Spin-current absorption by inhomogeneous spin-orbit coupling

We investigate the spin-current absorption induced by an inhomogeneous spin-orbit coupling due to impurities in metals. We consider the system with spin currents driven by the electric field or the spin accumulation. The resulting diffusive spin currents, including the gradient of the spin-orbit coupling strength, indicate the spin-current absorption at the interface, which is exemplified with experimentally relevant setups.

Figure 1

Diagrams of the spin current for the total Hamiltonian. Bold lines denote the total Green's function. A wavy line denotes the vector potential [$\mathit{A}(\mathit{q},\Omega )$]. A broken line and double broken line represent the SOC strength [${\lambda }_{\mathrm{SO}}\left(\mathit{u}\right)$] and the impurity potential [$U\left(\mathit{p}\right)$], respectively.

Figure 2

Diagrams of the leading contribution of the local spin current induced by the inhomogeneity of the SOC strength. Solid lines denote the free Green's function. Broken lines denote the SOC strength [${\lambda }_{\mathrm{SO}}\left(\mathit{u}\right)$] and double broken lines denote the impurity potential [$U\left(\mathit{p}\right)$]. Wavy lines represent the vector potential [$A_l(\mathit{q},\Omega )$].

Figure 3

Diagrams of the leading contribution of the diffusive spin current induced by the inhomogeneity of the SOC strength. (a) Contributions with the vertex correction to the spin-current operator. (b) Contributions with the vertex correction to the vector potential. (c) The vertex correction is defined as the summation of the ladder diagram with respect to the normal impurity potential.

Figure 4

Schematic illustration of the spin-current absorption by the input spin-polarized current. The external electric field ${\mathit{E}}_{\mathrm{em}}$ is applied to a ferromagnetic metal and therefore a spin-polarized current ${\mathit{j}}_s^z$ flows in the direction of the $y$ axis. The nonmagnetic metal attached to the ferromagnetic metal has a strong SOC ${\lambda }_{\mathrm{SO}}$ due to impurities. The diffusive spin currents ${\mathit{j}}_s^{x,y}$ are induced perpendicularly to the interface and then decay exponentially in an oscillatory fashion.

Figure 5

Schematic illustration of the absorption of spin currents driven by the spin accumulation. The arrows in the bottom metal denote input spin currents ${\left(j_{s,\mathrm{in}}^z\right)}_y$ by the spin accumulation. In the top metal, the left arrows are normal diffusion of the input spin currents ${\left(j_{s,\mathrm{in}}^z\right)}_z$, which is irrelevant to the variation of a SOC, whereas the middle and the right arrows are diffusion of spin currents ${\left(j_{s,\mathrm{abs}.}^y\right)}_z$ induced in the vicinity of the interface, relevant to the difference of SOCs.