Numerical Analysis and Innovative Simulation
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7.2 Computational Setup
The simulations in this chapter employ the coupled micromagnetic and spin-charge drift-diffusion framework developed in Chapter 6 , integrated with the FE
discretization described in Section 4.2 . The spin-transfer torque density at each spatial point is computed using the drift-diffusion formalism (Section 6.1 ) [32], with tunneling boundary conditions at each MgO barrier derived from the NEGFs analysis in Section 6.3 [118].
Table 7.1: Standard simulation parameters for the switching studies presented in this chapter. Values are representative of CoFeB\(|\)MgO-based magnetic tunnel junctions and are sourced from the Boris Computational
Spintronics material database and published experimental literature.
.
Parameter Symbol Value
Gilbert damping constant
\(\alpha \)
0.015–0.02
Gyromagnetic ratio
\(\gamma \)
1.76 · 1011 rad/(s T)
Saturation magnetization (CoFeB)
\(M_S\)
0.81 · 106 A/m to 1.2 · 106 A/m
Exchange stiffness constant (CoFeB)
\(A\)
1 · 10−11 J/m to 2 · 10−11 J/m
Interface anisotropy
\(K_i\)
1.53 · 10−3 J/m2
Current spin polarization
\(\beta _\sigma \)
0.52–0.7
Diffusion spin polarization
\(\beta _D\)
0.7–1.0
Electron diffusion coefficient (CoFeB)
\(D_{e,\text {FM}}\)
1 · 10−2 m2 /s to 2 · 10−2 m2 /s
Exchange length
\(\lambda _{J}\)
0.8 nm to 1 nm
Spin dephasing length
\(\lambda _{\varphi }\)
0.4 nm
Spin-flip length (CoFeB)
\(\lambda _{sf,\text {CoFeB}}\)
10 nm
Spin-flip length (Ta)
\(\lambda _{sf,\text {Ta}}\)
1.9 nm
Spin-flip length (Ru)
\(\lambda _{sf,\text {Ru}}\)
4 nm
Resistance (P state)
\(R_P\)
410 kΩ (ultra-scaled)
Resistance (AP state)
\(R_{AP}\)
750 kΩ (ultra-scaled)
Interlayer exchange coupling between non-adjacent magnetic layers is incorporated via the weak-form interface mapping described in Section 6.4 . Magnetization
dynamics are solved using the tangent-plane scheme (Section 5.3 ). Demagnetization fields are computed with the hybrid FE-BEM approach from Chapter 4 .
The standard simulation parameters employed throughout this chapter are summarized in Table 7.1 . Material set differences are detailed in Table 7.2 .
The simulated resistance values yield tunnel magnetoresistance ratios consistent with experimental CoFeB\(|\)MgO devices: \(\text {TMR} = (R_{AP} - R_P)/R_P = 83\%\) for the ultra-scaled composite FL (\(R_P = \SI
{410}{\kilo \ohm }\), \(R_{AP} = \SI {750}{\kilo \ohm }\)), and for the 70 nm SAF-enhanced stacks the input \(\text {RA} = \SI
{6.5}{\ohm \micro \meter \squared }\) with \(\text {TMR} = 150\%\) lies within the experimentally reported range (\(\text {RA} \approx \SIrange {3.5}{7}{\ohm \micro \meter \squared }\), \(\text {TMR}
\approx 100\)–\(250\%\)) for optimized MgO barriers at room-temperature [185, 186, 187]. The Gilbert damping constant \(\alpha = 0.015\)–\(0.02\) is representative of sputtered CoFeB thin
films [53]. In the macrospin limit, the critical switching current scales linearly with damping (\(I_{c0} \propto \alpha \)) [31], so the absolute switching times scale accordingly. However, the qualitative features
identified in this chapter, namely the sequential switching order, the back-hopping threshold, and the IEC role progression, are governed by the torque hierarchy and coupling geometry rather than the damping magnitude.
A critical insight emerges from investigating multi-layer composite FL structures: the spin-transfer torques in a multi-segment stack are inherently non-additive. The coupled partial differential equation system governing spin
transport exhibits essential non-linearity: the spin-accumulation profile in each magnetic layer is modified by the magnetization state of neighboring layers through spin-transport coupling across the shared interfaces, creating strong
feedback that cannot be decomposed into independent contributions. This non-additivity has direct practical implications: superposing single-layer simulation results will produce systematically incorrect switching times and
potentially miss qualitative features such as the sequential switching mechanism [154, 188].
All simulations presented in this chapter are performed at zero temperature (\(T = 0\) K). This deliberate choice isolates the deterministic switching dynamics from thermally activated processes, enabling distinct
identification of torque-driven phenomena. In particular, the back-hopping mechanism characterized in Section 7.4 is identified as a deterministic consequence of the
composite FL torque hierarchy precisely because stochastic thermal fluctuations are excluded from the simulations. The approximation is justified quantitatively: the characteristic switching times of 1 ns to 3 ns and back-hopping cycling periods of approximately
0.5 ns are many orders of magnitude shorter than the thermally activated reversal time \(\tau _{\text {th}} = \tau _0 \exp (\Delta )\),
which for \(\Delta > 60\) exceeds \(10^{17}\) s. Thermal fluctuations would add stochastic jitter to the trajectories without altering the qualitative switching sequence or the back-hopping mechanism.