J'affirme être informé que le plagiat est une faute grave susceptible de mener à des sanctions administratives et disciplinaires pouvant aller jusqu'au renvoi de l'Université de Strasbourg et passible de poursuites devant les tribunaux de la République Française.

Je suis conscient(e) que l'absence de citation claire et transparente d'une source empruntée à un tiers (texte, idée, raisonnement ou autre création) est constitutive de plagiat.

I affirm that I am aware that plagiarism is a serious misconduct that may lead to administrative and disciplinary sanctions up to dismissal from the University of Strasbourg and fiable to prosecution in the courts of the French Republic. I am aware that the absence of a clear and transparent citation of a source borrowed from a third party (text, idea, reasoning or other creation) is constitutive of plagiarism. helped me and supported me throughput this time. First of all I would like to express my gratitude to the thesis supervisor Riccardo Hertel for accepting me as his student and the excellent mentorship offered. I consider having received the opportunity to work with him a great honour and moving from India to a completely foreign environment was made much easier with his support and help. Riccardo was a patient supervisor throughout the three years and he always found time and interest for discussions and was patient enough to explain things in detail and to make corrections whenever necessary. From his guidance I learnt a lot: about magnetism and computation in particular and a wide range of other things. I would also like to thank Salia CheriĄ-Hertel for the discussions and valuable suggestions at various important points of the thesis. I would like to express my thankfulness to the jury; Daniel Lacour, Dietmar Weinmann, Dirk Grundler, Liliana Buda-Prejbeanu and Sam Ladak for accepting the request to be a part of the jury, carefully reading the thesis and for the valuable discussions during the defense and also for the careful corrections.

I express my appreciation for the experimental collaboration with the group of David Schmool regarding the study of magnonic modes in Cobalt nanodots.

The studies in this thesis was carried out at the Institut de Physique et de Chimie des Matériaux de Strasbourg (IPCMS-CNRS). I would like to express my sincere thanks to our institute director Pierre Rabu, the Doctroral School (ED-182) director Aziz Dinia, the IPCMS secretary Catherine Bonnin, the IPCMS librarian Beatrice Masson, the head of department (DMONS) Yves Henry, the department secretary Veronique Wernher and all the other people in the IPCMS for hosting me for these years and all the academic and legislative services provided. I would like to express my gratitude towards my colleagues and friends in IPCMS and in Strasbourg who helped me and supported me throughout these years: Aleena, Anatoli, Ankita, Aravind, Dheeksha, Gyandeep, Jinu, i Jose, Mathiew, Matias, Mehrdad, Mohammed Soliman, Nivedita, Sambith, Satakshi, Sharmin, Suvidya, and Swapneel. IŠm grateful to have such nice colleagues and the PhD was made fun working with these people. I would like to express my gratitude to Julien for proof reading the French abstract.

I would like to thank Senoy Thomas, my former professor for introducing me to the Ąeld of micromagnetism. I would like to express my love and gratitude towards my friends and my family for all the support. Finally I would like to express my gratitude and love towards all my friends whom I made in Strasbourg, who made my life here memorable and amusing.

Abstract

Le nanomagnétisme est la sous-branche du magnétisme qui traite de lŠétude des phénom' enes magnétiques ' a lŠéchelle du nanom' etre. LŠun des principaux moteurs de lŠétude des syst' emes de plus petite taille est le désir dŠobtenir un moyen de manipuler les propriétés magnétiques telles que la structure de lŠaimantation et ses propriétés ' a haute fréquence par conĄnement géométrique. Des exemples de cette capacité ' a contrôler la structure de lŠaimantation par conĄnement géométrique incluent lŠobservation dŠétats aimantés axialement dans des géométries allongées et lŠobservation de tourbillons magnétiques dans des structures planes minces. LŠexistence de ces structures de spin particuli' eres dans les particules de faible dimension leur attribue des propriétés statiques et dynamiques distinctes par rapport ' a lŠétat massif.

Dans le passé, les effets de ce modelage géométrique étaient principalement limités aux structures bidimensionnelles. Récemment, une nouvelle tendance a émergé dans le nanomagnétisme pour étendre ces effets ' a la troisi' eme dimension et, par conséquent, le nanomagnétisme tridimensionnel est apparu comme une nouvelle fronti' ere de la recherche. Des progr' es considérables ont été réalisés dans les techniques de mesure et de fabrication pour répondre ' a cette tendance. Les progr' es des techniques dŠimagerie par rayons X et par électrons ont facilité la visualisation des structures magnétiques tridimensionnelles avec une précision nanométrique, même si elles sont enfouies profondément dans le matériau.

Parall' element, les progr' es récents dans les techniques de fabrication telles que le dépôt induit par faisceau dŠélectrons focalisé (FEBID), également connu sous le nom de nanoimpression 3D, et dŠautres techniques de nanofabrication telles que la lithographie ' a deux photons (TPL) en combinaison avec diverses autres techniques de dépôt ont permis de fabriquer des matériaux ferromagnétiques avec des géométries de forme arbitraire a lŠéchelle nanométrique.

Par conséquent, les nanoarchitectures tridimensionnelles sont apparues comme une nouvelle catégorie de nanostructures. LŠacc' es ' a un degré de liberté supplémentaire dans ces structures pourrait faire apparaître des phénom' enes physiques nouveaux et intéressants qui sont absents dans les structures bidimensionnelles. Parall' element aux techniques de fabrication et de mesure iii de pointe, les simulations par éléments Ąnis constituent un excellent outil théorique pour étudier les propriétés fondamentales de ces types de structures. Dans cette th' ese, nous avons étudié la structure magnétique statique et les propriétés haute fréquence de différents types dŠarchitectures 3D telles que les structures fractales 3D de type Sierpinski, les nanoarchitectures Buckyball, les nanoarchitectures de type diamant et les nanoarchitectures cubiques en utilisant des simulations micromagnétiques par éléments Ąnis.

La th' ese est divisée en huit chapitres. Dans le premier chapitre, le contexte général et la motivation scientiĄque de lŠétude sont présentés. Dans le deuxi' eme chapitre, la théorie de base du micromagnétisme est introduite, y compris les différents termes énergétiques pertinents pour les simulations numériques. Le micromagnétisme est la théorie du continu du ferromagnétism qui décrit les phénom' enes magnétiques ' a une échelle de longueur mésoscopique. La principale quantité dŠintérêt dans la théorie du micromagnétisme est lŠaimantation M (r, t) qui est déĄnie comme la densité des moments magnétiques par unité de volume. LŠhypoth' ese centrale de la théorie du micromagnétisme est que la magnitude du vecteur M (r, t) reste constante tout au long du processus et que seule sa direction peut varier. La théorie micromagnétique décrit lŠénergie totale du syst' eme comme la somme de plusieurs termes énergétiques différents, tels que lŠénergie dŠéchange, lŠénergie de démagnétisation, lŠénergie dŠanisotropie cristalline et lŠénergie de Zeeman. LŠénergie dŠéchange est la contribution de la force fondamentale qui favorise un alignement parall' ele des moments magnétiques voisins dans les matériaux ferromagnétiques. Dans le mod' ele continu du micromagnétisme, lŠexpression de lŠénergie dŠéchange prend la forme dŠune fonction impliquant les gradients spatiaux de lŠaimantation, qui est minimale dans le cas dŠune distribution uniforme de lŠaimantation, de sorte que toute inhomogénéité dans la structure de lŠaimantation entraîne une augmentation de lŠénergie dŠéchange. LŠénergie de démagnétisation est due ' a lŠinteraction de lŠaimantation avec le champ magnétostatique, cŠest-' adire le champ magnétique créé par lŠensemble des moments magnétiques de lŠéchantillon. Cette contribution énergétique favorise la formation de structures de fermeture du Ćux, comme les tourbillons, qui minimisent les charges magnétostatiques et réduisent ainsi le champ dipolaire. La magnétostatique est également ' a lŠorigine du phénom' ene dŠanisotropie de forme, qui décrit la tendance des nanoparticules allongées ' a aligner lŠaimantation le long de son axe. LŠénergie dŠanisotropie magnéto-cristalline résulte du couplage des moments de spin et dŠorbite (couplage spin-orbite) et de lŠinteraction des ions et du champ cristallin. Le cas le plus simple de lŠénergie dŠanisotropie magnétocristalline est lŠanisotropie uniaxiale dans laquelle il existe une direction dŠaimantation préférée et toute déviation de lŠaimantation par rapport ' a cette direction est énergiquement pénalisée. EnĄn, lŠénergie Zeeman décrit lŠinteraction de lŠaimantation avec un champ externe. Ce terme énergétique favorise lŠalignement de lŠaimantation parall' element au champ externe. LŠénergie totale est obtenue comme la somme de ces différents termes énergétiques. ' A chacun de ces termes énergétiques, on peut attribuer un champ effectif décrivant la force de lŠinteraction. Par conséquent, le champ effectif total peut être décomposé en champs effectifs individuels pour chaque terme énergétique.

Les simulations numériques micromagnétiques impliquent une subdivision de la géométrie de la particule en un grand nombre de cellules de discrétisation.

En plus de ces cellules, la subdivision implique un ensemble de points (nİuds) auxquels lŠaimantation M (r, t) est calculée et interpolée dans chaque cellule. Pour le cas statique, une version discrétisée dŠune structure de lŠaimantation dŠéquilibre peut être obtenue en intégrant numériquement les densités dŠénergie micromagnétique sur tout le volume et en minimisant lŠénergie totale du syst' eme qui en résulte. La dynamique de M (r, t) est régie par lŠéquation de Landau-Lifshitz-Gilbert (LLG), lŠéquation de mouvement de lŠaimantation, qui décrit une dynamique amortie-précessionnelle de M (r, t) par rapport au champ effectif local. Dans nos études, nous avons maintenu la taille globale de la fractale constante ' a 512 nm et simulé cinq itérations de la fractale. Ainsi, le premier vii étage de la fractale était composé de quatre tétra' edres, chacun de longueur de côté 256 nm, et le cinqui' eme étage était composé de 1024 tétra' edres de longueur de côté 16 nm. En raison de leur disposition géométrique particuli' ere, tous les étages des fractales ont la même surface, tandis que le volume total diminue exponentiellement avec chaque étage. Le mod' ele par éléments Ąnis du premier étage était composé de pr' es de 500 000 cellules dŠéléments Ąnis (qui sont aussi des tétra' edres, bien que de forme irréguli' ere) et lŠétage Ąnal contenait pr' es de 700 000 cellules. Pour la simulation, nous avons utilisé des param' etres de matériau correspondant ' a ceux du Permalloy avec une aimantation ' a saturation de M s = 8 × 10 5 A m -1 , une rigidité dŠéchange de A = 1.3 × 10 -11 J m -1 , et une anisotropie magnétocristalline nulle. Ces param' etres donnent une longueur dŠéchange de 5.6 nm. Toutes les mailles des éléments Ąnis avaient une taille de maille inférieure ' a cette longueur intrins' eque. Les états dŠaimantation relaxés qui se développent dans ces géométries ' a champ nul sont obtenus en saturant la structure le long dŠun des sommets, puis en relaxant le syst' eme ' a champ externe nul.

dM dt = γ L [M × H eff ] - α L M s [M × [M × H eff ]] (1)
Le premier stade de la fractale contient des tétra' edres individuels relativement grands par rapport ' a la longueur dŠéchange du matériau. Dans ce cas, la structure de lŠaimantation rémanente ' a champ nul, obtenue apr' es la relaxation de la saturation, donne une structure magnétique dans laquelle chaque tétra' edre est dans un état de vortex magnétique, dont le plan de circulation se trouve le long de la face perpendiculaire ' a la direction de saturation.

En se déplaçant davantage vers lŠintérieur du tétra' edre, la distribution de lŠaimantation converge pour former une structure en forme de ŤAŤ. Dans la deuxi' eme étape, même si les tétra' edres sont deux fois moins grands quŠavant, la structure tourbillonnaire reste la même, sauf dans trois des tétra' edres. Dans ces trois tétra' edres, au lieu de former des états de vortex individuels, lŠaimantation sŠarrange dans une conĄguration en forme de λ, et lŠaimantation dans les tétra' edres forme collectivement un vortex déconnecté. Une structure de fermeture de Ćux partielle est ainsi obtenue en alignant lŠaimantation des tétra' edres voisins, de sorte que lŠaimantation sŠenroule pour former une boucle macroscopique. ' A partir de la troisi' eme étape, lorsque la taille des tétra' edres est encore réduite, les unités individuelles de tétra' edres sont trop petites pour entretenir un vortex et elles forment ' a la place des structures de vortex déconnectées sŠétendant sur plusieurs tétra' edres.

Les propriétés hystérétiques de ces structures sont simulées en appliquant un champ de 500 mT puis en réduisant lŠintensité du champ ' a zéro par étapes de champ suffisamment petites. Nous observons des différences considérables dans la boucle M-H ' a chaque étape de la fractale. Dans les premiers et deuxi' emes stades, la structure est dans un état de saturation magnétique ' a un champ de 500 mT. Lorsque le champ est progressivement réduit, lŠaimantation moyenne diminue presque linéairement. ' A partir de la troisi' eme étape, nous observons un comportement différent dans lequel lŠaimantation reste presque constante jusquŠ' a une intensité de champ spéciĄque, au-del' a de laquelle un saut abrupt se produit. Ce point de saut abrupt est caractérisé par la nucléation des structures tourbillonnaires déconnectées susmentionnées.

Pour calculer les modes magnétiques haute fréquence des différents états dŠéquilibre relaxés, les structures magnétiques ont été excités par différentes méthodes ' a lŠaide dŠune perturbation externe. Le processus oscillatoire ŞringdownŤ magnétique se déroulant apr' es lŠapplication de cette petite perturbation a été enregistré et analysé. La réponse en fréquence des cinq étapes est affichée sur la Ągure 2. Dans la premi' ere étape, nous pouvons identiĄer deux modes importants : un mode basse fréquence ' a 700 MHz et un mode haute fréquence ' a 2.7 GHz. Ces modes correspondent ' a la gyration synchrone des noyaux de vortex. Dans la deuxi' eme étape, nous pouvons identiĄer deux régions principales : plusieurs modes basse fréquence dans la plage entre 1,1 et 1,9 GHz, qui peuvent être attribués ' a lŠactivité des tourbillons dans les tétra' edres, et des modes comparativement plus faibles autour de 4,7 GHz, qui sont localisés au niveau des trois tétra' edres formant un tourbillon déconnecté. ' A partir de la troisi' eme étape, au lieu dŠavoir plusieurs pics nets bien déĄnis, nous obtenons une large bande de modes intermixtes. La largeur de cette bande augmente au fur et ' a mesure que nous passons du troisi' eme stade au quatri' eme stade, même si la fréquence médiane se déplace vers une valeur inférieure. Au cinqui' eme stade, alors que la taille des unités tétraédriques est encore réduite vers la limite du domaine unique, nous observons une déviation de cette excitation ' a large bande. En étudiant les oscillations magnétiques haute fréquence dépendantes du champ ' a différents stades, nous avons observé que la fréquence des modes augmente linéairement avec lŠintensité du champ appliqué.

La prochaine catégorie de structures tridimensionnelles que nous avons étudiée est celle des nanoarchitectures artiĄcielles de type Buckyball. Ces Spectre dŠexcitation haute fréquence des différents états ; vert : état de charge nulle, rouge : état désordonné, bleu : état de défaut géométrique.

' a 83 points de sommet. Le mod' ele par éléments Ąnis a une taille de cellule maximale de 4 nm et contient plus de 700 000 éléments tétraédriques.

Chapter 1

Introduction

In the course of history, research in magnetism has contributed signiĄcantly to the development of human civilization. HumanityŠs fascination for magnetism started in ancient Greece with the discovery of the Lodestone and has evolved ever since and revolutionized the life of humans [START_REF] Mattis | History of Magnetism[END_REF]. The invention of the magnetic compass enabled the great voyages and thereby the age of discovery. The invention of the dynamo eventually led to the large-scale production of electricity and all the beneĄts electricity brought with it. High-density magnetic data storage development paved the way for todayŠs society with its abundance of information. Other applications like magnetocaloric refrigeration or high-power electromotors have also evolved from research in magnetism.

