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

Modeling Spin-Orbit Torques
in Advanced Magnetoresistive Devices


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

unter der Betreuung von

Associate Prof. Viktor Sverdlov, MSc PhD
O.Univ.Prof. Dipl.-Ing. Dr.techn. Dr.h.c. Siegfried Selberherr

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

Nils Petter Jørstad, MSc.
Matrikelnummer: 12142741

Wien, im März 2026  

Abstract

The continued scaling of conventional semiconductor devices increases standby power consumption, necessitating the development of more energy-efficient technologies. Non-volatile spin-orbit torque magnetoresistive random access memory (SOT-MRAM) addresses this by eliminating standby power while promising improved switching speeds, power consumption, and endurance. Its operation depends on utilizing current-induced spin-orbit torques (SOTs) to switch the magnetization of ferromagnetic (FM) layers. Switching simulations can provide valuable insight, which is essential for optimizing device performance and reducing research and development costs. This work focuses on developing advanced methods for modeling SOTs and the resulting magnetization dynamics.

In this work, a spin and charge transport model is generalized for the computation of SOTs by treating the spin Hall effect and spin currents at nonmagnetic (NM)/FM interfaces. Boundary conditions based on a perturbative treatment of the Rashba spin-orbit coupling were developed, verified against experimental data, and applied to analyze the thickness dependence of SOTs in NM/FM bilayers and FM/NM/FM trilayers. The results show that thin NM layers offer practical advantages over conventionally thick layers due to enhanced interfacial effects. A non-perturbative treatment of interfacial spin-orbit coupling was also developed to capture the complex angular dependence of SOTs. The resulting boundary conditions were validated against published results, showing good agreement and that the Rashba-Edelstein effect is primarily responsible for the pronounced angular dependence and field-like SOT.

Moreover, a finite element method simulation framework, coupling charge, spin and magnetization dynamics, was developed for modeling switching in SOT-MRAM devices. The simulation framework is demonstrated for several SOT-induced switching mechanisms, including in-plane switching of a FM with shape anisotropy, and switching of a FM with perpendicular magnetic anisotropy assisted by either an external magnetic field or spin-transfer torque. The simulations reveal complex domain wall dynamics consistent with experimental observations. Finally, two advanced field-free switching concepts employing either built-in stray fields or trilayer SOTs were investigated, showing promising applications in SOT-MRAM.

Kurzfassung

Die fortlaufende Verkleinerung herkömmlicher Halbleiterbauelemente erhöht den Standby-Stromverbrauch, was die Entwicklung energieeffizienterer Technologien erforderlich macht. In nichtflüchtigen "Spin-Orbit-Torque Magnetoresistive Random-Access Memory" (SOT-MRAM) Speichern wird diese Problem gelöst, indem der Standby-Stromverbrauch eliminiert und gleichzeitig die Umschaltgeschwindigkeiten verbessert werden, was einen geringeren Stromverbrauch und eine höhere Lebensdauer verspricht. Die Funktionsweise basiert auf der Nutzung von strominduzierten Spin-Bahn-Drehmomenten (SBDs) zum Umschalten der Magnetisierung von ferromagnetischen (FM) Schichten. Umschalt-Simulationen können wertvolle Erkenntnisse liefern, die zur Optimierung der Geräteleistung und Senkung der Forschungs- und Entwicklungskosten unerlässlich sind. Diese Arbeit fokussiert sich auf die Entwicklung fortschrittlicher Methoden zur Modellierung von SBDs und der daraus resultierenden Magnetisierungsdynamiken.

In dieser Arbeit wird ein Spin- und Ladungstransportmodell für die Berechnung von SOTs verallgemeinert, indem sowohl der Spin-Hall-Effekt als auch Spinströme an nichtmagnetischen (NM)/FM-Grenzflächen behandelt werden. Es wurden Randbedingungen auf der Grundlage einer perturbativen Behandlung der Rashba-Spin-Bahn-Kopplung entwickelt, anhand von experimentellen Daten verifiziert und zur Analyse der Dickenabhängigkeit von SBDs in NM/FM-Doppelschichten und FM/NM/FM-Dreifachschichten angewandt. Die Ergebnisse zeigen, dass dünne NM-Schichten aufgrund verbesserter Grenzflächeneffekte praktische Vorteile gegenüber herkömmlichen dicken Schichten bieten. Darüber hinaus wurde eine nichtperturbative Behandlung der Grenzflächen-Spin-Bahn-Kopplung entwickelt, um die komplexe Winkelabhängigkeit der SBDs zu erfassen. Die resultierenden Randbedingungen wurden anhand publizierter Ergebnisse validiert und zeigen eine gute Übereinstimmung. Sie zeigen auch, dass in erster Linie der Rashba-Edelstein-Effekt für die ausgeprägte Winkelabhängigkeit und das feldähnliche SBD verantwortlich ist.

