Modeling Spin-Orbit Torques
in Advanced Magnetoresistive Devices

5.2 Perturbation Theory Model

The electrons experience the SOC as an effective magnetic field proportional to their momentum. Thus, for a NM/FM interface at \(z=0\), with an interfacial SOC, the carriers will experience a different field at the interface compared to the bulk of the FM region. In the case of a Rashba SOC, this can be captured by including the following spin-dependent potential barrier in the Hamiltonian [77]: \(\hat {H}_R = (\hbar ^2k_F/m_e)u_R\bm {\sigma }\cdot (\bm {k}\times \bm {z})\delta (z)\), where \(u_R\) is the dimensionless strength of the Rashba SOC, \((\bm {k}\times \bm {z})\) is the direction of the Rashba field the carriers experience, and the delta function places the interaction at the interface.

For such a system, it is difficult to determine the scattering matrices analytically, as there is no universal quantization axis for the entire system. However, if the SOC is weak compared to the bulk exchange splitting, it can be treated as a perturbation of the wave function at the interface. Using perturbation theory and considering only plane-wave scattering, Kim et al. showed that in the case of the interfacial Rashba SOC described by \(\hat {H}_R\), the interfacial scattering matrices up to first order of \(u_R\) can be expressed as follows [77]:

\begin{align} \hat {r}_{\bm {k}} & = \hat {r}_{\bm {k}}^{0} + \frac {u_R}{ik_z}\left (\hat {1} + \hat {r}_{\bm {k}}^{0}\right )\bm {\hat {\sigma }}\cdot \left (\bm {k}\times \bm {z}\right ) \left (\hat {1} + \hat {r}_{\bm {k}}^{0}\right ), \\ \hat {t}_{\bm {k}} & = \hat {t}_{\bm {k}}^{0} + \frac {u_R}{ik_z} \hat {t}_{\bm {k}}^{0}\bm {\hat {\sigma }}\cdot \left (\bm {k}\times \bm {z}\right ) \left (\hat {1} + \hat {r}_{\bm {k}}^{0}\right ), \\ \hat {r}^\prime _{\bm {k}} & = \hat {r}_{\bm {k}}^{\prime 0} + \frac {u_R}{ik_z}\hat {t}_{\bm {k}}^{0}\bm {\hat {\sigma }}\cdot \left (\bm {k}\times \bm {z}\right ) \hat {t}_{\bm {k}}^{\prime 0}, \\ \hat {t}^\prime _{\bm {k}} & = \hat {t}_{\bm {k}}^{\prime 0} + \frac {u_R}{ik_z}\left (\hat {1} + \hat {r}_{\bm {k}}^{0}\right )\bm {\hat {\sigma }}\cdot \left (\bm {k}\times \bm {z}\right ) \hat {t}_{\bm {k}}^{\prime 0}, \end{align} where \(\hat {r}^{0}_{\bm {k}}\) (\(\hat {r}^{\prime 0}_{\bm {k}}\)) and \(\hat {t}^{0}_{\bm {k}}\) (\(\hat {t}^{\prime 0}_{\bm {k}}\)) are the unperturbed reflection and transmission matrices for electrons incident from the NM (FM) side of the interface with wave vector \(\bm {k}\), respectively. Due to the Rashba SOC, the scattering matricies are no longer even in \(\bm {k_\|}\), which can give rise to new contributions to the currents at the interface from the anisotropic non-equilibrium distribution function term.

