  We describe how the spin Hall effect (SHE) can be studied from ab-initio by
combining density functional theory with the non-equilibrium Green's functions
technique for quantum transport into the so-called DFT+NEGF method. After
laying down our theoretical approach in particular discussing how to compute
charge and spin bond currents, DFT+NEGF calculations are carried out for ideal
clean systems. In these the transport is ballistic and the linear response
limit is met. The SHE emerges in a central region attached to two leads when we
apply a bias voltage so that electrons are accelerated by a uniform electric
field. As a result, we obtain a finite spin-Hall current and, by performing a
scaling analysis with respect to the system size, we estimate the ballistic
spin Hall conductivity (SHC). We consider 5d metals with fcc and bcc crystal
structures, finding that the SHC exhibits a rough qualitative dependence on the
d-band filling, and comment on these results in relation to existing
literature. Finally, within the same DFT+NEGF approach, we also predict the
appearance of a current-induced spin dipole moment inside the materials' unit
cell and estimate its magnitude.
