  We report on a theoretical study of the spin Hall Effect (SHE) and weak
antilocal-ization (WAL) in graphene/transition metal dichalcogenide (TMDC)
heterostructures, computed through efficient real-space quantum transport
methods, and using realistic tight-binding models parametrized from ab initio
calculations. The graphene/WS 2 system is found to maximize spin proximity
effects compared to graphene on MoS 2 , WSe 2 , or MoSe 2 , with a crucial role
played by disorder, given the disappearance of SHE signals in the presence of
strong intervalley scattering. Notably, we found that stronger WAL effects are
concomitant with weaker charge-to-spin conversion efficiency. For further
experimental studies of graphene/TMDC heterostructures, our findings provide
guidelines for reaching the upper limit of spin current formation and for fully
harvesting the potential of two-dimensional materials for spintronic
applications.
