NMR Investigation of How Free Composite Fermions Are at $\nu =\frac{1}{2}$

NMR measurements of the electron spin polarization ($\mathcal{P}$) have been performed on a 2D electron system at and around half-filled lowest Landau level. Comparing the magnetic field and the temperature dependence of $\mathcal{P}$ to models of free and interacting composite fermions (CF), the imbalance of spin-up and spin-down CF Fermi seas is mapped as a function of Zeeman energy. Independent measurements of the CF effective mass, $g$ factor, and Fermi energy are obtained from the thermal activation of $\mathcal{P}$ in tilted fields. The filling factor dependence of the $\mathcal{P}$ for $\frac{2}{5}<\nu <\frac{2}{3}$ reveals a broken particle-hole symmetry for the partially polarized CF Fermi sea.

Figure 1

The spin polarization (left) and Knight shift (right) in the limit of vanishing temperature versus the ratio of Zeeman and Coulomb energy $\eta =\frac{{\Delta }_{\mathrm{Z}}}{{\Delta }_{\mathrm{C}}}$ (bottom) or total magnetic field (top) at $\nu =\frac{1}{2}$. The error bars are of the order of the symbol size. The dashed line is a linear fit to the data, while the solid and dotted lines are the predictions of the Hamiltonian theory and the free CF model, respectively.

Figure 2

The $T$ dependence of the spin polarization at $\nu =\frac{1}{2}$ is compared to the free CF model [Eq. (1)] with fixed $m_{\mathrm{P}}^{\star }=1.67$ (a), to the prediction of the Hamiltonian theory (b), and the best fits to a simple two level model [Eq. (4)]. The corresponding spin gaps $\Delta$ are shown in Fig. 3. For better visibility each panel contains a selection of data and every other symbol is open.

Figure 3

(a) The polarization masses obtained from fits using Eq. (1) to our $\mathcal{P}(T)$ data (Fig. 2) versus magnetic field. The solid line corresponds to $m_{\mathrm{P}}^{\star }=1.67$, while the dashed line is the prediction of the free CF model [Eq. (2)]. (b) The magnetic field dependence of the spin-flip gap as extracted from the fits in Fig. 2c. The slope directly yields the CF $g$ factor.

Figure 4

The filling factor dependence of the polarization for $\frac{2}{5}<\nu <\frac{2}{3}$ at $T=100\mathrm{m}\mathrm{K}$ and various tilt angles $\theta$ (a) and at $\theta =0°$ and temperatures between 50 and 500 mK (b). The thin lines are to guide the eye.

Figure 5

The nuclear spin lattice relaxation time $T_1$ versus temperature for the untilted samples M280 and M242 at $\nu =\frac{1}{2}$. The solid and dashed lines are $T_1(T)$ in the free CF [Eq. (5)], and the dotted line in the Hamiltonian theory (only M280).