Current-induced nuclear spin depolarization at Landau level filling factor $\nu$ $=$ 1/2

In this work, we demonstrate that significant changes in electron temperature and nuclear spin polarization can be created by applying an electric current in a two-dimensional electron system at Landau level filling factor $\nu =1/2$. The current induced effects on nuclear spins can be attributed to electron heating and the efficient coupling between the nuclear and electron spin systems at $\nu =1/2$. The electron temperature, elevated by the current, can be measured with a thermometer based on the measurement of the nuclear spin relaxation rate. The electron temperature is found to be proportional to the square root of the current density at $\nu =1/2$. Electron spin transitions at $\nu =2/3$ and $1/2$ are utilized for the measurement of the current induced changes in nuclear spin polarization. Consistent results are obtained from these two different methods of nuclear magnetometry. The finite thickness of the electron wave function is found to be important for evaluation of the nuclear spin polarization even in a narrow quantum well. The nuclear spin polarization follows a Curie law dependence on the electron temperature. This work also allows us to evaluate the electron $g$ factor in high magnetic fields as well as the polarization mass of composite fermions.

Figure 1

(a) Energy level diagram for filling factor 2/3. The spin transition field $B_{\mathrm{tr}}$ is marked with the dotted line. When $B<B_{\mathrm{tr}}$, the ground state is spin unpolarized with the composite fermion Landau levels (0,$\uparrow$) and (0,$\downarrow$) occupied, while for $B>B_{\mathrm{tr}}$, the two filled levels are (0,$\uparrow$) and (1,$\uparrow$), and the ground state becomes fully polarized; (b) spin transition for a noninteracting and disorder-free composite fermion system at $\nu =1/2$. The density of states solely depends on the composite fermion effective mass. Transition from partial to full spin polarization takes place when the Zeeman energy surpasses the Fermi energy.

Figure 2

(Top) 2D plot of $R_{\mathrm{xx}}$ in the ($\nu ,B$) plane with $I_{\mathrm{ac}}=1$ nA at the base temperature. Regions labeled with $\uparrow \downarrow$ and $\uparrow \uparrow$ correspond to the incompressible $\nu =2/3$ fractional quantum Hall state with spin polarization $\mathcal{P}=0$ and 1, respectively. The two ground states of the $\nu =3/5$ quantum Hall fluid with different spin polarizations are also shown. (Bottom) Traces of $R_{\mathrm{xx}}$ and $R_{\mathrm{xy}}$ as a function of $B$ with $\nu$ fixed at 2/3. Dissipative transport takes place in the phase transition region.

Figure 3

(a) Measurement sequence for nuclear magnetometry based on the spin transition at $\nu =2/3$. (b) 2/3 transition peaks for ${\nu }_{\mathrm{rest}}=1/2$ measured at $T=17$, 54, and 134 mK. (c) Temperature dependence of the 2/3 spin transition field $B_{\mathrm{tr}}$ for ${\nu }_{\mathrm{rest}}$ $=$ 1/2. Plotted in solid diamonds are $B_{\mathrm{tr}}$ values extracted from single peak fits. Open circles represent the corrected $B_{\mathrm{tr}}$ values in which the offsets from the effects irrelevant to nuclear spins are removed. The dotted line marks the expected Curie law dependence of the transition field. See Sec. 4d for more details.

Figure 4

(a) Measurement of the spin transition position at $\nu =2/3$ for ${\nu }_{\mathrm{rest}}=1/2$ in titled magnetic fields. (b) 2/3 transition fields plotted as a function of ${\cos }^2\theta$ (squares) and its fit to Eq. (13) with finite thickness effect included (dotted lines). Also plotted for comparison is the expected angular dependence of $B_{\mathrm{tr}}$ for a zero-thickness 2DES (solid line. (c) $B_{\mathrm{tr}}$ as a function of $B_{\mathrm{extra}}$ (squares) and its linear fit (line), which can serve as a calibration curve for the nuclear magnetometry based on the 2/3 transition. The linear fit gives $\Delta B_{\mathrm{tr}}\approx -1.24\Delta B_N$ for the 16-nm quantum well. This is considerably different from $\Delta B_{\mathrm{tr}}\simeq -2\Delta B_N$ expected for the 2DES with zero thickness.