Darüber hinaus wurde ein Simulationsframework auf Basis der Finite-Elemente-Methode entwickelt, der Ladungs-, Spin- und Magnetisierungsdynamik koppelt, um das Umschalten in SOT-MRAM-Bauelementen zu modellieren. Das Simulationsframework wird für mehrere SBD-induzierte Umschaltmechanismen demonstriert, darunter das Umschalten eines FM mit Form-Anisotropie längs der Schichtebene sowie das Umschalten eines FM mit senkrechter magnetischer Anisotropie, unterstützt entweder durch ein externes Magnetfeld oder durch Spin-Transfer-Drehmoment. Die Simulationen zeigen komplexe Domänenwanddynamiken in Übereinstimmung mit experimentellen Beobachtungen. Schließlich wurden zwei fortschrittliche feldfreie Umschaltkonzepte untersucht, die entweder integrierte
Streufelder oder Dreifachschicht-SBDs verwenden, womit eine vielversprechende Anwendungen in SOT-MRAM aufgezeigt wird.

Acknowledgement

I would like to thank my supervisor, Prof. Viktor Sverdlov, for giving me the opportunity to work on this exciting topic and for his continuous support and guidance throughout my PhD journey at the Institute for Microelectronics. The success of the Christian Doppler Laboratory for Nonvolatile Magnetoresistive Memory and Logic would not have been possible without his steadfast vision and leadership.

I would like to express my gratitude to Prof. Siegfried Selberherr, the founder of the Institute, for his valuable feedback, support, and encouragement over the years and for fostering an excellent research environment.

I am grateful to all my colleagues at the Laboratory over the years: Roberto, Simone, Johannes, Tomáš, Mario, Wilton, and Bernhard, for their friendship, stimulating discussions, and fruitful collaborations. I would also like to thank Wolfgang Goes for the enjoyable and successful collaboration with our Laboratory.

My appreciation is also extended to all members of the Institute for Microelectronics for creating such a welcoming and inspiring workplace. Special thanks go to Diana and Petra for their tireless assistance with administrative matters, and to Manfred and Cerv for their invaluable technical support.

Finally, I owe my deepest gratitude to my family for their unconditional love, patience, and belief in me throughout this journey, and to my girlfriend and friends for their constant support and companionship. Their presence has made this endeavor even more meaningful and memorable.

