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Numerical Analysis and Innovative Simulation
Techniques for Designing Advanced MRAM

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3.3 Modelling Spin-Transfer Torque

A commonly used model for torque in STT-MRAM devices is the macro-spin model proposed by Slonczewski [117]. Here, the dynamics of the FL magnetization are described as behaving like a single spin influenced by the polarization occurring in a polarizing layer. The LLG equation is extended by such a torque term and takes the following form:

\begin{equation} \frac {\partial \mathbf {m}}{\partial t} = -\gamma _0 \mathbf {m} \times \mathbf {H}_\text {eff} + \alpha \mathbf {m} \times \frac {\partial \mathbf {m}}{\partial t} + \frac {1}{M_s} \mathbf {T}_\text {S}, \end{equation}

where \(\mathbf {T}_\text {S}\) is the spin torque originating from the angular momentum transferred from spin-polarized electrons in the reference layer.

The spin torque \(\mathbf {T}_\text {S}\) can be expressed as a combination of damping-like and field-like components:

\begin{equation} \mathbf {T}_\text {S} = \underbrace {a \mathbf {m} \times (\mathbf {m} \times \mathbf {p})}_{\text {damping-like}} + \underbrace {b (\mathbf {m} \times \mathbf {p})}_{\text {field-like}}, \end{equation}

where parameters \(a\) and \(b\) depend on the current density and \(\mathbf {p}\) represents the magnetization direction of the RL. Based on this, the LLG equation becomes:

\begin{align} \partial _t \mathbf {m} & = \frac {1}{1 + \alpha ^2} \Bigg [ -\gamma _0 (\mathbf {m} \times \mathbf {H}_\text {eff}) - \gamma _0 \alpha (\mathbf {m} \times (\mathbf {m} \times \mathbf {H}_\text {eff})) \nonumber \\ & \quad - \frac {\gamma \hbar J_C}{e M_S d_{FL}} \eta (\theta ) \left ( \mathbf {m} \times (\mathbf {m} \times \mathbf {p}) - \beta (\mathbf {m} \times \mathbf {p}) \right ) \Bigg ], \label {eq::macrospin_model} \end{align} where \(J_C\) is the current density, \(d_{FL}\) is the thickness of the free layer, \(\beta \) quantifies the field-like torque strength, and \(\eta (\theta )\) is the torque efficiency function that depends on the angular relationship between \(\mathbf {m}\) and \(\mathbf {p}\) [35, 118], further on whether a spin-valve or an MTJ is being described [119].

While this macro-spin model provides insight into STT dynamics, it assumes spatially uniform magnetization, making it less accurate for larger devices. To improve this, both \(\mathbf {m}\) and \(\mathbf {p}\) can be treated as spatially dependent fields. However, the model in Equation (3.59) neglects spin diffusion effects, which are crucial for specific materials and configurations [90].

A more general formulation considers the spin-accumulation vector \(\mathbf {S}\), representing the non-equilibrium polarization of conducting electrons. The torque due to \(\mathbf {S}\) is:

\begin{equation} \mathbf {T}_\text {S} = -D_e \frac {\mathbf {m} \times \mathbf {S}}{\lambda _J^2} - D_e \frac {\mathbf {m} \times (\mathbf {m} \times \mathbf {S})}{\lambda _\varphi ^2}, \label {eq::Ts} \end{equation}

where \(D_e\) is the electron diffusion coefficient, \(\lambda _J\) is the exchange length, and \(\lambda _\varphi \) is the spin dephasing length. The first term models the precession of spins in the local exchange field, while the second represents dephasing [32, 120, 121, 122].

The steady-state spin-accumulation satisfies the spin current continuity equation:

\begin{equation} \partial _t \mathbf {S} = 0 = -\nabla \cdot \mathbf {J}_S - D_e \left ( \frac {\mathbf {S}}{\lambda _{sf}^2} + \frac {\mathbf {S} \times \mathbf {m}}{\lambda _J^2} + \frac {\mathbf {m} \times (\mathbf {S} \times \mathbf {m})}{\lambda _\varphi ^2} \right ), \end{equation}

where \(\lambda _{sf}\) is the spin-flip length and \(\mathbf {J}_S\) is the spin current. The finite element solver implemented in this work solves this equation in both ferromagnetic and nonmagnetic regions [120, 123, 124, 125]. The drift-diffusion formalism introduced here provides the theoretical foundation for computing spin-transfer torques from first principles. However, extending this semi-classical approach to magnetic tunnel junctions requires special treatment of the tunneling process via appropriate boundary conditions at the tunnel-barrier interfaces. The complete derivation of the spin and charge drift-diffusion equations, their finite element implementation, the treatment of tunneling spin currents, and validation against analytical and experimental results are presented in detail in Section 6.1.