5.2.1 Rashba Currents in the Nonmagnetic Layer

Modifications to the MCT BCs by the Rashba SOC can be obtained by inserting the perturbed scattering matrices into Eq. (5.6) and limiting the contributions to the first order in \(u_R\), which yields

\begin{equation} \hat {j}_z(0^-) = -\frac {e}{h A}\sum _{\bm {k_\|} \in \mathrm {FS}} \left [ \hat {\mu }^N - \hat {r}_{\bm {k}}\hat {\mu }^N(\hat {r}_{\bm {k}})^\dagger - \hat {t}^{\prime }_{\bm {k}}\hat {\mu }^F(\hat {t}^{\prime }_{\bm {k}})^\dagger \right ] +\frac {e^2 u_R}{\pi m_e A}\operatorname {Im}\sum _{\bm {k_\|} \in \mathrm {FS}}\frac {k_F}{k_z}(\bm {k}\cdot \bm {E})\left (\hat {1} + \hat {r}_{\bm {k}}^{0}\right )\bm {\hat {\sigma }}\cdot \left (\bm {k}\times \bm {z}\right )\hat {t}^{\prime }_{\bm {k}}(\hat {\tau }^F - \hat {\tau }^N)(\hat {t}^{\prime }_{\bm {k}})^\dagger . \end{equation}

To simplify the expression, the identity \(\operatorname {Im}[M] = (M-M^\dagger )/2i\) for a given Hermitian matrix \(M\) was used, and that \(\hat {r}_{\bm {k}},\hat {t}_{\bm {k}}^{\prime },\) and \(\hat {\mu }^{F/N}\) are even in \(\bm {k_\|}\), while \((\bm {k}\cdot \bm {E})\) and \(\left (\bm {k}\times \bm {z}\right )\) are odd in \(\bm {k_\|}\), resulting in the odd terms vanishing after averaging over the momentum. The first term is nothing but the expression for the current at the interface from the MCT, while the second term is a new contribution arising from the interfacial SOC, which is proportional to the applied electric field. As the first term has already been discussed in detail in the previous chapter, the focus will be on the new contribution to the current.

Averaging over all the directions of \(\bm {k_\|}\) gives \(\sum _{\bm {k_\|}\in \mathrm {FS}} (\bm {k}\cdot \bm {u})\bm {\hat {\sigma }}\cdot (\bm {k}\times \bm {z}) = \sum _{\bm {k_\|}\in \mathrm {FS}}(k_\|^2/2)\bm {\hat {\sigma }}\cdot (\bm {u}\times \bm {z})\) for any unit vector \(\bm {u}\). Thus, the new current contribution due to the Rashba SOC can be rewritten as

\begin{equation} \hat {j}^E_z(0^-) = \frac {e^{2} u_R }{2\pi m_e A}\operatorname {Im}\sum _{\bm {k_\|} \in \mathrm {FS}}\frac {k_\|^2}{k_z}\left (\hat {1} + \hat {r}_{\bm {k}}^{0}\right )\bm {\hat {\sigma }}\cdot \left (\bm {E}\times \bm {z}\right )\hat {t}^{\prime }_{\bm {k}}( \hat {\tau }^F - \hat {\tau }^N)(\hat {t}^{\prime }_{\bm {k}})^\dagger . \end{equation}

Using the identity \(\hat {1} = \hat {p}^\uparrow + \hat {p}^\downarrow \), one can write \(( \hat {\tau }^F - \hat {\tau }^N) = (\tau ^\uparrow -\tau ^N) \hat {p}^\uparrow + (\tau ^\downarrow -\tau ^N)\hat {p}^\downarrow \). Expressing the scattering matrices in terms of the spin projection matrices and using the following projection identities: \(\hat {p}^s\hat {p}^s = \hat {p}^s \) and \(\hat {p}^s\hat {p}^{-s} = 0\), yields

