Manifestation of two-channel nonlocal spin transport in the shapes of Hanle curves

The dynamics of charge-density fluctuations in a system of two tunnel-coupled wires contains two diffusion modes with dispersion $i\omega =D{q}^2$ and $i\omega =D{q}^2+\frac{2}{{\tau }_t}$, where $D$ is the diffusion coefficient and ${\tau }_t$ is the tunneling time between the wires. The dispersion of corresponding spin-density modes depends on magnetic field as a result of the spin precession with Larmour frequency ${\omega }_L$. The presence of two modes affects the shape of the Hanle curve describing the spin-dependent resistance $R$ between the ferromagnetic strips covering the nonmagnetic wires. We demonstrate that the relative shapes of the $R\left({\omega }_L\right)$ curves, one measured within the same wire and the other measured between the wires, depends on the ratio ${\tau }_t/{\tau }_s$, where ${\tau }_s$ is the spin-diffusion time. If the coupling between the wires is local, i.e., only at the point $x=0$, then the difference of the shapes of intrawire and interwire Hanle curves reflects the difference in statistics of diffusive trajectories, which “switch” or do not switch near $x=0$. When one of the coupled wires is bent into a loop with a radius $a$, the shape of the Hanle curve reflects the statistics of random walks on the loop. This statistics is governed by the dimensionless parameter $a/\sqrt{D{\tau }_s}$.

Figure 1

(a) Schematic view of a four-terminal device used for nonlocal spin-transport measurements [17]. Epitaxial ZnO film of thickness $200\mathrm{nm}$, deposited on a sapphire substrate, is spaced from a NiFe layer by a thin MgO barrier; (b) and (c) Hanle curves measured for the channel length $L=90$ and $650\mathrm{nm}$, respectively. The insets show the theoretical fits plotted from Eqs. (17), (18), (24), and (26). The values ${\tau }_s$ used for both fits are the same, so that the dimensionless lengths $\mathcal{L}=L/\sqrt{4D{\tau }_s}$ differ seven times.

Figure 2

(a) Two-channel spin-transport device. The injector is located in the left channel. Two detectors in the left and right channels are located at the same distance $L$ from the injector. An electron reaches the first detector by diffusion and the second detector by a combined diffusion-tunneling process. (b) In contrast to (a), two channels are coupled locally at the point $x=0$; (c) the second wire is bent into a loop. Electron diffusion trajectories encircle the loop several times before the spin polarization is “forgotten”; (d) two branches, $E\left(k_x\right)$, of the energy spectrum of the tunnel-coupled wires. At small momenta, the splitting ${\Delta }_t$ is determined by tunneling, while at large $k_x$ the splitting ${\Delta }_s$ is dominated by the spin-orbit coupling in the wires.

Figure 3

Hanle curves ${\mathcal{R}}_{11}\left({\omega }_L\right)$ (blue) and ${\mathcal{R}}_{12}\left({\omega }_L\right)$ (green) measured by two detectors, one located in the same wire as the injector and the other located in the neighboring wire. Dashed curves are plotted directly from Eq. (19) using the definition (17); the solid curves are plotted from the asymptotic expansions, ${\mathcal{R}}_{11}$ is given by Eq. (18) and ${\mathcal{R}}_{12}$ is given by Eq. (22). In all graphs, the tunneling time ${\tau }_t$ is $10{\tau }_s$. The three panels correspond to different dimensionless lengths, $\mathcal{L}=L/\sqrt{4D{\tau }_s}$: $\mathcal{L}=3$ (a), $\mathcal{L}=0.45$ (b), and $\mathcal{L}=0.1$ (c). All curves for ${\mathcal{R}}_{12}$ are multiplied by ${\tau }_t/{\tau }_s=10$.

Figure 4

The difference in the diffusion trajectories in a single wire and in two wires coupled via a bridge manifests itself in the shapes of the Hanle curves. (a) The Hanle curves ${\mathcal{R}}_{11}\left({\omega }_L\right)$ (blue) and ${\mathcal{R}}_{12}\left({\omega }_L\right)$ (green, dashed) are plotted from Eqs. (38) and (39) for a small dimensionless length $\mathcal{L}=L/{\left(4D{\tau }_s\right)}^{1/2}=0.1$. The solid green curve is the Lorentzian asymptote, Eq. (40). The inset shows a comparison of ${\mathcal{R}}_{12}\left({\omega }_L\right)$ for the case of local tunneling (yellow), Eq. (40), and for the case of homogeneous tunneling (brown), Eq. (30); (b) same geometry as in (a). Nondiagonal components of nonlocal resistance corresponding to the perpendicular magnetizations of injector and detector are plotted; (c) the Hanle curve ${\mathcal{R}}_{11}\left({\omega }_L\right)$ (blue) and ${\mathcal{R}}_{12}\left({\omega }_L\right)$ (green) are plotted for the dimensionless length $\mathcal{L}=0.45$. The shapes are much closer than in (a).

Figure 5

(a) Evolution of the Hanle curves $\mathcal{R}\left({\omega }_L\right)$ with radius $a$. The curves are plotted from Eq. (46) for three values of the dimensionless loop radius ${\lambda }^{-1}=a/{\left(D{\tau }_s\right)}^{1/2}$. Dotted, dashed, and solid lines correspond to $\lambda =1.7$, 1, and $0.7$, respectively. (b) Same as (a) for the high-temperature domain $\lambda =0.35$ (dotted), $0.32$ (dashed), and $0.27$ (solid).