Contents

Abstract

Kurzfassung

Acknowledgement

List of Figures

List of Tables

1 Motivation and Research Goals

1.1 Outline of the Thesis

1.2 Research Setting

2 Fundamentals of Spintronics

2.1 The Electron’s Spin

2.2 Magnetoresistance

2.3 Spin-Transfer Torque (STT)

2.4 Spin-Orbit Torque (SOT)

2.4.1 The Spin Hall Effect (SHE)

2.4.2 The Rashba-Edelstein Effect

3 Micromagnetic Modeling

3.1 The Landau-Lifshitz-Gilbert (LLG) Equation

3.2 Effective Field

3.2.1 External Field

3.2.2 Exchange Field

3.2.3 The Dzyaloshinskii-Moriya Interaction (DMI)

3.2.4 Anisotropy Field

3.2.5 Demagnetizing Field

3.3 Current-Induced Torques

4 Charge and Spin Transport

4.1 Charge and Spin Currents in a Ferromagnet

4.2 Charge and Spin Drift-Diffusion

4.2.1 Continuity Equations

4.2.2 Spin Dephasing

4.2.3 Spin Torque

4.2.4 The Direct and Inverse SHE

4.3 Boundary Conditions

4.3.1 Nonmagnetic/Ferromagnetic Interface Boundary Conditions

4.3.2 Magnetic Tunneling Junctions

4.4 Partial Differential Equations (PDEs) for Charge and Spin Transport

4.4.1 Coupled Charge and Spin Drift-Diffusion PDEs

4.4.2 Decoupled Charge and Spin Drift-Diffusion PDEs

5 Interfacial Spin-Orbit Coupling

5.1 The Nonequilibrium Drift Term

5.2 Perturbation Theory Model

5.2.1 Rashba Currents in the Nonmagnetic Layer

5.2.2 Rashba Currents in the Ferromagnetic Layer

5.2.3 Interface Conductivities and Spin Torques

5.3 Effective Interface Field Model

5.3.1 Interface Model

5.3.2 Charge and Spin Currents

5.3.3 Interfacial and Bulk Spin Torques

5.3.4 Vanishing Interfacial Spin-Orbit Coupling Limit

5.3.5 Vanishing Interfacial Exchange Interaction Limit

6 Numerical Methods and Implementations

6.1 Fermi Surface Quadratures

6.2 One-Dimensional Charge and Spin Drift-Diffusion

6.2.1 General Solution of the Charge and Spin Drift-Diffusion Equations in One Dimension

6.2.2 Boundary Conditions

6.2.3 Linear System of Equations

6.2.4 Implementation and Validation of the One-Dimensional Charge and Spin Drift-Diffusion Solver

6.3 The Finite Element Method (FEM)

6.3.1 The Weak Formulation

6.3.2 The Finite Element Approximation

6.3.3 Internal Boundaries

6.3.4 Effective Interface Layer Approximation

6.3.5 Connected Interlayer Boundaries

6.4 Weak Formulation of the Drift-Diffusion Equations

6.4.1 Electrical Potential

6.4.2 Spin Accumulation

6.4.3 Magnetic Tunneling Junctions

6.4.4 Magnetoelectronic Circuit Theory Boundary Conditions

6.4.5 Interfacial Spin-Orbit Coupling Boundary Conditions

6.4.6 FEM Implementation

6.4.7 One-Dimensional Bilayer System Test

6.5 Weak Formulation of the LLG Equation

6.5.1 Exchange Term and DMI Boundary Conditions

6.5.2 Bulk and Interface Spin Torque

6.5.3 First Order Tangent-Plane Scheme

6.5.4 Implementation of the LLG Equation

6.5.5 One-Dimensional Problem

6.5.6 Three-Dimensional Problem

7 SOTs in Ferromagnetic Systems

7.1 Spin Currents, Accumulation and Torques

7.1.1 SHE in a Nonmagnetic Wire

7.1.2 SHE in Nonmagnetic/Ferromagnetic Bilayers

7.1.3 Rashba Effect in Nonmagnetic/Ferromagnetic Bilayers

7.1.4 Validity of the Perturbation Theory Approach to Interfacial Rashba Spin-Orbit Coupling

7.2 Thickness Dependence of SOTs

7.2.1 Bulk and Interface Contributions to SOTs

7.2.2 Comparison with Experimental Data

7.2.3 Rashba Interface Conductivities

7.3 Angular Dependence of SOTs

7.3.1 Higher-Order SOTs

7.3.2 Modification of SOTs by Interfacial Rashba Spin-Orbit
Coupling

7.4 Trilayer Spin Currents and Torques

7.4.1 Interface Generated Spin Currents in Trilayers

7.4.2 Unconventional Trilayer SOTs

8 SOT Driven Magnetization Dynamics

8.1 Simulation Parameters

8.2 In-Plane Switching

8.3 Perpendicular Switching with External Field

8.4 Field-Free Switching with STT Assist

8.5 Field-Free Switching with Magnetic Hardmask

8.6 Field-Free Switching with Trilayer Torques

9 Summary and Outlook

A Analytical Expressions for the Interface Conductances

A.1 Interface Conductances

A.2 Rashba Interface Conductances

B Analytical Solution for a One-Dimensional
Nonmagnetic/Ferromagnetic Bilayer

B.1 Lower Boundary

B.2 Upper Boundary

B.3 Nonmagnetic/Ferromagnetic Interface

B.4 Analytical Solution for Spin Currents

B.5 Analytical Solution for Spin Accumulation

Bibliography

List of Publications

Curriculum Vitae

List of Figures

1.1 Schematics of the typical STT- and SOT-MRAM cell structures.

2.1 Visual depiction of the GMR effect in a spin valve.

2.2 Schematic of the TMR effect in a MTJ.

2.3 Schematization of the spin-transfer mechanism.

2.4 Illustration of the SHE in a CIP NM/FM bilayer.

2.5 Illustration of the REE in a CIP NM/FM bilayer.

2.6 The Fermi-level spin texture for different SOCs.

3.1 The path of a single magnetic spin (red arrow) subjected to an effective magnetic field \(\bm {H_\mathrm {eff}}\) (black arrow) as described by the LLG equation.