Today, magnetism has penetrated practically all branches of science and technology.

Nanomagnetism [START_REF] Vaz | Magnetism in ultrathin Ąlm structures[END_REF] is the sub-branch of magnetism that concerns the study of the properties of magnetic materials at ever increasingly smaller dimensions. Along with scientiĄc curiosity, nanomagnetism research has traditionally been motivated by the rapid progress in information technology and its demand for low-power, high-density [START_REF] Tsang | fabrication and testing of spin-valve read heads for high density recording[END_REF], and non-volatile storage media. This demand skyrocketed with the discovery of the Giant magneto-resistance effect [START_REF] Baibich | Giant magnetoresistance of (001)Fe/(001)Cr magnetic superlattices[END_REF][START_REF] Binasch | Enhanced magnetoresistance in layered magnetic structures with antiferromagnetic interlayer exchange[END_REF] (GMR), for which P. Grünberg and A. Fert were awarded the Nobel Prize in Physics in 2007. This effect, along with the Tunnel magneto-resistance effect [START_REF] Yuasa | Giant roomtemperature magnetoresistance in single-crystal Fe/MgO/Fe magnetic tunnel junctions[END_REF][START_REF] Parkin | Giant tunnelling magnetoresistance at room temperature with MgO (100) tunnel barriers[END_REF] describes a signiĄcant change of the electric resistance in a magnetic multilayer system depending on the relative orientation of the magnetization direction of the layers. These discoveries started the Ąeld of spintronics [8,[START_REF] Wolf | Spintronics: A Spin-Based Electronics Vision for the Future[END_REF][START_REF] Žutić | Spintronics: Fundamentals and applications[END_REF], in which the interaction of the electron spin and the magnetization is of central importance. These studies led to a trend in nanomagnetism where much of the research was focused on thin Ąlms and interface effects.

In addition to its relevance for spintronic applications, the scientiĄc interest in nanomagnetism extends far beyond the desire to improve device miniaturization and other technological aspects. On the sub-micron length scale, magnetic materials often develop unique and characteristic features, such as the formation of magnetic vortices [START_REF] Shinjo | Magnetic Vortex Core Observation in Circular Dots of Permalloy[END_REF], domain walls, skyrmions [START_REF] Bogdanov | Chiral symmetry breaking in magnetic thin Ąlms and multilayers[END_REF][START_REF] Heinze | Spontaneous atomic-scale magnetic skyrmion lattice in two dimensions[END_REF][START_REF] Fert | Magnetic skyrmions: advances in physics and potential applications[END_REF], or other topological structures. These fundamental magnetic structures can display particle-like properties and high stability. Their typical size, which is governed by material-speciĄc length scales (the exchange lengths), is in the range of a few nanometers for usual ferromagnetic materials. These properties render such structures exciting candidates for shift-register type applications. Consequently, much effort has been made to understand how these magnetic structures are affected by Ąnite-size effects. A prominent example is the magnetic racetrack memory [START_REF] Parkin | Memory on the racetrack[END_REF][START_REF] Parkin | Magnetic domain-wall racetrack memory[END_REF] device concept, whose basic principle is to use magnetic domain walls in nanowires as units of information that can be displaced within the device in a controlled way through electrical currents [START_REF] Allwood | Magnetic domain-wall logic[END_REF]. In this speciĄc case, the nanopatterning of a magnetic material serves to guide particle-like magnetic structures (a more recent version of the racetrack memory device uses skyrmions [START_REF] Kang | Voltage controlled magnetic skyrmion motion for racetrack memory[END_REF][START_REF] Tomasello | A strategy for the design of skyrmion racetrack memories[END_REF][START_REF] Koshibae | Memory functions of magnetic skyrmions[END_REF] instead of domain walls) along a predeĄned path. But there are many other cases where Ąnite-size effects on the submicron scale can signiĄcantly impact a nanomagnetŠs properties. Over more than two decades, the inĆuence of the size and shape of magnetic nanoparticles has been a central topic in magnetism research [START_REF] Vaz | Magnetism in ultrathin Ąlm structures[END_REF]. Experimental and simulation studies have evidenced the delicate impact that details of a magnetic particleŠs geometry can have on its static, dynamic, and hysteretic properties. The patterning of nanomagnets is also important in magnonics, where suitably shaped magnets can guide high-frequency magnetic excitations (spin waves) to exploit interference effects for logical operations and data processing [START_REF] Grundler | Nanomagnonics around the corner[END_REF].

Most of these studies on the shape-and size dependence in nanomagnetism concerned patterned thin-Ąlm elements of varying thickness [START_REF] Vaz | Magnetism in ultrathin Ąlm structures[END_REF][START_REF] Guimarães | Principles of Nanomagnetism[END_REF]. However, recent advancements in direct write techniques such as Focused-electron-beam-induced deposition (FEBID) 1 [START_REF] Fernández-Pacheco | Writing 3D Nanomagnets Using Focused Electron Beams[END_REF][START_REF] Teresa | Review of magnetic nanostructures grown by focused electron beam induced deposition (FEBID)[END_REF][START_REF] Utke | Gas-assisted focused electron beam and ion beam processing and fabrication[END_REF][START_REF] Fernández-Pacheco | Three dimensional nanomagnetism[END_REF] or lithographic techniques such as two-photonlithography (TPL) 2 [START_REF] Maruo | Three-dimensional microfabrication with two-photon-absorbed photopolymerization[END_REF][START_REF] Askey | Use of two-photon lithography with a negative resist and processing to realise cylindrical magnetic nanowires[END_REF][START_REF] Williams | Two-photon lithography for 3D magnetic nanostructure fabrication[END_REF], in combination with various deposition techniques 1 Focused electron beam induced deposition: this is a direct write technique in which a precursor material injected as a gas is deposited on the surface of a substrate on the focus of an electron beam.

Thus controlling the position of the beam arbitrary three dimensional patterns could be written.

2 TTwo photon lithography: In this technique a femtosecond laser on the infrared range is focused on a photosensitive resist. the high intensity of the focused beam allows the absorption of two photons which results in a photosensitive polymerisation of the resist. This technique can thus be employed to fabricate three dimensional nanostructures [START_REF] Donnelly | Element-SpeciĄc X-Ray Phase Tomography of 3D Structures at the Nanoscale[END_REF][START_REF] Williams | Two-photon lithography for 3D magnetic nanostructure fabrication[END_REF][START_REF] Meng | Fabrication of a 3D nanomagnetic circuit with multi-layered materials for applications in spintronics[END_REF] have made it possible to fabricate an entirely new category of magnetic nanostructures by extending the patterning into three dimensions instead of two. The fabrication techniques offer possibilities to generate magnetic materials with nanometric feature size and of essentially arbitrary shape. These techniques can be used, for instance, to fabricate magnetic nano-architectures consisting of interconnected magnetic nanowires. In parallel to this development in nano-patterning techniques, there has also been spectacular progress in measurement and imaging techniques such as neutron tomography [START_REF] Manke | Three-dimensional imaging of magnetic domains[END_REF][START_REF] Hilger | Tensorial neutron tomography of three-dimensional magnetic vector Ąelds in bulk materials[END_REF], electron tomography [START_REF] Tanigaki | Three-dimensional observation of magnetic vortex cores in stacked ferromagnetic discs[END_REF], X-ray nano-tomography [START_REF] Streubel | Retrieving spin textures on curved magnetic thin Ąlms with full-Ąeld soft x-ray microscopies[END_REF][START_REF] Donnelly | Tomographic reconstruction of a three-dimensional magnetization vector Ąeld[END_REF] and laminography [START_REF] Donnelly | Time-resolved imaging of three-dimensional nanoscale magnetization dynamics[END_REF]. These techniques make it possible to probe and image complicated three-dimensional magnetization structures and their dynamical properties, even if they are buried deep inside the volume of a three-dimensional magnetic sample [START_REF] Donnelly | Experimental observation of vortex rings in a bulk magnet[END_REF][START_REF] Donnelly | Tomographic reconstruction of a three-dimensional magnetization vector Ąeld[END_REF]. In the past, such information was accessible only indirectly through comparisons with micromagnetic simulations [START_REF] Ha | Micromagnetic study of magnetic conĄgurations in submicron permalloy disks[END_REF], which were often combined with surface-sensitive experimental imaging techniques [START_REF] Hertel | Three-dimensional magnetic-Ćux-closure patterns in mesoscopic fe islands[END_REF], or by reconstructing the contrast obtained from grazing-incidence XMCD transmission measurements. Due to this two-pronged development in fabrication and measurement techniques, three-dimensional (3D) nanomagnetism has recently evolved into an important research topic in magnetism [START_REF] Fernández-Pacheco | Three dimensional nanomagnetism[END_REF][START_REF] Fischer | Launching a new dimension with 3D magnetic nanostructures[END_REF].

Similar to how the properties of the bulk material change under the inĆuence of various size-, shape-and interface effects in two-dimensional magnetic structures, the patterning of these novel three-dimensional nano-architectures could bring out new magnetic phenomena and physical effects absent in two-dimensional systems. Such new features could result in new functionalities for various applications, such as high-density data storage, magnonics, non-conventional computing [START_REF] Fernández-Pacheco | Three dimensional nanomagnetism[END_REF]. These developments could also result in a new type of magnetic metamaterials [START_REF] Liu | Metamaterials: a new frontier of science and technology[END_REF][START_REF] Rangu | Magnetic metamaterials: A comparative study of resonator geometry and metal conductivity[END_REF]: ferromagnets with artiĄcial properties generated by the peculiar three-dimensional structure and that do not occur naturally in the magnetic material. Such magnetic metamaterials would be a logical extension of existing optical and acoustic metamaterials made of suitably shaped periodic 3D nanoarchitectures.

The extension of nanostructures into the third dimension often gives rise to new types of magnetization structures such as, e.g., 3D vortex domain walls [START_REF] Donnelly | Time-resolved imaging of three-dimensional nanoscale magnetization dynamics[END_REF][START_REF] Ivanov | Direct observation of current-induced motion of a 3d vortex domain wall in cylindrical nanowires[END_REF][START_REF] Donnelly | Experimental observation of vortex rings in a bulk magnet[END_REF],

Bloch point structures [START_REF] Da Col | Observation of bloch-point domain walls in cylindrical magnetic nanowires[END_REF], 3D skyrmions [START_REF] Pathak | Three-dimensional chiral magnetization structures in FeGe nanospheres[END_REF][START_REF] Pathak | Geometrically constrained skyrmions[END_REF][START_REF] Liu | Nucleation and stability of skyrmions in three-dimensional chiral nanostructures[END_REF], or HopĄons [START_REF] Tai | Static Hopf solitons and knotted emergent Ąelds in solid-state noncentrosymmetric magnetic nanostructures[END_REF][START_REF] Liu | Binding a hopĄon in a chiral magnet nanodisk[END_REF]. These 3D spin structures with non-trivial topologies often have higher stability and improved susceptibility to magnonic excitations. Moreover, they may be manipulated through spin-polarized currents and therefore have the potential to be exploited for applications in three-dimensional spintronics. HopĄons, for example, could be ideal candidates for racetrack-type devices because owing to their vanishing gyrovector, they should not experience deĆections by the Hall effect, which affects the motion of current-driven skyrmions [START_REF] Wang | Current-driven dynamics of magnetic hopĄons[END_REF]. Even conventional spin structures such as domain walls can have different dynamical properties in 3D nanostructures when compared to 2D. For example, a domain wall propagating in a 2D strip under the inĆuence of an external Ąeld would become unstable above a particular speed due to Walker breakdown [START_REF] Schryer | The motion of 180°domain walls in uniform dc magnetic Ąelds[END_REF]. In contrast to this, previous studies with micromagnetic simulations [START_REF] Yan | Fast domain wall dynamics in magnetic nanotubes: Suppression of Walker breakdown and Cherenkov-like spin wave emission[END_REF][START_REF] Hertel | Ultrafast domain wall dynamics in magnetic nanotubes and nanowires[END_REF] have shown that winding the nanostrips in the form of a tube can suppress the Walker breakdown [START_REF] Schryer | The motion of 180°domain walls in uniform dc magnetic Ąelds[END_REF] and thereby permit domain-wall velocities as fast at 1000 m s -1 . These ultra-fast domainwalls can even reach the phase velocity of spin waves and thus create a Spin-Cherenkov effect [START_REF] Yan | Fast domain wall dynamics in magnetic nanotubes: Suppression of Walker breakdown and Cherenkov-like spin wave emission[END_REF][START_REF] Yan | Spin-Cherenkov effect and magnonic Mach cones[END_REF]. This phenomenons is similar to the existence of a Şsonic-boomŤ in the case of supersonic projectiles. In addition, when compared to planar thin Ąlms, the curved surfaces of nanotubes lack space inversion symmetry, which can lead to various curvature-induced effects [START_REF] Streubel | Magnetism in curved geometries[END_REF][START_REF] Saxena | Deformable curved magnetic surfaces[END_REF][START_REF] Sheka | A perspective on curvilinear magnetism[END_REF], such as curvature-induced DzyaloshinskiiŰMoriya interaction (DMI) [START_REF] Gaididei | Curvature effects in thin magnetic shells[END_REF] and curvature induced effective anisotropy. It has been suggested that these curvature-induced phenomena, whose strength could be controlled by the particle geometry, could provide a new method for manipulating the magnetic properties of the nanostructures [START_REF] Sanz-Hernández | ArtiĄcial double-helix for geometrical control of magnetic chirality[END_REF]. Such effects also have the potential to be exploited in three-dimensional high-density data storage (3D-HDD) media, in which information can be stored in the form of magnetic entities such as domain-walls, skyrmions or vortices. This Ąeld of research, named cuvilinear magnetism [START_REF] Streubel | Magnetism in curved geometries[END_REF][START_REF] Sheka | Curvature effects in statics and dynamics of low dimensional magnets[END_REF][START_REF] Gaididei | Curvature effects in thin magnetic shells[END_REF], is still in the developing stage and one can assume that several interesting physical phenomena related to these effects and the associated applications are yet to be discovered.

An interesting sub-category of three-dimensional magnetic nanostructures are interconnected nanowire networks. The aforementioned sophisticated nanopatterning techniques allow for the assembly of nanowire networks in the form of complex nanoarchitectures, e.g., magnetic buckyballs [START_REF] Donnelly | Element-SpeciĄc X-Ray Phase Tomography of 3D Structures at the Nanoscale[END_REF] or arrays of Şnano-treesŤ [START_REF] Keller | Direct-write of free-form building blocks for artiĄcial magnetic 3D lattices[END_REF][START_REF] May | Realisation of a frustrated 3d magnetic nanowire lattice[END_REF][START_REF] May | Magnetic charge propagation upon a 3D artiĄcial spinice[END_REF]. In such nanoscale architectural forms, a particular geometric situation develops at the vertices, where three or more nanowires meet. This arrangement of nanowires leads to magnetic frustration, as in the case of artiĄcial spin ices (ASI) 2.4. Hence, these structures can be regarded as three-dimensional artiĄcial spin ice (3D-ASI) systems. Along with novel advanced fabrication and measurement techniques, micromagnetic simulations are necessary to understand the physics of these three-dimensional nanoarchitectures.

In this thesis, we investigate the three-dimensional magnetic structures formed in different types of such nanoarchitectures and study their dynamic properties, like the response to external Ąelds and their high-frequency dynamics. We investigate these effects through Ąnite-element micromagnetic simulations. Principally, we investigated artiĄcial Buckyball nanoarchitectures, diamond-lattice-like architectures, and cubiclattice-like architectures. The thesis is divided into six chapters. The Ąrst chapter is the introduction, which provides the studyŠs context, motivation, and relevance. The second chapter introduces the theory of micromagnetism. In particular, it discusses the micromagnetic energy terms and the equation describing the magnetization dynamics.

