NMR study of the electron spin polarization in the fractional quantum Hall effect of a single quantum well: Spectroscopic evidence for domain formation

The authors have developed a technique to determine the electron spin polarization in a single GaAs quantum well by combining resistance-detected (RD) nuclear magnetic resonance (NMR) with conventional NMR. We use this method to study the unpolarized-polarized spin transition at filling factor $\nu =2∕3$ for small and large currents. The formation of domains with these two polarizations at the transition was observed spectroscopically. Domains with full, partial, or no polarization were also uncovered at $\nu =3∕5$ and $4∕7$. An anomalous NMR line shape around filling factor $\nu =1∕3$, 1, and $2∕3$ was measured. This pecularity of the NMR line is probably not due to skyrmions, as previously reported for $\nu =1$.

Figure 1

Plot of the longitudinal resistivity vs magnetic field $B$ for a $15\mathrm{nm}$ quantum well with a mobility of $1.6\times {10}^6{\mathrm{cm}}^2∕\mathrm{V}\mathrm{s}$ at a density of $n=1.55\times {10}^{11}{\mathrm{cm}}^{-2}$ (a). We applied a current of $I=75\mathrm{nA}$ and the magnetic field is swept slowly upwards ($dB∕dt=0.02\mathrm{T}∕\min$ at the fixed gate voltage $V_g=-0.3\mathrm{V}$). The left panel (b) shows a gate voltage sweep at fixed magnetic field $\left(9.25\mathrm{T}\right)$ around $\nu =2∕3$. The right panel (c) shows a magnetic field sweep for small currents $(I=1\mathrm{nA})$. The experiments were performed in a ${}^3\mathrm{He}∕{}^4\mathrm{He}$ dilution refrigerator at $T=50\mathrm{mK}$.

Figure 2

The solid line represents ${\rho }_{xx}$ vs RF at $\nu =1∕2$. The dashed line shows the Fourier transform of the FID signal from the substrate. The experiments were performed at $B=9.25\mathrm{T}$ and $T=55\mathrm{mK}$. The density was held constant at $n=1.12\times {10}^{11}{\mathrm{cm}}^{-2}$ for the RDNMR measurements. Inset, schematic diagram of a Hall bar inserted into an NMR coil.

Figure 3

Knight shift $K_s$ vs density for constant filling $\nu =1∕2$. The straight line indicates the expected Knight shift for a fully polarized electron system. Below $B_c\approx 10\mathrm{T}$, the system is no longer completely polarized. This “calibration curve” was used to study the spin phase transition at $\nu =2∕3$, $3∕5$, and $4∕7$. The inset shows ${\rho }_{xx}$ vs $\nu$ at $B=15\mathrm{T}$.

Figure 4

Plot of ${\rho }_{xx}$ vs filling factor around $\nu =2∕3$. The thin traces represent a sweep from higher to lower filling factor (solid line) and from lower to higher filling factor (dashed line) at $I=1\mathrm{nA}$ while the thick traces show a plot at $I=40\mathrm{nA}$. In the latter case, the small resistance peak of the spin transition develops into the huge longitudinal resistance (HLR) peak and a pronounced hysteresis is observed.

Figure 5

RDNMR/NMR technique applied to the high current spin phase transition at $\nu =2∕3$. The solid line shows ${\rho }_{xx}$ vs NMR frequency shift (polarization) and the dashed line the substrate reference signal. In the solid line, two resonance lines are observed at $\mathcal{P}=0$ and $\mathcal{P}=1$. These measurements were carried out at $B=8.1\mathrm{T}$, $I=40\mathrm{nA}$, $n=1.35\times {10}^{11}{\mathrm{cm}}^{-2}$, and $T=55\mathrm{mK}$.

Figure 6

Study of the RDNMR resonance lines at different gate voltages ($B=8.1\mathrm{T}$, $T=55\mathrm{mK}$ and $I=40\mathrm{nA}$). In (a) the gate voltage has been swept from low to high filling factor and in (b) from high to low. The main figures represent RDNMR measurements for several constant gate voltage values. The upper left panels of both figures show the gate voltage sweeps also shown in Fig. 4. The numbers indicate the position on the peak, where the RDNMR was carried out (main figure). The upper right panels compare the ${\rho }_{xx}$ values for the on- and off-resonance case.

Figure 7

RDNMR experiments at the $\nu =2∕3$ spin transition in the low current regime for a gate sweep from low to high filling factor (a) and from high to low (b). The top left panels of both figures show the gate voltage sweeps and the right panels show the off- and on-resonance signals of ${\rho }_{xx}$ (see also Fig 6). These measurements were performed at $B=8.1\mathrm{T}$ and $T=55\mathrm{mK}$.

Figure 8

Gate voltage sweeps (top panels) and RDNMR measurements (main graphs) at $B=10\mathrm{T}$ (a) and $B=12\mathrm{T}$ (b). Spin transitions for $\nu =3∕5$ and $\nu =4∕7$, respectively, are observed in transport (top panels). The expected $\mathcal{P}=1∕3$ to $\mathcal{P}=1$ transition for $\nu =3∕5$ $({\nu }_{\mathrm{CF}}=3)$ and $\mathcal{P}=1∕2$ to $\mathcal{P}=1$ for $\nu =4∕7$ $({\nu }_{\mathrm{CF}}=4)$ are measured by RDNMR (solid lines)/NMR (dashed lines).

Figure 9

Gate voltage sweeps at $B=8\mathrm{T}$ (a) and $B=15\mathrm{T}$ (b), respectively. The black dots indicate the position where RDNMR experiments were performed. An anomalous line shape was observed in the RDNMR experiments at the high filling factor side of $\nu =2∕3$ [at $\nu =0.69$ (b)], at the high filling factor side of $\nu =1∕3$ (at $\nu =0.355$, plotted as a dashed curve in (e), and at the low filling factor side of $\nu =1$ (c). A conventional line shape is seen at the low filling factor side of $\nu =1∕3$ [at $\nu =0.323$, plotted as a solid line in (e)], and of $\nu =2∕3$ (not shown), at the spin-phase transitions and at other filling factors, e.g., $\nu =0.5$ and 0.54 (f).