Origin and enhancement of the spin Hall angle in the Weyl semimetals LaAlSi and LaAlGe

We study the origin of the strong spin Hall effect (SHE) in a recently discovered family of Weyl semimetals, $\mathrm{LaAl}X$ ($X$ = Si, Ge) via first-principles with maximally localized Wannier functions. We show that the strong intrinsic spin Hall effect in $\mathrm{LaAl}X$ originates from the multiple slight anticrossings of nodal lines and points near to the Fermi energy due to their high-mirror symmetry and large spin-orbit interaction. It is further found that hole doping and increasing the temperature can enhance the spin Hall conductivity (${\ensuremath{\sigma}}_{\mathrm{SH}}$). However, the former also increases the electrical conductivity (${\ensuremath{\sigma}}_{c}$), while the latter decreases it. As a result, the independent tuning of ${\ensuremath{\sigma}}_{\mathrm{SH}}$ and ${\ensuremath{\sigma}}_{c}$ by increasing the temperature can enhance the spin Hall angle (proportional to $\frac{{\ensuremath{\sigma}}_{\mathrm{SH}}}{{\ensuremath{\sigma}}_{c}}$), a figure of merit of charge-to-spin current interconversion of spin-orbit torque devices. The underlying physics of such independent changes of the spin Hall and electrical conductivities by thermal means is revealed through the band-resolved and $k$-resolved spin Berry curvature. Our finding offers a way in the search of high spin Hall angle materials for room temperature spin-orbitronics applications.