3.2 A sketch of Bloch and Néel type domain walls.

3.3 The uniaxial anisotropy energy of a single spin projected onto the unit sphere.

4.1 Illustration of the SHE (a) and ISHE (b).

5.1 Visualization of the SOF, SOP, REE, and SREE mechanisms in CIP NM/FM bilayers.

6.1 The ARE between the analytical and numerical solution for the spin-mixing conductance, obtained from Eq. (6.4) using Eq. (5.29) with \(u_R =0\). The numerical solution was computed using the spherical Lebedev, Gauss-Legendre, and the midpoint method quadrature rules for an increasing number of integration points. The analytical solution is given by Eq. (A.1).

6.2 The distribution of integration points for the Lebedev (a), Gauss-Legendre (b), and midpoint (c) spherical quadrature rules, with \(1800\), \(1730\), and \(1800\) points respectively.

6.3 Sketch of the 1D multilayer system and the labeling scheme for the layers and interfaces.

6.4 1D analytical and 1D-CSDD solver solutions for the spin accumulation (a) and spin current density (b) distributions in a NM (20 nm)/FM (20 nm) bilayer system.

6.5 (a) Example sketch of a 2D domain \(\Omega \) with Dirichlet, Neumann and Robin BCs.

6.6 Sketch of a function, the corresponding FEM solution, and the basis functions for a 1D system with 6 elements and 7 nodes.

6.7 Sketch of a 2D domain with two subdomains \(\omega _1\) and \(\omega _2\) separated by an internal boundary \(\Gamma \).

6.8 Flow chart of the ViennaSpinMag FEM 3D charge and spin drift-diffusion module.

6.9 Quasi-1D mesh of the NM(20 nm)/FM(20 nm) bilayer system with \(2603\) elements.

6.10 Comparison of the quasi-1D FEM and analytical solutions for the spin accumulation for varying spin dephasing lengths \(\phi \) in a NM(20 nm)/FM(20 nm) bilayer system.

6.11 Same as Fig. 6.10, but with a fixed spin dephasing length \(\lambda _\phi = 0.1\) nm and an increasingly refined mesh.

6.12 NMAE \(\eta \) between the FEM and analytical solutions for the spin accumulation in a NM(20 nm)/FM(20 nm) bilayer system.

6.13 Flow chart of the LLG module in the ViennaSpinMag software.

6.14 The minimum energy state of a 1D magnetic system with interfacial (a) or bulk DMI (b).

6.15 The relaxed magnetization state of a \(9\,\mathrm {nm}\) thick helimagnetic nanodisk.

6.16 Time evolution of the magnetization (a) and the relaxed state (b) of a \(9\,\mathrm {nm}\) thick helimagnetic nanodisk with a diameter of \(140\,\mathrm {nm}\).

7.1 Spin accumulation generated by the SHE in a cylindrical (a) and rectangular (b) wire.

7.2 Spin accumulation and spin current density generated by the SHE in a \(10\,\mathrm {nm}\) thick layer for various spin-flip lengths.

7.3 The normalized magnetization (a), spin accumulation generated by the SHE (b), and the resulting spin torque acting on the magnetization (c), in a NM/FM bilayer system.

7.4 The distribution of the spin accumulation (a-b), spin current density (c-d), and spin torque (e-f) generated by the SHE in a NM/FM bilayer system.

7.5 The same as Fig. 7.3 except with an interfacial Rashba SOC and without SHE (\(\alpha _\mathrm {SH} = 0\)).

7.6 Same as Fig. 7.4 except with an interfacial Rashba SOC and \(\alpha _\mathrm {SH} = 0\).

7.7 The total spin torque generated by the interfacial Rashba SOC in a NM/FM bilayer as a function of the coupling strength \(u_R\) for various interfacial exchange interaction strengths \(u_m\).

7.8 The dependence of the total torque on the NM thickness in a NM/FM bilayer system for SHE (a), RE (b), and a combination of SHE and RE generated torques (c).

7.9 The Ir thickness dependence of the DL (a) and FL (b) effective spin torque conductivities for an Ir(\(t_\mathrm {Ir}\))/CoFeB(\(2.3\,\mathrm {nm}\)) bilayer structure with an in-plane current and magnetization along \(x\).

