Patterns formation in axially symmetric Landau-Lifshitz-Gilbert-Slonczewski equations

The Landau-Lifshitz-Gilbert-Slonczewski equation describes magnetization dynamics in the presence of an applied field and a spin polarized current. In the case of axial symmetry and with focus on one space dimension, we investigate the emergence of space-time patterns in the form of wavetrains and coherent structures, whose local wavenumber varies in space. A major part of this study concerns existence and stability of wavetrains and of front- and domain wall-type coherent structures whose profiles asymptote to wavetrains or the constant up-/down-magnetizations. For certain polarization the Slonczewski term can be removed which allows for a more complete charaterization, including soliton-type solutions. Decisive for the solution structure is the polarization parameter as well as size of anisotropy compared with the difference of field intensity and current intensity normalized by the damping

Traveling domain walls in chiral ferromagnets

We show that chiral symmetry breaking enables traveling domain wall solution for the conservative Landau-Lifshitz equation of a uniaxial ferromagnet with Dzyaloshinskii-Moriya interaction. In contrast to related domain wall models including stray-field based anisotropy, traveling wave solutions are not found in closed form. For the construction we follow a topological approach and provide details of solutions by means of numerical calculations.Comment: 16 pages, 4 figure

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

Vectorial Kerr magnetometry

Moderní magnetické materiály a na nich postavená zařízení se v poslední době stávají značně komplexními s vysokými nároky jak na výrobu, tak na charakterizaci jejich vlastností. Z malého výčtu jevů, které sou schopny charakterizace takových struktur, v práci využíváme tu, jenž nese název magneto-optický Kerrův jev. V teoretické části řešíme odezvu elektromagnetické vlny při odrazu od rozhraní materiálu s nenulovou magnetizací. Dále se zabýváme návrhem a konstrukcí zařízení, které je schopno tuto odezvu detekovat. V experimentální části je funkčnost zařízení prokázána na různých magnetických systémech: metastabilních vrstvách železa, pro případ Stonerovy–Wohlfarthovy částice a v neposlední řadě na magnetických vortexech. Navržený rastrovací vektorový Kerrův magnetometer nám umožnil charakterizovat magnetické vzorky s rozlišením 500 nm.Increased complexity of novel magnetic materials in the last decade has placed high demands on the manufacturing process as well as on the characterization. One of the possibilities for characterization of magnetic samples is to exploit the magneto-optical effects. The presented work uses the magneto-optical Kerr effect as a major characterization technique to probe the magnetic properties of samples. We have developed a mathematical model describing the effect of the magnetization on the polarized light and present an apparatus capable of measuring the response given by the light-matter interaction. The experimental results show the performance of the apparatus on the various magnetic systems including meta-stable iron layers, Stoner-Wohlfarth particles and magnetic vortices. The scanning vectorial Kerr magnetometer allowed us to probe the vector of magnetization with diffraction limited resolution below 500 nm.

Ga(^+) focused Ion beam irradiated Ni(_81)Fe(_19) thin films and Planar nanostructures investigated by the Magneto-Optical Kerr Effect

Patterned magnetic films are of interest for storing and sensing information, and possible logic applications, and find commercial applications in consumer goods such as personal computers. This thesis addresses the fast magnetic patterning of capped ultrathin Ni(_81)Fe(_19) films in Chapter 5, and the patterning and controlled magnetic switching of planar nanowires in Chapter 4. Controlled domain wall switching of complex wire geometries with comer structures, artificial trapping sites, 3-terminal junctions and more complex wire circuits is described in Chapters 6-7.The magnetic switching of planar Ni(_81)Fe(_19) nanowires fabricated by 30 keV, focused ion beam Ga(^+) ions was investigated, in the width range 60-500 nm. Experimentally measured wire easy axis coercivity is inversely proportional to width, similar to Stoner-Wohlfarth switching behaviour. Angular switching data for wires is presented. Significantly, wire coercivity and anisotropy field are shown to be strongly dependent on the ion beam raster direction during wire fabrication. The controlled propagation of head-to-head domain walls in a 27 Hz anticlockwise rotating magnetic field, through smoothly rounded comers is experimentally demonstrated. Domain wall propagation fields, 7 ± 3 Oe, just above the intrinsic domain wall coercivity were found. Using an L-shaped rounded comer geometry, the magnetic fields at which domain walls are introduced into wires and the domain wall propagation field were separated. Reproducible pinning and depinning of single domain walls on artificial domain wall traps with depths from 35-125 nm is demonshated.3-Terminal continuous Ni(_81)Fe(_19) wire junctions, suitable for AND/OR domain wall logic operations are described, in which the magnetic switching field of the device output is strongly dependent on the number of domain walls (0, 1, or 2), at the junction. An operating field phase diagram is presented in the context of junction integration with existing domain wall logic elements. Capped NigiFei9 films were ferromagnetically quenched by radiation induced transport of bilayer interfacial atoms. For Si/ Ni(_81)Fe(_19)/Al or Si/ Ni(_81)Fe(_19)/Au bilayers, the critical Ga(^+) ion dose to quench ferromagnetic ordering (Φ), measured by the magneto-optical Kerr (MOKE) effect, is demonstrated to be linearly proportional to the square of NigiFei9 thickness, (tNiFe)(^2) Therefore ultrathin-capped Ni(_81)Fe(_19) films can be magnetically quenched at ion doses ~ an order of magnitude lower then Ni(_81)Fe(_19)/Si samples, which are typically patterned by radiation sputtering from the vacuum- surface interface. Bilayer coercivity, uniaxial anisotropy field, remanent magnetization, and saturation magnetization as measured by MOKE, were tailored by controlled localized ion doses. Ga(^+) ion doses as low as 8 x 10(^13) ions.cm(^-2) reproducibly quenched measured room temperature ferromagnetism in 2 nm thick buried (_81)Fe(_19) films. Patterning of 200 nm wide in-plane magnetized wires embedded between a non magnetic cap and substrate is demonstrated

Exciting and steering propagating spin waves using a graded magnonic index

In this thesis, the results of time-resolved magneto-optical imaging experiments, micromagnetic simulations and analytical theory will be presented. These three approaches were used in order to understand how spin waves can be both excited and steered using magnetic non-uniformities (which gives rise to the graded magnonic index). The results presented in this thesis, as a whole, reveal that the graded magnonic index not only can be exploited either to excite or steer propagating spin waves in a deliberate manner, but is in fact a ubiquitous feature that needs to be taken into account when considering any dynamical phenomena in nano- and micro-magnetism.Engineering and Physical Sciences Research Council (EPSRC