Effective spin Hall properties of a mixture of materials with and without spin-orbit coupling: Tailoring the effective spin-diffusion length

We study theoretically the effective spin Hall properties of a composite consisting of two materials with and without spin-orbit (SO) coupling. In particular, we assume that SO material represents a system of grains of radius, $a$, and density, $n$, in a matrix with no SO. We calculate the effective spin Hall angle, $\theta^{\scriptstyle{SH}}_{\scriptstyle{\text{eff}}}$, and the effective spin diffusion length, $\lambda_{\scriptstyle{\text{eff}}}$, of the mixture. Our main qualitative finding is that, if the bare spin diffusion length, $\lambda$, is much smaller than $a$, then $\lambda_{\scriptstyle{\text{eff}}}$ is strongly {\em enhanced}, well beyond $\lambda/(na^3)^{1/2}$, which can be expected from purely ``geometrical" consideration. The physical origin of this additional enhancement is that, with small diffusion length, $\lambda \ll a$, the spin current mostly flows {\em around the grain} without suffering much loss. We also demonstrate that the voltage, created by a spin current, is sensitive to a very weak magnetic field directed along the spin current, and even reverses sign in a certain domain of fields. The origin of this sensitivity is that the spin precession, caused by magnetic field, takes place outside the grains where SO is absent.