-spin density, and the dotted lines the minority-spin density. Inset: Fermi contours corresponding to the $V_{xc}=0$ (red) and $V_{xc}\ne 0$ (blue) calculations at $B_{\parallel }=20$ T.Reuse & Permissions