  A key challenge in manipulating the magnetization in
heavy-metal/ferromagnetic bilayers via the spin-orbit torque is to identify
materials that exhibit an efficient charge-to-spin current conversion. Ab
initio electronic structure calculations reveal that the intrinsic spin Hall
conductivity (SHC) for pristine $\beta$-W is about sixty percent larger than
that of $\alpha$-W. More importantly, we demonstrate that the SHC of $\beta$-W
can be enhanced via Ta alloying. This is corroborated by spin Berry curvature
calculations of W$_{1-x}$Ta$_x$ ($x$ $\sim$ 12.5%) alloys which show a giant
enhancement of spin Hall angle of up to $\approx$ $-0.5$. The underlying
mechanism is the synergistic behavior of the SHC and longitudinal conductivity
with Fermi level position. These findings, not only pave the way for enhancing
the intrinsic spin Hall effect in $\beta$-W, but also provide new guidelines to
exploit substitutional alloying to tailor the spin Hall effect in various
materials.
