Finite Element Formalism for Micromagnetism

The aim of this work is to present the details of the finite element approach we developed for solving the Landau-Lifschitz-Gilbert equations in order to be able to treat problems involving complex geometries. There are several possibilities to solve the complex Landau-Lifschitz-Gilbert equations numerically. Our method is based on a Galerkin-type finite element approach. We start with the dynamic Landau-Lifschitz-Gilbert equations, the associated boundary condition and the constraint on the magnetization norm. We derive the weak form required by the finite element method. This weak form is afterwards integrated on the domain of calculus. We compared the results obtained with our finite element approach with the ones obtained by a finite difference method. The results being in very good agreement, we can state that our approach is well adapted for 2D micromagnetic systems.Comment: Proceedings of conference EMF200

Multiscale and multimodel simulation of Bloch point dynamics

We present simulation results on the structure and dynamics of micromagnetic point singularities with atomistic resolution. This is achieved by embedding an atomistic computational region into a standard micromagnetic algorithm. Several length scales are bridged by means of an adaptive mesh refinement and a seamless coupling between the continuum theory and a Heisenberg formulation for the atomistic region. The code operates on graphical processing units and is able to detect and track the position of strongly inhomogeneous magnetic regions. This enables us to reliably simulate the dynamics of Bloch points, which means that a fundamental class of micromagnetic switching processes can be analyzed with unprecedented accuracy. We test the code by comparing it with established results and present its functionality with the example of a simulated field-driven Bloch point motion in a soft-magnetic cylinder