In the third chapter, the numerical implementation of the micromagnetic theory is brieĆy introduced, along with the numerical details of a post-processing Fourier analysis tool developed in this thesis. The fourth chapter consists of the micromagnetic investigation of magnetic structures and high-frequency modes formed in Cobalt nanodots.

The Ąfth chapter summarizes our studies of the Buckyball nanoarchitectures where their magnetic conĄgurations, hysteretic properties, and high-frequency response are discussed in detail. The sixth chapter reports on the studies on two types of artiĄcial magnonic crystals (AMC): the diamond-type crystal and the cubic-lattice-type crystal.

Chapter 2

Fundamental aspects of micromagnetism

Theory of micromagnetism

Micromagnetism is a continuum theory for ferromagnetic materials. It describes the magnetization structure on a mesoscopic length scale and captures the typical magnetic features developing in the sub-micron range. It thereby provides the appropriate theoretical basis to study the physical properties of typical ferromagnetic nanostructures, including the more recent three-dimensional types. Micromagnetic theory bridges the gap between atomistic and quantum-mechanical models on the one hand and a macroscopic description through MaxwellŠs theory of electromagnetism on the other hand. In the latter, material-speciĄc electromagnetic phenomena are described through volumeaveraged quantities, such as permeabilities and susceptibilities. Instead of this macroscopic description, the ŞmicroscopicŤ features of the magnetization (hence the name) are central to micromagnetic theory, even though their size is typically rather in the nanometer than in the micrometer range. In addition to providing a framework for calculating the spatial structure of the magnetization, the theory of micromagnetism also addresses aspects of the magnetization dynamics unfolding typically on the pico-and nanosecond time scale. The theory of micromagnetism was pioneered by W. F. Brown

Jr in 1940-41 when he published two papers [START_REF] Brown | Theory of the approach to magnetic saturation[END_REF][START_REF] Brown | The effect of dislocations on magnetization near saturation[END_REF] on the theory of micromagnetism of ferromagnetic materials building. It builds upon the previous works of Akulov [START_REF] Akulov | Zur Theorie der Magnetisierungskurve von Einkristallen[END_REF],

Bloch [START_REF] Bloch | Zur Theorie des Austauschproblems und der Remanenzerscheinung der Ferromagnetika[END_REF] and, perhaps most importantly, by Landau and Lifshitz in 1935 [START_REF] Landau | On the theory of the dispersion of magnetic permeability in ferromagnetic bodies[END_REF]. Later contributions by Kittel [START_REF] Kittel | Ferromagnetic domain theory[END_REF], Stoner-Wohlfarth [START_REF] Stoner | A mechanism of magnetic hysteresis in heterogeneous alloys[END_REF], Néel [START_REF] Néel | Energie des parois de bloch dans les couches minces[END_REF], Aharoni [START_REF] Aharoni | Magnetization curve of the inĄnite cylinder[END_REF][START_REF] Aharoni | International series of monographs on physics[END_REF], and Shtrikman have further contributed to the foundations of the theoretical framework.

More recently, the rapid development in computation technology has established the micromagnetic theory as an important branch of modern magnetism. Nowadays, computational micromagnetism is an indispensable tool for the fundamental investigation of nanomagnetic phenomena such as magnetization structures, energetics and dynamic properties of nano-and mesoscale systems.

Theory of micromagnetism

The central quantity of interest in micromagnetic theory is a ferromagnetŠs spontaneous magnetization Ąeld. The magnetization is deĄned as the density of magnetic moments per unit volume, and it is represented as a continuous vector Ąeld M (r, t). A central assumption of micromagentic theory is that the vector Ąeld of the magnetization is directional, i.e., the magnetization vector has constant length ♣M ♣ = M s = const. within the magnetic material, where M s is the saturation magnetization of the material. Only

Theory of micromagnetism

its direction is allowed to vary in time and space, not its magnitude. Micromagnetism, in general, deals with the determination of M (x, t) or the reduced magnetization m(x) = M (x)/M s at each point x. For the static state, the basis for this involves the minimization of the free energy of the system. The free energy (or the total energy) of a ferromagnet depends on the structure M (r) of the magnetization. The most important contributions to a magnetŠs total energy arise from various components, the most important of which are usually the exchange energy, the magneto-crystalline anisotropy energy, magnetostatic-dipolar energy, and the Zeeman energy. The theory of micromagnetism uses continuum expressions of these intrinsic energy terms, which all depend on the magnetic structure M (r). Each of these expressions can be formally derived from more microscopic pictures involving discrete spins S z by taking the continuum limit. The transition to the continuum picture is based on the deĄnition of the magnetization M (r) as the density of magnetic moments. In the following section we brieĆy introduce the theoretical expressions used to calculate each of the energies.

Exchange energy

The exchange interaction describes the fundamental tendency of ferromagnetic materials to align neighboring magnetic moments parallel to each other. The microscopic origin of the ferromagnetic exchange interaction is purely quantum-mechanical and arises from PauliŠs exclusion principle. It is a short-range interaction that, by favoring a parallel arrangement of neighboring spins, gives rise to the phenomenon of ferromagnetism.

There are different ways to derive the micromagnetic form of the ferromagnetic exchange energy density. One possibility is to take the classical atomistic Heisebenberg interaction as a microscopic starting point and to perform a transition to the continuum limit. In the Heisenberg model, the exchange interaction between two magnetic moments S i (r i ) and S j (r j ) is given by

H ex = -2 i̸ =j J ij (r ij ) S i (r i ) • S j (r j ) (2.1)
where J i,j is the quantum-mechanical exchange integral [START_REF] Heisenberg | Zur Theorie des Ferromagnetismus[END_REF]. Assuming that the angle ϕ ij between neighboring spins S i (r i ) and S j (r j ) is small, one can apply a Taylor expansion of eq. (2.1) which, after a few manipulations [START_REF] Kittel | Physical Theory of Ferromagnetic Domains[END_REF], leads to the form

Theory of micromagnetism

e exc = A (∇m x ) 2 + (∇m y ) 2 + (∇m z ) 2 (2.2)
Where A is the exchange stiffness constant of the material expressed in J m -1 . In this derivation, A is related to the quantum-mechanical exchange integral J i,j [START_REF] Heisenberg | Zur Theorie des Ferromagnetismus[END_REF] and depends, moreover, on the atomic lattice constant and the crystalline structure [START_REF] Kittel | Physical Theory of Ferromagnetic Domains[END_REF]. In the theory of micromagnetism, A is assumed to be a position-independent constant of the material.

A different approach consists in applying a phenomenological principle based on the fact that the exchange interaction tends to preserve a homogeneous structure of the magnetization Ąeld M (r). Therefore, any inhomogeneity in the magnetic structure should lead to an increase in exchange energy. In this perspective, one can assume that the expression for the exchange energy density should be a function of the spatial derivatives of the reduced magnetization ∂m i /∂x j in order to describe the degree of local inhomogeneity in the vector Ąeld of the magnetization. Several categories of combinations of gradients can be discarded for symmetry reasons. For instance, one can assume that the exchange energy is invariant under time-inversion symmetry, i.e., an operation of the type M → -M . Based on such considerations, one can derive the term

e xc = i,k,l A kl ∂m i ∂x k ∂m i ∂x l (2.3)
as a generalized expression of such a phenomenological Şinhomogeneity energy densityŤ [START_REF] Hubert | Magnetic Domains -The Analysis of Magnetic Microstructures[END_REF][START_REF] Landau | Electrodynamics of continuous media[END_REF]. In practice, the symmetric tensor A kl can be represented as a constant A, and the expression for the ferromagnetic exchange simpliĄes to eq. (2.2), which describes the exchange energy per unit volume in the continuum representation. According to eq. (2.2), any deviation from a perfectly homogeneous magnetization state leads to an increase of the exchange energy density and is thus penalized, while the energy term vanishes for a perfect homogeneous state.

Crystalline anisotropy energy

The magnetocrystalline energy is derived from the coupling between spin and orbital moments (L-S coupling) and the interaction of the ions and the crystal Ąeld [START_REF] Birss | Symmetry and magnetism[END_REF]. In a crystalline ferromagnet, this relativistic effect results in the formation of one or more

Theory of micromagnetism

easy axes along which the magnetization preferably aligns, and hard axes which represent unfavorable orientations of the magnetization. This anisotropic effect is accounted for in an energy term in which any angular deviation of m from the easy axis in energetically penalized. The simplest case of magneto-crystalline anisotropy is that of uniaxial anisotropy, in which there exists one preferred direction of magnetization (the easy axis). The direction orthogonal to this axis is called the hard-axis. The uniaxial anisotropic energy density e ku can be expressed as follows as the function of the relative orientation the magnetization and the direction of the easy axis k 4 (2.4)

e ku = -K u1 (m • k) 2 + K u2 (m • k)
The constants K u1 and K u2 are the anisotropy constants of Ąrst and second nonvanishing order, respectively, which in SI units are expressed as J m -3 . A preferential magnetization direction can also arise from the effect of the so-called shape anisotropy, which favors magnetization along the long axis of elongated particles. The shape anisotropy, however, is a purely magnetostatic effect that is unrelated to the crystalline structure of the material.

Zeeman energy

The magnetostatic energy of a ferromagnet in an magnetic ĄeldŮthe Zeeman energyŮ is the energy connected with the interaction of an externally applied Ąeld H ext with the magnetization M (r). The Zeeman energy density e zee can be expressed as follows.

e Zee = -µ 0 H Zee • M (2.5)
According to eq. (2.5), the value of Zeeman energy is minimal (most negative) when the magnetization is aligned parallel to the direction of the external Ąeld, while the orientation anti-parallel to the applied Ąeld is energetically penalized.

Demagnetizing energy

The demagnetizing energy is the magnetostatic energy originating from the interaction of the magnetization with its own dipolar Ąeld. The latter is also called the stray Ąeld or the demagnetizing Ąeld. This interaction is often the most difficult term to calculate in micromagnetism. The magnetostatic dipolar Ąeld results from the sum

Theory of micromagnetism

of all magnetic dipole moments that constitute the magnetic structure. Within the discrete or atomistic description, each magnetic moment represents a magnetic point dipole that contributes to the sampleŠs dipolar Ąeld. The individual magnetic moments can be expressed as

µ i (r i ) = g L µ B S(r i ) (2.6)
where g L is the Lande g-factor

Then the dipolar Ąeld can be expressed as the sum of individual dipoles,

H dip. (r) = 1 4π i  µ i (r i ) R 3 - 3 (µ i (r i ) • R) • R R 5  (2.7)
where R = rr i and R = ♣R♣.

The micromagnetic continuum theory does not use such dipole sums. Instead, the continuum expression of the dipolar Ąeld involves integrals over the so-called magnetostatic charge distribution [START_REF] Brown | Magnetostatic Principles in Ferromagnetism[END_REF][START_REF] Kronmüller | Micromagnetism and the Microstructure of Ferromagnetic Solids[END_REF][START_REF] Hubert | Magnetic Domains -The Analysis of Magnetic Microstructures[END_REF]. A suitable starting point for deriving this expression is the constitutive equation [START_REF] Jackson | Classical electrodynamics[END_REF][START_REF] Griffiths | Introduction to Electrodynamics[END_REF], which relates the magnetic Ąeld H, the magnetic induction B, and the magnetization M :

B(r) = µ 0 H(r) + µ 0 M (r) (2.8)
In the absence of electrical charge and displacement currents, j = 0 and ∂D/∂t = 0, MaxwellŠs equations yield that the magnetostatic Ąeld H is irrotational [START_REF] Jackson | Classical electrodynamics[END_REF]:

∇ × H = 0 (2.9)
Hence, H can be expressed as the gradient of a scalar potential U (r)

H(r) = -∇U (r) (2.10)
where U (r) is the magnetostatic scalar potential. Equating for H in equation (2.8) and taking the divergence yields

∇ • B = µ 0 (-∇ 2 U + ∇ • M ) (2.11)
The left-hand side vanishes according to MaxwellŠs equation about the absence of magnetic monopoles ∇ • B = 0. Inserting this in 2.11 we obtain the Poisson equation

Theory of micromagnetism

∇ 2 U = -ρ(r) (2.12)
Where ρ(r) = -∇•M is the magnetostatic volume charge density. From eq. ( 2.12), it becomes clear that the dipolar Ąeld vanishes in an inĄnitely extended sample with homogeneous magnetization. The dipolar Ąeld arises as a result of spatial inhomogeneities of M s , which can be related either to divergences of the magnetic orientation or to a change of the modulus of ♣M s ♣, as it occurs, e.g., across the surface of a magnetic sample. The general solution of the Poisson equation (2.12) is given by

U (r) = 1 4π Ω ρ (r ′ ) ♣r -r ′ ♣ d 3 r ′ + 1 4π S σ (r ′ ) • df ′ ♣r -r ′ ♣ (2.13)
Where Ω is the volume the magnetic body is occupying and S is the surface of this volume. Equation (2.13) shows that the magnetostatic potential U (r) can be decomposed into the sum of two integrals; one representing the contributions of the volume charges and the other those from the surface charges:

ρ = -∇ • M volume charge density σ = M • n surface charge density.
These charge density distributions are the sources of the magnetostatic Ąeld H dip = -∇U . The magnetostatic energy contribution arising from the dipolar Ąeld is similar to the Zeeman term, where the external Ąeld H Zee is replaced by the internal Ąeld H dip , but divided by two:

e dip = - 1 2 µ 0 H dip (r) • M (r) (2.14)
The factor1 2 is typical for self-energy terms. In this case, it reĆects the fact that the magnetic moments of the sample are not only the sources of the dipolar Ąeld, but that they are at the same time inĆuenced by it. In the context of dipole sums, the origin of this factor 1/2 can thus be understood from the necessity to exclude a double summation 1 .

Theory of micromagnetism

The total dipolar energy E dip can be computed by integrating e dip over the sample volume Ω:

E dip = - µ 0 2 Ω M (r) • H dip (r)dV (2.15)
The energy density of the magnetostatic Ąeld is

e dip = µ 0 2 H 2 dip (2.16)
It can be shown [START_REF] Brown | [END_REF][START_REF] Aharoni | International series of monographs on physics[END_REF] that the total energy stored in the dipolar Ąeld is equal to the sampleŠs magnetostatic self-energy:

- µ 0 2 Ω M (r) • H dip (r)dV = (entire space) µ 0 2 H 2 dip dV (2.17)
From equation 2.15 it becomes obvious that the total dipolar energy of a ferromagnetic body is always positive. Considering the negative sign on the left-hand side, this also means that the dipolar Ąeld H dip is oriented mainly antiparallel to the magnetization M throughout the sample. For this reason, the dipolar Ąeld is also called the demagnetizing Ąeld.

Since the dipolar energy is non-negative, it is minimized by eliminating its sources, i.e., by arranging the magnetization Ąeld in such a way to minimize surface charges and volume charges. This is known as BrownŠs pole avoidance principle. It is the fundamental principle which leads to the formation of Ćux-closure structures like magnetic vortices or other divergence-free domain patterns in thin-Ąlm elements [START_REF] Van Den Berg | Self-consistent domain theory in soft-ferromagnetic media. II. Basic domain structures in thin-Ąlm objects[END_REF], and it provides the reason for the effect known as shape anisotropy. The latter describes the tendency of a homogeneously magnetized elongated particle to orient the magnetization along the longest axis, thereby minimizing the amount of magnetic surface charges.

As we further observe in this thesis, the pole-avoidance principle also appears in the tendency to form ice-rule obeying vertices in artiĄcial spin ice structures.

Total energy

From all these energy contributions, the total energy of the ferromagnetic body can be calculated by integrating the sum of the individual energy densities over the sample volume.

