Direct Measurement of the $g$ Factor of Composite Fermions

The activation gap $\Delta$ of the fractional quantum Hall states at constant fillings $\nu =2/3$ and $2/5$ has been measured as a function of the perpendicular magnetic field $B$. A linear dependence of $\Delta$ on $B$ is observed while approaching the spin-polarization transition. This feature allows a direct measurement of the $g$ factor of composite fermions which appears to be heavily renormalized by interactions and strongly sensitive to the electronic filling factor.

Figure 1

The qualitative magnetic field scaling of CF Landau levels $(n,s)$, with $n$ the Landau level index and $s=\uparrow ,\downarrow$ the spin. The zero temperature ground states at $\nu =2/3$ and $2/5$ are obtained by occupying the lowest two CF Landau levels with spin (thick black lines), with the Fermi energy (dashed line) lying midway between the ground state and the first excitation (thin line). A quantum phase transition occurs between a spin-unpolarized and fully polarized ground state at the critical magnetic field $B_c$, with a corresponding activation gap $\Delta$ vanishing linearly as $|B-B_c|$. The grey regions show the disorder broadening of the CF Landau levels. Transport measurements are affected by this disorder and the measured gap ${\Delta }_{\mathrm{transport}}$ is systematically smaller than $\Delta$. However, the broadening depends only weakly on the magnetic field, and therefore the slope of both gaps versus $B$ is the same.

Figure 2

Resistivity ${\rho }_{xx}(B)$ without illumination ($n_e=0.89\times {10}^{15}{\mathrm{m}}^{-2}$, black line) and for maximal illumination ($n_e=1.50\times {10}^{15}{\mathrm{m}}^{-2}$, grey line). Arrows mark the positions of the filling factors $\nu =2/3$ and $\nu =2/5$ at which activation measurements are performed.

Figure 3

Temperature dependence of the resistivity ${\rho }_{xx}(T)$ for filling factors $\nu =2/3$ and $\nu =2/5$ at different densities, respectively, different magnetic fields shown as an Arrhenius plot. The lines show fits of activated transport ${\rho }_{xx}(T)\propto \exp (-\Delta /2T)$ to our data. (a) Filling factor $\nu =2/3$ at magnetic fields ranging from $B=5.64\mathrm{T}$ (topmost) to $B=9.08\mathrm{T}$ (bottom curve). (b) Filling factor $\nu =2/5$ at magnetic fields ranging from $B=9.23\mathrm{T}$ (topmost) to $B=12.55\mathrm{T}$ (bottom curve).

Figure 4

Activation gaps $\Delta (B)$ at filling factors $\nu =2/3$ and $\nu =2/5$ as a function of the magnetic field. $\Delta$ is determined from the fits displayed in Fig. 3. The activation gaps are nicely fitted by a linear dependence as shown by the solid lines. The slopes of these lines are directly related to the CF $g$ factor by $0.5|g|{\mu }_{\mathrm{B}}$. Inset: $g$ factor extracted from the linear dependence of the activation energy $\Delta$. The two values of $|g|=0.92$ and 2.80 are obtained at $\nu =2/5$ and $\nu =2/3$. The dashed line shows a linear interpolation [23].