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D I S S E R T A T I O N

Numerical Analysis and Innovative Simulation
Techniques for Designing Advanced MRAM

ausgeführt zum Zwecke der Erlangung des akademischen Grades
eines Doktors der technischen Wissenschaften

unter der Betreuung von

Associate Prof. Victor Sverdlov, Ph.D. MSc.
O.Univ.Prof. Dipl.-Ing. Dr.techn. Dr.h.c. Siegfried Selberherr

eingereicht an der Technischen Universität Wien
Fakultät für Elektrotechnik und Informationstechnik
von

Dipl.-Ing. Mario Bendra
Matrikelnummer: 51820045

Wien, im Juni 2026  

Abstract

Scaling spin-transfer torque magnetoresistive random access memory (STT-MRAM) to sub-20 nm dimensions poses major challenges for thermal stability, switching reliability, and energy efficiency. As devices shrink, the macro-spin approximation breaks down, requiring a rigorous treatment of non-uniform magnetization dynamics and transport phenomena. This thesis presents a computational framework for modeling and optimizing ultra-scaled, multi-layered magnetoresistive random access memory (MRAM) devices.

First, a three-dimensional micromagnetic solver based on finite element method (FEM) is developed. To address the computationally expensive demagnetization field in complex geometries, a hybrid finite element-boundary element method (FE-BEM) is used. Numerical stiffness from the exchange interaction is handled by a tangent-plane time integration scheme, which reformulates the nonlinear Landau–Lifshitz–Gilbert equation as a linear saddle-point problem. This approach is benchmarked against adaptive higher-order backward differentiation formula (BDF) and implicit-explicit (IMEX) methods, demonstrating superior stability and efficiency for the stiff dynamics of ultra-scaled magnetic elements.

Central to this work is the extension of the micromagnetic framework to include a coupled spin and charge drift-diffusion formalism. Unlike standard models that assume continuity of spin currents, this work derives and implements specialized boundary conditions. These are used to rigorously describe spin filtering across magnesium oxide (MgO) tunnel barriers and spin dephasing at metallic interfaces. Furthermore, a novel numerical approach for interlayer exchange coupling (IEC) is introduced. This leverages an interface-mapping algorithm to efficiently simulate synthetic antiferromagnet (SAF) without the prohibitive computational cost of volumetric meshing of angstrom-scale spacers.

Applying this unified solver reveals that reliability issues in ultra-scaled devices, such as back-hopping, are deterministic consequences of the inter-segment torque hierarchy within composite free layers. They are not stochastic thermal effects. Crucially, this thesis demonstrates that by engineering the relative tunnel barrier polarizations and free layer geometry, this instability can be either suppressed for reliable binary switching or deliberately exploited. This allows realization of multi-level cell (MLC) capable of storing multiple bits per physical cell through distinct intermediate magnetization configurations. Additionally, double-spin torque magnetic tunnel junction (ds-MTJ) architectures are investigated. In these, the cooperation between giant magnetoresistance (GMR) and tunneling magnetoresistance (TMR) torques enables sub-nanosecond switching. When IEC is introduced in advanced multi-layer stacks, it emerges as a unifying design parameter. Specific coupling windows govern the stability of the SAF-enhanced reference layer. Hybrid free layers with metallic spacers exploit the spacer material’s spin-flip length to enhance switching speed. The joint optimization of spacer material and thickness determines write efficiency.

Kurzfassung

Die Skalierung von Spin-Transfer Torque Magnetoresistive Random Access Memory (STT-MRAM) in den Sub-20 nm-Bereich bringt tiefgreifende Herausforderungen hinsichtlich thermischer Stabilität, Schaltzuverlässigkeit und Energieeffizienz mit sich. Mit schrumpfenden Bauteilabmessungen wird die Makro-Spin-Näherung unzureichend, was eine rigorose Behandlung der räumlich nicht-uniformen Magnetisierungsdynamik und Transportphänomene erfordert. Diese Arbeit präsentiert ein umfassendes computergestütztes Framework für die Modellierung und Optimierung von ultra-skalierten, mehrschichtigen MRAM-Bauteilen.

Zunächst wird ein dreidimensionaler mikromagnetischer Löser basierend auf der Finiten-Elemente-Methode (FEM) entwickelt. Um das rechenintensive Entmagnetisierungsfeld in komplexen Geometrien aufzulösen, wird eine hybride Finite-Elemente-Randelemente-Methode (FE-BEM) eingesetzt. Der numerischen Steifheit, die der Austauschwechselwirkung inhärent ist, wird durch die Implementierung eines Tangential-ebenen-Zeitintegrationsschemas begegnet, welches die nichtlineare Landau–Lifshitz–Gilbert Gleichung als lineares Sattelpunktproblem neu formuliert. Dieser Ansatz wird gegen adaptive Backward Differentiation Formulas (BDF) höherer Ordnung und implizit-explizite (IMEX) Methoden verglichen und zeigt überlegene Stabilität und Effizienz für die steife Dynamik ultra-skalierter magnetischer Elemente.

Im Zentrum dieser Arbeit steht die Erweiterung des mikromagnetischen Frameworks um einen gekoppelten Spin- und Ladungs-Drift-Diffusions-Formalismus. Anders als Standardmodelle, die eine Kontinuität der Spinströme annehmen, leitet diese Arbeit spezialisierte Randbedingungen her und implementiert diese, um die Spinfilterung durch Magnesiumoxid (MgO)-Tunnelbarrieren und die Spindephasierung an metallischen Grenzflächen rigoros zu beschreiben. Darüber hinaus wird eine neuartige numerische Behandlung der Interlayer Exchange Coupling (IEC) eingeführt, die einen Algorithmus zur Grenzflächenabbildung nutzt. Dies ermöglicht die effiziente Simulation von Synthetic Antiferromagnets (SAF) ohne den prohibitiven Rechenaufwand einer volumetrischen Vernetzung von Abstandshaltern im Angström-Bereich.

Die Anwendung dieses vereinheitlichten Lösers zeigt, dass Zuverlässigkeitsprobleme in ultra-skalierten Bauteilen, wie etwa das „Back-Hopping", deterministische Konsequenzen der Inter-Segment-Torque-Hierarchie innerhalb zusammengesetzter freier Schichten sind und nicht auf stochastische thermische Effekte zurückzuführen sind. Entscheidend ist der Nachweis, dass diese Instabilität durch gezielte Abstimmung der relativen Tunnelbarrieren-Polarisationen und der Geometrie der freien Schicht entweder für zuverlässiges binäres Schalten unterdrückt oder gezielt ausgenutzt werden kann, um Multi-Level Cells (MLC) zu realisieren, die über unterschiedliche intermediäre Magnetisierungskonfigurationen mehrere Bits pro physischer Zelle speichern können. Zusätzlich werden Double Spin-Torque (dsMTJ)-Architekturen untersucht, bei denen die Synergie zwischen Giant Magnetoresistance (GMR) und Tunneling Magnetoresistance (TMR) Torques ein Schalten im Sub-Nanosekundenbereich ermöglicht. Wird Interlayer Exchange Coupling (IEC) in fortgeschrittenen Mehrschichtstrukturen eingeführt, erweist es sich als vereinheitlichender Designparameter: Spezifische Kopplungsfenster bestimmen die Stabilität SAF-verstärkter Referenzschichten, hybride freie Schichten mit metallischen Spacern nutzen die Spin-Flip-Länge des Spacer-Materials zur Verbesserung der Schaltgeschwindigkeit, und die gemeinsame Optimierung von Spacer-Material und -Dicke bestimmt die Schreibeffizienz.

