Scaling Law of Confined Spin-Hall Effect

We incorporate quantum size effect to investigate the extrinsic spin-Hall effect in ultrathin metal films. A Lippmann-Schwinger formalism based theoretical method, accounting for quantum confinement and surface roughness scattering, is developed to calculate both spin-Hall and longitudinal resistivities and spin-Hall angle. The presence of quantum confinement gives rise to a linear relation $\rho_{sH}=\alpha \rho+\beta$ between the extrinsic spin-Hall resistivity $\rho_{sH}$ and longitudinal charge resistivity $\rho$. The linear term $\alpha\rho$ originates from side jump, and the constant $\beta$ is due to skew scattering. This deviates significantly from the commonly accepted scaling law $\rho_{sH}=a\rho^2+b\rho$ in a bulk conductor. Thus we call for cautious interpretation of experimental data when applying the scaling law.