E tot =

Ω (e exc + e ani + e zee + e dip ) dV (2.18)

Theory of micromagnetism

This term E tot = E[m(r)] takes into account the main micromagnetic contributions, and it depends on the spatial distribution of the magnetization Ąeld M (r). An equilibrium state of the magnetization M 0 is given when the energy functional E tot is a minimum, i.e., δE(M ) = 0, such that any variation δm of the magnetization

M = M 0 + δm results in an increase in E tot .
Taking into consideration only variations δm consistent with the constraint of constant norm of the magnetization, Brown derived the equilibrium condition of vanishing torque [START_REF] Brown | [END_REF][START_REF] Miltat | Domains and Domain Walls in Soft Magnetic Materials[END_REF],

m × H eff = 0 (2.19)
in combination with a surface condition, which in its simplest form is

n • ∇m = ∂m ∂n = 0 (2.20)
where n is the surface normal vector. In a more general case, BrownŠs static boundary condition (2.20) should also include torques arising from surface anisotropy contributions, which we neglect here [START_REF] Rado | Spin-wave resonance in a ferromagnetic metal[END_REF]. This condition has been further extended for the stationary high-frequency dynamic case by Guslienko and Slavin [START_REF] Guslienko | Boundary conditions for magnetization in magnetic nanoelements[END_REF]. The effective Ąeld H eff is deĄned through the variational derivative of the energy functional with respect to the magnetization:

H eff = - 1 µ 0 M s δE tot δm (2.21)
In an explicit form, the effective Ąeld H eff can be written as

H eff = 2A µ 0 M s ∇ 2 m -∇U dip - 1 µ 0 M s ∂e an ∂m + H ext (2.22)
where the speciĄc expression for the effective Ąeld of the magnetocrystalline anisotropy depends on the Thus the system is in an equilibrium when the magnetization m(r) is aligned with the effective Ąeld, resulting in the torque to vanish. Solving micromagnetic problemsŰ in generalŰinvolves the calculation of the effective Ąeld H eff in order to determine the the distribution of the magnetic vector Ąeld, i.e., the magnetization structure. This is typically done by solving the Landau-Lifshitz-Gilbert equation discussed in the next section. Explicit solutions based on energy minimization have been obtained for a few fundamental examples such as domain-walls [START_REF] Kronmüller | Micromagnetism and the Microstructure of Ferromagnetic Solids[END_REF][START_REF] Riedel | Micromagnetic treatment of Néel walls[END_REF][START_REF] Lilley | Energies and widths of domain boundaries in ferromagnetics[END_REF], nucleation problems [START_REF] Frei | Critical size and nucleation Ąeld of ideal ferromagnetic particles[END_REF][START_REF] Kronmüller | Theory of nucleation Ąelds in inhomogeneous ferromagnets[END_REF],

domain patterns [START_REF] Van Den Berg | Self-consistent domain theory in soft-ferromagnetic media. II. Basic domain structures in thin-Ąlm objects[END_REF][START_REF] Hubert | Stray-Field-Free Magnetization ConĄgurations[END_REF], and the law of approach to ferromagnetic saturation [96,[START_REF] Seeger | Die Einmündung in die ferromagnetische SättigungŮI: Allgemeine theorie des EinĆusses innerer Spannungen auf das Einmündungsgesetz, mit Anwendungen auf Zwischengitteratome und abgeschreckte Metalle[END_REF].

Magnetization dynamics: The Landau-Lifshitz-Gilbert equation

Generally, more than one state exists which satisĄes 2.19 (meaning that the minimum can be a local minimum) and unique solutions exist only for certain simple examples.

The different energy contributions describe competing interactions, and the minimization of the total energy usually represents a compromise between the different tendencies arising from the individual energy terms. For instance, the tendency to form magnetic Ćux-closure structures, favored by the magnetostatic interaction, is in competition with the ferromagnetic exchange interaction, which tries to prevent any inhomogeneity. Since one interaction is of short and the other of long range, the balance between these effects can be described by characteristic lengths, the so-called exchange lengths [START_REF] Kronmüller | Micromagnetism and the Microstructure of Ferromagnetic Solids[END_REF], which will be discussed later.

Magnetization dynamics: The Landau-Lifshitz-Gilbert equation

In micromagnetism, the time-dependent dynamics of the magnetization vector is described by the Landau-Lifshitz-Gilbert (LLG) equation. The derivation of the LLG equation starts from the classical equation describing the rotational motion of a rigid body of angular momentum P under an applied torque τ ,

dP dt = τ (2.23)
The magnetic analogue of this equation is given by,

τ = dL dt = M × H eff (2.24)
The gyromagnetic ratio γ 0 , given by

γ 0 = g L ♣e♣µ 0 2m = 2.212 76 × 10 5 m A -1 s -1 (2.25)
connects the angular momentum L = -M γ 0 to the magnetization M . Here, e and m are the charge and the rest mass of electron, respectively, and g L is the Landé g-factor [START_REF] Landé | Über den anomalen Zeemaneffekt (Teil I)[END_REF].

Substituting for L, we obtain the following equation which describes the undamped, time-dependent precession of M in the case of a torque exerted by According to this equation, M precesses around the effective Ąeld with a frequency ω = -γH eff without any dissipation. The negative sign in the frequency implies that the sense of rotation is anti-clockwise with respect to a positive H eff . In a realistic scenario, this motion is subjected to various dissipative interactions such as orbital couplings, spin-diffusion, or interactions with impurities. To account for such dissipative effects, L.D. Landau and E.M. Lifshitz [START_REF] Landau | On the theory of the dispersion of magnetic permeability in ferromagnetic bodies[END_REF] introduced a phenomenological damping constant α L and added a corresponding damping term to the precession term to obtain the Landau-Lifshitz (LL) equation

H eff dM dt = -γ 0 τ = -γ[M × H eff ] (2.26)

Magnetization dynamics: The Landau-Lifshitz-Gilbert equation

dM dt = γ L [M × H eff ] - α L M s [M × [M × H eff ]] (2.27)
The second term in equation (2.27) represents the relaxation motion of the magnetization towards H eff , so that after a Ąnite time the motion comes to a stop as the magnetization aligns with the effective Ąeld and the torque vanishes. Note that both the relaxation and the precession term describe changes dM /dt of the magnetization that is perpendicular to M , which ensures the constraint of constant modulus ♣M ♣.

In 1955, Gilbert [START_REF] Gilbert | A Lagrangian formulation of the gyromagnetic equation of the magnetization Ąeld[END_REF][START_REF] Gilbert | A phenomenological theory of damping in ferromagnetic materials[END_REF] suggested an alternative form of the Landau-Lifshitz

Basic micromagnetic structures and effects

equation (2.27), motivated by the reasoning that not only the relaxation motion, but also the precessional part of the dynamics should be affected by dissipation. In the case of strong damping, it appears particularly important to consider damping effects also in the precessional part of the dynamics. By introducing a velocity-dependent dissipation term and the Gilbert damping constant α G , he presented an alternative, implicit equation which accounts for the general effect of damping on the magnetization dynamics. This equation allows to accommodate also the case of strong damping

dM dt = γ G [M × H eff ] + α G M s  M × dM dt  (2.28)
The seemingly different equations for the magnetization given by Landau-Lifshitz

dM dt = γ 1 + α 2 G [M × H eff ] + α G γ 1 + α 2 G M s [M × [M × H eff ]] (2.29)
In spite of the formal equivalence of the equations, the Gilbert formulation is often considered to be more realistic, and in the modern literature only the dimensionless value α G , not α L , is used to describe a magnetic materialŠs damping properties. One can therefore drop the subscript G when referring to the the damping constant α. In dynamic processes, one can distinguish low-damping cases, which are characterized by values of α ≪ 1, while values of α ≳ 0.5 describe cases of high damping. Assuming such unrealistically large damping values can be useful in numerical simulations if the goal is to quickly calculate an equilibrium structure.

The general approach to Ąnding a solution to a micromagnetic problem involves the numerical integration of the LLG equation in time, given an initial conĄguration and a number of input parameters describing the material properties, the geometry of the ferromagnetic body, and additional simulation parameters such as external Ąelds.

Details of the numerical methods are discussed in the next chapter.

Basic micromagnetic structures and effects

In the case of an equilibrium structure, as brieĆy mentioned before, the magnetization distribution of a ferromagnetic body develops a state where a compromise is established

Basic micromagnetic structures and effects

between the different competing energy terms so as to minimize the total energy as much as possible. This behavior results in the formation of a few types of fundamental micromagnetic structures. In the following section, a few fundamental structures which will be later used in the thesis are brieĆy presented.

Magnetic domain walls

A magnetic domain wall is the interface region between two oppositely magnetized neighboring domains. The rotation of the magnetization direction occurring across the domain wall ensures a smooth transition of the magnetization within a speciĄc distance known as the domain wall width. Based on the type of transition, several different kinds of domain walls can be distinguished. The Ąrst kind is the Bloch-domain wall which describes the smooth transition between two oppositely magnetized domains so that within the domain wall the magnetization is gradually rotated in a plane separating the two domains. The width b of a Bloch wall is given by

b = πl k (2.30)
Where l k is the exchange length of the anisotropy, an intrinsic length scale of the material. Further below we will discuss another exchange length, l s , which is due to the competition between ferromagnetic exchange and magnetostatics. In an anisotropic ferromagnetic material, the value of l k depends on the competition between the exchange interaction and the magnetocrystaline anisotropy energy. Accordingly, l k can be expressed as [START_REF] Lilley | Energies and widths of domain boundaries in ferromagnetics[END_REF]

l k = A K u (2.31)
The energy density of a Bloch-wall, that is, the energy stored in a unit area of the wall, can be expressed as

ϵ b = 4 √ AK (2.32)
The magnetic structure of a Bloch wall evades the formation of volume charges (ρ = -∇ • m) in expense of the formation of surface charges (σ = n • m). Bloch walls are commonly observed in anisotropic magnetic particles with lateral dimensions considerably exceeding the domain wall width.

Basic micromagnetic structures and effects

In soft-magnetic thin Ąlms, the magnetostatic energy contribution by the surface charge development plays a decisive role. Hence a smooth 180 degree rotation of the magnetization between two oppositely magnetized in-plane domains develops differently than the previously described Bloch wall. In such cases, the tendency to suppress magnetic surface charges results in the formation of Néel, walls [START_REF] Néel | Energie des parois de bloch dans les couches minces[END_REF], in which the magnetization rotates in the Ąlm plane. The central part of the domain is thereby magnetized perpendicular in the Ąlm plane, perpendicular to the domain wall. In the case of Néel walls, the energy density and the wall width are determined by the competition between demagnetizing energy and exchange energy. The domain wall width n of Néel wall is also characterized by an exchange length:

n = πl s (2.33) l s = 2A µ 0 M 2 s (2.34)
and the energy density of the Néel wall is given by In the case of thin elongated nanostructures, such as soft-magnetic thin cylindrical nanowires, a different type of domain wall is observed. In these systems, the magnetic structure is characterized by domains in which the magnetization aligns parallel to the wire axis because of the dominant effect of shape anisotropy. The transition regions from in such wires are called head-to-head (tail-to-tail) domain walls [START_REF] Kläui | Head-to-head domain walls in magnetic nanostructures[END_REF][START_REF] Mcmichael | Head to head domain wall structures in thin magnetic strips[END_REF]. If

ϵ n = 2A M s (2.35)

Basic micromagnetic structures and effects

parts of the nanowire are magnetized in opposite axial directions, a head-to-head or tail-to-tail domain wall structure is formed at the interface. In such elongated structures, different types of head-to-head walls can form. If the thickness or diameter remains below above a critical limit, the head-to-head domain wall is of transverse type, whereas in the case of larger widths or thicknesses, vortextype head-to-head domain walls are formed. In the case of cylindrical nanowires, there is also the case of head-to-head walls in the form of Bloch-point domain walls [START_REF] Da Col | Observation of Bloch-point domain walls in cylindrical magnetic nanowires[END_REF][START_REF] Arrott | Point singularities and magnetization reversal in ideally soft ferromagnetic cylinders[END_REF][START_REF] Andreas | Multiscale and multimodel simulation of Bloch-point dynamics[END_REF][START_REF] Hertel | Magnetic drops in a soft-magnetic cylinder[END_REF], which typically form in the case of wire diameters exceeding about 60 nm.

The exchange length of a material is not only useful to describe the typical extension of micromagnetic structures, but is also an important parameter in the context of micromagnetic numerical simulations. To perform accurate simulations of magnetization structures that can develop within a ferromagnetic system, it is necessary to use a sufficiently small spatial discretization. This is necessary in order to avoid numerical discretization errors that can lead to completely erroneous results like, e.g. the Şdomain wall collapseŤ described by Donahue [START_REF] Donahue | A variational approach to exchange energy calculations in micromagnetics[END_REF]. As a rule of thumb, the maximum distance between two neighboring descretization points (the cell size) should always remain below the smallest of the set of exchange lengths of the modeled material. For the materials used in the simulations of this thesis, the exchange lengths are in the order of 4 nm to 5 nm, and consequently the Ąnite element models which we used are ensured to have maximum cell sizes below these limits.

Basic micromagnetic structures and effects

Magnetic vortices

Magnetic vortices are swirling structures of the magnetization that efficiently close the magnetic Ćux. Accordingly, they usually form as a result of the magnetic systemŠs tendency to minimize the demagnetizing energy by forming divergence-free structures.

In two-dimensional Ąlms and nanodots, the magnetic structure of a vortex is characterized by the in-plane circulation of the magnetization around the core of the vortex. At the center of the vortex, the magnetization points normal to the plane of the vortex.

For topological reasons, magnetic vortices are highly stable structures once they are formed. The size of the vortex core is deĄned by the interplay between the demagnetizing and exchange energy terms and is thus typically depends on the exchange length l s , although the precise value can change with the thickness of the structure. The core radius of a vortex in a Ąlm of thickness h can be approximated as [START_REF] Gliga | Ultrafast vortex core dynamics investigated by Ąnite-element micromagnetic simulations[END_REF],

r core = 0.68l s  h l s  1/3
(2.36)

Shape anisotropy: axial magnetization of elongated structures

Similar to how the magneto-crystalline anisotropy of a ferromagnetic body can result in a preferred direction of magnetization, the geometric shape of the body can also give rise to a preferred orientation of magnetization. This effect in which elongated objects exhibit a tendency to be magnetized along the long direction is called shape anisotropy. Even though the effects are similar, shape anisotropy has a completely different origin from that of magneto-crystalline anisotropy. While the latter is attributable to the spinorbit coupling, the shape anisotropy is caused by the tendency of the body to minimize the demagnetizing energy by reducing the formation of magnetic surface charges (σ = n • m). Consider an elongated ferromagnetic body, as visualized in Ągure 2.5. The preferred orientation of the magnetization is along the long axis of the body as depicted in (a) because this arrangement results in the formation of comparatively less surface charges than the structure shown in (b), where the magnetization is oriented along the short axis. Therefore, the magnetic conĄguration (a) has a lower demagnetization energy than the second case. In soft-magnetic materials, this effect results in the formation of a shape dependent direction of preferred magnetization along the long axis.

Basic micromagnetic structures and effects

Because of this effect, elongated elements such as cylindrical nanowires, nanotubes, or rectangular strips and wires with high aspect ratio are preferably magnetized along the long axis in the absence of a strong external Ąelds. This behavior is important for the studies discussed in this thesis concerning interconnected nanowire networks. The competing interaction of axially magnetized nanowires at the meeting points at the vertices gives rise to emergent effects and artiĄcial spine ice (ASI) behavior.

The shape anisotropy effect also occurs in extended Ąlms, where it favors an in-plane magnetization. This tendency can be in competition with, e.g., a unixaial magnetocrystalline anisotropy, if the crystalline easy axis is perpendicular to the surface normal, thus favoring an out-of-plane magnetic orientation. This type of competition can be used to quantify the relative strength of the magnetocrystalline anisotropy, and it is at the basis of the deĄnition of the dimensionless quality factor Q = 2K u /(µ 0 M s ).