Sažetak

Skaliranje Spin-Transfer Torque Magnetoresistive Random Access Memory (STT-MRAM) u područje ispod 20 nm donosi duboke izazove u pogledu termalne stabilnosti, pouzdanosti prebacivanja i energetske učinkovitosti. Kako se dimenzije uređaja smanjuju, aproksimacija makro-spina postaje nedostatna, što zahtijeva rigorozan tretman prostorno neuniformne dinamike magnetizacije i transportnih fenomena. Ova disertacija predstavlja sveobuhvatan računalni okvir za modeliranje i optimizaciju ultra-skaliranih, višeslojnih MRAM uređaja.

Prvo, razvijen je trodimenzionalni mikromagnetski rješavač temeljen na metodi konačnih elemenata (FEM). Za rješavanje računalno zahtjevnog polja demagnetizacije u složenim geometrijama koristi se hibridna metoda konačnih elemenata i rubnih elemenata (FE-BEM). Numerička krutost svojstvena interakciji izmjene rješava se implementacijom sheme vremenske integracije u tangentnoj ravnini, koja reformulira nelinearnu Landau–Lifshitz–Gilbertovu jednadžbu kao linearni problem sedlaste točke. Ovaj pristup uspoređen je s adaptivnim formulama diferencijacije unatrag (BDF) višeg reda i implicitno-eksplicitnim (IMEX) metodama, pokazujući superiornu stabilnost i učinkovitost za krutu dinamiku ultra-skaliranih magnetskih elemenata.

Središnji dio ovog rada je proširenje mikromagnetskog okvira uključivanjem spregnutog formalizma drifta i difuzije spina i naboja. Za razliku od standardnih modela koji pretpostavljaju kontinuitet spinskih struja, ovaj rad izvodi i implementira specijalizirane rubne uvjete za rigorozan opis filtriranja spina kroz tunelske barijere od magnezijevog oksida (MgO) i defaziranja spina na metalnim sučeljima. Nadalje, uveden je novi numerički tretman za Interlayer Exchange Coupling (IEC), koristeći algoritam mapiranja sučelja koji omogućuje učinkovitu simulaciju sintetičkih antiferomagneta (SAF) bez prohibitivnog računalnog troška volumetrijskog umrežavanja razmačnika na razini angstrema.

Primjena ovog objedinjenog rješavača otkriva da su problemi pouzdanosti u ultra-skaliranim uređajima, poput „back-hoppinga", determinističke posljedice hijerarhije inter-segmentnih momenata unutar kompozitnih slobodnih slojeva, a ne stohastičkih termalnih efekata. Ključno je da ova teza pokazuje kako se preciznim podešavanjem relativnih polarizacija tunelskih barijera i geometrije slobodnog sloja ova nestabilnost može ili potisnuti za pouzdano binarno prebacivanje ili ciljano iskoristiti za realizaciju višerazinskih ćelija (MLC) sposobnih za pohranu više bitova po fizičkoj ćeliji putem različitih intermedijarnih konfiguracija magnetizacije. Dodatno, istražene su arhitekture Double Spin-Torque (dsMTJ), gdje sinergija između momenata divovskog magnetootpora (GMR) i tunelskog magnetootpora (TMR) omogućuje prebacivanje ispod jedne nanosekunde. Kada se Interlayer Exchange Coupling (IEC) uvede u napredne višeslojne strukture, on postaje objedinjujući projektni parametar: specifični prozori vezanja određuju stabilnost SAF-pojačanih referentnih slojeva, hibridni slobodni slojevi s metalnim razmačnicima iskorištavaju duljinu spin-flipa materijala razmačnika za poboljšanje brzine prebacivanja, a zajednička optimizacija materijala i debljine razmačnika određuje učinkovitost zapisivanja.

Acknowledgement

First and foremost, I would like to express my deepest gratitude to my supervisor, Prof. Viktor Sverdlov, for the opportunity to pursue my doctorate under his guidance. I greatly value the freedom he gave me to explore different directions in my research. His genuine trust in letting me find my own path stood out. His unwavering support extended beyond academia. When I became a father, his understanding, patience, and encouragement meant more to me than I can put into words. I am deeply grateful for having a supervisor who cared not just about my scientific growth but also about my well-being. It has been a privilege to work with him within the Christian Doppler Laboratory for Nonvolatile Magnetoresistive Memory and Logic.

I would also like to sincerely thank Prof. Siegfried Selberherr, who founded the Institute for Microelectronics more than three decades ago, for his steady support of my work, for fostering a culture of open scientific exchange, especially through the weekly seminars, and for cultivating an excellent and inspiring working environment.

I am grateful for the opportunity to work at the Christian Doppler Laboratory for Nonvolatile Magnetoresistive Memory and Logic, which provided an outstanding research environment. My sincere thanks go to my colleagues Roberto, Johannes, Simone, Tomáš, Nils, Bernhard, and all other members of the laboratory for their collaboration, friendship, and the many stimulating discussions we shared. I would also like to thank Wolfgang Goes for the fruitful collaboration with our laboratory.

Furthermore, I thank all my colleagues at the Institute for Microelectronics for the warm atmosphere and many enriching conversations. In particular, I would like to thank Diana for her tireless help with organizational and administrative matters. I am also grateful to Manfred, Cerv, and the entire technical and administrative staff for ensuring the institute’s infrastructure has always run smoothly.

My progress relied on a wide and invaluable network of family and friends. Without them, science would be far less meaningful and certainly far less enjoyable. I am deeply grateful to my parents and my entire family for their unconditional support at every turn throughout my life and studies.

My heartfelt thanks also go to our closest friends, Stjepan and Ana. Your steadfast support, trust, and friendship have accompanied us through every stage of this journey. Your presence has meant more than words can express.

Above all, I would like to express my deepest and most personal gratitude to my beloved wife, Gabrijela, for her boundless love, patience, and encouragement throughout this demanding journey. She supported me, especially during the long and difficult months of the pandemic. Her strength, understanding, and unwavering belief in me made this work possible in ways that cannot be measured.