\begin{multline} \hat {j}^E_z(0^-) = v_F\operatorname {Im} \left [ G_{Rt}^{\uparrow \uparrow \uparrow }\hat {p}^\uparrow \bm {\hat {\sigma }}\cdot (\bm {E}\times \bm {z})\hat {p}^\uparrow (\tau ^\uparrow -\tau ^N) + G_{Rt}^{\uparrow \downarrow \downarrow }\hat {p}^\uparrow \bm {\hat {\sigma }}\cdot (\bm {E}\times \bm {z})\hat {p}^\downarrow (\tau ^\downarrow -\tau ^N) \right . \\ \left . +G_{Rt}^{\downarrow \uparrow \uparrow }\hat {p}^\downarrow \bm {\hat {\sigma }}\cdot (\bm {E}\times \bm {z})\hat {p}^\uparrow (\tau ^\uparrow -\tau ^N) + G_{Rt}^{\downarrow \downarrow \downarrow }\hat {p}^\downarrow \bm {\hat {\sigma }}\cdot (\bm {E}\times \bm {z})\hat {p}^\downarrow (\tau ^\downarrow -\tau ^N) \right ], \end{multline} where \(v_F = \hbar k_F/m_e\) is the Fermi velocity, and the new Rashba interface spin conductances are defined as

\begin{equation} G_{Rt}^{ss^\prime s^{\prime \prime }} = \frac {u_R}{A}\frac {e^2}{h}\sum _{\bm {k_\|} \in \mathrm {FS}} \frac {k_\|^2}{k_z k_F} (1+r^s_{\bm {k}}) t^{\prime s^\prime }(t^{\prime s^{\prime \prime }})^*. \end{equation}

To obtain the charge and spin current contribution from the Rashba SOC in real space, the following trace relations for an arbitrary vector \(\bm {b}\) are required:

\begin{align} \operatorname {Tr}[\hat {p}^s(\bm {\hat {\sigma }}\cdot \bm {b})\hat {p}^{s^\prime }] = & \frac {1}{2}(s+s^\prime )(\bm {m}\cdot \bm {b}), \\ \operatorname {Tr}[\bm {\hat {\sigma }}\hat {p}^s(\bm {\hat {\sigma }}\cdot \bm {b})\hat {p}^{s^\prime }] = & \frac {1}{2}[(1-ss^\prime )\bm {b} + i(s-s^\prime )(\bm {m}\times \bm {b})+2ss^\prime (\bm {m}\cdot \bm {b})\bm {m}]. \end{align}

Using the first trace relation yields the following expression for the charge current at the interface:

\begin{equation} j^E_{zc}(0^-) = \operatorname {Tr}[\hat {j}^E_z(0^-)] = -v_F\operatorname {Im}\left [G_{Rt}^{\uparrow \uparrow \uparrow }\tau ^\uparrow - G_{Rt}^{\downarrow \downarrow \downarrow }\tau ^\downarrow - G_{Rt}^{\uparrow \uparrow \uparrow }\tau ^N + G_{Rt}^{\downarrow \downarrow \downarrow }\tau ^N \right ]\bm {m}\cdot (\bm {E}\times \bm {z}) \end{equation}

The spin current is obtained by using the second trace relation:

\begin{multline} \label {eq:perturbation_spin_current_NM} \bm {j^E_{zs}}(0^-) = \operatorname {Tr}[\bm {\hat {\sigma }}\hat {j}^E_z(0^-)] = v_F\operatorname {Im}[G_{Rt}^{\uparrow \uparrow \uparrow }\tau ^\uparrow + G_{Rt}^{\downarrow \downarrow \downarrow }\tau ^\downarrow - G_{Rt}^{\uparrow \uparrow \uparrow }\tau ^N - G_{Rt}^{\downarrow \downarrow \downarrow }\tau ^N][\bm {m}\cdot (\bm {E}\times \bm {z})]\bm {m} \\ -v_F\operatorname {Re}[G_{Rt}^{\downarrow \uparrow \uparrow }\tau ^\uparrow - (G_{Rt}^{\uparrow \downarrow \downarrow })^*\tau ^\downarrow -G_{Rt}^{\downarrow \uparrow \uparrow }\tau ^N + (G_{Rt}^{\uparrow \downarrow \downarrow })^*\tau ^N ][(\bm {E}\times \bm {z})\times \bm {m}] \\ + v_F\operatorname {Im}[G_{Rt}^{\downarrow \uparrow \uparrow }\tau ^\uparrow - (G_{Rt}^{\uparrow \downarrow \downarrow })^*\tau ^\downarrow -G_{Rt}^{\downarrow \uparrow \uparrow }\tau ^N + (G_{Rt}^{\uparrow \downarrow \downarrow })^*\tau ^N ]\bm {m}\times [(\bm {E}\times \bm {z})\times \bm {m}], \end{multline} where the identity \(\operatorname {Im}[iA] = \operatorname {Re}[M]\) was used and that \(\operatorname {Im}[M^*] = -\operatorname {Im}[M]\).

