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We describe how the spin Hall effect (SHE) can be studied from ab-initio bycombining density functional theory with the non-equilibrium Green's functionstechnique for quantum transport into the so-called DFT+NEGF method. Afterlaying down our theoretical approach in particular discussing how to computecharge and spin bond currents, DFT+NEGF calculations are carried out for idealclean systems. In these the transport is ballistic and the linear responselimit is met. The SHE emerges in a central region attached to two leads when weapply a bias voltage so that electrons are accelerated by a uniform electricfield. As a result, we obtain a finite spin-Hall current and, by performing ascaling analysis with respect to the system size, we estimate the ballisticspin Hall conductivity (SHC). We consider 5d metals with fcc and bcc crystalstructures, finding that the SHC exhibits a rough qualitative dependence on thed-band filling, and comment on these results in relation to existingliterature. Finally, within the same DFT+NEGF approach, we also predict theappearance of a current-induced spin dipole moment inside the materials' unitcell and estimate its magnitude.

Ab-initio transport theory for the intrinsic spin Hall effect applied to 5$d$ metals