Finally, I dedicate this achievement to my daughter, Aurelia-Izabel. You are my greatest motivation and my greatest joy. I hope that one day you will look at this work and see not only a scientific accomplishment, but also a reminder that curiosity, perseverance, and kindness can go hand in hand. I strive to be a role model for you and to encourage you to follow your own path with confidence and courage.

Without the love and support of my family and friends, none of this would have been the same. I am deeply thankful to all of you.

Contents

Abstract

Kurzfassung

Sažetak

Acknowledgement

List of Figures

List of Tables

1 Introduction

1.1 Outline of the Thesis

1.2 Research Setting

2 Spin-Transfer Torque Magnetic Random Access Memory

2.1 Introduction to Magnetoresistive Random Access Memory

2.1.1 Historical Context

2.2 Magnetoresistance Phenomena

2.2.1 Giant Magnetoresistance

2.2.2 Tunnel Magnetoresistance

2.3 Spin-Transfer Torque Switching

3 Micromagnetic Modelling

3.1 Landau-Lifshitz-Gilbert Equation

3.2 Effective Magnetic Field

3.2.1 Exchange Field

3.2.2 Zeeman Field

3.2.3 Anisotropy Field

3.2.3.1 Uniaxial Anisotropy

3.2.3.2 Planar Anisotropy

3.2.3.3 Cubic Anisotropy

3.2.4 Demagnetizing Field

3.2.5 Ampère Field

3.2.6 Thermal Field

3.2.7 Interlayer Exchange Coupling

3.3 Modelling Spin-Transfer Torque

4 Computational Methods

4.1 Finite Difference Method

4.1.1 Discretization of the Landau-Lifshitz-Gilbert Equation

4.1.1.1 External Field

4.1.1.2 Exchange Field

4.1.1.3 Anisotropy Field

4.1.1.4 Demagnetizing Field

4.1.1.5 Ampère Field

4.1.1.6 Thermal Field

4.1.1.7 Time Integration Scheme

4.2 Finite Element Method

4.2.1 Weak Formulation and Variational Form

4.2.2 Finite Element Discretization

4.3 Finite Element Micromagnetic Solver

4.3.1 Efficient Algorithms for Demagnetizing Field Computation

4.3.1.1 Truncation Approach

4.3.1.2 Hybrid Finite Element–Boundary Element Method

4.3.1.3 Verification and Benchmarking

5 Numerical Time Integration of the LLG Equation

5.1 Challenges in LLG Time Integration

5.1.1 Stiffness and Disparate Time Scales

5.1.2 The Unit-Sphere Constraint

5.1.3 Structural Properties

5.2 Classification of Time Integrators

5.2.1 Explicit Runge–Kutta Methods

5.2.2 Implicit Methods and A-Stability

5.2.3 Multistep Methods: BDF Schemes

5.2.4 Implicit-Explicit Method

5.3 The Tangent-Plane Scheme

5.3.1 Reformulation of the LLG Equation

5.3.2 Weak Formulation

5.3.3 Complete Weak Form with Spin-Transfer Torque

5.3.4 Time Discretization: The \(\theta \)-Method

5.3.5 The Saddle-Point System

5.3.6 Constraint Enforcement and Magnetization Update

5.4 Adaptive Multistep Methods via SUNDIALS

5.4.1 CVODE: Variable-Order BDF

5.4.2 ARKStep: IMEX Runge–Kutta

5.5 Verification and Validation

5.5.1 The \(\mu \)MAG Standard Problem 4

5.5.2 Solver Comparison on Standard Problem 4

5.5.3 Constraint Preservation and Solver Accuracy

5.5.4 Adaptive Time Stepping

5.6 Summary

6 Spin-Transfer Torques in Ultra-Scaled MRAM Cells

6.1 Spin and Charge Drift-Diffusion Formalism

6.1.1 Derivation of the Transport Equations

6.1.2 Three-Dimensional Formulation

6.1.3 Spin-Accumulation Equation

6.1.4 Spin Dephasing and Complete Transport Equations

6.2 Finite Element Implementation

6.2.1 Coupled LLG and Drift-Diffusion Framework

6.2.2 Treatment of the Tunnel Barrier

6.3 Tunneling Spin-Current Boundary Conditions

6.3.1 Tunneling Current Expressions

6.3.2 Finite Element Boundary Terms

6.3.3 Implementation of Interface Mapping

6.4 Numerical Implementation of Interlayer Exchange Coupling

6.4.1 IEC Energy and Effective Field

6.4.2 IEC Boundary Term in Weak Formulation

6.4.3 Mapping Strategy for IEC

6.5 Validation Results

6.5.1 Spin-Accumulation and Spin Torque Solutions

6.5.2 Effect of Dephasing Length on Torque Localization

6.6 GMR Effect in Spin-Valves

6.7 Torques in Elongated Structures

6.7.1 Comparison with Drift-Diffusion Solution

6.7.2 Interaction Between Tunneling and Bulk Torques

6.8 Preconditioner Evaluation

6.9 Conclusion

7 Advanced Switching Dynamics and Novel Device Structures

7.1 Introduction

7.2 Computational Setup

7.3 Ultra-Scaled MRAM Cell Switching

7.3.1 Device Architecture and Geometry

7.3.2 Elongated FL: Domain Wall Failure

7.3.3 Composite FL Solution

7.3.4 Switching Mechanisms in Composite FLs

7.3.5 FL Length Optimization

7.3.6 Material Parameter Robustness

7.3.7 Switching Characteristics in Asymmetric FLs

7.4 Back-Hopping and Multi-Level Cell Operation

7.4.1 Torque Analysis

7.4.1.1 AP\(\to \)P: Cooperative Torques

7.4.1.2 P\(\to \)AP: Competing Torques and Back-Hopping Origin

7.4.2 Cyclic Switching Visualization

7.4.3 Multi-Level Cell Realization

7.4.4 FL Thickness Variation

7.4.5 Voltage Polarity Robustness

7.5 Interlayer Exchange Coupling Effects

7.5.1 Interlayer Exchange Coupling as Control Parameter

7.5.2 Systematic IEC Sweep Across Three Stack Configurations

7.6 Advanced Structures

7.6.1 SAF Architecture and Operating Principle

7.6.2 Reliability Requirements: The Critical Role of Coupling Strength

7.6.3 Spin-Torque Analysis and IEC Design Criterion

7.6.4 Hybrid FL with Metallic Spacers

7.6.5 Four-Stage Torque Decomposition

7.6.6 Write Efficiency Analysis

7.6.7 Device Concept and Dual Torque Mechanism

7.6.8 Influence of Spacer Material

7.6.9 Role of Interlayer Exchange Coupling

7.6.10 SAF-Enhanced Double Spin-Torque MTJ

7.7 Synthesis

7.7.1 Design Guidelines for Ultra-Scaled Composite FL Structures

7.7.2 Design Guidelines for Multi-Level Cell Implementation

7.7.3 Design Guidelines for ds-MTJ and SAF-Enhanced Structures

7.7.4 Cross-Architecture Performance Comparison

8 Summary and Outlook

8.1 Summary

8.1.1 Micromagnetic Solver and Time Integration

8.1.2 Coupled Spin and Charge Drift-Diffusion Transport

8.1.3 Interlayer Exchange Coupling and Interface-Mapping

8.1.4 Application to Advanced MRAM Architectures

8.2 Outlook: Advanced Numerical Methods for Micromagnetics

8.2.1 Implicit Treatment of Spin-Transfer Torque in IMEX Schemes

8.2.2 Higher-Order Finite Elements for Exchange Resolution

8.2.3 Parallel-in-Time Integration

8.2.4 Unified Space-Time Finite Element Methods

8.3 Conclusion

Bibliography