Thus, the inclusion of interfacial Rashba SOC at the interface yields new contributions to the BCs describing out-of-plane charge and spin currents generated by an in-plane electric field. Particle conservation demands that the charge current is continuous across the interface, i.e., \(j^E_{cz}(0^-) = j^E_{cz}(0^+)\). However, the SOC opens up a channel for angular momentum transfer between the lattice and the spin at the interface; hence, the spin current can be discontinuous and must therefore be accounted for at the FM side of the interface.

5.2.2 Rashba Currents in the Ferromagnetic Layer

At the interface, angular momentum is absorbed and emitted due to the SOC, resulting in a difference in the current densities across the interface. Assuming spin angular momentum is only transferred to and from the spin-orbit field at the interface, the difference in the current densities across the interface is given to the first order in \(u_R\) by [77]

\begin{equation} \Delta \hat {j}^E_z = \hat {j}^E_z(0^+) - \hat {j}^E_z(0^-) = -\frac {e^2 u_R}{2\pi m_e A} \operatorname {Im}\sum _{\bm {k_\|} \in \mathrm {FS}}\frac {k_\|^2}{\vert k_z \vert } \bm {\hat {\sigma }} \cdot (\bm {E}\times \bm {z}) [(\hat {1} + \hat {r}^0_{\bm {k}})\tau ^N(\hat {1} + (\hat {r}^0_{\bm {k}})^\dagger ) + \hat {t}^{\prime 0}_{\bm {k}}\tau ^F(\hat {t}^{\prime 0}_{\bm {k}})^\dagger ]. \end{equation}

Expanding the scattering matrices in terms of the spin projection matrices yields

\begin{equation} \Delta \hat {j}^E_z =-v_F\operatorname {Im}\left [G^{\uparrow \uparrow }_{Rt}\bm {\hat {\sigma }} \cdot (\bm {E}\times \bm {z})\hat {p}^\uparrow \tau ^\uparrow + G^{\downarrow \downarrow }_{Rt}\bm {\hat {\sigma }} \cdot (\bm {E}\times \bm {z})\hat {p}^\downarrow \tau ^\downarrow + G^{\uparrow \uparrow }_{Rr}\bm {\hat {\sigma }} \cdot (\bm {E}\times \bm {z})\hat {p}^\uparrow \tau ^N+ G^{\downarrow \downarrow }_{Rr}\bm {\hat {\sigma }} \cdot (\bm {E}\times \bm {z})\hat {p}^\downarrow \tau ^N \right ], \end{equation}

where Rashba interface conductances for transmission and reflection were introduced:

\begin{equation} G^{ss^\prime }_{Rt} = \frac {u_R}{A}\frac {e^2}{h}\sum _{\bm {k_\|} \in \mathrm {FS}} \frac {k_\|^2}{\vert k_z \vert k_F} t^{\prime s}(t^{\prime s^\prime } )^*, \end{equation}

and

\begin{equation} G^{ss^\prime }_{Rr} = \frac {u_R}{A}\frac {e^2}{h}\sum _{\bm {k_\|} \in \mathrm {FS}} \frac {k_\|^2}{\vert k_z \vert k_F}(1+r^s)(1+r^{s^\prime })^*, \end{equation}

respectively.