Spin waves in curved magnetic shells

This thesis aims to theoretically explore the geometrical effects on spin waves, the fundamental low-energy excitations of ferromagnets, propagating in curved magnetic shells. Supported by an efficient numerical technique developed for this thesis, several aspects of curvilinear spin-wave dynamics involving magnetic pseudo-charges, the topology of curved magnets, symmetry-breaking effects, and dynamics of spin textures are studied. In recent years, geometrical and curvature effects on mesoscale ferromagnets have attracted the attention of fundamental and applied research. Exciting curvature-induced phenomena include chiral symmetry breaking, the stabilization of magnetic skyrmions on Gaussian bumps, or topologically induced domain walls in Möbius ribbons. Spin waves in vortex-state magnetic nanotubes exhibit a curvature-induced dispersion asymmetry due to geometric contributions to the magnetic volume pseudo-charges. However, previous theoretical studies were limited to simple and thin curved shells due to the complexity of analytical models and the time-consuming nature of existing numerical techniques. For a systematic study of spin-wave propagation in curved shells, the first of five thematic parts of this thesis deals with developing a numerical method to calculate spin-wave spectra in waveguides with arbitrarily shaped cross-sections efficiently. For this, a finite-element/boundary-element method to calculate dynamic dipolar fields, the Fredkin-Koehler method, was extended for propagating waves. The technique is implemented in the micromagnetic modeling package TetraX developed and made available as open source to the scientific community. Equipped with this method, the second part of the thesis studies the influence of geometric contributions to the magnetic charges leading to nonlocal chiral symmetry breaking. Introducing the toroidal moment to spin-wave dynamics allows us to predict whether this symmetry breaking is present even in complicated systems with spatially inhomogeneous equilibria or shells with gradient curvatures. The theoretical study of curvilinear magnetism is extended to thick shells, uncovering a curvature-induced nonreciprocity in the spatial mode profiles of the spin waves. Consequently, nonreciprocal dipole-dipole hybridization between different modes leads to asymmetric level gaps enabling spin-wave diode behavior. Besides unidirectional transport, curvature modifies the weakly nonlinear spin-wave interactions. The third part of this thesis focuses on topological effects. A topological Berry phase of spin waves in helical-state nanotubes is studied and connected to a local curvature-induced chiral interaction of exchange origin. The topology of more complicated systems, such as magnetic Möbius ribbons, is shown to impose selection rules on the spectrum of possible spin waves and split it into modes with half and full-integer indices. To understand the effects of achiral symmetry breaking, the fourth part of this thesis focuses on the deformation of symmetric shells, here, cylindrical nanotubes, to polygonal and elliptical shapes. Lowering rotational symmetry leads to splitting spin-wave dispersions into singlet and doublets branches, which is explained using a simple group theory approach and is analogous to the electron band structure in crystals. Apart from mode splitting, this symmetry breaking allows hybridization between different spin-wave modes and modifies their microwave absorption. While this hybridization appears discretely in polygonal tubes, tuning the eccentricity of elliptical tubes allows controlling the level gaps appearing from hybridization. Finally, the last part focuses on the dynamics of spin waves in the vicinity of spin textures in curvilinear systems. The dynamics of topological meron strings are shown to exhibit dipole-induced chiral symmetry breaking like spin waves in curved shells. Moreover, modulational instability is predicted from the softening of their gyrotropic modes, similar to the formation of stripe domains in flat systems. This stripe domain formation can also be observed in curved shells but leads to tilted or helix domains. Overall, this thesis contributes to the fundamental understanding of spin-wave dynamics on the mesoscale but also advertises these for possible magnonic applications.:Abstract Acknowledgements Contents 1 Introduction Theoretical Foundations 2 Micromagnetic continuum theory 3 Spin waves Numerical methods in micromagnetism 4 Overview 5 Finite-element dynamic-matrix method for propagating spin waves 6 Numerical reverse-engineering of spin-wave dispersions 7 TetraX: A micromagnetic modeling package Aspects of curvilinear magnetization dynamics 8 Magnetic charges 9 Topology 10 Achiral symmetry breaking 11 Spin textures Closing remarks 12 Summary and outlook 13 Publications and conference contributions Appendix A Extended derivations and proofs B Supplementary data and discussion List of Figures List of Tables Bibliography Alphabetical IndexZiel dieser Arbeit ist es, die geometrischen Effekte auf Spinwellen (Magnonen), die fundamentalen niederenergetischen Anregungen von Ferromagneten, die sich in gekrümmten magnetischen Schalen ausbreiten, theoretisch zu untersuchen. Unterstützt durch ein effizientes numerisches Verfahren, das für diese Arbeit