List of Publications

List of Figures

1.1 Schematics of different existing memory types (gray pyramid) and prospective spintronic technologies based on MTJs that could replace them (colored pyramid). L1-L4 denote various cache levels; SRAM and DRAM represent static and dynamic random-access memory, respectively; and HDD represents magnetic hard disk drives. Mechanisms for writing bit information in MTJ devices include the Oersted field, SOT-MRAM, STT-MRAM, and DW motion.

2.1 Illustration of spin-dependent electron scattering in a metallic multilayer structure demonstrating the GMR effect. Red arrows represent spin-up electrons, blue arrows denote spin-down electrons, and black arrows indicate the magnetization direction. In the AP configuration (left), both spin channels experience greater scattering, leading to higher resistance. In the P configuration (right), majority-spin electrons encounter less scattering, resulting in lower resistance.

2.2 Illustration of spin-dependent tunneling responsible for the TMR effect. In the P configuration (left), spin-polarized electrons tunnel into available states of the same spin orientation in the second ferromagnetic layer, resulting in high conductance. In the AP configuration (right), spin mismatch reduces the tunneling probability, leading to low conductance.

2.3 Illustration of magnetization orientations in MTJs, showing the difference between in-plane (iMTJ) and perpendicular (pMTJ) configurations.

3.1 Magnetization dynamics on the unit sphere. (a) Pure precession: \(\mathbf {m}\) rotates around \(\mathbf {H}_{\mathrm {eff}}\) at constant polar angle (no damping, \(\alpha = 0\)). (b) Pure damping: \(\mathbf {m}\) relaxes directly toward \(\mathbf {H}_{\mathrm {eff}}\) along a meridian (no precession). (c) Combined motion: damped precession spirals toward equilibrium.

3.2 Anisotropy functions visualized as energy surfaces. (a) Uniaxial anisotropy with \(K_u = 1\) and \(\mathbf {a} = \mathbf {e}_3\). (b) Planar anisotropy with \(K_p = 1\) and \(\mathbf {a} = \mathbf {e}_3\). (c) Cubic anisotropy with \(K_{c1} = K_{c2} = 1\), and \(\mathbf {a}_i = \mathbf {e}_i\) for all \(1 \leq i \leq 3\).

3.3 Schematic depiction of a trilayer structure composed of left and right semi-infinite FM regions, separated by a NMS layer. The interface magnetization on the left FM interface points into an arbitrary direction, whereas the interface magnetization in the right FM forms an angle \(\Delta \phi \) relative to the magnetization of the left FM. The magnetization vectors originate at the respective interface regions \(\partial \Omega _1\) and \(\partial \Omega _2\), which are highlighted by hatching. Figure based on the figure originally published in  [116].

4.1 Discretization of a unit sphere using two common methods in micromagnetics: (a) structured Cartesian mesh used in FDM, and (b) unstructured tetrahedral mesh used in FEM.

4.2 Mesh points for the FDM grid in one (a) and two (b) spatial dimensions. The grid spacing is uniform with \(h\) between adjacent points. In (b), the 2D indexing convention for a \(3 \times 3\) grid is shown.

4.3 Representation of the original solution \(u\) and the finite element approximation \(u_h\) in one (a) and two (b) spatial dimensions. The basis functions for all nodes are shown at the bottom of the 1D plot. The highlighted basis function \(\varphi _2(x)\) and corresponding solution value \(u_2\) at node \(x_2\) are shown in orange. The 2D plot shows the smooth function \(u(x, y)\), its piecewise approximation \(u_h(x, y)\), and the basis function \(\varphi _{5,6}(x, y)\) centered at node \((x_5, y_6)\).

4.4 Flowcharts of the micromagnetic simulation implementations.

4.5 Finite-element mesh used for computing the demagnetization field with the truncation approach. (a) The magnetic region is shown in blue, while the surrounding external region is shown in light blue. The external domain is chosen to be approximately five times larger than the magnetic region in each spatial direction. (b) The magnetic region represents the area of interest and is embedded within the larger external domain used to model open boundary conditions.

4.6 Comparison of demagnetizing energy for three distinct magnetization states: (a) uniform, (b) flower, and (c) vortex. For each case, the top subfigures show the corresponding magnetic potential, and the bottom subfigures show the magnetization distribution.

4.7 Comparison of demagnetizing energy for unit cubes with different magnetization configurations (uniform, flower, vortex), calculated using the truncation approach and the hybrid FE-BEM approach against analytical reference values  [153]. The hybrid FE-BEM shows excellent agreement with analytical results, while the truncation approach significantly underestimates the energy. Based on the figure originally published in  [154].

4.8 Two-disc configuration with parallel magnetizations. Left: demagnetizing field \(H_{d,x}\) along the symmetry axis. Right: relative error with respect to the analytical reference (). The FE-BEM maintains accuracy across the axis, while the truncation approach shows significant deviations.

4.9 Two-disc configuration with anti-parallel magnetizations. Left: demagnetizing field \(H_{d,x}\) along the symmetry axis. Right: relative error with respect to the analytical reference. The FE-BEM reproduces the analytical field within a few percent, while the truncation approach yields larger errors.

4.10 Demagnetizing field in an STT-MRAM cell with P magnetization configuration. Arrows indicate the demagnetization field direction, while color-coding represents the magnetic scalar potential \(u\). Left: hybrid FE-BEM. Right: truncation approach. Both methods yield similar results, validating the implementation for disconnected magnetic domains. Figure originally published in  [154].

4.11 Demagnetizing field in an STT-MRAM cell with AP magnetization configuration. Arrows indicate the demagnetization field direction, while color-coding represents the magnetic scalar potential \(u\). Left: hybrid FE-BEM. Right: truncation approach. Both methods yield similar results, validating the implementation for disconnected magnetic domains. Figure originally published in  [154].

