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

TL;DR

The paper addresses how current-induced torques in light NM/FM bilayers arise and vary with FM species and NM thickness, focusing on orbital torque (OT) as a mechanism that does not rely on strong SOC in the NM. It develops semi-realistic tight-binding models derived from ab initio band structures and computes the damping-like torkance $t_{yx}$ via the Kubo formalism, separating intraband and interband contributions and emphasizing the intrinsic interband (Berry-curvature) part. Key findings show that Ti/FM torques are larger for Ni than Co, while Cu/FM torques are larger for Co than Ni in the clean limit, with disorder and interface effects playing significant roles; NM thickness analyses reveal a bulk-origin OT with long characteristic lengths, yet the sign and magnitude of OT do not always align with bulk orbital Hall conductivity. The work provides microscopic insight into OT, highlights the nonuniversality of FM dependence, and offers guidelines for designing light-metal–based orbitronic devices by tuning NM material, FM species, and layer thickness.

Abstract

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 semi-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.

Figures (5)

• Figure 1: Illustration of the tight-binding model for the NM/FM bilayer structure with fcc(111) orientation. The presented system is drawn for $N_\mathrm{NM}=30$ and $N_\mathrm{FM}=10$. The system is periodic along $x$ and $y$ directions. The FM layer has an out-of-plane magnetization $\mathbf{M}$ along $\hat{\mathbf{z}}$.
• Figure 2: Schematic illustration of the (a) OT and (b) self-induced SOT in a NM/FM bilayer. The red and blue arrows represent the OAM ($L$) and spin ($S$), respectively. For each case, two possible contributions associated with the bulk NM and the NM/FM interface are shown.
• Figure 3: (a) Torkance $t_{yx}$ in Ti/FM and Cu/FM bilayers (FM = Co, Ni) with $N_\mathrm{NM}=20$ and $N_\mathrm{FM}=10$. Chemical potential dependence of $t_{yx}$ in (b) Ti/FM and (c) Cu/FM bilayers.
• Figure 4: (a) Torkance $t_{yx}$ as a function of the broadening energy $\Gamma$ in (a) Ti/FM and (b) Cu/FM bilayers (FM = Co, Ni) with $N_\mathrm{NM}=20$ and $N_\mathrm{FM}=10$.
• Figure 5: Torkance $t_{yx}$ as a function of the broadening energy $\Gamma$ in (a) Ti/Co, (b) Ti/Ni, (c) Cu/Co, and (d) Cu/Ni bilayers, with $N_\mathrm{NM}=1,5,10,20,30,40$ and $N_\mathrm{FM}=10$. (e),(f) $N_\mathrm{Ti}$ dependence of $t_{yx}$ in Ti/FM bilayers for $\Gamma = 10$ meV and $\Gamma = 100$ meV. (g),(h) $N_\mathrm{Cu}$ dependence of $t_{yx}$ in Cu/FM bilayers for $\Gamma = 10$ meV and $\Gamma = 100$ meV.