Figure 5

(a) Longitudinal resistance $R_{\mathrm{xx}}$ at $\nu =1/2$ as a function of $B_{\mathrm{tot}}$ with the perpendicular field $B$ fixed at 6, 7, 8, 10, and 12 T. (b) $R_{\mathrm{xx}}$ plotted as a function of the extra Zeeman field. Triangles are experimental points obtained by directly converting $B_{\mathrm{tot}}$ to $B_{\mathrm{tot}}-B$. The line is the result of subtracting the contribution from the orbital effect of the in-plane magnetic field. (c) An example of the time evolution of $R_{\mathrm{xx}}$ (open dots) as a consequence of nuclear spin relaxation at $\nu =1/2$ and the corresponding change in the Zeeman field $\Delta E_Z/\left(g_e{\mu }_B\right)$ (line) as a function of time. The conversion from $R_{\mathrm{xx}}$ to $\Delta E_Z/\left(g_e{\mu }_B\right)$ is based on the calibration data in (b).

Figure 6

(Top) The phase transition diagrams of $R_{\mathrm{xx}}$ (defined as $\frac{d{V}_{\mathrm{xx}}}{dI}$) plotted in the (${\nu }_{\mathbf{meas}}$, $B$) plane with ${\nu }_{\mathrm{rest}}=1/2$ and an applied $I_{\mathrm{dc}}$ of 0, 5, 20, and 400 nA from left to right. The detection filling factor ${\nu }_{\mathrm{meas}}$ is varied from 0.575–0.755 for each of the diagrams. Bottom: (a) the $\nu =2/3$ spin transition peaks for an applied $I_{\mathrm{dc}}=0$, 5, 20, and 400 nA during the time the system is left at ${\nu }_{\mathrm{rest}}=1/2$. (b) The dc current dependence of the phase transition field $B_{\mathrm{tr}}$ at $T=18$, 45, and 71 mK. See text for details.

Figure 7

The ac current dependence of the phase transition field $B_{\mathrm{tr}}$ for ${\nu }_{\mathrm{rest}}=1/2$ at various bath temperatures. The values of $B_{\mathrm{tr}}$ are extracted from the spin transition peaks of $R_{\mathrm{xx}}$ with $I_{\mathrm{ac}}=1$ nA. The lock-in amplifier is locked to $I_{\mathrm{ac}}$. The frequency of the second ac current $I_{\mathrm{ac}}2$ is chosen to be different from $I_{\mathrm{ac}}$ so that it does not interfere with the measurement of the differential resistance.

Figure 8

The current induced effects on transport properties at $\nu =1/2$ measured at the various bath temperatures (less than 20 to 194 mK). The differential resistance $R_{\mathrm{xx}}$ is plotted as a function of the second ac current $I_{\mathrm{ac}}2$. A perpendicular magnetic field of $B=8$ T was applied in all of the measurements.

Figure 9

(a) An example of the measurement sequence for extracting the nuclear spin relaxation rate $1/T_1$. A small ac current, $I_{\mathrm{ac}}$ is always turned on to measure the (differential) resistance $R_{\mathrm{xx}}$, and the second ac current $I_{\mathrm{ac}2}$ (usually much larger than $I_{\mathrm{ac}}$) is turned on and off alternately to drive the nuclear spin polarization into different values. An averaging over multiple runs and small changes in the nuclear spin polarization were used in the measurements in order to obtain reliable $1/T_1$ data. (b) Temperature dependence of $1/T_1$. The squares are the experimental data and the solid line is a linear fit down to $\sim 45$ mK. In the lower inset, $1/T_1$ at the base temperature is plotted as a function of $I_{\mathrm{ac}}2$. (c) $I_{\mathrm{ac}}2$ dependence of the electron temperature $T_e$ (diamonds). Also plotted for comparison are the fits to the square root of the current, i.e., $T_e\propto \sqrt{I_{\mathrm{ac}}2}$ (thicker line), as well as the predicted values of the parameter-free hydrodynamic model (thinner, red line).

Figure 10

The $\nu =2/3$ transition field $B_{\mathrm{tr}}$ for ${\nu }_{\mathrm{rest}}=1/2$ plotted as a function of electron temperature $T_e$. The upper inset shows the raw data of the $I_{\mathrm{ac}}2$ dependence of $B_{\mathrm{tr}}$ measured at the base temperature. The electron temperature is converted from $I_{\mathrm{ac}}2$ using the thermometry based on the nuclear spin relaxation rates shown in Fig. 9. In the lower inset, the corresponding $|B_N|/B$, namely, the ratio between the magnitude of the nuclear and external magnetic field is plotted as a function of $1/T_e$. It follows the Curie law $B_N=\frac{0.87\mathrm{mK}}{g_e}\frac{B}{T_e}$.

Figure 11

Comparison of the two methods of nuclear magnetometry, which are based on the spin transitions at $\nu =2/3$ (open symbols) and $\nu =1/2$ (solid squares). The ac current ($I_{\mathrm{ac}}2$) induced change in nuclear field $\Delta B_N$ at $\nu =1/2$ is plotted as a function of $I_{\mathrm{ac}}{2}^{-1/2}$, which is proportional to $1/T_e$. See text for details.