Numerical Analysis and Innovative Simulation
Techniques for Designing Advanced MRAM
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6.2 Finite Element Implementation
6.2.1 Coupled LLG and Drift-Diffusion Framework
The LLG equation, coupled with the spin and charge drift-diffusion formalism for computing torques, enables the description of the magnetization dynamics of structures containing an arbitrary number of layers. The LLG equation
is given by:
\(\seteqnumber{0}{6.}{20}\)
\begin{equation}
\frac {\partial \mathbf {m}}{\partial t} = -\gamma \mathbf {m}\times \mathbf {H}_\mathrm {eff} + \alpha \mathbf {m}\times \frac {\partial \mathbf {m}}{\partial t} + \frac {1}{M_S}\mathbf
{T}_\mathrm {S}, \label {eq:llg}
\end{equation}
where \(\gamma \) is the gyromagnetic ratio, \(\alpha \) is the Gilbert damping parameter, \(M_S\) is the saturation magnetization, and \(\mathbf {H}_\mathrm {eff}\) is the effective field. The effective field includes the
magnetic anisotropy field, the exchange field, and the demagnetization field. As detailed in Chapter 4 , the demagnetization field contribution is evaluated only on the
disconnected magnetic domains by using the hybrid approach FE-BEM. The TPS described in Chapter 5 handles the temporal evolution, ensuring numerical stability
even with the stiff exchange terms.
Analytical solutions to these sets of partial differential equations are possible only in simplified scenarios, thus numerical methods are necessary to resolve the dynamics in general settings. The FEM naturally handles meshes with
complex geometries and multiple material domains. Therefore, it serves to implement a solver capable of simulating charge, spin-accumulation, and magnetization dynamics. The implementation utilizes the open-source C++ FE library MFEM [145].
6.2.2 Treatment of the Tunnel Barrier
Figure 6.1: (Left) Current density distribution in an MTJ biased under a constant voltage for a non-uniform magnetization configuration of the FL. The magnetization varies from parallel to the reference layer in the center to
anti-parallel on the sides. (Right) The current density is larger in the center, where the FL magnetization is parallel to that of the RL and the magnetization-dependent conductivity is highest. Based on the figure originally published
in [182].
A key challenge in applying the drift-diffusion formalism to MTJs is the treatment of the TB. While the TMR effect can be reproduced by describing the TB conductivity through an angle-dependent expression, using effective
material parameters in the spin diffusion equation is insufficient to capture all expected torque properties. The TB is modeled as a poor conductor with a local resistance that depends on the relative orientation of the
magnetization [123]. Accordingly, this dependence is introduced phenomenologically by expressing the TB conductivity as an explicit function of the relative magnetization angle, given by:
\(\seteqnumber{0}{6.}{21}\)
\begin{equation}
\sigma (\theta ) = \frac {\sigma _\mathrm {P} + \sigma _\mathrm {AP}}{2}\left (1 + \frac {\mathrm {TMR}}{2 + \mathrm {TMR}}\cos \theta \right ), \label {eq:tmr-conductivity}
\end{equation}
where \(\sigma _\mathrm {P}\) and \(\sigma _\mathrm {AP}\) are the conductivities in the parallel and anti-parallel configurations, respectively, TMR is the tunneling magnetoresistance ratio, and \(\theta \) is the angle
between the RL and FL magnetizations. This is a manifestation of Ohm’s law relating the voltage and the charge-current through a structure with many transverse modes [32]. Figure 6.1 shows the charge-current density computed in an MTJ structure with non-uniform magnetization in the FL. The magnetization varies from parallel to the RL in the
center to anti-parallel at the edges. Due to the angular dependence of the TB conductivity, the current density is highest in the center, where the magnetizations are parallel, demonstrating the expected TMR-induced current
redistribution.