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

We describe how the spin Hall effect (SHE) can be studied from ab initio by combining density functional theory with the nonequilibrium Green's functions technique for quantum transport into the so-called $\mathrm{DFT}+\mathrm{NEGF}$ method. After laying down our theoretical approach, in particular discussing how to compute charge and spin bond currents, $\mathrm{DFT}+\mathrm{NEGF}$ 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 the existing literature. Finally, within the same $\mathrm{DFT}+\mathrm{NEGF}$ approach, we also predict the appearance of a current-induced spin dipole moment inside the materials' unit cell and estimate its magnitude.