Modeling Spin-Orbit Torques
in Advanced Magnetoresistive Devices
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Chapter 9 Summary and Outlook
Devices exploiting the electron’s spin have revolutionized data storage and magnetic sensing. Current research focuses on MRAM as a potential non-volatile replacement for SRAM and DRAM. Among these, SOT-MRAM is
especially promising as a cache memory due to its sub-ns switching speed and near-unlimited endurance resulting from separate read and write paths. However, its large-scale adoption remains limited by the lack of reliable
deterministic field-free switching, high switching current densities, and incompatibility with BEOL CMOS processes, particularly annealing temperatures above \(400^\circ \,\si {C}\).
Several promising approaches have been proposed to address these challenges, each requiring careful optimization of device design and material composition. Fast, flexible, and accurate TCAD tools are essential for guiding
experimental efforts and reducing the costs of research and development. Hence, this thesis was dedicated to the formulation and implementation of theoretical models and numerical methods for the accurate computation of SOTs
and the simulation of the resulting magnetization dynamics in various FM systems.
A unique FEM framework that combines the LLG equation with the charge and spin drift-diffusion model has already been successfully employed by S. Fiorentini to model STTs and switching in MTJs [74], providing a natural
starting point for this work. First, the drift-diffusion model was extended to capture the SHE and ISHE. The resulting FEM implementation enabled basic yet efficient modeling of SOTs in bilayer HM/FM systems. However, since
SOTs are highly dependent on the spin transport properties of the HM/FM interface, a more detailed description of the interface was required.
Thus, the NM/FM BCs from the MCT were employed to model the HM/FM interface. To implement the MCT BCs in the FEM solver, the framework had to be extended to support discontinuities in the solution across material
interfaces. This was accomplished through the introduction of effective interface layers, enabling standard FEM methods to capture the discontinuities. The implementation was validated against the analytical solution for a
NM/FM bilayer system. Additionally, a 1D-CSDD solver was developed to facilitate the study and analysis of typical experimental SOT systems, aiding both the development of new BCs and their subsequent integration into the
3D FEM solver.
The combination of the SHE and the MCT BCs successfully captured the reported thickness dependence of the DL torques; however, the large FL torques observed in thin HM layers remained unaccounted for. Previous studies
have attributed these FL torques to interfacial SOC effects, such as the REE. Since the MCT BCs do not include interfacial SOC contributions, an extension was required. By treating the Rashba SOC as a perturbation of the
Hamiltonian located at the interface, an extension of the BCs compatible with the 3D drift-diffusion formalism was derived. Using these extended BCs, both DL and FL torques were accurately reproduced, as demonstrated for
Pt/CoFeB and W/CoFeB bilayer systems.
The SOTs are typically described in terms of the conventional DL and FL components; however, experimental studies reveal a more complex dependence on the magnetization direction. To capture this nontrivial angular
dependence, interfacial SOC must be treated beyond the perturbative regime. Accordingly, BCs were derived based on 3D scattering from a spin-dependent delta-function potential barrier, where the combined exchange interaction
and SOC at the interface is represented as an effective magnetic field, allowing for the strong SOC regime to be considered. Using these BCs, the angular dependence of SOTs in Pt/Co and Ta/CoFeB systems could be accurately
reproduced with Rashba SOC. Furthermore, it was shown that the angular dependence is most pronounced when the SOC strength is comparable to the exchange interaction, and that the REE is the primary contributor to both
the strong FL torque and the observed angular dependence, in agreement with prior results for 2D-Rashba systems.
Both approaches to treating the interfacial SOC account for spin currents generated through the SOF and SOP mechanisms, which offer practical applications in FM/NM/FM trilayers. It was demonstrated that, with an in-plane
magnetization of the bottom FM layer, out-of-plane polarized spin currents can be generated, producing unconventional torques on the upper layer. Furthermore, calculations based on the fitted bilayer SOTs of W/CoFeB and
Pt/CoFeB indicate that the SOT efficiency can be enhanced or maintained by employing ultra-thin NM SLs, while simultaneously increasing the out-of-plane spin polarization component.
To simulate SOT-induced magnetization dynamics, the FEM framework was extended to include the DMI, which plays an essential role in systems with strong SOC. The DMI implementation was validated against analytical
solutions and previously reported numerical results. Combined with the full 3D drift-diffusion computed SOTs, the magnetization dynamics in typical SOT-MRAM devices could be studied in detail. This was demonstrated through
simulations of in-plane switching, perpendicular switching with an external field, and field-free switching assisted by an STT pulse through the MTJ. The resulting magnetization dynamics reveal a complex switching process,
dominated by domain nucleation and domain wall propagation in agreement with experimental observations.
Simulations of recently proposed practical switching schemes were performed using either the stray field of a MHM or the unconventional trilayer SOTs to achieve FFS. The stray field of the MHM, computed with a hybrid
FEM-BEM method, was shown to enable reliable FFS, though with limited performance gains and scalability. In contrast, switching using the unconventional trilayer SOTs demonstrated highly reliable FFS with low switching
times and current densities. The switching performance was shown to improve as the NM SL thickness decreased, due to enhanced SOT efficiency and increased out-of-plane spin polarization.
In summary, a comprehensive theoretical and numerical framework for computing SOTs and simulating the resulting magnetization dynamics in FM systems with strong interfacial SOC was developed. The implemented models
were verified against analytical solutions and experimental results, demonstrating their numerical and physical accuracy. The developed TCAD tools were employed to study various SOT systems and switching schemes,
demonstrating their versatility and effectiveness in providing valuable insights into the underlying physics and guiding future experimental efforts towards the realization of efficient and reliable SOT-MRAM devices.
Recently, new materials such as topological insulators, Weyl semimetals, AFMs, and altermagnets have shown great promise due to their large SOT efficiencies and intrinsic ability to generate out-of-plane spin polarization without
requiring additional FM layers. Future work could focus on extending the simulation framework to accurately model the spin currents generated in these novel materials and the resulting SOTs in adjacent FMs. Moreover, including
VCMA in the modeling framework could enable the study of voltage-assisted SOT switching, which has the potential to significantly reduce switching energy in SOT-MRAM devices, and allow for high-density multipillar devices to
be modeled.
Additionally, interfacial and interlayer exchange and DMI effects can have a significant impact on the magnetization dynamics in multilayer systems. Investigating their influence on the magnetization dynamics could help improve
reliability or lead to the discovery of new switching mechanisms and device architectures. Finally, exploring the effects of temperature, defects, and material inhomogeneities on SOT generation and magnetization dynamics could
provide further insights into optimizing device performance for practical applications.