7.10 Thickness dependence of the Pt(\(t_{Pt}\))/CoFeB(3 nm)/MgO(1 nm) DL (a) and FL (b) spin torque efficiencies.

7.11 Thickness dependence of the W(\(t_{W}\))/CoFeB(1 nm)/MgO(1 nm) DL (a) and FL (b) spin torque efficiencies.

7.12 A schematic of a NM/FM bilayer system with an electric field \(\bm {E}\) applied along \(x\).

7.13 Computed and measured SOTs for two different CIP NM/FM bilayer systems as a function of the polar angle of the magnetization.

7.14 The spin torque as a function of the polar angle of the magnetization for different strengths of the: Rashba SOC \(u_R\) (a-b), magnetic exchange interaction \(u_m\) (c-d).

7.15 The spin torque as a function of the Rashba SOC strength (a-b) and the polar angle of the magnetization (c-d) showing the different contributions to the total torque.

7.16 Out-of-plane spin current generated at the NM side of a FM/NM interface as a function of the magnetization direction.

7.17 Interface generated spin currents in CIP FM/NM/FM trilayers.

7.18 The full angular dependence of the SOTs in a NM(\(1\,\si {nm}\) )/FM(1 nm) bilayer (a) and a FM(4 nm)/NM(1 nm)/FM(1 nm) trilayer (b).

7.19 The spin torque efficiency (a) and the \(z/y\)-polarization component ratio (b) as a function of the NM spacer thickness for a FM(\(4\,\si {nm}\))/NM(\(t_\mathrm {NM}\))/FM(\(1\,\si {nm}\)) trilayer.

8.1 Schematic of an in-plane SOT-MRAM device with IMA (a), and the corresponding mesh used for the simulations (b).

8.2 The \(y\)-component of the volume averaged magnetization and the applied electrical current density as a function of time.

8.3 The spatial distribution of the \(y\)-component of the magnetization in the center of the FL at different time steps during the switching process.

8.4 Schematic of an PMA SOT-MRAM device with an external magnetic field (a), and the corresponding mesh used for the simulations (b).

8.5 The \(z\)-component of the volume averaged magnetization and the applied current density as a function of time for four possible switching modes.

8.6 The spatial distribution of the \(z\)-component of the magnetization in the center of the FL at different time steps during the switching process.

8.7 Schematic of a hybrid STT-SOT-MRAM device with PMA (a), and the corresponding mesh used for the simulations (b).

8.8 The \(z\)-component of the volume averaged magnetization and the applied current as a function of time.

8.9 The spatial distribution of the \(z\)-component of the magnetization in the center of the FL at different time steps during the switching process.

8.10 Schematic of a MHM-SOT-MRAM device with PMA (a), and the corresponding mesh used for the simulations (b).

8.11 The \(z\)-component of the volume averaged magnetization and the applied current density as a function of time.

8.12 The spatial distribution of the \(z\)-component of the magnetization in the center of the FL at different time steps during the switching process.

8.13 Applied current density versus switching time for FFS with MHM.

8.14 Schematic of a Trilayer-SOT-MRAM device with PMA (a), and the corresponding mesh used for the simulations (b).

8.15 The \(z\)-component of the volume averaged magnetization and the applied current density as a function of time.

8.16 The spatial distribution of the \(z\)-component of the magnetization in the center of the FL at different time steps during the switching process.

8.17 Applied current density versus switching for FFS with trilayer SOTs.

List of Tables

6.1 The bulk and interface parameters used for the NM/FM bilayer.

6.2 Material parameters for the 1D system and FeGe. The values for FeGe are taken from Ref. [94].

7.1 The interface parameters used to compute the BCs for the NM/FM interface taken from [70].

7.2 Bulk material parameters used for the fitting of the bilayer SOT dependence on the NM thickness.

7.3 Interface parameters for the Pt/CoFeB and W/CoFeB systems. The real part of the spin-mixing conductances was taken from [105] and [106], for Pt/CoFeB and W/CoFeB, respectively.

7.4 Material parameters taken from reported values in the literature or extracted from experimental or ab initio data.

7.5 Interface parameters obtained by fitting the angular dependence of the computed torques with the experimental data from Garello et al. [113].

8.1 Typical micromagnetic material parameters for CoFeB.

8.2 Tunneling spin current parameters used to model the CoFeB/MgO/CoFeB MTJ. The parameters are taken from [74].