Numerical Analysis and Innovative Simulation
Techniques for Designing Advanced MRAM
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6.4 Numerical Implementation of Interlayer Exchange Coupling
As discussed in Section 6.3.3, the IEC is crucial for SAFs. While Chapter 3 defined the physics, this section
details the numerical implementation in the FEM solver, which follows the same logic as the TB mapping described above.
6.4.1 IEC Energy and Effective Field
The interlayer exchange coupling energy density for two ferromagnetic layers separated by a thin non-magnetic spacer (such as Ru or Ta) is given by:
\(\seteqnumber{0}{6.}{28}\)
\begin{equation}
\mathcal {E}_\mathrm {iec} = -J_\mathrm {iec} (\mathbf {m}_1 \cdot \mathbf {m}_2), \label {eq:iec-energy}
\end{equation}
where \(J_\mathrm {iec}\) is the interlayer exchange coupling constant (in \(\si {\joule \per \meter \squared }\)), and \(\mathbf {m}_1\) and \(\mathbf {m}_2\) are the unit magnetization vectors at the two interfaces.
The sign of \(J_\mathrm {iec}\) determines the preferred alignment: positive values favor parallel alignment (ferromagnetic coupling), while negative values favor anti-parallel alignment (antiferromagnetic coupling), as required
for SAF structures.
In dynamic micromagnetics, representing energy contributions in terms of effective fields rather than boundary conditions is often more convenient, as it facilitates direct integration into the LLG equation. The effective field
contribution from the IEC acting on a layer at the interface is obtained via \(\mathbf {H}_\mathrm {eff} = -\frac {1}{\mu _0 M_s} \frac {\partial \mathcal {E}}{\partial \mathbf {m}}\), yielding:
\(\seteqnumber{0}{6.}{29}\)
\begin{equation}
\mathbf {H}_\mathrm {iec} = \frac {J_\mathrm {iec}}{\mu _0 M_s t} \mathbf {m}_\mathrm {neighbor}, \label {eq:iec-field}
\end{equation}
where \(t\) is the thickness of the ferromagnetic layer.
6.4.2 IEC Boundary Term in Weak Formulation
In the FE weak formulation of the LLG equation, the IEC contribution appears as boundary integrals at the ferromagnet-spacer interfaces. For a structure with left (L) and right (R) ferromagnetic layers separated by a spacer (see
Figure 3.3), the following boundary terms must be added to the RHS of the weak formulation (5.25):
\(\seteqnumber{0}{6.}{30}\)
\begin{equation}
\frac {J_\mathrm {iec} \gamma }{\mu _0 M_{s,\mathrm {L}}} \int _{\Gamma _\mathrm {R|spacer}} \mathbf {m}_\mathrm {L} \cdot \mathbf {w} \, d\Gamma + \frac {J_\mathrm {iec} \gamma }{\mu _0
M_{s,\mathrm {R}}} \int _{\Gamma _\mathrm {spacer|L}} \mathbf {m}_\mathrm {R} \cdot \mathbf {w} \, d\Gamma , \label {eq:iec-weak-form}
\end{equation}
where \(\Gamma _\mathrm {R|spacer}\) and \(\Gamma _\mathrm {spacer|L}\) denote the interfaces between the spacer and the right and left ferromagnetic layers, respectively, and \(\mathbf {w}\) is the test function.
6.4.3 Mapping Strategy for IEC
To evaluate Equation (6.31), the solver employs the same mapping strategy as described in Section 6.3.3 for the tunneling spin-current:
-
(i) During initialization, for each integration point on one side of the IEC interface, the solver identifies the nearest integration point on the opposite side.
-
(ii) The element ID and local integration point index are stored in a mapping table.
-
(iii) During time-stepping, the magnetization values \(\mathbf {m}_\mathrm {neighbor}\) are accessed through these pre-computed maps with \(O(1)\) complexity.
This unified approach allows the simulation of arbitrary stacks (e.g., Co/Ru/Co SAFs).