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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 $ρ_{sH}=αρ+β$ between the extrinsic spin-Hall resistivity $ρ_{sH}$ and longitudinal charge resistivity $ρ$. The linear term $αρ$ originates from side jump, and the constant $β$ is due to skew scattering. This deviates significantly from the commonly accepted scaling law $ρ_{sH}=aρ^2+bρ$ in a bulk conductor. Thus we call for cautious interpretation of experimental data when applying the scaling law.