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Numerical Analysis and Innovative Simulation
Techniques for Designing Advanced MRAM

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6.9 Conclusion

This chapter has presented a comprehensive three-dimensional finite-element framework for computing spin-transfer torques and simulating magnetization dynamics in ultra-scaled MRAM cells. The approach is founded on the coupled spin and charge drift-diffusion formalism, which provides a unified description of both interface-induced Slonczewski torques and bulk Zhang–Li torques arising from magnetization textures within a single computational framework, eliminating the need for separate analytical expressions or ad hoc superposition of torque contributions.

A central achievement of this work is the extension of the drift-diffusion formalism to MTJs through a physically motivated treatment of the tunnel barrier. By modeling the TB as a poor conductor with magnetization-dependent conductivity and incorporating spin-current boundary conditions derived from NEGFs and circuit theory, the framework accurately reproduces the characteristic angular and voltage dependencies expected in MTJs. The implementation employs a unified interface mapping strategy that enables consistent treatment of both tunneling spin-currents at TB interfaces and interlayer exchange coupling at SAF interfaces, facilitating the simulation of complex multi-layered stacks representative of modern MRAM architectures.

Validation against analytical solutions confirms that the finite-element implementation with the tunneling boundary conditions yields quantitative agreement for the spin-accumulation profile, the damping-like and field-like torque components, and the sinusoidal angular dependence characteristic of MTJs. The investigation of elongated ferromagnetic structures with magnetization textures reveals that the Slonczewski and Zhang–Li torque contributions are not simply additive: the polarized tunneling spin-current modifies the bulk spin-accumulation in a manner that cannot be captured by superposing independently computed torque terms. This finding underscores the necessity of a self-consistent treatment when simulating STT-driven dynamics in devices where domain walls or non-uniform magnetization states may develop during switching.

The systematic evaluation of time integration strategies demonstrates that the fully implicit BDF method, equipped with a preconditioner incorporating the linearized spin-torque Jacobian, achieves time steps that are two orders of magnitude larger than those of explicit or IMEX schemes, offering substantial computational savings for large-scale simulations. Nevertheless, the TPS method remains attractive due to its algorithmic simplicity, robust preservation of the unit-sphere constraint, and freedom from the need to tune a preconditioner. For the device simulations presented in Chapter 7, the TPS method is employed: its computational overhead is acceptable for the moderate problem sizes considered, and its unconditional stability simplifies the analysis of STT-driven switching phenomena without introducing numerical artifacts that could obscure the underlying physics.