5.1 Hierarchy of time scales in micromagnetic simulations. The exchange interaction imposes the CFL stability limit (\(\tau _\mathrm {exch} \sim 10^{-12}\) s for \(h = 2\) nm mesh). Demagnetization dynamics occur at \(\tau _\mathrm {demag} \sim 10^{-11}\) s. Precession due to external fields operates at \(\tau _\mathrm {prec} \sim 10^{-9}\) s. Damping and switching occur at \(10^{-9}\) – \(10^{-8}\) s, far above the explicit stability limit. Material: Permalloy (\(A = 1.3 \times 10^{-11}\) J/m, \(M_s = 8 \times 10^5\) A/m, \(\alpha = 0.02\)).

5.2 Stability regions in the complex plane \(z = \lambda \Delta t\) with physical eigenvalues. The Physics of Interest (Precession, Damping) lies near the origin and could be stable for explicit methods. The exchange eigenvalue position depends on the time step: for \(\Delta t = 10^{-13}\,\si {\second }\), \(\lambda _\mathrm {exch} \Delta t \approx -1\) lies within the RK4 stability region (black dot). Increasing \(\Delta t\) moves it left (gray dot), eventually exiting explicit stability regions. Explicit solvers (Red/Blue) are forced to take tiny time steps (\(\Delta t \to 0\)) just to keep the black dot inside their stability boundary. The Implicit solver (Green) remains stable for any \(\Delta t\), allowing us to step over the stiff exchange modes. IMEX methods combine both approaches: exchange modes are treated implicitly (Green), while precession and damping remain explicit (Red/Blue).

5.3 Block structure of the tangent-plane scheme saddle-point system corresponding to (). The \((1,1)\)-block \(A\) combines the damping, precession, and implicit exchange terms. The off-diagonal blocks \(B\) and \(B^T\) enforce the tangent-plane constraint \(\mathbf {v} \cdot \mathbf {m} = 0\), while the solution vector contains both the velocity \(\mathbf {v}\) and Lagrange multipliers \(\bm {\lambda }\).

5.4 Setup for the \(\mu \)MAG Standard Problem 4. (a) Permalloy film geometry (\(500 \times 125 \times 3\) \(\si {\nano \meter \cubed }\)) discretized by an unstructured tetrahedral mesh with 45391 elements. (b) Initial S-state magnetization configuration showing the characteristic S pattern in the \(xy\)-plane, colored by the \(x\)-component of magnetization.

5.5 Magnetization dynamics for Standard Problem 4 (Field 1) computed with different time integrators. All solvers agree with the \(\mu \)MAG reference solutions  [174, 175].

5.6 Deviation of computed magnetization trajectories from the \(\mu \)MAG reference solutions: (left) d’Aquino et al.  [174], (right) Albuquerque et al.  [175]. All solvers exhibit comparable deviation from both references, with differences attributable to the finite element discretization rather than time integration errors.

5.7 (a) Deviation from the unit-sphere constraint \(|\mathbf {m}|=1\) before renormalization. (b) Difference between solver trajectories and the TPS reference solution. (c) Time step evolution for adaptive solvers (BDF, IMEX) compared to the fixed time step used by TPS and RK4.

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].

6.2 Validation of the spin drift-diffusion model with tunneling spin-current boundary conditions for semi-infinite ferromagnetic layers. (a) spin-accumulation and (b) spin-transfer torque components computed along the stack direction, comparing the analytical solution with the FE implementation. The magnetization is oriented along \(x\) in the RL and along \(z\) in the FL. (c) Angular dependence of the damping-like torque for several values of the RL spin polarization (\(P_\mathrm {RL} = 0.4\), \(0.7\), \(0.9\)), reproducing the expected sinusoidal behavior characteristic of MTJs. Figure (c) adapted from  [182].

6.3 Spin-accumulation and torque in a symmetric MTJ structure including NM contacts. (a) and (b) present results for \(\lambda _\varphi = 0.4\) \(\si {\nano \meter }\). The shorter dephasing length guarantees faster decay of the transverse spin-accumulation components, so that the torque acts only in the proximity of the TB interface. Adapted from the figure  [182].

6.4 Angular dependence of the charge current for (a) an MTJ with MgO tunnel barrier at two TMR ratios (\(V = \SI {-1.3}{\volt }\)) and (b) a spin valve with metallic spacer at two GMR ratios (\(V = \SI {-0.2}{\volt }\)). Markers represent simulation data and solid/dashed lines show analytical fits using Equation () and Equation (), respectively.

6.5 (a) Non-uniform magnetization texture with the magnetization orientation changing from \(x\) to \(-z\). (b) Comparison of the spin torque \(\mathbf {T}_\mathrm {S}\) to the ZL torque \(\mathbf {T}_\mathrm {ZL}\) for an approximately 100 nm long magnetization texture, with the parameters of Table . (c) Comparison for an approximately 3 nm long magnetization texture with \(\lambda _\varphi = 0.4\) nm. The shorter dephasing length quickly dissipates the transverse spin-accumulation, ensuring agreement between the ZL approximation and the drift-diffusion solution. The two approaches show good agreement in both cases.

6.6 (a) Torque computed for an MRAM cell with elongated RL and FL and a magnetization profile in the FL similar to that of Figure  (a), with a width of approximately 3 \(\si {\nano \meter }\). The brown vectors indicate the magnetization direction in the RL and in two parts of the FL. (b) Close-up of the spin torque \(\mathbf {T}_\mathrm {S}\) compared to the ZL torque \(\mathbf {T}_\mathrm {ZL}\). The presence of the MTJ also influences the bulk torque component, making the unified approach the most suitable for ultra-scaled MTJs with elongated ferromagnetic layers. Figure (a) originally published in  [182].

6.7 Spatially-averaged magnetization dynamics during STT-driven switching computed with different time integrators. The TPS solution with \(\Delta t = 0.01\,\si {\pico \second }\) serves as the reference.

6.8 Quantitative solver analysis: (a) deviation from the unit-sphere constraint before renormalization; (b) trajectory deviation from the TPS reference solution; (c) time step evolution for the adaptive solvers compared to the fixed \(\Delta t = 0.01\,\si {\pico \second }\) baseline.

7.1 Three-dimensional renderings of (a)–(b) conventional large-diameter MTJ cells and (c)–(e) ultra-scaled 2.3 nm diameter pillar geometries with different FL configurations: elongated single FL, composite two-segment FL with middle TB, and composite FL with non-magnetic metallic spacer  [182].

7.2 Magnetization dynamics in an ultra-scaled cell with a 15 nm elongated single FL (cf. Figure  (c)) at ±2 V bias. (a), (b) Time evolution of magnetization components for AP to P and P to AP initial configurations showing failed switching. (c) Three-dimensional magnetization snapshot at 2 ns revealing the domain wall that prevents reversal. Figures based on  [182].

7.3 Successful switching of composite FL structures at −2 V bias: (a) two 5 nm FL segments separated by a 0.9 nm MgO TB (cf. Figure  (d)), reaching switching after approximately 2 ns, (b) two 5 nm FL segments separated by a 2 nm non-magnetic metallic spacer (cf. Figure  (e)), achieving switching after approximately 1 ns. Figures based on  [182].