entwickelt wurde, werden verschiedene Aspekte der krummlinigen Spinwellen-Dynamik untersucht: magnetische Pseudoladungen, die Topologie gekrümmter Magnete, Symmetriebrechungseffekte und die Dynamik von Spin-Texturen. In den letzten Jahren haben Geometrie- und Krümmungseffekte auf mesoskaligen Ferromagneten die Aufmerksamkeit der Grundlagen- und angewandten Forschung auf sich gezogen. Zu den spannenden krümmungsinduzierten Phänomenen gehören chirale Symmetriebrechung, die Stabilisierung magnetischer Skyrmionen auf Gaußschen Unebenheiten oder topologisch induzierte Domänenwände in Möbiusbändern. Spinwellen in magnetischen Nanoröhren im Vortex-Zustand zeigen eine krümmungsinduzierte Dispersionsasymmetrie aufgrund geometrischer Beiträge zu den magnetischen Volumen-Pseudoladungen. Bisherige theoretische Studien beschränkten sich jedoch auf einfache und dünne gekrümmte Schalen, da die analytischen Modelle zu komplex und die bestehenden numerischen Verfahren zu zeitaufwändig waren. Für eine systematische Untersuchung der Spinwellenausbreitung in gekrümmten Schalen befasst sich der erste von fünf thematischen Teilen dieser Arbeit mit der Entwicklung einer numerischen Methode zur effizienten Berechnung von Spinwellenspektren in Wellenleitern mit beliebig geformten Querschnitten. Dazu wurde eine Finite-Elemente/Grenzelement-Methode zur Berechnung dynamischer Dipolfelder, die Fredkin-Köhler-Methode, für propagierende Wellen erweitert. Die Technik ist in dem mikromagnetischen Modellierungspaket TetraX implementiert, das während dieser Arbeit entwickelt und der wissenschaftlichen Gemeinschaft als Open Source zur Verfügung gestellt wurde. Ausgestattet mit dieser Methode untersucht der zweite Teil der Arbeit den Einfluss von geometrischen Beiträgen zu den magnetischen Ladungen, die zu nichtlokaler chiraler Symmetriebrechung führen. Durch die Einführung des toroidalen Moments in die Spin-Wellen-Dynamik lässt sich vorhersagen, ob diese Symmetriebrechung auch in komplizierten Systemen mit räumlich inhomogenen Gleichgewichtszuständen oder magnetischen Schalen mit Gradientenkrümmungen vorhanden ist. Die theoretische Untersuchung des krummlinigen Magnetismus wird auf dicke Schalen ausgedehnt, für die eine krümmungsbedingte Nichtreziprozität in den räumlichen Modenprofilen der Spinwellen gefunden wird. Als Konsequenz führt nicht-reziproke Dipol-Dipol-Hybridisierung zwischen verschiedenen Moden zu asymmetrischen Niveaulücken, die die Konstruktion von Spinwellen-Dioden ermöglichen. Neben unidirektionalem Transport modifiziert die Krümmung auch die schwach nichtlinearen Spin-Wellen-Wechselwirkungen. Der dritte Teil dieser Arbeit befasst sich mit topologischen Effekten. So wird eine topologische Berry-Phase von Spinwellen in Nanoröhren im Helix-Zustand untersucht, die mit einer lokalen krümmungsinduzierten chiralen Wechselwirkung in Verbindung gebracht wird. Es wird gezeigt, dass die Topologie komplizierterer Systeme, wie z.B. magnetischer Möbiusbänder, dem Spektrum möglicher Spinwellen Auswahlsregeln auferlegt, das damit in Moden mit halb- und ganzzahligen Indizes aufspaltet. Um die Auswirkungen der achiralen Symmetriebrechung zu verstehen, konzentriert sich der vierte Teil dieser Arbeit auf die Verformung symmetrischer Schalen, hier zylindrischer Nanoröhren, zu polygonalen und elliptischen Formen. Die Verringerung der Rotationssymmetrie führt zu einer Aufspaltung der Spin-Wellen-Dispersionen in Singlets Dublets, was mit einem einfachen gruppentheoretischen Ansatz erklärt wird und analog zur Elektronenbandstruktur in Kristallen ist. Abgesehen von der Modenaufspaltung ermöglicht diese Symmetriebrechung eine Hybridisierung zwischen verschiedenen Spin-Wellen-Moden und verändert zudem deren Mikrowellenabsorption. Während diese Hybridisierung in polygonalen Röhren diskret auftritt, kann die Exzentrizität elliptischer Röhren genutzt werden um die durch Hybridisierung entstehenden Niveaulücken kontinuierlich einzustellen. Schließlich konzentriert sich der letzte Teil auf die Dynamik von Spinwellen in der Umgebung von Spinstrukturen in krummlinigen Systemen. Es wird gezeigt, dass die Dynamik topologischer Meron-Strings dipol-induzierte chirale Symmetriebrechungen wie Spinwellen in gekrümmten Schalen aufweist. Darüber hinaus wird eine Instabilität der gyrotropen Mode vorhergesagt, ähnlich der Bildung von Streifendomänen in flachen Systemen. Diese Bildung von Streifendomänen kann auch in gekrümmten Schalen beobachtet werden, führt aber zu gekippten oder spiralförmigen Domänen. Insgesamt trägt diese Arbeit zum grundlegenden Verständnis der Spinnwellen-Dynamik auf der Mesoskala bei, aber diskutiert auch mögliche magnonische Anwendungen.:Abstract Acknowledgements Contents 1 Introduction Theoretical Foundations 2 Micromagnetic continuum theory 3 Spin waves Numerical methods in micromagnetism 4 Overview 5 Finite-element dynamic-matrix method for propagating spin waves 6 Numerical reverse-engineering of spin-wave dispersions 7 TetraX: A micromagnetic modeling package Aspects of curvilinear magnetization dynamics 8 Magnetic charges 9 Topology 10 Achiral symmetry breaking 11 Spin textures Closing remarks 12 Summary and outlook 13 Publications and conference contributions Appendix A Extended derivations and proofs B Supplementary data and discussion List of Figures List of Tables Bibliography Alphabetical Inde