To obtain the difference in the charge and spin current densities across the interface, the following trace relations are required:

\begin{align} \operatorname {Tr}[(\bm {\hat {\sigma }}\cdot \bm {b})\hat {p}^s] & = s(\bm {b}\cdot \bm {m}), \\ \operatorname {Tr}[\bm {\hat {\sigma }}(\bm {\hat {\sigma }}\cdot \bm {b})\hat {p}^s] & = \bm {b} + is(\bm {b}\times \bm {m}). \end{align} The difference in charge current density across the interface is then given by

\begin{equation} \Delta j^E_{cz} = -v_F\operatorname {Im}\left [-G^{\uparrow \uparrow }_{Rt}\tau ^\uparrow + G^{\downarrow \downarrow }_{Rt}\tau ^\downarrow -G^{\uparrow \uparrow }_{Rr}\tau ^N + G^{\downarrow \downarrow }_{Rr}\tau ^N\right ][(\bm {E}\times \bm {z})\cdot \bm {m}] = 0, \end{equation}

which ensures that the charge current is conserved across the interface. The difference in the spin current density across the interface reads

\begin{multline} \label {eq:perturbation_spin_current_difference} \Delta \bm {j^E_{sz}} = \\ -v_F\operatorname {Im}\left [G^{\uparrow \uparrow }_{Rt}((\bm {E}\times \bm {z}) - i((\bm {E}\times \bm {z})\times \bm {m}))\tau ^\uparrow + G^{\downarrow \downarrow }_{Rt}((\bm {E}\times \bm {z}) + i((\bm {E}\times \bm {z})\times \bm {m}))\tau ^\downarrow \right . \\ \left . + G^{\uparrow \uparrow }_{Rr}((\bm {E}\times \bm {z}) - i((\bm {E}\times \bm {z})\times \bm {m}))\tau ^N + G^{\downarrow \downarrow }_{Rr}((\bm {E}\times \bm {z}) + i((\bm {E}\times \bm {z})\times \bm {m}))\tau ^N \right ] \\ = v_F\operatorname {Re}\left [G^{\uparrow \uparrow }_{Rt}\tau ^\uparrow - G^{\downarrow \downarrow }_{Rt}\tau ^\downarrow + G^{\uparrow \uparrow }_{Rr}\tau ^N - G^{\downarrow \downarrow }_{Rr}\tau ^N \right ][(\bm {E}\times \bm {z})\times \bm {m}], \end{multline} which shows that part of the FL spin current is lost at the interface, while the longitudinal and DL components are conserved. This loss of spin current can be interpreted as a torque on the spin-orbit field at the interface, describing the transfer of angular momentum from the spin current to the crystal lattice mediated through the spin-orbit interaction [76]. Since the lattice effectively serves as an unlimited reservoir of angular momentum, this torque manifests as a parasitic loss channel for the spin current. Another loss of spin current can be introduced by considering an interfacial exchange interaction [76, 77], which is not parasitic as it yields an interfacial torque acting on the local magnetic moments.

The Rashba spin current at the FM side of the interface is obtained from the sum of Eqs. (5.14) and (5.21):