7.4 Three-dimensional magnetization snapshots of the sequential switching process in ultra-scaled composite FL cells (cf. Figure  (d)), showing initial, intermediate (plateau), and final states for (a) AP\(\to \)P and (b) P\(\to \)AP transitions. The switching order is direction-dependent: FL\(_1\) switches first in AP\(\to \)P, while FL\(_2\) switches first in P\(\to \)AP, reflecting the asymmetric torque configurations at each interface  [123].

7.5 Magnetization trajectories during (a) AP\(\to \)P and (b) P\(\to \)AP switching for FL segment lengths of 2, 3, 4, 5, and 7.5 nm with standard polarization (\(P_1 = 0.62\), \(P_2 = 0.57\), \(P_3 = 0.2\)) and Set A material parameters (Table ). Critical design constraints emerge: segments shorter than approximately 5 nm exhibit back-hopping instability during P\(\to \)AP switching, because the anisotropy barrier scales with segment length (\(\mathcal {E}_{\mathrm {B}} \propto K_\text {eff} \cdot l\)) while the interfacial spin-transfer torque is approximately length-independent, so shorter segments satisfy the instability condition () more readily. Segments longer than approximately 6 nm fail to complete switching at moderate bias. The optimal range of 3 nm to 5 nm provides reliable switching in both directions. Based on Figure  [193].

7.6 Same configuration as Figure  but with Set B material parameters (Table ), confirming robustness of the FL length constraints across material parameter sets. Based on Figure  [194].

7.7 Magnetization trajectories for FL segment lengths of 2, 3, 4, and 5 nm with enhanced TB\(_2\) polarization (\(P_1 = 0.5\), \(P_2 = 0.9\), \(P_3 = 0.2\)) and Set A material parameters. The enhanced TB\(_2\) polarization modifies the back-hopping threshold compared to the standard case (Figure ). Based on Figure  [193].

7.8 Same enhanced TB\(_2\) polarization as Figure  but with Set B material parameters, demonstrating that the polarization-dependent switching behavior persists across material choices. Based on Figure  [194].

7.9 Magnetization trajectories with standard polarization (\(P_1 = 0.62\), \(P_2 = 0.57\), \(P_3 = 0.2\)) and asymmetric FL segment lengths for (a) AP\(\to \)P and (b) P\(\to \)AP switching across FL lengths of 2 nm to 5 nm. Based on Figure  [188].

7.10 Magnetization trajectories with enhanced TB\(_2\) polarization (\(P_1 = 0.5\), \(P_2 = 0.9\), \(P_3 = 0.2\)) and asymmetric FL segment lengths for (a) AP\(\to \)P and (b) P\(\to \)AP switching across FL lengths of 2 nm to 5 nm. The enhanced TB\(_2\) polarization intensifies the cyclic transitions compared to the standard polarization case (Figure ), confirming deterministic back-hopping. Based on Figure  [188].

7.11 Spin-transfer torque heatmap showing \(T_{S,x}\) (top) and \(T_{S,z}\) (bottom) as a function of stack position and magnetization angle for standard polarization (\(P_1 = 0.62\), \(P_2 = 0.57\), \(P_3 = 0.2\)). Left column: FL\(_2\) rotation, right column: FL\(_1\) rotation. Contour lines indicate torque magnitude within ferromagnetic layers.

7.12 Spin-transfer torque heatmap for enhanced TB\(_2\) polarization (\(P_1 = 0.5\), \(P_2 = 0.9\), \(P_3 = 0.2\)). Same layout as Figure . The enhanced polarization introduces a pronounced torque imbalance: \(T_{S,x}\) on FL\(_1\) is significantly enhanced by the high \(P_2 = 0.9\), while the torque on FL\(_2\) is reduced by the lower \(P_3 = 0.2\).

7.13 Three-dimensional surface representation of \(T_{S,x}\) (a, b) and \(T_{S,z}\) (c, d) for standard polarization (\(P_1 = 0.62\), \(P_2 = 0.57\), \(P_3 = 0.2\)). Panels (a, c): FL\(_2\) rotation, (b, d): FL\(_1\) rotation. The in-plane torque \(T_{S,x}\) is negative throughout the angular range with a minimum near \(90^\circ \). A Gaussian filter suppresses numerical noise from the FE mesh.

7.14 Three-dimensional torque surfaces for enhanced TB\(_2\) polarization (\(P_1 = 0.5\), \(P_2 = 0.9\), \(P_3 = 0.2\)). Compared to the standard case (Figure ), \(T_{S,x}\) on FL\(_1\) (b) reaches approximately twice the magnitude of FL\(_2\) (a), reflecting the \(P_2/P_3 = 0.9/0.2\) ratio. This imbalance drives sequential switching and back-hopping.

7.15 Decomposition of damping-like and field-like spin-transfer torques on FL\(_1\) and FL\(_2\) during (a) AP\(\to \)P and (b) P\(\to \)AP switching. The torque hierarchy between TB\(_1\) and TB\(_2\) governs the sequential switching order and determines back-hopping onset  [199].

7.16 Cyclic switching visualization during P\(\to \)AP transition: three-dimensional magnetization snapshots at five characteristic stages (i)–(vi) of the FL reversal cycle, colored by the \(m_x\) magnetization component. Based on  [199].

7.17 Resistance (purple, left axis) and average magnetization \(m_x\) (orange dashed, right axis) as a function of time for four pulse durations: (a) \(t_\mathrm {pulse} = \SI {1.0}{\nano \second }\), (b) 1.3 ns, (c) 2.0 ns, and (d) 3.0 ns. The vertical dotted line marks pulse termination \(t_\mathrm {off}\). The four resistance levels are clearly distinguishable with \({>}20\%\) separation between adjacent states. Based on Figure  [194].

7.18 Average magnetization \(\langle m_x \rangle \) versus time for P\(\to \)AP switching at 1.5 V bias with write pulse durations ranging from 0 to 3.5 ns (color-coded from red for short pulses to blue for long pulses). By terminating the write pulse at different times, distinct final magnetization configurations are accessed: short pulses (\(t_\mathrm {pulse} \lesssim \SI {0.8}{\nano \second }\)) return the composite FL to its initial state, intermediate pulses stabilize partial-switching configurations, and the longest pulses (\(t_\mathrm {pulse} > \SI {2.5}{\nano \second }\)) drive a complete back-hopping cycle. Based on  [199].