自旋波驱动畴壁运动动力学的微磁学研究

학위논문 (박사) -- 서울대학교 대학원 : 공과대학 재료공학부, 2021. 2. Chan Park.자벽 이동은 오랫동안 차세대 논리 및 메모리 장치를 개발하는 데에 가능한 해결책으로 여겨져 왔다. 자벽 이동을 구동하기 위하여, 최근 스핀파가 새로운 원동력으로 제안되고 있다. 그러나, 자벽이동의 기구와 원리 관련 이해가 부족하며, 스핀파를 이용하여 자벽이동을 정밀하게 제어하는 것은 많은 해결되지 못한 문제를 가지고 있다. 이 논문에서는 자성 나노스트립 （magnetic nanostrip） 에서 스핀파로 인한 자벽 이동의 동역학을 미시 자기 시뮬레이션 (micromagnetic simulation) 을 이용하여 아래와 같이 세 가지 문제를 중심으로 조사하였다. 첫째, 스핀파가 구동된 자벽 이동의 물리적 메커니즘; 둘째, 스핀파로 인한 자벽 이동의 관성 변위; 셋째, 임의의 스핀파 (arbitrary spin waves) 와 여러 종류의 자벽이 포함된 시스템에서의 자벽 이동 거동; 첫 번째 문제와 관련하여, 스핀파의 흡수를 계산하였고 스핀파 펄스를 사용했다는 점에서 기존 연구와 차별화된다. 계산된 스핀파 흡수는 자벽 이동 속도와 동일한 경향을 가지며, 자벽 이동은 spin-transfer torque (STT) 를 제공하기 위하여 스핀파 흡수를 필요로 한다는 것이 확인되었다. 두 번째 문제와 관련하여, 유발된 스핀파 펄스가 자벽 이동을 구동할 수 있는 것과 자벽 이동의 가속과 감속 현상이 관찰되었다. 스핀파 펄스가 가해지면, 자벽이 가속과 감속을 한다는 것을, 1차원 모델을 이용하여, 설명하였다. 특히, 감속 과정은 자벽의 이완 (domain wall relaxation) 의 결과로 발생하는 것을 확인하였다 세 번째 문제와 관련하여, 서로 다른 파형의 스핀파와 다양한 형태의스택형 자벽 구조가 사용되었다. 그리고 임의의 스핀파에 의한 자벽이동을 푸리에 분석을 이용하여 정량화하였으며, 다양한 형태의 자벽이 포함된 자벽이동은 resonant 픽의 움직임이 변형된다는 것이 확인되었다. 이외에, 스택형 자벽 구조의 움직임은 속도 스펙트럼 (velocity spectrum) 에 변화를 나타내는 것을 확인하였다. 이 연구는 스핀파와 자벽 이동의 상호작용에 대한 이해를 높이고 다양한 구조의 자벽이 포함된 시스템에서의 자벽이동을 제어하는 것에 실질적으로 활용될 수 있으며, 자벽 이동을 이용하는 장치의 개발에 큰 도움을 줄 수 있을 것이다.Magnetic domain wall motion has long been considered a feasible solution to developing next-generation logic and memory devices. Recently spin wave has been proposed as a new driving force for the domain wall motion. Due to the unclear physics, however, it is currently still immature to achieve reliable control of domain wall motion using spin wave. In this thesis, the dynamics of spin wave-induced domain wall motion in a magnetic nanostrip is investigated using micromagnetic simulation. Particularly, three important problems are studied: (1) mechanism of spin wave-induced domain wall motion, (2) spin wave-induced domain wall inertial displacements, and (3) domain wall motion in cases with arbitrary spin waves and multiple domain walls. As regards the first problem, spin wave absorption by domain wall is for the first time calculated and is compared with the forward domain wall velocity. The excellent agreement between the two quantities suggests that forward domain wall motion necessarily consumes spin wave absorption for the required magnonic spin-transfer torque. Concerning the second problem, a spin wave pulse is generated to drive domain wall motion. Negligible acceleration and inevitable deceleration are observed. Such inertial displacements can be understood based on a 1-D model developed and used in this study. Particularly, the deceleration process is found to be a result of domain wall relaxation which includes the release of domain wall internal energy and reduction of the out-of-plane tilting of domain wall. Concerning the third problem, spin waves of different waveforms are generated and stacked domain wall structures are formed. It is found that spin wave harmonic is the basic element when interacting with domain wall and an arbitrary spin wave-induced domain wall motion can be quantified based on the Fourier analysis. The motion of the stacked domain walls is shown to exhibit modifications in the velocity spectrum, which can be ascribed to a changed property of spin wave reflection. This thesis aims to shed further light on the interaction between spin waves and domain walls and pave the way for future development of domain wall motion-based applications.Abstract i Acknowledgement ii Lsit of Figures iii List of Tables xv Chapter 1. Introduction 1 1.1 Motivation 1 1.1.1 Novel data storage based on domain wall motion 1 1.1.2 Other applications based on domain wall motion 7 1.2 Background 10 1.2.1 Domain wall 10 1.2.2 Domain wall motion 16 1.2.3 Spin wave-induced domain wall motion 22 1.3 Research objectives 28 1.4 Scope of this thesis 29 Reference 31 Chapter 2. Theoretical fundamentals 36 2.1 Basics of magnetism 36 2.1.1 Magnetic field 38 2.1.2 Magnetic moment 41 2.1.3 Magnetic interactions 52 2.1.4 Magnetic order 65 2.2 Theory of micromagnetism 77 2.2.1 Assumptions in the continuum theory of micromagnetism 79 2.2.2 Thermodynamics in micromagnetism 80 2.2.3 Landau free energy and effective field 81 2.2.4 Static micromagnetism 95 2.2.5 Dynamic micromagnetism 100 2.2.6 Micromagnetic simulation 135 Reference 138 Chapter 3. Mechanism of spin wave-induced domain wall motion 145 3.1 Introduction 145 3.2 Micromagnetic simulation 147 3.3 Results and discussion 147 3.4 Conclusion 153 Reference 155 Chapter 4. Spin wave-induced domain wall inertial displacements 157 4.1 Introduction 157 4.2 Micromagnetic simulation 159 4.3 Results and discussion 160 4.4 Conclusion 172 Reference 173 Chapter 5. Domain wall motion in cases with arbitrary spin wave and multiple domain walls 177 5.1 Introduction 177 5.2 Micromagnetic simulation 179 5.3 Results and discussion 180 5.4 Conclusion 195 Reference 197 Chapter 6. Conclusion and future works 200 6.1 Conclusion 200 6.2 Future works 201 List of Publications 203 Abstract in Korean 204Docto

