The broken inversion symmetry at the surface of a metallic film (or, more generally, at the interface between a metallic film and a different metallic or insulating material) greatly amplifies the influence of the spin-orbit interaction on the surface properties. The best known manifestation of this effect is the momentum-dependent splitting of the surface state energies (Rashba effect). Here we show that the same interaction also generates a spin-polarization of the bulk states when an electric current is driven through the bulk of the film. For a semi-infinite jellium model, which is representative of metals with a closed Fermi surface, we prove as a theorem that, regardless of the shape of the confinement potential, the induced surface spin density at each surface is given by ${bf S} =-gamma hbar {bf hat z}times {bf j}$, where ${bf j}$ is the particle current density in the bulk, ${bf hat z}$ the unit vector normal to the surface, and $gamma=frac{hbar}{4mc^2}$ contains only fundamental constants. For a general metallic solid $gamma$ becomes a material-specific parameter that controls the strength of the interfacial spin-orbit coupling. Our theorem, combined with an {it ab initio} calculation of the spin polarization of the current-carrying film, enables a determination of $gamma$, which should be useful in modeling the spin-dependent scattering of quasiparticles at the interface.
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