Theoretical study of orbital torque: Dependence on ferromagnet species and nonmagnetic layer thickness

The manipulation of magnetization in ferromagnetic metals (FMs) through orbital torque (OT) has emerged as a promising route for energy-efficient magnetic devices without relying on heavy metals. While Ti and Cu are among the most extensively studied light nonmagnetic metals (NMs) for OT devices, theoretical calculations of the resulting torque have remained limited. Here, we present a systematic and quantitative theoretical study of current-induced torques in Ti/FM and Cu/FM (FM = Co, Ni) bilayers using realistic tight-binding models derived from ab initio electronic structures. We find that the torque in Ti/FM is larger for Ni than for Co, but this trend does not necessarily hold in Cu/FM, revealing that the FM dependence of OT is not universal but varies with the orbital current source. Moreover, the dependence of OT on NM thickness clearly indicates its NM bulk origin in both Ti- and Cu-based systems. Notwithstanding, the quantitative characteristics of OT cannot be explained by a simplified picture based on the individual bulk properties of the NM or FM layers. These results provide microscopic insight and practical guidance for designing light-metal-based orbitronic devices.