A value of Q ≃ 1 denotes a strongly anisotropic material, while low-anisotropy, i.e., soft-magnetic materials have Q ≪ 1.

ArtiĄcial spin ices

ArtiĄcial spin ices

Water ice exists in about 18 different crystalline structures [START_REF] Petrenko | Physics of ice[END_REF]. In all these structures, neighboring water molecules are linked together by strong hydrogen bonds. That is, each oxygen atom in a water-ice is surrounded by four hydrogen atoms, out of which two are bonded by covalent bonds and the other two realize a hydrogen bond with the neighboring water molecules2.6. This state, in which two of the four hydrogen atoms are closer to and two are farther away from the oxygen atom result in a Ştwoin/two-outŤ type of arrangement, which is known as the Bernal-Fouler rule or simply as the ice-rule [START_REF] Bernal | A theory of water and ionic solution, with particular reference to hydrogen and hydroxyl ions[END_REF]. Each oxygen atom in the crystal can exist in six different conĄgurations obeying the two-in/two-out rule, resulting in a sixfold degeneracy per oxygen atom. The collective degeneracy of the system grows exponentially with increasing number of oxygen atoms. In 1935, [START_REF] Nisoli | The concept of spin ice graphs and a Ąeld theory for their charges[END_REF][START_REF] Pauling | The structure and entropy of ice and of other crystals with some randomness of atomic arrangement[END_REF] Linus Pauling made use of this ice-rule to explain the zero-point entropy of water [START_REF] Giauque | Molecular rotation in ice at 10 K. Free energy of formation and entropy of water[END_REF][START_REF] Giauque | The entropy of water and the third law of thermodynamics. the heat capacity of ice from 15 to 273°K[END_REF]. The ice-rule gained broader signi-Ącance since the appearance of spin ice systems [START_REF] Bramwell | The history of spin ice[END_REF]. Magnetic spin ices are materials consisting of a tetrahedral arrangement of magnetic ions, whose low energy state is a Ştwo-in/two-outŤ state analogous to the water ice. This ground state arrangement in magnetic spin ices gives rise to the inability to satisfy the energy minimization conditions for all the competing interactions from the neighbors, which results in frustration.

ArtiĄcial spin ices are [START_REF] Heyderman | ArtiĄcial ferroic systems novel functionality from structure, interactions and dynamics[END_REF][START_REF] Nisoli | Colloquium: ArtiĄcial spin ice: Designing and imaging magnetic frustration[END_REF][START_REF] Skjžrvø | Advances in artiĄcial spin ice[END_REF][START_REF] Wang | ArtiĄcial spin ice in a geometrically frustrated lattice of nanoscale ferromagnetic islands[END_REF][START_REF] Schiffer | ArtiĄcial spin ice: Paths forward[END_REF] patterned magnetic structures, which attempt to mimic these frustrated interactions. ArtiĄcial spin ices can be regarded as

ArtiĄcial spin ices

metamaterials made of elongated single domain units, arranged in a particular way so that their magnetostatic interaction at the vertex points can lead to frustration. Two well-known examples are the square ASI [START_REF] Mól | Magnetic monopole and string excitations in two-dimensional spin ice[END_REF], in which four single-domain units are arranged in a Ş+Ť shape to form a square grid, and the Kagome lattice [START_REF] Qi | Direct observation of the ice rule in an artiĄcial kagome spin ice[END_REF][START_REF] Rougemaille | ArtiĄcial kagome arrays of nanomagnets: A frozen dipolar spin ice[END_REF] in which three such units are arranged at each vertex to form a hexagonal grid. A geometric description of the square and Kagome lattices is given in Ągure 2.7. Since all the individual units in such an ASI are of single-domain type, they can be assigned an Ising-like dipole moment. We can identify various different vertex conĄgurations based on the total number of moments oriented towards or away from the vertex. The square ASI can have 16 different vertex conĄgurations, out of which the Ąrst six obey the ice rule. The vertex conĄgurations which does not follow the ice rule are known as magnetic defects. Such magnetic defects are also sometimes referred to as emergent monopoles [START_REF] Castelnovo | Magnetic monopoles in spin ice[END_REF][START_REF] Ladak | Direct observation of magnetic monopole defects in an artiĄcial spin-ice system[END_REF][START_REF] Ladak | Direct observation and control of magnetic monopole defects in an artiĄcial spin-ice material[END_REF][START_REF] Mengotti | Real-space observation of emergent magnetic monopoles and associated Dirac strings in artiĄcial kagome spin ice[END_REF][START_REF] Mól | Magnetic monopole and string excitations in two-dimensional spin ice[END_REF][START_REF] Loreto | Emergence and mobility of monopoles in a unidirectional arrangement of magnetic nanoislands[END_REF][START_REF] Rougemaille | Chiral nature of magnetic monopoles in artiĄcial spin ice[END_REF]. ArtiĄcial spin ices are one of the most advanced sub-branches of nanomagnetism, and owing to their re-conĄgurability [START_REF] Iacocca | ReconĄgurable wave band structure of an artiĄcial square ice[END_REF][START_REF] Wang | Switchable geometric frustration in an artiĄcial-spin-iceŰsuperconductor heterosystem[END_REF] they have potential applications in a large number of Ąelds such as magnonic logic computation [START_REF] Arava | Engineering relaxation pathways in building blocks of artiĄcial spin ice for computation[END_REF][START_REF] Gypens | Balanced magnetic logic gates in a kagome spin ice[END_REF][START_REF] Caravelli | Logical gates embedding in artiĄcial spin ice[END_REF][START_REF] Sanz-Hernández | A nano-magnetic Galton board[END_REF][START_REF] Kaffash | Nanomagnonics with artiĄcial spin ice[END_REF], data storage [START_REF] Gilbert | Direct visualization of memory effects in artiĄcial spin ice[END_REF], reservoir computation [START_REF] Hon | Numerical simulation of artiĄcial spin ice for reservoir computing[END_REF][START_REF] Jensen | Reservoir Computing in ArtiĄcial Spin Ice[END_REF], stochastic computation [START_REF] Sanz-Hernández | Tunable stochasticity in an artiĄcial spin network[END_REF], re-conĄgurable magnonic devices [START_REF] Iacocca | ReconĄgurable wave band structure of an artiĄcial square ice[END_REF] and platforms for re-programmable magnonic resonators and crystals [START_REF] Skjžrvø | Advances in artiĄcial spin ice[END_REF]. Until recent years, research on artiĄcial spin ice systems was limited to two dimensional structures [START_REF] Heyderman | ArtiĄcial ferroic systems novel functionality from structure, interactions and dynamics[END_REF][START_REF] Nisoli | Colloquium: ArtiĄcial spin ice: Designing and imaging magnetic frustration[END_REF][START_REF] Wang | ArtiĄcial spin ice in a geometrically frustrated lattice of nanoscale ferromagnetic islands[END_REF][START_REF] Skjžrvø | Advances in artiĄcial spin ice[END_REF][START_REF] Lendinez | Magnetization dynamics in artiĄcial spin ice[END_REF][START_REF] Iacocca | ReconĄgurable wave band structure of an artiĄcial square ice[END_REF]. Accordingly, much attention was focused on systems where patterned nano-elements are arranged in a single plane or in multiple two-dimensional planes [START_REF] Chern | Realizing three-dimensional artiĄcial spin ice by stacking planar nano-arrays[END_REF]. With the recent advancements in three-dimensional fabrication techniques, such as FEBID [START_REF] Fernández-Pacheco | Writing 3D Nanomagnets Using Focused Electron Beams[END_REF][START_REF] Teresa | Review of magnetic nanostructures grown by focused electron beam induced deposition (FEBID)[END_REF][START_REF] Utke | Gas-assisted focused electron beam and ion beam processing and fabrication[END_REF][START_REF] Fernández-Pacheco | Three dimensional nanomagnetism[END_REF][START_REF] Magén | Focused-electron-beam engineering of 3D magnetic nanowires[END_REF] and TPL [START_REF] Maruo | Three-dimensional microfabrication with two-photon-absorbed photopolymerization[END_REF][START_REF] Williams | Two-photon lithography for 3D magnetic nanostructure fabrication[END_REF][START_REF] Askey | Use of two-photon lithography with a negative resist and processing to realise cylindrical magnetic nanowires[END_REF], a new generation of nanostructures can be formed -three dimensional networks of interconnected nanowires such the Buckyball network fabricated by Donnelly et al. [START_REF] Donnelly | Element-SpeciĄc X-Ray Phase Tomography of 3D Structures at the Nanoscale[END_REF] or the three dimensional diamond lattice fabricated by Keller et al. [START_REF] Keller | Direct-write of free-form building blocks for artiĄcial magnetic 3D lattices[END_REF] and Andrew May et al. [START_REF] May | Realisation of a frustrated 3d magnetic nanowire lattice[END_REF]. These structures consist of a three dimensional network of interconnected nanowires and they can be fabricated in a way so that these individual nanowires exist in a single domain-state, magnetized along the wire axes. Thus, they can have the basic properties of artiĄcial spin ices and these structures can be classiĄed as the Ąrst generation of three-dimensional artiĄcial spin ice structures [START_REF] May | Magnetic charge propagation upon a 3D artiĄcial spinice[END_REF]. 3.1 Introduction

Introduction

One of the Ąrst works in computational micromagnetic simulation was published in 1969 by A.E LaBonte [START_REF] Labonte | Two-dimensional Bloch-type domain walls in ferromagnetic Ąlms[END_REF], in which he numerically calculated the domain-wall formation in a ferromagnetic Ąlm. Five decades after this start of computational studies on micromagnetic structures, following an enormous leap in computation technology as well as signiĄcant advances in the Ąeld of numerical techniques and applied mathematics, micromagnetic simulations have evolved into an indispensable and branch of modern day magnetism. Today, computational micromagentism is a powerful and highly reliable tool for studying magnetization structures and their dynamical properties in nanoscale systems. Micromagnetic simulations have the advantage that they can be used to obtain information which is often inaccessible with standard experimental techniques, such as the internal magnetization structure in complicated three-dimensional systems, comparisons of the energy of different magnetization states, or the study of ultra-fast dynamic process with high spatial and temporal resolution.

Today, more than a dozen of free and closed micromagnetic simulation tools exist [START_REF] Leliaert | TomorrowŠs micromagnetic simulations[END_REF]. They can be divided into two major categories based on the numerical technique implemented as Ąnite difference method (FDM) and Ąnite element method (FEM) [START_REF] Zienkiewicz | The Finite Element Method Ů Its Basis and Fundamentals[END_REF].

In Ąnite-difference methods, the geometry is subdivided into equidistant cubic cells.

This regular arrangement of the unit cells allows for the calculation of the demagnetizing Ąelds by fast Fourier transform techniques, which can be computed very efficiently and accurately. This type of numerical technique is well suited for the simulation of extended Ąlms, rectangular strips and other regular geometries, as the equidistant Ąnite-difference grid can accurately model these kinds of shapes. Although these methods are usually fast and accurate, they are not very efficient when complicated geometries such as curved surfaces are to be treated. Due to the effect of magnetic surface charges, an accurate representation of the sample shape is essential in micromagnetic simulations.

In the case of irregular geometries, the Şstaircase approximationŤ needed in the FDM can introduce errors in micromagnetic simulations that persist even at small cell sizes [START_REF] Garcıá-Cervera | Accurate numerical methods for micromagnetics simulations with general geometries[END_REF].

Another difficulty of FDM methods is that they require the simulation of the entire prism of the volume in which a discretized structure is embedded. This is inefficient, e.g., in the case of interconnected wire networks with a lot of empty space, which occupy only a very small fraction of the volume over which they extend. In such cases, for a more accurate description of the curved surface while using a lower number of degrees of freedom (i.e., number of discretization cells) [START_REF] Hun | Generation of realistic particle structures and simulations of internal stress: A numerical/AFM study of LiMn 2 o 4 particles[END_REF].

Introduction

the Ąnite-element method based simulation tools becomes relevant and may in fact be the only way with which certain systems can be reasonably modeled. In FEM, the geometry is subdivided into irregular cells, which in the three-dimensional case are usually tetrahedral elements of varying sizes. In contrast to the regular grid of a FDM, the position and density of the discretization points in FEM are freely adjustable, which results in a much more accurate description of arbitrary geometries. In addition to the methodŠs intrinsic geometric Ćexibility, which allows for the precise modeling of complex geometries, there is no need in the FEM to include the non-magnetic volume between magnetic particles in the simulations. This superiority of FEM makes them the ideal method for the simulation of three-dimensional nanowire networks.

All simulations in this thesis are performed using the Ąnite element micromagnetic simulation tool tetmag [START_REF] Hertel | Large-scale magnetostatic Ąeld calculation in Ąnite element micromagnetics with H2-matrices[END_REF] developed by Riccardo Hertel at the IPCMS Strasbourg.

In this chapter, we brieĆy illustrate the basic working principle behind this software package. The tetmag code requires various input parameters for each simulation to deĄne the details of the problem. The input Ąles deĄne, e.g., the micromagentic material parameters including the saturation magnetization M s , exchange stiffness constant A, the value of crystalline anisotropic constants and, if applicable, external parameters such as external Ąeld, pulse parameters etc. Most fundamentally, the simulation requires a Ąnite-element mesh describing the geometry of the system. Open-source packages like GMSH [START_REF] Geuzaine | Gmsh: A 3-D Ąnite element mesh generator with built-in pre-and post-processing facilities[END_REF] or Netgen [START_REF] Schöberl | NETGEN an advancing front 2D/3D-mesh generator based on abstract rules[END_REF] are available for the generation of FE meshes according to the geometry of choice. These codes implement several advanced algorithms for the generation of unstructured grid meshes from the geometric description, such as the Delaunay and the Advancing-front method.

Finite element formulation

Finite element formulation

A fundamental aspect of the Ąnite element method consists in interpolating an unknown function inside each Ąnite element based on the computed values at the discretization nodes [START_REF] Fish | A First Course in Finite Elements[END_REF][START_REF] Ern | Theory and Practice of Finite Elements[END_REF]. Consider a tetrahedral element e with four nodes i = 1, 2, 3, 4. The value of a function u(r) inside the element can be approximated as,

u(x) ≃ û(x) = 4 i=1 N e i (x)ũ e i (3.1)
where ũe i is the value of the function at the nodes and N e i are the elementŠs shape functions. The shape functions serve as basis functions in the Ąnite-element formulation.

For each node i of an element e, there is a shape function with the property of having a value equal to one at the node, and zero at all other nodes of the element:

N e i (x j ) = δ ij (3.2)
where δ ij is the Kronecker symbol and x j is the position of the node j of element e. Outside the element e, the elementŠs shape functions are equal to zero. The mist common and simple choice are linear shape functions, in which the value varies linearly with the position within each element e. They can be expressed as

N e i (r) = 1 6V e (a i + b i x + c i y + d i z) (3.3)
Here V e is the volume of the element e. These coefficients depends only on the shape of the elements. For a tetrahedral element, the coefficients can be expressed as follows [START_REF] Kikuchi | Finite Element Methods in Mechanics[END_REF], where i = 1, 2, 3 and 4.

Finite element formulation

a i = 1 6Ve x i+1 y i+1 z i+1 x i+2 y i+2 z i+2 x i+3 y i+3 z i+3 b i = -1 6Ve 1 y i+1 z i+1 1 y i+2 z i+2 1 y i+3 z i+3 c i = -1 6Ve x i+1 1 z i+1 x i+2 1 z i+2 x i+3 1 z i+3 d i = -1 6Ve x i+1 y i+1 1 x i+2 y i+2 1 x i+3 y i+3 1
The volume V e of the element e can be calculated as

V e = 1 6
1 x 1 y 1 z 1 1 x 2 y 2 z 2 1 x 3 y 3 z 3 1 x 4 y 4 z 4 (3.4)
These operations can be interpreted as a coordinate transformation from the Cartesian to the local, barycentric system. One of the advantages of the Ąnite element method is the freedom of choice of the location of these discretization points, which allows for the accurate description of any type of geometries. The functions are represented by means of basis shape functions and mathematical operations such as differentiation and integration can be executed on those basis functions.

