Boundary spin Hall effect in a two-dimensional semiconductor system with Rashba spin-orbit coupling

We derive boundary conditions for the coupled spin-charge diffusion equations at a transmitting interface between two-dimensional electron systems with different strengths of the Rashba spin-orbit (SO) coupling $\alpha$, and an electric field parallel to the interface. We consider the limit where the spin-diffusion length l_s is long compared to the electron mean free path l, and assume that $\alpha$ changes discontinuously on the scale of l_s. We find that the spin density is also discontinuous on the scale of l_s. In the case where the electron mobility is constant across the interface, this leads to the complete suppression of the expected spin injection from a region with $\alpha \mathrm{\ne }0$ into a non-SO region with $\alpha =0$.