Modeling Spin-Orbit Torques
in Advanced Magnetoresistive Devices
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4.4 Partial Differential Equations (PDEs) for
Charge and Spin Transport
Calculating the spin torque acting on the local magnetization in the FM layers of a multilayer system requires solving the drift-diffusion equations for the entire system using the BCs described in the previous section. As the charge
and spin potential components are coupled, a self-consistent solution of the drift-diffusion equations is required. In this section, the set of second-order partial differential equations (PDEs) is presented.
4.4.1 Coupled Charge and Spin Drift-Diffusion PDEs
To solve the drift-diffusion equations for the electrical potential and spin accumulation self-consistently, it is convenient to express the charge and spin currents in terms of the spin potential \(\bm {V_s}\) such that the MCT BCs
can be applied directly, and that the total solution has the same unit. The currents expressed in terms of the charge and spin potentials read
\(\seteqnumber{1}{4.42}{0}\)
\begin{align}
\bm {j_{c}} & = -\sigma \nabla V_c + \beta _D \sigma (\nabla \bm {V_s})^T\bm {m} + \alpha _\mathrm {SH}\sigma (\nabla \times \bm {V_s}) , \\ \frac {e}{\mu _B}\tilde {J}_s & = -\sigma
\nabla \bm {V_s} + \beta _{\sigma }\bm {m} \otimes (\nabla V_c) + \alpha _\mathrm {SH}\sigma \varepsilon (\nabla V_c).
\end{align}
Inserting these expressions into the continuity equations for charge and spin Eqs. (4.17) and (4.18), respectively, the following coupled set of second-order PDEs for the charge and spin potentials is obtained:
\(\seteqnumber{1}{4.43}{0}\)
\begin{align}
-\sigma \nabla ^2V_c + \beta _D\sigma (\nabla ^2\bm {V_s})\cdot \bm {m} + \alpha _\mathrm {SH}\sigma (\nabla \times \nabla )\cdot \bm {V_s} & = 0, \\ -\sigma \nabla ^2\bm {V_s} +\beta
_\sigma \sigma \bm {m}\nabla ^2V_{c} + \alpha _\mathrm {SH}\sigma (\nabla \times \nabla )V_c +\sigma \left ({\frac {\bm {V_s}}{\lambda _{sf}^2} + \frac {\bm {V_s}\times \bm {m}}{\lambda _J^2}
+\frac {\bm {m}\times (\bm {V_s}\times \bm {m})}{\lambda _\phi ^2}}\right ) & = 0,
\end{align}
which must be solved simultaneously for the entire system using the BCs described in the previous section.
4.4.2 Decoupled Charge and Spin Drift-Diffusion PDEs
In some cases, it can be more convenient to solve the drift-diffusion equations for the charge and spin separately. For instance, when the charge current is known or can be assumed to be constant everywhere in the system, when the
effects of the spin accumulation on the charge current are not of interest, or simply to reduce the computational complexity of the problem by solving smaller systems of PDEs.
In such cases, to retain the effects of the charge current contributions from the spin accumulation on the spin current, the electric field has to be expressed in terms of the charge current using Eq. (4.21a). The resulting expression for the electric field is then inserted into Eq. (4.21b), which yields the following expression for the spin current:
\(\seteqnumber{0}{4.}{43}\)
\begin{equation}
\label {eq:decoupled_spin_current} \tilde {J}_s = -D_e\nabla \bm {S} - \frac {\mu _B}{e}\beta _{\sigma }\bm {m} \otimes \left [\bm {j_c} - \frac {e}{\mu _B}\beta _{D}D_e(\nabla \bm {S})^T\bm
{m} \right ] - \frac {\mu _B}{e}\alpha _\mathrm {SH}\varepsilon \left [\bm {j_c} -\frac {e}{\mu _B}\alpha _\mathrm {SH}D_e(\nabla \times \bm {S}) \right ].
\end{equation}
If the charge current is assumed to be constant, the spin accumulation can be directly obtained by solving Eq. (4.18) using Eq. (4.44).
If the charge current is not considered as constant throughout the system, the charge current has to be first calculated by solving the Laplace equation for the electrical potential:
\(\seteqnumber{0}{4.}{44}\)
\begin{equation}
\sigma \nabla ^2 V^0_c = 0,
\end{equation}
and then taking the gradient of the solution \(j^0_c = -\sigma \nabla V^0_c\) [74]. Here, the superscript \(^0\) denotes the absence of the spin accumulation in the computation. The effects of the spin accumulation on the
charge current can later be reinserted by using the computed spin accumulation, such that the charge current is given by
\(\seteqnumber{0}{4.}{45}\)
\begin{equation}
\bm {j^\prime _{c}} = -\sigma \nabla V^0_c + \beta _D D_e (\nabla \bm {S^\prime })^T\bm {m} + \frac {e}{\mu _B}\alpha _\mathrm {SH}D_e(\nabla \times \bm {S^\prime }) ,
\end{equation}
where the prime \(^\prime \) denotes that the spin accumulation was obtained from the previous step. The reinsertion of the spin accumulation can be done iteratively by solving
\(\seteqnumber{0}{4.}{46}\)
\begin{equation}
\nabla \bm {j^\prime _{c}} = 0 \\
\end{equation}
for the electrical potential, followed by computing the current and solving the spin drift-diffusion PDE for the spin accumulation, until the solution converges. This approach was employed by S. Fiorentini to capture the GMR effect
in spin valves [74], where it was demonstrated that \(n\leq 3\) iteration was sufficient to achieve good convergence with an analytical solution for the current.
Boundary conditions such as the ones from the MCT require a similar treatment in order to decouple the electrical potential from the computation of the spin accumulation. Expressing the drop in the electrical potential across the
interface in terms of the charge current gives
\(\seteqnumber{0}{4.}{47}\)
\begin{equation}
\Delta V_c = (G_+)^{-1}[\bm {j_c}\cdot \bm {n} - P G_+(\Delta \bm {V_s}\cdot \bm {m})].
\end{equation}
Inserting this expression into the BCs (4.35) gives the following decoupled BCs for the charge and spin currents:
\(\seteqnumber{1}{4.49}{0}\)
\begin{align}
\bm {j_c}\cdot \bm {n} & = G_+\Delta V_c + P G_+\Delta \bm {V^\prime _s}\cdot \bm {m} \\ \tilde {J}_s\bm {n} & = \frac {\mu _B}{e}\left [ P(\bm {j_c}\cdot \bm {n})\bm {m} +
(1-P^2)G_+(\Delta \bm {V_s}\cdot \bm {m})\bm {m} + aG_+\boldsymbol {m}\times (\Delta \bm {V_s}\times \boldsymbol {m}) - bG_+\Delta \bm {V_s}\times \boldsymbol {m}\right ]. \label
{eq:MCT_currents_haney_spin}
\end{align}