7.19 Magnetization response under reversed voltage polarity (opposite to the polarity required for switching): (a) AP state with positive voltage, stable at \(m_x \approx -1\) for \(V \leq \SI {2.5}{\volt }\), but cyclic oscillations between \(m_x = -1\) and \(+1\) emerge at \(V \geq \SI {3.0}{\volt }\) with increasing frequency, (b) P state with negative voltage, stable at \(m_x = +1\) across the entire range (2.0 V to 4.5 V). The asymmetry arises from the cooperative torque configuration of the P state: both TB\(_1\) and TB\(_2\) torques act to stabilize the parallel alignment (see Section ), whereas the AP state suffers from competing torques that enable destabilization at sufficient voltage.

7.20 Influence of IEC strengths \(J_{\text {iec},1}\) and \(J_{\text {iec},2}\) on the AP\(\to \)P switching dynamics in the ultra-scaled composite FL with MgO TB separator (Figure  (d)). The IEC is swept at the FL\(_1\)–FL\(_2\) interface through the middle MgO barrier: (a) negative \(J_{\text {iec},1}\) (AFM), (b) \(J_{\text {iec},1} = 0\), and (c) positive \(J_{\text {iec},1}\) (FM), each producing distinct switching dynamics. Based on Figures  [203].

7.21 Systematic IEC sweep for the two ultra-scaled composite FL structures from  [208]: (a–c) AP\(\to \)P and (d–f) P\(\to \)AP transitions for the composite FL with MgO TB separator (Figure  (d)), and (g–l) the same analysis for the composite FL with NMS separator (Figure  (e)). In each row, the left panel applies negative \(J_{\text {iec},1}\) (AFM), the middle panel sets \(J_{\text {iec},1} = 0\), and the right panel applies positive \(J_{\text {iec},1}\) (FM). The coupling strength range reflects experimentally reported values for CoFeB\(|\)MgO interfaces  [204, 205] and metallic spacers  [206, 207]. Based on Figures  [208].

7.22 Layer stacks of the three configurations studied in  [116]: (a) Stack A (ultra-scaled composite FL, same architecture as Figure  (d)), (b) Stack B (SAF without PL), and (c) Stack C (SAF with PL). IEC acts at different interfaces in each stack: FL\(_1\)–FL\(_2\) coupling through TB\(_2\) in Stack A, HL–RL coupling through Ru in Stack B, and both HL–RL (AFM) and RL–PL (FM) coupling in Stack C.

7.23 Switching trajectories during P\(\to \)AP transition for the three stack configurations: (a) Stack A (ultra-scaled composite FL), (b) Stack B (SAF without PL), and (c) Stack C (SAF with PL). Weak FM coupling suppresses back-hopping in Stack A, strong AFM coupling maintains RL stability in Stack B, and PL reversal in Stack C depends on the RL–PL coupling strength  [116].

7.24 Spin-torque distribution at the moment of back-hopping onset for (a) Stack A, (b) Stack B, and (c) Stack C. The field-like component \(T_{S,x}\) and damping-like component \(T_{S,z}\) reveal distinct torque patterns at each ferromagnetic interface, with IEC modifying both the magnitude and spatial distribution of the torques  [116].

7.25 Three-dimensional magnetization states during P\(\to \)AP switching in Stack B (SAF without PL), contrasting weak (\(J_{\text {iec}} = \SI {-0.5}{\milli \joule \per \meter \squared }\)) and strong (\(J_{\text {iec}} = \SI {-1.32}{\milli \joule \per \meter \squared }\)) AFM HL–RL coupling. (a)–(c) weak coupling, full view and (d)–(f) weak coupling, FL clipped to reveal RL, (g)–(i) strong coupling, full view and (j)–(l) strong coupling, FL clipped to reveal RL. Columns show the initial state, mid-switching (\({\sim }\SI {0.4}{\nano \second }\)), and final state (\({\sim }\SI {1}{\nano \second }\)). With strong coupling, the RL remains uniformly oriented throughout switching, with weak coupling, the RL develops domain walls and its magnetization reverses almost completely  [215].

7.26 Three-dimensional magnetization states during P\(\to \)AP switching in Stack C (SAF with PL), contrasting weak (\(J_{\text {iec}} = \SI {0.2}{\milli \joule \per \meter \squared }\)) and strong (\(J_{\text {iec}} = \SI {0.8}{\milli \joule \per \meter \squared }\)) FM RL–PL coupling. (a)–(c) weak coupling, full view and (d)–(f) weak coupling, FL clipped to reveal PL, (g)–(i) strong coupling, full view and (j)–(l) strong coupling, FL clipped to reveal PL. Columns show the initial state, mid-switching (\({\sim }\SI {0.4}{\nano \second }\)), and final state (\({\sim }\SI {1}{\nano \second }\)). With strong coupling, the PL remains stable, with weak coupling, the PL reverses almost completely due to parasitic spin-torques overcoming the IEC  [215].

7.27 Layer stack of the SAF-enhanced composite FL configuration from  [216]. Stack D and Stack E share the same architecture, differing only in the NMS material between the FL segments: tungsten (W, \(\lambda _{sf} = \SI {2.4}{\nano \meter }\)) for Stack D, tantalum (Ta, \(\lambda _{sf} = \SI {1.9}{\nano \meter }\)) for Stack E. IEC acts at the HL–RL interface (through Ru, AFM coupling) and at the FL\(_1\)–FL\(_2\) interface (through the metallic spacer, FM coupling). Stack C employs a single FL (same architecture as Figure  (c)).

7.28 Switching duration with different FM coupling for three stack configurations: (a) Stack C with single FL at strong (\(\SI {0.8}{\milli \joule \per \meter \squared }\)) and weak (\(\SI {0.2}{\milli \joule \per \meter \squared }\)) FM coupling, (b) Stack D with W spacer composite FL at \(J_{\text {iec}} = \SI {0.37}{\milli \joule \per \meter \squared }\) and \(\SI {0.8}{\milli \joule \per \meter \squared }\), and (c) Stack E with Ta spacer composite FL at \(J_{\text {iec}} = \SI {0.2}{\milli \joule \per \meter \squared }\) and \(\SI {0.8}{\milli \joule \per \meter \squared }\). Weaker coupling delays switching in all three configurations, with the Ta spacer (c) exhibiting the longest switching times  [216].

7.29 Three-dimensional magnetization states during P\(\to \)AP switching in Stack D (W spacer composite FL), contrasting weak (\(J_{\text {iec}} = \SI {0.37}{\milli \joule \per \meter \squared }\)) and strong (\(J_{\text {iec}} = \SI {0.8}{\milli \joule \per \meter \squared }\)) FM FL\(_1\)–FL\(_2\) coupling. (a)–(c) weak coupling, full view and (d)–(f) weak coupling, FL\(_2\) clipped to reveal FL\(_1\), (g)–(i) strong coupling, full view and (j)–(l) strong coupling, FL\(_2\) clipped to reveal FL\(_1\). Rows show the initial state, mid-switching (\({\sim }\SI {0.5}{\nano \second }\)), and final state (\({\sim }\SI {1.7}{\nano \second }\)). With weak coupling, FL\(_1\) reverses first producing a transient plateau at \(\langle m_x \rangle \approx 0.5\), with strong coupling, synchronized reversal eliminates the plateau  [216].

