Scaling Law of Confined Spin-Hall Effect

Abstract

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.

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