Spin-Hall conductivity and electric polarization in metallic thin films

We predict theoretically that when a normal metallic thin film (without bulk spin-orbit coupling, such as Cu or Al) is sandwiched by two insulators, two prominent effects arise due to the interfacial spin-orbit coupling: a giant spin-Hall conductivity due to the surface scattering and a transverse electric polarization due to the spin-dependent phase shift in the spinor wave functions.

Figure 1

Left: A metallic film of thickness $d$ is sandwiched by two dissimilar insulating materials, such as oxides or vacua. The electric fields (solid blue arrows) at the interfaces point inward ($\pm z$), and the resulting Rashba magnetic fields (dashed red arrows) point sideward ($\pm y$) for an electron moving along the $x$ direction. Right: The confining potential $V_C$ in Eq. (1) along the confinement direction (i.e., $z$ axis).

Figure 2

Left: Schematic view of the band structure in metallic thin film with interfacial Rashba spin-orbit coupling, majority (solid red) and minority (dashed blue). Right: Schematic picture of the spatial part of the wave function for the first two subbands; the stepwise curve is the confinement potential profile and the three different curves are the wave function from the lowest-order perturbation Eq. (3b) (solid black for both majority and minority) and the exact wave function from Eq. (10) (solid red for majority and dashed blue for minority) which shows the spatial separation of opposite spins. The discontinuity at the interfaces is due to the $\delta$-function-like potential in $H_R$.