\begin{multline} \bm {j^E_{sz}}(0^+) = \bm {j^E_{sz}}(0^-) + \Delta \bm {j^E_{sz}} \\ = v_F\operatorname {Im}[G_{Rt}^{\uparrow \uparrow \uparrow }\tau ^\uparrow + G_{Rt}^{\downarrow \downarrow \downarrow }\tau ^\downarrow - G_{Rt}^{\uparrow \uparrow \uparrow }\tau ^N + G_{Rt}^{\downarrow \downarrow \downarrow }\tau ^N][\bm {m}\cdot (\bm {E}\times \bm {z})]\bm {m} \\ -v_F\operatorname {Re}[\Gamma _{Rt}^{\downarrow \uparrow \uparrow }\tau ^\uparrow -(\Gamma _{Rt}^{\uparrow \downarrow \downarrow })^*\tau ^\downarrow - \Gamma _{Rr}^{\downarrow \uparrow \uparrow }\tau ^N+(\Gamma _{Rr}^{\uparrow \downarrow \downarrow })^*\tau ^N][(\bm {E}\times \bm {z})\times \bm {m}] \\ + v_F\operatorname {Im}[\Gamma _{Rt}^{\downarrow \uparrow \uparrow }\tau ^\uparrow -(\Gamma _{Rt}^{\uparrow \downarrow \downarrow })^*\tau ^\downarrow - \Gamma _{Rr}^{\downarrow \uparrow \uparrow }\tau ^N+(\Gamma _{Rr}^{\uparrow \downarrow \downarrow })^*\tau ^N]\bm {m}\times [(\bm {E}\times \bm {z})\times \bm {m}]. \end{multline} The new Rashba interface conductances are defined as

\begin{equation} \label {eq:Gamma_Rt} \Gamma ^{(-s)ss}_{Rt} = G^{(-s)ss}_{Rt} - G^{ss}_{Rt} = \frac {u_R}{A}\frac {e^2}{h}\sum _{\bm {k_\|}} \frac {k_\|^2}{k_z k_F} r^{-s}_{\bm {k}} \vert t^{\prime s} \vert ^2 - \frac {u_R}{A}\frac {e^2}{2\pi \hbar }\sum _{\bm {k_\|}, k_{z}^2 < 0 } \frac {k_\|^2}{\vert k_z \vert k_F} \vert t^{\prime s} \vert ^2 , \end{equation}

and \(\Gamma ^{\downarrow \uparrow \uparrow }_{Rr} - (\Gamma ^{\uparrow \downarrow \downarrow }_{Rr})^* = G_{Rt}^{\downarrow \uparrow \uparrow }+G^{\uparrow \uparrow }_{Rr} -(G_{Rt}^{\uparrow \downarrow \downarrow }+G^{\downarrow \downarrow }_{Rr})^*\), where

\begin{equation} \Gamma ^{(-s)ss}_{Rr} = \frac {u_R}{A}\frac {e^2}{h}\sum _{\bm {k_\|}} \frac {k_\|^2}{k_z k_F}[1- r_{\bm {k}}^{\prime (-s)}(r_{\bm {k}}^{\prime s})^*]r_{\bm {k}}^{\prime s}. \end{equation}

The second term in Eq. (5.23) describes a contribution from evanescent states with \(k_{z}^2 < 0\), which can give rise to a finite torque [77].

5.2.3 Interface Conductivities and Spin Torques

To further simplify the expressions for the currents, they are expressed in terms of effective Rashba interface conductivities:

\begin{equation} j^E_{zc}(0^{{\mathbin {\textpm }}}) = (\sigma ^\uparrow _{R} -\sigma ^\downarrow _{R}) [\bm {m}\cdot (\bm {E}\times \bm {z})], \end{equation}

\begin{equation} \bm {J}^E_{sz}(0^+) =-\frac {\mu _B}{e}(\sigma ^\uparrow _{R} +\sigma ^\downarrow _{R})[\bm {m}\cdot (\bm {E}\times \bm {z})]\bm {m} -\frac {\mu _B}{e}\operatorname {Im}[\gamma ^{\uparrow \downarrow }_{R}]\bm {m}\times [(\bm {E}\times \bm {z})\times \bm {m}] +\frac {\mu _B}{e}\operatorname {Re}[\gamma ^{\uparrow \downarrow }_{R}][(\bm {E}\times \bm {z})\times \bm {m}], \end{equation}