7.30 Three-dimensional magnetization states during P\(\to \)AP switching in Stack E (Ta spacer composite FL), contrasting weak (\(J_{\text {iec}} = \SI {0.2}{\milli \joule \per \meter \squared }\)) and strong (\(J_{\text {iec}} = \SI {0.8}{\milli \joule \per \meter \squared }\)) FM FL\(_1\)–FL\(_2\) coupling. (a)–(c) weak coupling, full view and (d)–(f) weak coupling, FL\(_2\) clipped to reveal FL\(_1\), (g)–(i) strong coupling, full view and (j)–(l) strong coupling, FL\(_2\) clipped to reveal FL\(_1\). Rows show the initial state, mid-switching (\({\sim }\SI {0.75}{\nano \second }\)), and final state (\({\sim }\SI {1.7}{\nano \second }\)). The shorter spin-flip length of Ta (\(\SI {1.9}{\nano \meter }\)) compared to W (\(\SI {2.4}{\nano \meter }\)) produces weaker inter-segment torques, eliminating the transient plateau and slowing overall switching  [216].

7.31 Four-stage torque decomposition during P\(\to \)AP switching for (a)–(d) Stack C with single FL, (e)–(h) Stack D with W spacer composite FL, and (i)–(l) Stack E with Ta spacer composite FL. Rows show the spin-transfer torque distribution at the initial P state (\(\theta \approx 0^\circ \)), \(\theta = 45^\circ \), \(\theta = 135^\circ \), and the final AP state (\(\theta \approx 180^\circ \)). The composite FL stacks show additional torque peaks at the NMS interfaces absent in the single-FL configuration  [216].

7.32 Layer stack of the double spin-torque MTJ (ds-MTJ) architecture, in which the FL is sandwiched between two reference layers with anti-parallel magnetizations separated by a TB and a non-magnetic metallic spacer. The anti-parallel RL configuration produces cooperative torques from both interfaces that constructively add, and the NMS material choice (Ru vs. Ta) controls the secondary torque magnitude via its spin-flip length  [218].

7.33 Switching performance of ds-MTJ vs. conventional pMTJ structures, showing (a) the switching time ratio and (b) absolute switching times for both transition directions. The ds-MTJ with Ru-NMS achieves a 2–3\(\times \) speedup and sub-nanosecond switching at moderate voltages, while Ta-NMS provides a 1.2–1.5\(\times \) enhancement, consistent with the ratio of their spin-flip lengths (4 nm vs. 1.9 nm)  [218].

7.34 Influence of IEC on ds-MTJ switching dynamics with Ru-NMS, exploring a \(3 \times 5\) parameter matrix of AFM, zero, and FM coupling through TB\(_1\) (columns) with varying NMS coupling (curves within each panel) for (a) AP\(\to \)P and (b) P\(\to \)AP transitions. Zero coupling (\(J_{\text {iec},1} = J_{\text {iec},2} = 0\)) generally yields the fastest switching, while strong coupling can introduce energy landscape distortions that trap the magnetization in unintended states  [218].

7.35 Influence of IEC on ds-MTJ switching dynamics with Ta-NMS, exploring a \(3 \times 5\) parameter matrix of AFM, zero, and FM coupling through TB\(_1\) (columns) with varying NMS coupling (curves within each panel) for (a) AP\(\to \)P and (b) P\(\to \)AP transitions. Zero coupling (\(J_{\text {iec},1} = J_{\text {iec},2} = 0\)) generally yields the fastest switching, while strong coupling can introduce energy landscape distortions that trap the magnetization in unintended states  [218].

7.36 Switching trajectories for SAF-enhanced ds-MTJ structures comparing Ru and Ta NMS at varying thicknesses: (a) P\(\to \)AP and (b) AP\(\to \)P transitions. Optimal performance at Ru 1.0 nm (\(J_{\text {iec}} = \SI {-0.65}{\milli \joule \per \meter \squared }\), AFM) and Ta 0.25 nm (\(J_{\text {iec}} = \SI {0.4}{\milli \joule \per \meter \squared }\), FM). Ru at 0.4 nm (\(\SI {-2.1}{\milli \joule \per \meter \squared }\)) exhibits oscillatory behavior from excessively strong AFM coupling  [217].

7.37 Spin-transfer torque distribution across the full multilayer stack for (a) single SAF ds-MTJ and (b) double SAF ds-MTJ, comparing Ru (AFM) and Ta (FM) NMS. The Ru spacer generates stronger torques on RL\(_2\) due to its longer spin-flip length, while the double SAF configuration provides enhanced reference layer stability with minimal switching speed penalty  [217].

List of Tables

5.1 Comparison of preconditioner strategies for the C-language Variable-coefficient Ordinary Differential Equation solver (CVODE) Newton system.

5.2 Summary of implemented time integrators.

5.3 Quantitative comparison of solvers on \(\mu \)MAG Standard Problem 4. Errors are computed against both reference solutions.

6.1 Parameters used in the drift-diffusion simulations with the tunnel barrier (TB) boundary terms.

6.2 Quantitative performance comparison for spin-transfer torque (STT)-driven switching. The trajectory error is measured by \(e_\infty \) relative to the tangent-plane scheme (TPS) reference (baseline row). Speedup is computed as \(\text {Speedup} = \text {CPU}_{\text {TPS,ref}}/\text {CPU}_{\text {method}}\).

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.

7.2 Comparison of material parameter sets used to verify robustness of the back-hopping mechanism.

7.3 Multi-level cell state definitions and characteristics.

7.4 Material and geometry parameters for the three MDPI 2024 stack configurations  [116] and the MME 2025 synthetic antiferromagnet (SAF)-enhanced stacks  [216]. Common parameters are listed in Table .

7.5 Layer parameters of the double-spin torque magnetic tunnel junction (ds-MTJ) architecture  [218]. The free layer (FL) is sandwiched between RL\(_1\) (MgO tunnel barrier) and RL\(_2\) (metallic spacer). Two nonmagnetic spacer (NMS) materials are compared.

7.6 SAF-enhanced ds-MTJ spacer parameters and coupling values  [217]. The optimal thickness for each spacer material is highlighted.

7.7 Cross-architecture comparison of the seven device configurations studied in this chapter. Each column represents a distinct architecture family, and rows compare key design metrics and the role of interlayer exchange coupling (IEC). Switching speedup is relative to a conventional single-RL perpendicular magnetic tunnel junction (pMTJ). Dashes indicate metrics not applicable or not directly quantified for that architecture. The ultra-scaled and multi-level cell (MLC) studies (Sections  and ) use \(J_{\text {iec}} = 0\), the IEC role listed refers to the effect observed when coupling is subsequently introduced (Section ).