FIG. 4.

Spin-orbit torque (SOT) to the bulk magnetism in magnetic bilayer described by Eq. (17). The red lines are contributions from a current flowing in the normal metal, and the other lines are those from a current flowing in the ferromagnet. Among these, the blue lines are contributions from majority electrons flow and the cyan line represents contributions from minority electron flow. The dashed and dot-dashed lines are the real part, thus representing fieldlike torque (FLT), while the solid lines are the imaginary part, thus representing dampinglike torque (DLT). The dot-dashed line represents the third term in $Re[{\mathcal{T}}_R^{\mathrm{F}}]$ in Eq. (20), which originates from a nonzero value of J and transitions to resulting the evanescent states. The spin-orbit torque from electrons in normal metal, majority electrons, and minority electrons are divided by $2h_R\hslash e{E}_x\tau A{k}_F^3/\pi m_e,2h_R\hslash e{E}_x{\tau }^{\uparrow }A{k}_F^3/\pi m_e$, and $2h_R\hslash e{E}_x{\tau }^{\downarrow }A{k}_F^3/\pi m_e$, respectively, and the results are dimensionless. Thus, the total spin-orbit torque is given by the weighted sum of all the contributions with the weighting factors (τ, τ^↑, τ^↓).