Valley Polarization and Susceptibility of Composite Fermions around a Filling Factor $\nu =\frac{3}{2}$

We report magnetotransport measurements of fractional quantum Hall states in an AlAs quantum well around a Landau level filling factor $\nu =\frac{3}{2}$, demonstrating that the quasiparticles are composite fermions (CFs) with a valley degree of freedom. By monitoring the valley level crossings for these states as a function of applied symmetry-breaking strain, we determine the CF valley susceptibility and polarization. The data can be explained well by a simple Landau level fan diagram for CFs, and are in nearly quantitative agreement with the results reported for CF spin polarization.

Figure 1

(a) Schematic of the sample, piezoelectric stack, and strain gauge. (b) Diagram showing transfer of electrons from the [100] to the [010] valley as a function of applied strain, at constant density. (c) Magnetoresistance traces at a density of $5.7\times {10}^{11}{\mathrm{cm}}^{-2}$ for different values of applied strain at $T\simeq 50\mathrm{mK}$. The applied strain for each trace, from bottom to top, is $-0.2$, 0.0, 0.3, 0.6, 1.1, 3.4, 3.6, $4.1\times {10}^{-4}$ with zero strain shown in gray (red). The traces are vertically offset for clarity. The area in the shaded box is magnified in Fig. 2.

Figure 2

High-field magnetoresistance data for different values of applied strain at $T\simeq 50\mathrm{mK}$. The thick trace shown in gray (red) denotes minimum strain. Solid vertical lines mark FQHE minima; positions are calculated from the periodicity of the low field Shubnikov–de Haas oscillations. The numbers shown on the right-hand side give the value of applied strain for each trace.

Figure 3

(a) Energy fan diagram for electrons showing Landau level crossings as a function of strain for constant magnetic field. (b) Piezoresistance trace taken at $\nu =10$ at $T\approx 50\mathrm{mK}$. Note that the small diamonds shown in (a) are not resolved in this measurement. (c) Piezoresistance traces taken at FQHE minima at $T\approx 50\mathrm{mK}$. Traces have been offset for clarity, and data for $\nu =\frac{8}{5}$ and $\frac{7}{5}$ have been magnified vertically to improve contrast. (d) Proposed energy fan diagram for CF Landau levels.

Figure 4

(a) Valley susceptibility ${\chi }_v$, normalized to its band value, for various integer (triangles) and fractional (squares) filling factors. The horizontal dashed line represents ${\chi }_v$ for electrons measured from low field IQHE minima ($\nu =2,8,10$). (b) Ratio of polarization energy ($\Delta E$) to Coulomb energy ($e^2/4\pi \epsilon \ell$), for various FQH states, versus inverse CF filling factor. The polarization energy is the amount of splitting (valley or spin) necessary to fully polarize the system. Ratios for AlAs valley polarization are shown as (blue) squares, measurements of GaAs spin polarization (Ref. 12) as black stars, and calculations of spin polarization (Ref. 13) as (red) circles. Note that the calculations are for states around $\nu =\frac{1}{2}$, which are equivalent to states around $\nu =\frac{3}{2}$. The dashed line represents the theoretical boundary between the partially and fully polarized phases of the FQHE states. Note that the measured polarization energies for both valley and spin systems are higher than the calculated values.