Differentiation

Given a function u(r) such that,

u(r) = i=1 N e i (r)ũ e i (3.5)
The derivative of u with respect to x, y and z can be written as

Finite element formulation

∂u ∂x = 4 i=1 u i ∂N i (x, y, z) ∂x (3.6) ∂u ∂y = 4 i=1 u i ∂N i (x, y, z) ∂y (3.7) ∂u ∂z = 4 i=1 u i ∂N i (x, y, z) ∂z (3.8) (3.9)
Hence for the element e

∇u e (x, y, z) = 4 i=1 u e i ∇N e i (x, y, z) (3.10)
the derivatives of u(r) are constant within an element as the basis function N i are linear functions in space. From eq. (3.3), the derivatives of the shape function can be expressed as,

∂N i ∂x = b i , ∂N i ∂y = c i , ∂N i ∂z = d i (3.11)
Where a i , b i and c i are matrices given by 3.2. Substituting these in 3.10, we get the following expression for ∇u(r), This can be expressed as

∇ũ e (x, y, z) =        b e 1 c e 1 d
∇ũ e (x, y, z) = B e ũe (x, y, z) (3.12)
The matrix B e depends only on the spatial discretization of the element e and is independent of the function u(r). In the implementation of a Ąnite-element solver, B can be set-up and stored in the preprocessing stage, before starting the simulation, and used whenever such a mathematical operation is required.

Computation of the effective Ąeld

Integration

As in the case of differentiation, the integral of a function can be expressed by means of

Computation of the effective Ąeld

We can use these principles to numerically treat micromagnetic problems with the FEM. The methods described here are detailed in Ref. [START_REF] Hertel | Statische und dynamische Magnetisierung in Nanostrukturen[END_REF]. They have been originally implemented by R. Hertel in the code TetraMag [START_REF] Kakay | Speedup of FEM Micromagnetic Simulations With Graphical Processing Units[END_REF], which preceded the tetmag software used in this thesis.

The effective Ąeld, which is deĄned in eq. (2.22), is calculated by taking the variational derivative of the energy functional with respect to the reduced magnetization,

H eff = 2A µ 0 M s ∇ 2 m - 1 µ 0 M s ∂e an ∂m + H ext -∇U (3.15)
The effective Ąeld can be expressed as

H eff = H ani + H exc + H ext + H dem (3.16)
where the exchange Ąeld H exc is given as

H exc = 2A µ 0 M s ∇ 2 m (3.17)
The anisotropic Ąeld H ani is given as

H ani = - 1 µ 0 M s ∂e an ∂m (3.18)
Each component of the effective Ąeld is computed individually at the N discretization nodes.

Computation of the effective Ąeld

Exchange Ąeld: weak formulation

The exchange Ąeld is given by eq. (3.17). In addition, speciĄc conditions at the surface boundary ∂S need to be considered which, as derived by Brown [START_REF] Brown | Theory of the approach to magnetic saturation[END_REF] and Rado-Weertman [START_REF] Rado | Spin-wave resonance in a ferromagnetic metal[END_REF] are i

∂m ∂n ∂S = 0 (3.19)
n the case of vanishing surface torques (zero surface anisotropy). In the FEM representation the exchange Ąeld is calculated by using the Galerkin method. As a Ąrst step, both sides of eq. (3.17) are multiplied by test functions ψ i (r) and integrated over the problem domain. Furthermore, the test functions are represented in the basis of the element shape functions. This procedure yields, after some manipulation, a set of 3N equations for the exchange Ąeld H exc,i,r . Here the index i = 0, 1, 2 . . . N represents the node number and r = x, y, z are the Cartesian components.

V

ψ i H exc,r dV = V 2A µ 0 M s ψ i ∇ 2 m r dV (3.20)
Integrating by parts gives

V ψ i H exc,r dV = - 2A µ 0 M s V ∇ψ i • ∇m r dV + 2A µ 0 M s ∂S ψ i ∇m r • n r dS (3.21)
The partial integration removes the second derivatives involved in the original equation and yields an equation, called the weak form, which contains only Ąrst derivatives in space. The surface integral in the second part of the equation vanishes according to the condition (3.19) as the normal derivative of the magnetization vanishes at the surface boundary. Thus, the remaining part can be written as the sum of x, y, and z components and the volume integration can be written as the sum over the total number of elements E. For simplicity, we consider only the x component

E e=1 Ve ψ x (x)H exc,x (x)dV = - 2A µ 0 M s E e=1 Ve ∇ψ x (x)∇m x (x)dV (3.22)
where the integration is now carried over the volume V e of the e th element. As showed in the previous section, the test functin ψ, reduced magnetization m(x) and H exc can be expressed in terms of the shape functions.

Computation of the effective Ąeld

ψ e x (x) ≃ 4 i=1 ψ i x N e i (x) m e x (x) ≃ 4 j=1 m j
x N e j (x)

H e exc,x (x) ≃ 4 l=1 H exc,l (x)N e l (x) (3.23)
Then equation 3.22 can be written as

E e=1 4 i=1 4 l=1 ψ i x H x,l Ve N e,i N e,l dV = - 2A µ 0 M s E e=1 4 i=1 4 j=1 ψ i x m x,j
Vc ∇N e,i ∇N e,j dV

(3.24)
The expression can be further simpliĄed by the use of two characteristic integrals.

The element matrix

M (e) i,j = Ve ψ i ψ j dV (3.25)
Is called the mass matrix. The matrix element of the mass matrix are given by the equation

Ve N i N j dV = (1 + δ i,j ) V e 20 (3.26)
We now approximate the M e i,j with a diagonal matrix by means of the Şmass-lumpingŤ approximation

M e i,j = δ i,j 4 j=1 M e i,j (3.27)
The diagonal elements of M e i,j are obtained by summing the elements of the corresponding row. Thus,

M e i,j = V e 20 4 j=1 (1 + δ i,j ) = V e 20 (2 + 1 + 1 + 1) = V e 4 ( 3

Computation of the effective Ąeld

contains the stiffness matrix, K, which is of central importance in FEM formulations.

Substituting for these integrals and re-arranging, the x component of the exchange Ąeld H exc,x can be written as

H x,l = - 2A µ 0 M s 4 k e=1 V e k e=1 x,y,z α c n iα c n jα m x,j V e (3 .30)
In matrix form, this can be expressed as

H i α = A ij • mα j (3.31)
where α = x, y, z for each of the three Cartesian components and A ij is given by

A ij = - 2A µ 0 M s 4 k e=1 V e k e=1
x,y,z α c e iα c e jα V e (3.32)

Anisotropy Ąeld

The anisotropic Ąeld H ani due to a second order uni-axial anisotropy can be expressed as follows,

H ani,i = - -2K u µ 0 M s • k u,i • (m i • k u,i ) (3.33)
Where i = 0, 1, 2...N is the node index and k u.i is the uniaxial anisotropy axis direction at the i th node.

Demagnetizing Ąeld: FEM/BEM method

The dipolar Ąeld or demagnetizing Ąeld, which accounts for the long-range magnetostatic interactions, is usually the most complicated component of the effective Ąeld to compute in terms of both computation time and mathematical complexity. The demagnetizing Ąeld H dem can be expressed as H dem = -∇U (r), where U (r) is the magnetostatic scalar potential at the point r. In principle, the potential U (r) can be calculated from the explicit formula [START_REF] Jackson | Classical electrodynamics[END_REF][START_REF] Brown | Magnetostatic Principles in Ferromagnetism[END_REF]]

U (r) = Ω M (r) • ∇ r ′ G(r, r ′ )dV (3.34)
where Ω is the volume containing the ferromagnetic material and G(r, r ′ ) is the GreenŠs function

Computation of the effective Ąeld

G(r, r ′ ) = 1 r -r ′ (3.35)
The direct integration of equation 3.34 requires O(N 2 ) matrix elements, where N is the number of discretization points. For a very small problem with 10,000 nodes this approach would already require nearly one GB of memory, and the quadratic dependence on N makes it impossible to use such an approach in modern simulations like those of this thesis, which may involve millions of Ąnite elements. The calculation and storage of such large matrices is prohibitively expensive and slow. For a typical time-dependent simulation, the demagnetizing Ąeld has to be calculated several tens of thousand times, and thus the direct integration approach would be much too slow and is never used in simulations.

Instead of a direct integration, U (r) can be determined by solving the Poisson equation. In order to account for the boundary conditions of the magnetostatic potential, the FEM can be combined with the boundary element method in the form of a Ąnite-element/boundary-element (FEM-BEM) scheme as described by Fredkin and Koehler [START_REF] Fredkin | Hybrid method for computing demagnetizing Ąelds[END_REF][START_REF] Koehler | Finite element methods for micromagnetics[END_REF]. Solving the Poisson equation is numerically less expensive than a direct integration. Instead of having to store the O(N 2 ) discretized GreenŠs function, the formalism requires the solution of a linear set of equations described by the O(N ) stiffness matrix introduced in section 3.3.1 is stored.

A practical difficulty of solving the magnetostatic problem in this way is connected with the boundary conditions of U (r), which are deĄned at inĄnity [START_REF] Chen | A review of Ąnite element open boundary techniques for static and quasi-static electromagnetic Ąeld problems[END_REF] according to U (r) → 0 for r → ∞. This open-boundary condition suggests a requirement to calculate the potential also in elements far outside the volume containing the magnetic material (Ω), hoping that a suitable cut-off radius can be deĄned at which the potential can be set to zero. Such an approach would be inaccurate an undesirable, as it would introduce the need to include a large number of uninteresting external elements. This number of unwanted elements would grow rapidly with size in the three-dimensional case.

Such a situation can be circumvented by FEM-BEM schemes like the Fredkin-Koehler method [START_REF] Fredkin | Hybrid method for computing demagnetizing Ąelds[END_REF]. Essentially, the boundary element method maps the boundary conditions in the outer region onto a surface integral, which needs to be coupled to the FEM solution in the inner region. The method is accurate and removes the need to extend the computational region beyond the ferromagnetic volume. The disadvantage

Computation of the effective Ąeld

of this method is that the surface integral involves a densely populated matrix, which in spite of being much smaller than in the case of a direct integration can still limit the applicability of the method in the case of large problems. The size of the BEM matrix scales approximately with the order of O(N 4/3 ) in the case of a spherical particle, or more generally with O(N 2 b ) if N b is the number of nodes at the surface Ű the boundary nodes. In our Ąnite-element simulation tool, the obstacle related with the size of the BEM matrix is removed by the use of H 2 -type hierarchical matrices [START_REF] Hackbusch | Hierarchical matrices: algorithms and analysis[END_REF][START_REF] Hackbusch | On H2-matrices[END_REF][START_REF] Börm | Efficient numerical methods for non-local operator: H2-matrix compression, algorithms and analysis[END_REF]. By using H 2 matrices one can convert the quadratic complexity of the surface integral to a linear one. As demonstrated recently, we can thereby achieve a reduction of the matrix size by about 99 % in problems with over 10 6 surface nodes [START_REF] Hertel | Large-scale magnetostatic Ąeld calculation in Ąnite element micromagnetics with H2-matrices[END_REF]. This signiĄcant reduction of the numerical costs, obtained while preserving a high accuracy, enables us to calculate problems with an extensive size on machines with modest speciĄcations.

The Poisson equation for the magnetic scalar potential U (r) is given by

∆U (r) = -ρ (3.36)
∇ 2 U 1 (r) = -∇ • M , r ∈ Ω (3.38)
with Neumann-type boundary condition at the surface ∂Ω

∂U 1 ∂n ∂Ω = M • n (3.39)
The second part U 2 solves the Laplace equation

∇ 2 U 2 = 0 (3.40)

Computation of the effective Ąeld

.The normal derivative of U 2 at the boundary ∂Ω is continuous: Integrating by parts,

∂U in 2 (r) ∂n - ∂U out 2 (r) ∂n = 0 (3.
V ∇ • ψ • ∇U 1 dV - dS ψ(∇U 1 -M ) • ndS = V ∇ψ • M dV (3.43)
Using the Neumann boundary condition (3.39) for the gradient of U 1 at the boundary ∂Ω one obtains

v ∇ψ • ∇U 1 dV = V ∇ψ • M dV (3.44)
From this, U 1 can be computed by expanding the terms in the basis deĄned by the shape functions. Next, we consider the Laplace equation of the second part U 2 :

∇ 2 U 2 = 0 (3.45)
A unique solution of the Laplace equation is obtained ifhe Dirichlet boundary conditions are provided, i.e., the value of U 2 (r) at r ∈ ∂Ω. This information can be obtained from the following relation

U 2 (r) = 1 4π δΩ U 1 (r ′ ) ∂G(r, r ′ ) ∂n ds ( 3

Computation of the effective Ąeld

U 2 (r) = 1 4π ∂Ω U 1 (r ′ ) ∂G(r, r ′ ) ∂n ds +  Θ(r) 4π -1  (3.47)
Here Θ(r) is the solid angle subtended at the surface point r. The integration according to eq. 3.47 is implemented numerically by a collocation approach, as described in Ref. [START_REF] Hertel | Large-scale magnetostatic Ąeld calculation in Ąnite element micromagnetics with H2-matrices[END_REF].

With this collocation scheme, the discretized representation of the integral (3.47) takes the form

U 2 = P • U 1 (3. 48
)
where P is a densely populated matrix of size N b ×N b (N b is the number of nodes at the surface). The O(N 2 b ) scaling of this dense matrix is the aforementioned bottleneck of the Fredkin & Koehler approach. For large problems, the memory requirements for the storage of this matrix can become prohibitively high. To circumvent this, a hierarchical matrix compression method is employed.

H 2 matrix compression

Sub-matrices of rank t × s can be approximated by low rank matrices. In factorized form,

K t×s ≈ AB T , A ∈ R t×k , B ∈ R s×k (3.49)
There exist several different types of algorithms for matrix compression. Analytical methods such as by Taylor expansion, interpolation etc. are highly reliable but they result in matrices with ranks higher than necessary [START_REF] Börm | Low-rank approximation of integral operators by interpolation[END_REF][START_REF] Börm | Approximation of integral operators by Green quadrature and nested cross approximation[END_REF]. On the other hand algebraic methods offer high efficiency [START_REF] Bebendorf | Approximation of boundary element matrices[END_REF] but may lead to unreliable results. Hybrid matrix compression [START_REF] Börm | Hybrid cross approximation of integral operators[END_REF] combines an initial analytic approximation and then an algebraic compression in order to get the beneĄts of both techniques. Our micromagnetic code uses H 2 -matrix compression [START_REF] Hertel | Large-scale magnetostatic Ąeld calculation in Ąnite element micromagnetics with H2-matrices[END_REF] provided by the open-source matrix compression algorithm H2Lib [START_REF] Börm | H2lib, a library for hierarchical matrices[END_REF]. Using this method, a remarkable reduction of the memory requirements by up to about 99 % can be achieved. This reduction moreover increases the computation time of the matrix-vector product. It offers a nearly linear O(N b ) scaling of memory with the number of surface nodes N b . We can thereby rapidly calculate magnetostatic Ąelds even for large problems that, without such a compression, would theoretically require RAM sizes in the TB range. Using the H 2 compression scheme, the size of the matrices required to treat such problems can be reduced to a few GB.

Time integration of the LLG equation

Time integration of the LLG equation

In

∂m i,l ∂t ′ = - 1 1 + α 2 m i,l × h i,l - α 1 + α 2 m i,l × (m i,l × h i,l ) (3.50)
where the subscripts i = 1, 2, 3...N are the node index numbers, l = x, y, z are the three Cartesian components, m = M /M s is the reduced magnetization, h = µ 0 H eff is the reduced effective Ąeld, and t ′ = ♣γ♣ µ 0 t is the reduced time. Once this system of ordinary differential equations (ODE) is set up, numerical ODE solvers can be used to integrate for m(r, t) in time. Various ODE solvers, which we will discuss later, are available as open-source in the form of libraries.

