Nature of the Spin Transition in the Half-filled Landau Level

We report transport and nuclear spin relaxation studies of a density tunable two-dimensional electron system at filling $\nu =1/2$ in tilted magnetic fields. The transition from partial to full spin polarization with an in-plane field leaves a clear signature in the resistance. Nuclear spin relaxation studies suggest that puddles of minority spins are responsible for an observed non-Korringa temperature dependence. This inhomogeneous spin polarization, similarly encountered in manganites where it strongly affects resistance, may help with understanding the spin dependent transport at $\nu =1/2$.

Figure 1

(a) Dependence of ${\rho }_{xx}(\nu =1/2)$ on $B_{\parallel }$ for various carrier densities (corresponding to $B_{\perp }=6$ to 12 T). (b) ${\rho }_{xx}(\nu =1/2)$ at $B_{\perp }=8\mathrm{T}$ as a function of $B_{\mathrm{tot}}$ for samples I (black squares) and II (red circles). Also shown are ${\rho }_{xx}$ traces which are corrected for the magneto-orbital contribution of $B_{\parallel }$ (solid lines, with arrows indicating values for ${\mathcal{P}}_{\mathrm{CF}}=1$). These traces are used to calculate the derivative $d{r}_{xx}/d{B}_{\mathrm{tot}}$ in panel (c). The normalized resistivity, $r_{xx}$, is defined as $r_{xx}={\rho }_{xx}/{\rho }_{xx}({\mathcal{P}}_{\mathrm{CF}}=1)$. All experiments were performed at a base temperature of less than 20 mK.

Figure 2

Nuclear spin relaxation measurement in titled magnetic fields with $B_{\perp }=8\mathrm{T}$ at $\nu =1/2$. (a) Top panel: The dc current sequence used for varying the nuclear spin polarization. Bottom panel: ${\rho }_{xx}$ traces in response to the nuclear spin depolarization procedure for four different $B_{\mathrm{tot}}$. (b) ${\rho }_{xx}$ and its derivative as a function of $B_{\mathrm{tot}}$ for sample I. Four points (A–D) corresponding to the traces in panel (a) are highlighted. (c) $B_{\mathrm{tot}}$ dependence of $T_1$ and $T_1^{-1}$.

Figure 3

(a) Nuclear spin relaxation rate $T_1^{-1}$ at $B_{\perp }=8\mathrm{T}$ and zero tilt as a function of electron temperature $T$; (b) $T$ dependence of $(T_{1}T{)}^{-1}$ and the upper limit of the contribution of the nuclear spin diffusion (grey line, with an offset for clearer view); (c) magnetic field dependence of $T_1^{-1}$ at the base temperature and zero tilt; (d) spin split CF band diagram with disorder effects included.