Crossover between spin swapping and Hall effect in disordered systems

We theoretically study the crossover between spin Hall effect and spin swapping, a recently predicted phenomenon that consists of the interchange between the current flow and its spin polarization directions [M. B. Lifshits and M. I. Dyakonov, Phys. Rev. Lett. 103, 186601 (2009)]. Using a tight-binding model with spin-orbit coupled disorder, spin Hall effect, spin relaxation, and spin swapping are treated on equal footing. We demonstrate that spin swapping and spin Hall effect present very different dependencies as a function of the spin-orbit coupling and disorder strengths and confirm that the former exceeds the latter in the parameter range considered. Three setups are proposed for the experimental observation of the spin swapping effect.

Figure 1

(a) Sketch of Mott scattering by a spin-orbit coupled impurity. Electrons with an out-of-plane spin polarization pointing up have a larger probability to scatter towards the right while electrons with an out-of-plane spin polarization pointing down have a larger probability to scatter towards the left. (b) Sketch of spin swapping mediated by a spin-orbit coupled impurity (see also Ref. [46]). When the spin of the carrier lies in the scattering plane ($\mathit{k},{\mathit{k}}^{\prime }$), it experiences a magnetic field ${\mathit{B}}_{\mathrm{so}}\propto {\mathit{k}}^{\prime }\times \mathit{k}$ that depends on the direction towards which the spin is scattered. This induces a scattering-induced precession giving rise to the spin swapping.

Figure 2

(a) An illustration of the crossover between SSW and SHE: SSW produces an edge spin accumulation that is polarized along the direction of injection (green arrows) and only survives close to the interface, while SHE builds up smoothly and results in a spin accumulation with a polarization normal to the plane. (b) Schematic of the tight-binding model of the system. The circles refer to the atomic sites and $\epsilon$ stands for the on-site energy. The right and left leads are free from disorder with $\epsilon ={\epsilon }_0$. The central region is composed of a ferromagnet with polarization parallel to $\mathit{y}$ and the disordered nonmagnetic layer (symbolized by atoms having different on-site energies ${\epsilon }_{ij}={\epsilon }_0+{\gamma }_{ij}, {\gamma }_{ij}$ being the random on-site potential). The disorder-induced SOC is symbolized by the $\mathit{k}$-dependent magnetic field $\mathit{B}(\mathit{k},{\mathit{k}}^{\prime })$. The red arrows in the nonmagnetic region stand for the carriers' spins which we chose to have random directions to emphasize the effect of their coupling with the deflected momenta.

Figure 3

Mean-free path as a function of the sample length for different disorder strengths. The lower curves correspond to the normal (localization free) diffusive regime whereas a small divergence is obtained for the weaker disorder due to size effects (blue curve). The inset shows the corresponding conductances.

Figure 4

Two-dimensional mapping of the (a), (c) $x$ and (b), (d) $z$ components of the spin density (in units of ${10}^{-3}\hslash /2$) demonstrating the crossover between SSW and SHE, for (a), (b) $\alpha =0.3$ and (c), (d) $\alpha =0.6$. The disorder strength is $\mathrm{\Gamma }=1\mathrm{eV}$. The first 20 layers compose the ferromagnetic edge whose magnetization direction is indicated by the black arrow.

Figure 5

Spatial profile of the ($x,y,z$) components of the spin density calculated at the upper edge of the sample ($y=L/2$). The blue and red curves represent the spatial profile of the spin accumulation induced by SSW and SHE, respectively, while the green curve shows the relaxation of the injected spin current. The parameters are the same as in Fig. 4. The vertical line represents the FM/NM interface.

Figure 6

Two-dimensional mapping of the $x$ component of the spin density for various SOC strengths, ranging from $\alpha =0.2$ to $\alpha =0.5$. We choose $\mathrm{\Gamma }=1$ eV in all the subplots.

Figure 7

$x$ and $z$ components of the spin density calculated for one chosen point at the upper edge of the sample with the coordinates ($x_0,L/2$). SSW-induced spin accumulation first increases and then decreases when increasing the SOC parameter $\alpha$, while SHE-induced spin accumulation monotonically increases. We have verified that the same behavior holds for different values of $x_0$. Here $\mathrm{\Gamma }=1$ eV.

Figure 8

Left panels: Two-dimensional mapping of the $x$ component of the spin density for (a) $\mathrm{\Gamma }=0.6\mathrm{eV}$ and (c) $\mathrm{\Gamma }=1\mathrm{eV}$. Right panels: Spatial profile of the (b) $x$ and (d) $z$ components of the spin density calculated at the upper edge of the sample ($y=L/2$), for different disorder strengths ($\mathrm{\Gamma }=0.6$, 0.8, and 1 eV). Here, we set $\alpha =0.4$.

Figure 9

Sketches of various setups for measuring spin swapping: (a) direct injection using a three-terminal device, (b) nonlocal injection in a one-dimensional “spin transistor” configuration, and (c) direct injection in a Hall cross. The blue layers represent ferromagnets and the red layer represents a normal metal. The thick red arrow denotes the direction of injection while the small blue (red) arrows indicate the direction of the electron spins before (after) spin swapping.