The simplest approach for Ąnding the magnetization at time t + δt would be ane explicit time integration of the LLG equation based on the value of m and h eff at the time t. In this approach, which does not account for the change in the effective Ąeld caused by the change in magnetization within the time step δt, the calculation can become unstable. The time integration therefore employs schemes in which the effective Ąeld is continuously updated with any change in m(r, t) at each integration step. This is especially important in the case of the exchange Ąeld, where even a small variation in the magnetization distribution can manifest as a considerable change in the effective Ąeld.

To be solved with standard library solvers, equation (3.50) can be expressed as an initial value problem (IVP) in the following form,

dy i dt = f i (y 1 , y 2 , y 3 ...y 3N , t), i = 1, 2, 3 . . . L, y i (t = 0) = W (3.51)
Where y i are the three Cartesian components of the magnetization m(r, t) and f i is the right-hand side of equation 3.50 which should be supplied as a routine to the solver. The solver takes the initial W value, that is the discretized magnetization state at t = 0, as an input. We should also provide the value of the time step δt,

Time integration of the LLG equation

which is the time-step after which the solver returns the computed magnetization to the user. In the case of adaptive time steps, the solver may internally sub-divide this user-deĄned time step δt further into Ąner intervals. The value selected by the user for the time step is therefore in fact the maximum time step that the routine performing the integration is allowed to take before an update of the effective Ąelds is enforced.

To save computational resources, tetmag provides the option to ŞfreezeŤ (i.e., not to update) the demagnetizing Ąeld within this time range δt. With time steps in the range of 100 fs this approximation is usually very accurate. In any case, the remaining effective Ąeld terms, in particular the exchange Ąeld, are updated continuously. Once the solver returns the value of m(r) at the time t + δt time, the demagnetizing Ąeld is calculated again and the effective Ąeld is updated. This process is repeated until convergence is obtained, or until a user-deĄned time range is simulated. During the simulation, it is important to chose a suitable value for the time-step δt. A too large time-step can lead to instabilities, while a time step too small unnecessarily increases the computation time required for the simulation.

The time integration of the LLG is a critical part of the code, and in spite of the internally performed adaptive adjustment of time steps, the task cannot be performed in a fully automated manner. The user must provide the maximum time step δt in order to deĄne the range of admissible time steps ∆t that are used for the LLG integration.

The optimum value for this input parameter can be determined based on the value of damping term α used for the simulation. Typically, if we are interested in Ąnding the minimum-energy relaxed magnetic state for a particular set of conditions and details of the precessional magnetization dynamics are unimportant, we may use a large value of alpha (0.5-0.6) and accordingly a large time step (δt = 1 ps). On the other hand, if we are interested in the time-dependent dynamics of the system with a realistic low value for the damping term (α = 0.01), an accordingly smaller time step (δt = 0.1 ps) must be chosen.

Our micromagnetic simulation software tetmag has an option for choosing between two different solvers to treat this initial-value problem with 3N coupled degrees of freedom. The Ąrst one is a Dormand-Prince 5 th order Runge-Kutta method based solver provided by the ODEINT package [START_REF] Ahnert | Odeint Ű Solving Ordinary Differential Equations in C++[END_REF] included in the C++ Boost libraries [START_REF] Schling | The Boost C++ Libraries[END_REF]. The second one is CVODE, an Adams-Moulton predictor-corrector method based solver provided by the Sundials library [START_REF] Hindmarsh | SUNDIALS: Suite of nonlinear and differential/algebraic equation solvers[END_REF]. The Ąrst solver is mostly used for fast

Simulation method: working of tetmag

All the simulations discussed in this thesis are are carried out with our Ąnite element simulation software tetmag, which is developed and maintained by Riccardo Hertel at the IPCMS Strasbourg. Although the development of the micromagnetic Ąnite-element code was not apart of this thesis, in this section we brieĆy discuss about the working and the capabilities of tetmag. The main part of the tetmag code is written in C++ [START_REF] Stroustrup | The C++ Programming Language[END_REF] in an object-oriented form. This modular structure of the code allows for a Ćexible development, making it possible to insert, improve or modify speciĄc classes of the code according to new simulations demands without having to re-write or to modify existing parts of the code. This approach makes the code robust and easily maintainable. For several central and performance-sensitive tasks, tetmag makes use of high-level libraries like Eigen [START_REF] Guennebaud | [END_REF] or Boost [START_REF] Schling | The Boost C++ Libraries[END_REF]. The code is thread-parallelized with OpenMP [START_REF] Chapman | Using OpenMP: Portable Shared Memory Parallel Programming (ScientiĄc and Engineering Computation[END_REF],

which makes tetmag suitable for large-scale computations on high-performance clusters.

A good part of the results discussed in this thesis have been carried out on the highperformance computer center of the University of Strasbourg [176]. Moreover, tetmag can also achieve a signiĄcant speedup through GPU acceleration. This is achieved by implementing certain classes in the CUDA language [START_REF] Nickolls | Scalable Parallel Programming with CUDA[END_REF] and making use of the Thrust framework [START_REF] Bell | Thrust: A Productivity-Oriented Library for CUDA[END_REF] for GPU acceleration. The choice of whether to use GPU acceleration or thread-based CPU parallelization can be made by the user through a simple interface. For large-scale problems, where the number of elements exceeds about 5×10 6 , the simulation rate can be increased considerably by enabling GPU acceleration.

The tetmag software takes as input the following parameters for each simulation

• an ASCII Ąle listing the material parameters

• a text Ąle containing details of the simulation conĄguration

• optional: a Ąle containing the initial magnetization structure of the problem

Post processing: Extraction of frequency modes with a windowed inverse Fourier transform

The Ąnite element mesh can be generated by means of free and open-source codes like Gmsh [START_REF] Geuzaine | Gmsh: A 3-D Ąnite element mesh generator with built-in pre-and post-processing facilities[END_REF], Netgen [START_REF] Schöberl | NETGEN an advancing front 2D/3D-mesh generator based on abstract rules[END_REF], FreeCAD [179] etc. In the text Ąle describing the simulation conĄgurations, the user can choose specify the input parameters such as the value of Gilbert damping (typically with values 0.01 ≤ α ≤ 1), the total simulation time, various types of external Ąelds, the type of solver to use, whether to use CPU or GPU for the calculation, and the initial magnetization state. The initial magnetization state is the conĄguration from which the time evolution of the simulation state starts. This can be a trivial state like a homogeneously magnetized state or artiĄcial conĄgurations like a randomized initial state, where the magnetization at each discretization point is oriented in a random direction in space. The initial state can also be user-prepared, for example one can use the relaxed state obtained from a previous simulation. Once these input ingredients are prepared, a simulation can be initiated. The tetmag software will Ąnd the value of the magnetization vector m(r, t) at each discretization point at each time-step by integrating the initial value problem in time, as discussed in the previous section. The output is a time series of magnetization states for each time step. The standard output for tetmag is given in Ąles of ŞvtuŤ format, which contain information about the Ąnite element mesh and on the magnetization vector Ąeld at each point.

These Ąles can be viewed and analyzed with the powerful ParaView software [START_REF] Henderson | ParaView guide, a parallel visualization application[END_REF],

which is free and open-source. Details about the post processing of the simulated data are brieĆy discussed in the next section.

Post processing: Extraction of frequency modes with a windowed inverse Fourier transform

Oscillatory magnetization processes, at which the magnetization vibrates in the GHz range, play an important role in magnonics and in the analysis of the intrinsic modes of a nanomagnet. Such magnetization oscillations can be excited by applying a short perturbation to a relaxed magnetic structure. The signals obtained from the ensuing simulated dynamics may contain contributions from two or more distinct and superposed frequencies. To disentangle such oscillations, and in general, to get a deeper understanding about the dynamical properties of the system, it is necessary to Ąlter out the individual modes and to analyze their spatial proĄle separately.

The Ąrst part of this thesis was involved with the development of post-processing

Post processing: Extraction of frequency modes with a windowed inverse Fourier transform

tools to identify and isolate individual high-frequency modes from the simulated results of an oscillatory magnetization dynamics through Fourier analysis. In this subsection, the theory and implementation of this technique is briefed. The Fourier transform g(ω)

of a function f (t) is deĄned as

g(ω) = F (f (t)) = 1 √ 2π ∞ -∞ f (t)e iωt dt (3.52)
And the inverse function, inverse Fourier transform is deĄned as Where x(t n ) is the input signal, X(ω k ) is the complex-valued spectrum and N is the number of samples in the input signal.

f (t) = F -1 (g(Ω)) = 1 √ 2π ∞ -∞ g(ω)e -iωt dω (3.
The original signal, or any particular part of the signal, can be re-created from X(ω k ) by the inverse function, i.e., the inverse discrete Fourier transform (IDFT), which is deĄned as

x (t n ) = 1 N N -1 k=0 X (ω k ) e -jω k tn , n = 0, 1, 2, . . . , N -1 (3.55)
To investigate the magnonic modes of any system, the relaxed magnetization conĄgurations are excited with various techniques and the resulting magnetic ring-down; that is the oscillatory, small angle precessional relaxation from the excited to equilibrium state is recorded as a function of time t. The recorded time-dependent magnetization usually contains a superposition of several frequency modes, and it can be difficult to understand the dynamic processes associated with each frequency through a simple

Post processing: Extraction of frequency modes with a windowed inverse Fourier transform

visual inspection. Hence, in the initial stages of the studies of this thesis, a Fourier analysis tool was developed which could identify and Ąlter out the individual modes from these superposed oscillations. The tool was written in Python [START_REF] Van Rossum | Python reference manual[END_REF] in an objectoriented way. For the numerical calculation of the Fourier transform and the inverse Forier transform, the FFT library of the Numpy package [START_REF] Harris | Array programming with NumPy[END_REF] was employed. The tool takes as input the VTU Ąles produced by the tetmag simulation software. The time evolution signal of the magnetization at each node point is obtained by reading the corresponding data from all the VTU Ąles. The code will then extract the frequency modes contained in these signals as explained above. The local power spectra at each node i can be deĄned as

p i (ω) = ♣g i (ω)♣ = ♣F (m i (t))♣, i = 1, 2, 3 . . . N, l = x, y, z, (3.56)
where F (f ) is the discretized Fourier transform. It contains the information about the different frequency components and their relative intensity at each node i = 1, 2, 3..N .

To obtain a global view of the entire structure, the power spectra at all the discretization nodes are summed, yielding an average spectrum

P (ω) = N i=1 p i (ω) (3.57)
It is to be noted that the average of the discretized power-spectra can be completely different from the power-spectrum of the averaged magnetization. The spatially averaged power spectrum P (ω) contains information about the frequency of various modes and its average intensity in the whole structure. This can be displayed as a frequency vs. intensity plot, and the various frequency components and their relative intensity can be visually identiĄed. The dynamical proĄle of a desired individual frequency can then be Ąltered by means of discretized windowed-inverse Fourier transform at each node. This is achieved by suppressing the remaining frequency contributions in the Fourier transform g i (ω) and then carrying out an inverse Fourier transform (iDFT) back to the time domain. This results in the extraction of the oscillation of m(r, t) at a particular frequency at each node. This process is repeated for N nodes to get the overall dynamics at the desired frequency.

The Forier analysis tool was modiĄed and improved continuously throughout the thesis period such as to carry out various problem-speciĄc and advanced analyses.

For example, to look into the modes developing at artiĄcial nanowire networks, the

Post processing: Extraction of frequency modes with a windowed inverse Fourier transform

tool was modiĄed to include an option which could extract some speciĄc geometrical locations of a lattice. This made it possible to selectively analyze the behavior of nanowires, spheres and other units isolated from the rest of the lattice. The Ćexibility of the python language made these modiĄcations effortless and efficient. The working principle of the Fourier extraction tool is illustrated in Ągure 3.3.

Post processing: Extraction of frequency modes with a windowed inverse Fourier transform

Finite element modeling and simulation

Finite element modeling and simulation

The Ąnite-element mesh of the nanodot was prepared using the open-source mesh generation software GMSH [START_REF] Geuzaine | Gmsh: A 3-D Ąnite element mesh generator with built-in pre-and post-processing facilities[END_REF]. The radius of the disc was varied from 75 to 100 nm and the thickness was varied from 20 to 50 nm. The maximum mesh-size (lc) was maintained below 4 nm to make sure that the cell size was always below the exchange length of the material. was used for the exchange stiffness constant A. The value of uniaxial anisotropy was varied from 0 to 18 kJ m -3 .

The relaxed magnetization structures at various Ąelds were calculated by simulating the magnetization structure until the value of maximum torque falls below a pre-deĄned value. These relaxed states were then excited with various methods:

1. With a Gaussian Ąeld pulse [START_REF] Yan | Calculations of three-dimensional magnetic normal modes in mesoscopic permalloy prisms with vortex structure[END_REF] or 2. By applying a small enough Ąeld at a small angle to slightly lift the magnetization from the equilibrium and then letting it relax to the original Ąeld [START_REF] Baker | Proposal of a micromagnetic standard problem for ferromagnetic resonance simulations[END_REF].

Magnetization structure

The magnetic ring-down of these excited states are recorded and the individual frequency modes are isolated an analyzed using the techniques described in the previous chapter. The results of the FMR were simulated by varying the in-plane Ąeld from 100 mT to 900 mT.

Magnetization structure

A simulated hysteresis loop of the nanodisc with thickness 50 nm and radius 100 nm is shown in Fig. Ąrst mode slightly decreases and that the frequency of the second mode increases lin-

Magnetization dynamics

early with an increase in M s . Based on the frequency values and the relative separation between the two modes, we could assume that the most probable value of the saturation magnetization is about 1100 mT.

Based on these results, the value of saturation magnetization was Ąxed and various other parameters such as anisotropy constant are varied. It was observed that a thickness of 20 nm, radius 100 nm, and a uniaxial anisotropy of 18 kJ m -3 resulted in the closest agreement to the experimental results. Using these parameters, the Ąeld dependence of the frequency response shwon in Fig. 4.8 was simulated.

Conclusion

Conclusion

To summarize, two dimensional arrays of Cobalt nanodots were fabricated by the collaborators and the Ąeld dependent frequency response was measured. Two prominent modes were identiĄed in this and to investigate the dynamical proĄle of these modes Ąnite element simulations were carried out. Owing to limitations in computational resources simulations were carried out on a single nanodot, instead of the whole array.

We could identify the micromagnetic structure formed in these nanodots at various Ąelds and also could identify the magnetic proĄle of the two modes. The Ąeld dependent frequency response was re-created through simulation which had qualitative agreement with the simulation results. Also, due to some constraints in experimental techniques, there were minor uncertainties in the material parameters. By comparing the frequency of the modes obtained in the simulation with that of the experimental values the material parameters could be indirectly obtained. As a future extension to this study, the whole array of nanodots could be simulated instead of a single one.

