Intrinsic spin Hall effect in topological semimetals with single Dirac nodal ring

The spin Hall effect (SHE) maintains a distinct, pivotal role within spintronics. In this study, we systematically investigate the intrinsic SHE of topological semimetals with a single Dirac nodal ring (DNR) in the half-Heusler ABC family of compounds (HfCoAs, HfCoP, HfNiAs, ZrCoP, TaCoGe, TiSiNi, NbCoSi). Taking HfCoAs as an example, we show that an ideal single DNR lies within the $M_y$ mirror plane adjacent to the Fermi surface when spin-orbit coupling (SOC) is absent, as protected by combined space-time inversion ($\mathrm{PT}$) symmetry and $M_y$ mirror-reflection symmetry. With the inclusion of the SOC, the DNR is gapped out resulting in plenty of minimal band gap near the Fermi level, which yields a large spin Hall conductivity owing to the large spin Berry curvature hotspots around the nodal ring. Our research provides a comprehensive understanding of spin transport properties inherent in the simplest topological DNR semimetals, providing a solid foundation for exploring and engineering more complicated nodal line semimetals. This work may also offer practical applications in advancing spintronics device development with suitable SHE materials.