Intrinsic Spin Hall Effect in the presence of Extrinsic Spin-Orbit Scattering

Intrinsic and extrinsic spin Hall effects are considered together on an equal theoretical footing for the Rashba spin-orbit coupling in two-dimensional (2D) electron and hole systems, using the diagrammatic method for calculating the spin Hall conductivity. Our analytic theory for the 2D holes shows the expected lowest-order additive result for the spin Hall conductivity. But, the 2D electrons manifest a very surprising result, exhibiting a non-analyticity in the Rashba coupling strength $\alpha$ where the strictly extrinsic spin Hall conductivity (for $\alpha = 0$) cannot be recovered from the $\alpha \to 0$ limit of the combined theory. The theoretical results are discussed in the context of existing experimental results.