Characterization of magnetic nanostructures by magnetic force microscopy

Práce pojednává o mikroskopii magnetických sil magneticky měkkých nanostruktur, zejména NiFe nanodrátů a různě tvarovaných tenkých vrstev - například disků. Práce se zaměřuje na téměř vše, co s touto mikroskopickou technikou souvisí: přípravu měřicích sond a vzorků, samotná pozorování a mikromagnetické simulace magnetického stavu vzorků. Byla pozorována jádra magnetických vírů, jak s komerčními, tak s námi připravenými sondami. Podařilo se zobrazit i magnetické doménové stěny v nanodrátech o průměru pouhých 50 nm. Připravili jsme fungující sondy s různými magnetickými vrstvami: magneticky tvrdého kobaltu, slitiny CoCr a magneticky měkké slitiny NiFe. Magneticky tvrdé sondy poskytovaly lepší signál, zatímco magneticky měkké byly vhodnější pro pozorování magneticky měkkých vzorků, protože je příliš neovlivňují. Námi připravené sondy jsou přinejmenším srovnatelné se standardními komerčními sondami. Simulace se ve většině případů shodují jak s měřením, tak teorií. Dále představujeme také naše prvotní výsledky modelování interakce vzorku s magnetickou sondou, které mohou složit k simulaci měření pomocí mikroskopie magnetických sil, a to i v případě, kdy sonda ovlivňuje magnetický stav vzorku.The thesis deals with magnetic force microscopy of soft magnetic nanostructures, mainly NiFe nanowires and thin-film elements such as discs. The thesis covers almost all aspects related to this technique - i.e. from preparation of magnetic probes and magnetic nanowires, through the measurement itself to micromagnetic simulations of the investigated samples. We observed the cores of magnetic vortices, tiny objects, both with commercial and our home-coated probes. Even domain walls in nanowires 50 nm in diameter were captured with this technique. We prepared functional probes with various magnetic coatings: hard magnetic Co, CoCr and soft NiFe. Hard probes give better signal, whereas the soft ones are more suitable for the measurement of soft magnetic structures as they do not influence significantly the imaged sample. Our probes are at least comparable with the standard commercial probes. The simulations are in most cases in a good agreement with the measurement and the theory. Further, we present our preliminary results of the probe-sample interaction modelling, which can be exploited for the simulation of magnetic force microscopy image even in the case of probe induced perturbations of the sample.

Numerical study of magnetic processes: extending the Landau-Lifshitz-Gilbert approach from nanoscale to microscale

The micromagnetic theory describes the magnetic processes in magnetic materials on a microscopic time and space length. Therefore, micromagnetic models are since long employed in the design of for instants magnetic storage media as magnetic tapes and Random Access Memory elements, used in computers. The use of efficient numerical techniques and the availability of powerful computers now make it possible to apply the same micromagnetic models on larger and more complex material systems with the aim of increasing our insight in the experimentally observed magnetic phenomena. In this PhD research, an efficient numerical micromagnetic model is developed that enables the analysis of magnetic processes starting from the nanometer space scale up to the micrometer space scale. Therefore, efficient algorithms are presented on the one hand to simulate the ultra fast dynamics of the magnetic processes as described by the Landau-Lifshitz-Gilbert equation. On the other hand, powerful numerical techniques are developed to evaluate the magnetic fields, characteristic to the micromagnetic description, in a fast way. The developed micromagnetic model is validated extensively in comparative studies with other micromagnetic and macroscopic magnetic material models. Moreover, the model is successfully applied in different magnetic research domains: magnetic switching processes in classical samples with nanometer dimensions are analysed, magnetic domains are studied in structures with order micrometer dimensions and magnetic hysteresis properties are investigated