Anatomy of spin Hall and anomalous Hall conductivities in Co/$5d$-heavy-metal heterostructures from band filling and interfacial orbital hybridization

Intrinsic spin Hall and anomalous Hall conductivities (SHC and AHC) for heterostructures of Co and $5d$ heavy metal ($\mathrm{HM}=\mathrm{Ta},W,\mathrm{Re},\mathrm{Os},\mathrm{Ir},\mathrm{Pt}$) are investigated by comparing chemical (HM element) trend and different HM thickness by means of first-principles calculations. We find that the HM-element dependencies of SHC and AHC are caused by the $5d$ band filling but importantly originated from different band characteristic. The SHC associates with the spin Berry curvature in $5d$ bands inside the HM atomic layers. In contrast, the AHC is induced by the interfacial bands at the Co/HM, in which the origin can be decomposed into two effects; the $5d$ band filling of interfacial HM atomic layer and a proximity effect of neighboring Co atomic layer inducing the magnetic moment at the top HM. The former determines the overall distribution of Berry curvature in Brillouin zone, while the latter, asymmetric distribution of Berry curvature by a time-reversal symmetry breaking. As a consequence, different thickness dependencies of the HM atomic layers to the SHC and AHC appear, being significant and absent, respectively. The findings provide guiding principles for controllable SHC and AHC in device developments.