\begin{equation} \bm {J}^E_{sz}(0^-) =-\frac {\mu _B}{e}(\sigma ^\uparrow _{R} +\sigma ^\downarrow _{R})[\bm {m}\cdot (\bm {E}\times \bm {z})]\bm {m} -\frac {\mu _B}{e}\operatorname {Im}[\sigma ^{\uparrow \downarrow }_{R}]\bm {m}\times [(\bm {E}\times \bm {z})\times \bm {m}] +\frac {\mu _B}{e}\operatorname {Re}[\sigma ^{\uparrow \downarrow }_{R}][(\bm {E}\times \bm {z})\times \bm {m}], \end{equation}

where the spin currents densities were converted into units of A/s with the conversion factor \(-\mu _B/e\). The interface Rashba conductivities are defined as follows:

\begin{align} \sigma _{R}^{\uparrow } & = v_F \operatorname {Im}[G_{Rt}^{\uparrow \uparrow \uparrow }] (\tau ^\uparrow - \tau ^N), \\ \sigma _{R}^{\downarrow } & = v_F \operatorname {Im}[G_{Rt}^{\downarrow \downarrow \downarrow }] (\tau ^\downarrow - \tau ^N), \\ \sigma ^{\uparrow \downarrow }_{R} & = v_F G_{Rt}^{\downarrow \uparrow \uparrow }(\tau ^\uparrow - \tau ^N) - v_F(G_{Rt}^{\uparrow \downarrow \downarrow })^*(\tau ^\downarrow - \tau ^N), \\ \gamma ^{\uparrow \downarrow }_{R} & = v_F \Gamma _{Rt}^{\downarrow \uparrow \uparrow }\tau ^\uparrow - v_F(\Gamma _{Rt}^{\uparrow \downarrow \downarrow })^*\tau ^\downarrow - v_F[\Gamma _{Rr}^{\downarrow \uparrow \uparrow } - (\Gamma _{Rr}^{\uparrow \downarrow \downarrow })^*]\tau ^N. \end{align} Thus, the contribution from the interfacial Rashba SOC to the currents at either side of the interface can be described in terms of two real and two complex interface conductivities. Similar to the interface conductances from the MCT BCs, the interface conductivities can be either computed or treated as fitting parameters. It should be noted that \(\operatorname {Im}[\gamma ^{\uparrow \downarrow }_{R}] = \operatorname {Im}[\sigma ^{\uparrow \downarrow }_{R}]\), thus, since only the imaginary part of \(\sigma ^{\uparrow /\downarrow }\) contributes to the currents, only five real parameters are required to describe the new contributions.

On the FM side of the interface, the transverse spin current is absorbed within a few nm from the interface, resulting in a spin torque acting on the magnetization. Assuming all the transverse angular momentum is absorbed, the total spin torque from the Rashba SOC is given by the transverse component of the Rashba spin current at the FM side of the interface:

\begin{equation} \label {eq:perturbation_spin_torque} \bm {\tau ^E_s} = (I-\bm {m}\otimes \bm {m})\bm {J}^E_{sz}(0^+) = -\frac {\mu _B}{e}\operatorname {Im}[\gamma ^{\uparrow \downarrow }_{R}]\bm {m}\times [(\bm {E}\times \bm {z})\times \bm {m}] +\frac {\mu _B}{e}\operatorname {Re}[\gamma ^{\uparrow \downarrow }_{R}][(\bm {E}\times \bm {z})\times \bm {m}]. \end{equation}

The operator \((I-\bm {m}\otimes \bm {m})\) removes the longitudinal components of a spin quantity. The spin torque \(\bm {\tau ^E_s}\) is given here in units of \(\si {A/s}\), and can be converted into units of \(\si {A/m}\), compatible with the LLG equation, through division by the FM thickness \(d_\mathrm {FM}\), i.e., \(\bm {T^E_s} = \bm {\tau ^E_s}/d_\mathrm {FM}\). If the torque is not fully absorbed or one wants to obtain the spatial profile of the torque, the drift-diffusion equations must be solved for the entire system using the extended BCs, followed by computing the torque with Eq. (4.19).