Scaling Law of Confined Spin-Hall Effect

TL;DR

This paper develops a quantum confinement model for the extrinsic spin-Hall effect in ultrathin metal films, revealing a linear resistivity relation that challenges the traditional quadratic scaling law used in bulk conductors.

Contribution

It introduces a new theoretical framework incorporating quantum size effects and surface roughness, showing a linear relation between spin-Hall and charge resistivities in ultrathin films.

Findings

Quantum confinement induces a linear relation between resistivities.

The linear relation is due to side jump and skew scattering.

Challenges the traditional quadratic scaling law in bulk materials.

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.