Current-induced spin torque is essential and crucial in spintronics. In this work, we systematically investigate the spin torque in transition metal (TM)/ferromagnet (FM) bilayers by using first-principles calculations and taking into account the phonon scattering at room temperature. To examine the spin and orbital Hall contributions, the studied transition metals include 5d heavy metals $\mathrm{Pt},\mathrm{W},\mathrm{Au},$ and $\mathrm{Ta}$ as well as 3d light metals $\mathrm{Ti},\mathrm{V},\mathrm{Cr},\mathrm{Cu},$ etc. We found that in TM$/\mathrm{Co}\mathrm{Fe}$ bilayers with typical 3d and 5d transition metals, the spin torque on $\mathrm{Co}\mathrm{Fe}$ mainly originates from the spin Hall mechanism, with the magnitude and sign of the damping-like torque efficiency consistent with those of the spin Hall conductivity (SHC). In TM$/\mathrm{Ni}$ bilayers, the spin torque is contributed by three mechanisms, including the spin and orbital Hall effect in the TM, as well as self-torque in $\mathrm{Ni}$. For TM$/\mathrm{Ni}$ bilayers, when SHCs are large in TMs, for instance in $\mathrm{W}$ and $\mathrm{Pt}$, the spin torques are mainly contributed by the spin Hall effect rather than the orbital Hall effect due to the partial cancellation of orbital and self-torque. For other cases, including $\mathrm{Ti}/\mathrm{Ni}$, $\mathrm{V}/\mathrm{Ni}$, and $\mathrm{Ta}/\mathrm{Ni}$, the orbital Hall torques also compete with either self-torque in $\mathrm{Ni}$ or the spin Hall effect in the TMs, which leads to the resultant spin torque efficiency not being pronounced. Our work reveals the less efficient contribution of the orbital Hall effect than the spin Hall effect on the spin torque in TM/FM bilayers.

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• Hall effect
• Spin current
• Spin torque
• Spin-orbit coupling
• Ferromagnets
• Magnetic multilayers
• Green's function methods
• LDA