Chapter 5

Three-dimensional Sierpinski fractal structures

Introduction

Introduction

Geometric fractals are self-similar structures which are scale invariant [START_REF] Mandelbrot | Fractals: Form, Chance and Dimension[END_REF]. From the crystalline structure of snowĆakes to the distribution of veins on plant leaves, to the surface structure of mountains Ű fractals are abundant in nature. Usually, fractal structures are naturally formed as a result of a growth processes [START_REF] Peitgen | Chaos and Fractals: New Frontiers of Science[END_REF]. In the Ąeld of magnetism, it has been observed that, under speciĄc conditions, magnetic domains and domain walls can form fractal structures [START_REF] Sayko | Fractal domain structures in thin amorphous Ąlms[END_REF][START_REF] Han | Fractal study of magnetic domain patterns[END_REF][START_REF] Kreyssig | Probing fractal magnetic domains on multiple length scales in Nd 2 Fe 14 B[END_REF][START_REF] Manke | Three-dimensional imaging of magnetic domains[END_REF]. Numerous studies have been carried out in the past to investigate the effects of the formation of fractal domain walls on the magnetization reversal [START_REF] Kim | Correlation between fractal dimension and reversal behavior of magnetic domain in Co/Pd nanomultilayers[END_REF][START_REF] Attané | Invasion percolation universality class and fractal geometry of magnetic domains[END_REF][START_REF] Lee | Fractal dimension of magnetic domain walls in CoFe/Pt multilayers[END_REF]. Additionally, the magnetic properties of 2D ferromagnetic nanostructures with self-similar fractal geometries have gained some attention [START_REF] Swoboda | Control of spin-wave excitations in deterministic fractals[END_REF][START_REF] Dai | Controlled magnetization reversal and magnetic spectra of artiĄcial Sierpinski-fractal structure[END_REF][START_REF] Monceau | Spin waves in deterministic fractals[END_REF][START_REF] Monceau | Effects of deterministic and random discrete scale invariance on spin wave spectra[END_REF]. The interaction of coupled, self-similar oscillators in such structures can potentially give rise to interesting collective phenomena, such as a wide band absorption in fractal antennae [START_REF] Puente-Baliarda | On the behavior of the Sierpinski multiband fractal antenna[END_REF]. In this section, we discuss magnetization structures formed in a three-dimensional Sierpinski gasket structure, their magnetization reversal, and their oscillatory high-frequency dynamics.

Methodology

Methodology

A 3D Sierpinski structure is made of four tetrahedrons recursively arranged on the four corners of another tetrahedron. This can be considered as a three-dimensional variation of the more familiar two-dimensional Sierpinski triangles shown in Fig. 5.1(a). Our stage one fractal structure was a large tetrahedron of side length 512 nm which was, in turn, made of 4 smaller tetrahedrons of side length 256 nm. The successive stages of 2,3,4 and 5 are made by recursively dividing the individual tetrahedrons into further self-similar structures. Thus, the n th iteration of the fractal had 4 n tetrahedrons of side length 512 2 n . Our Ąnal iteration, that is stage 5 of the fractal, contains 1024 tetrahedrons of side length 16 nm. The Ąnite-element models of each iteration of the fractal are displayed in Figs. [START_REF] Binasch | Enhanced magnetoresistance in layered magnetic structures with antiferromagnetic interlayer exchange[END_REF]

.1(b)-(f).

For the numerical simulation we, used material parameters corresponding to those of Permalloy, with a saturation magnetization of M s = 8 × 10 5 A m -1 and an exchange stiffness of A = 1.3 × 10 -11 J m -1 and zero crystalline anisotropy. These material parameters results in an exchange length of 5.6 nm and, correspondingly, all Ąniteelement meshes are made with a mesh-size remaining below this value. The maximum cell size was set to 4.0 nm. The stage one structure consisted of nearly 100,000 nodes and nearly 500,000 Ąnite element cells (which are also tetrahedrons), and the stage Ąve structure had over 36,000 nodes and more than 70,000 Ąnite element cells. Since the total side length of the whole structure was kept constant at 512 nm and the because of the particular way in which the tetrahedrons are distributed, the total surface area of the structure remains constant at √ 3 • 512 2 for all the stages. The total volume decreases logarithmically with each stage, resulting in a corresponding increase of the surface-to-volume ratio, see Fig. 5.2.

Stable magnetic states developing in these structures are obtained by Ąrst saturating the structure along the z direction under a sufficiently strong magnetic Ąeld, which is then gradually relaxed to zero. The hysteretic properties of these structures were obtained by applying an external Ąeld of 500 mT along the z direction and then reducing the Ąeld in appropriate small steps. The magnonic response of the various equilibrium states are obtained by exciting the relaxed states by various techniques and recording the magnonic ring-down. The frequency components are then investigated by means of an inverse windowed Fourier transform. From the third stage onwards, the size of the individual tetrahedrons are too small for the formation of magnetic vortices in a single tetrahedron and they all exist in a Λ like structure and neighboring tetrahedrons collectively forming disconnected vortices observed in the second stage. In this case, the local cluster of tetrahedrons with a size corresponding to the side length of the second stage acts as the container of these vortex units. In the fourth and Ąfth stages, the local cluster is further sub-divided into smaller units which individually exists in almost a single domain state and they collectively form the disconnected vortex structure.

Magnetization structure at zero Ąeld

Hysteretic properties

Hysteretic properties

To study the quasistatic evolution of the magnetic fractal structures in a magnetic Ąeld of varying strength we simulated the hysteresis loops of these systems. An external Ąeld of 500 mT is applied along the z axis, thereby saturating the structure along the z direction. The Ąeld is then reduced in small steps till -500 mT and the volume-averaged magnetization component of the magnetization vector along the applied Ąeld direction is recorded. The compiled results of the hysteretic simulations of all the stages are summarized in Fig. 5.8. One can observe that a very diverse shape of hysteresis loops can be obtained depending on changes in the stage of the fractal. For the Ąrst stage, the structure exists in a saturated state at a Ąeld of 500 mT, as the external Ąeld is lowered we observe a gradual, smooth decrease of the reduced magnetization ⟨M ⟩/M s This deviation is caused by the nucleation of the disconnected vortex structures at the respective Ąelds. The Ąeld at which these vortices are nucleated can be called as the

Hysteretic properties

Magnonic excitations

Magnonic excitations

To investigate the high-frequency properties of these fractal structures, we simulated the small-angle precession modes of these structures at various Ąelds by exciting the relaxed magnetization states by appropriate methods and then recording the magnetic ring-down. A comparison of the frequency response stages one to Ąve is given in Ągure 5.9. In the Ąrst stage we can identify two major peaks: one low frequency mode at 700 MHz and a higher mode at 2.4 GHz. These modes are caused by the synchronous gyration of the vortex cores. A time evolution of the variation in the M z component at a frequency of 700 MHz which is extracted using the inverse windowed Fourier transform is given in Ągure 5.10.

In the second stage, we can identify two regions of oscillation. Several low frequency modes from 1.1 GHz to 1.9 GHz and a comparatively weaker modes around 4.7 GHz.

Similar to the Ąrst case, the lower modes are caused by the activity of the vortex structures and the higher frequency modes are caused by the activity of the three To investigate the Ąeld dependence of frequency response, an external Ąeld was applied along one of the corners of the tetrahedron and the magnonic response investigated. The procedure is repeated for various Ąeld strengths from 100 mT to 900 mT and the results are compared, see Fig. 5.11. In the presence of a sufficiently strong external Ąeld, the tetrahedrons are magnetized along the Ąeld direction and they no longer are in a vortex like state we discussed before. This shift in magnetic structure is reĆected in the frequency response.

Magnonic excitations

Chapter summary

Chapter summary

A three-dimensional Sierpinski fractal structure is made of four tetrahedrons arranged at the four corners of a larger tetrahedron, higher stages of the fractal are generated by dividing each tetrahedron into four sub-tetrahedrons. We simulated Ąve stages (1-5) of fractal with a side length of 512 nm. The individual tetrahedron units in the Ąrst two stages of the fractal were large enough so that their equilibrium magnetization structure at zero Ąeld consisted of the formation of a three -dimensional vortex structures. Each higher iteration results in the side length of the tetrahedrons getting halved. From third stage onward, the individual tetrahedrons are too small to individually contain a magnetic vortex and all the units exist in a single domain state. In third stage onward, Ćux closure is achieved by the formation of a vortex-like structure where the magnetization of the neighboring tetrahedral units circles to form a closed structure. The

Chapter summary

Ąeld response of these structures were studied by simulating the hysteresis loops. The stage of the fractal had a considerable impact on the magnetization reversal and the frequency response of these structures. The magnonic response of the relaxed states were studied by exciting these states by a Gaussian pulse and analyzing the resulting magnetic ring-down for a time period of 10 ns or more. The self-similar geometry and the existence of a wide range of length scales in these fractals had a considerable effect on their magnonic signature. The Ąrst stage of the fractal exhibited a typical frequency spectrum that included a pronounced sub-GHz peak corresponding to the gyration of the vortex core. In addition, it contained a few weaker high-frequency modes corresponding to the oscillation of the vertices of the tetrahedrons. From the third stage onward, we could see the emergence of a wide band of intertwined peaks corresponding to the activity of magnetization structures of varying length scales replacing the individual, well deĄned peaks found at lower stages. On the fourth stage, this wide-band nature of the response was retained and the median frequency of the band was shifted towards a lower frequency region. In the Ąfth stage, due to the small size of individual units, we can see a departure from the fractal behavior seen in third and fourth stages as the width of the band is considerably decreased.

In short, we could see that the fractal nature of these structures had a considerable impact on their magnetization structures, and most importantly their wide-band magnonic response. Even though experimental techniques at this stage has not evolved to a point where these kinds of three dimensional structures can be fabricated, we believe the results of this short study will be helpful for future investigations into these types of geometries.

Geometry and Ąnite element modelling

Introduction

As already discussed in the introduction, three-dimensional magnetism has recently emerged as a major topic of interest in magnetism [START_REF] Fernández-Pacheco | Three dimensional nanomagnetism[END_REF][START_REF] Fischer | Launching a new dimension with 3D magnetic nanostructures[END_REF], producing novel phenomena which were absent in two-dimension. This includes curvature-induced effects [START_REF] Hertel | Curvature-induced magnetochirality[END_REF][START_REF] Volkov | Experimental observation of exchange-driven chiral effects in curvilinear magnetism[END_REF] and three-dimensional magnetic structures such as skyrmion tubes [START_REF] Birch | Real-space imaging of conĄned magnetic skyrmion tubes[END_REF], Bloch points [START_REF] Da Col | Observation of Bloch-point domain walls in cylindrical magnetic nanowires[END_REF], or HopĄons [START_REF] Kent | Creation and observation of HopĄons in magnetic multilayer systems[END_REF]. Recent advancements in three-dimensional patterning techniques such as focused-ion-beam-induced deposition (FEBID) [START_REF] Utke | Gas-assisted focused electron beam and ion beam processing and fabrication[END_REF][START_REF] Fernández-Pacheco | Writing 3D Nanomagnets Using Focused Electron Beams[END_REF][START_REF] Teresa | Review of magnetic nanostructures grown by focused electron beam induced deposition (FEBID)[END_REF][START_REF] Keller | Direct-write of free-form building blocks for artiĄcial magnetic 3D lattices[END_REF] or twophoton lithography (TPL) [START_REF] Maruo | Three-dimensional microfabrication with two-photon-absorbed photopolymerization[END_REF][START_REF] Askey | Use of two-photon lithography with a negative resist and processing to realise cylindrical magnetic nanowires[END_REF], in combination with various other deposition methods, has made it possible to fabricate complex three-dimensional nanoarchitectures.

An interesting category of three-dimensional nanostructures are artiĄcial ferromagnetic Buckyball architectures [START_REF] Donnelly | Element-SpeciĄc X-Ray Phase Tomography of 3D Structures at the Nanoscale[END_REF][START_REF] Gliga | Architectural structures open new dimensions in magnetism: Magnetic buckyballs[END_REF]. Their peculiar geometric arrangement leads to competing magnetic interaction at the vertices, resulting in frustration as already known in the case of artiĄcial spin ice structures [START_REF] Wang | ArtiĄcial spin ice in a geometrically frustrated lattice of nanoscale ferromagnetic islands[END_REF][START_REF] Heyderman | ArtiĄcial ferroic systems novel functionality from structure, interactions and dynamics[END_REF][START_REF] Nisoli | Colloquium: ArtiĄcial spin ice: Designing and imaging magnetic frustration[END_REF][START_REF] Skjžrvø | Advances in artiĄcial spin ice[END_REF]. The artiĄcial buckyball geometries can be considered as a three-dimensional and interconnected variation of the familiar Kagome lattice [START_REF] Qi | Direct observation of the ice rule in an artiĄcial kagome spin ice[END_REF], which has vertex arrangements quite similar to those of the buckyballs in the sense that in both cases the vertices represent the intersection of three wires meeting at equal angles. In contrast to the buckyball, the hexagonal lattice has a hexagonal arrangement whereas the Buckyball contains both hexagons and pentagons. Because of this similarity with the hexagonal lattice, the buckyball geometry can be considered as a model system which represents a transition from twodimensional artiĄcial spin ices (2D-ASI), which were extensively studied to a third dimension. Three-dimensional artiĄcial spin ice structures (3D-ASI) hold the potential for many applications such as data-storage, neuromorphic computing [START_REF] Grollier | Neuromorphic spintronics[END_REF][START_REF] Marković | Physics for neuromorphic computing[END_REF], spintronic devices etc. Therefore, the understanding about the physics of such structures is of great importance. In this section, we present results of extensive micromagnetic simulations, especially regarding the magnetization structure, the frustrated states at vertices, the hysteretic properties, and their high-frequency magnetization dynamics.

Geometry and Ąnite element modelling

The structure is made of 90 cylindrical nanowires interconnected at 60 vertices. At the vertices, spheres are added to form a smooth and interconnected three-dimensional spherical network of hexagons and pentagons. Three cylindrical nanowires meet at each and sphere diameter 16 nm, while the largest structure had nanowires of length 250 nm and thickness 60 nm, and a sphere size 80 nm. To ensure a smooth geometric approximation of the curved surfaces and to accurately model the magnetization structure, the largest cell size in the Ąnite element mesh (lc) was always Ąxed below R/2, and moreover remained below the exchange length of the material used. Thus a maximum cell size of 3 nm was used for the smaller structures and a cell size of 5 nm was used

Material parameters

for the larger models. As a result of this, the smallest structure had over 120,000 Ąnite elements and the largest had nearly 2 million elements.

Material parameters

Since structures of this type are are usually patterned by means of FEBID, to get a realistic result we used material parameters corresponding to FEBID-deposited Cobalt (FEBID-Co) [START_REF] Fernandez-Pacheco | Private Communication[END_REF] with an exchange stiffness constant A = 1.5 × 10 -11 J m -1 and saturation magnetization µ 0 M s = 1.2 T where µ 0 is the vacuum permeability, we assumed zero crystalline anisotopy for the material. These material parameters result in an exchange length l exc = 2A/µ 0 20M 2 s = 5.1 nm. As mentioned in the previous section, the mesh size was always made sure to be smaller than this length. To save simulation time, a value of 0.5 was used for the GilbertŠs damping parameter α for the simulation of the static structures, where dynamic effects are not of interest. In contrast, a value of α = 0.01 was used for simulations on the high-frequency magnetization dynamics in theses systems.

Simulation method and post-processing

We have investigated principally the static equilibrium magnetization structure at zero Ąeld, their dynamic high-frequency modes and their hysteretic behaviour. The equilibrium magnetization structure is obtained by integrating the LLG equation in time, as explained in the previous chapter, until a predeĄned convergence criterion is reached. This convergence criterion can either be reached when the total energy ceases to change as the simulation proceeds, a total simulation time exceeding a particular limit (we often limit the calculations of the magnetic ring-down simulations to a total simulation time of 10 ns to 20 ns), or a decline of the magnitude of the effective torque in the structure falling below a user-deĄned threshold value at every discretization point. For our simulations, we took the third approach to simulate static structures, such that the simulation was allowed to run until the maximum value of the effective torque 1 1 This is a numerical value which is proportional to the local torque experienced by the magnetization under the effective Ąeld. The magnetization vector and the effective Ąelds are normalized for numerical convenience and hence the absolute value of this torque does not have any real physical signiĄcance and this is only used as an internal numerical control parameter.

Static magnetic conĄgurations

was below 1.0 × 10 -4 . From our experience with tetmag we know that this parameter, which is proportional to the magnitude of the maximum local torque ♣m × h eff ♣, is suf-Ąciently low to guarantee that the static structure obtained represents an energetically minimum state.

In the next step, high-frequency magnetic oscillations are excited with the methods already described in section 4.1: The Ąrst option is to apply a suitable low-amplitude Gaussian Ąeld pulse in the ps range. Exposing the sample to such a pulse represents a small perturbation that temporarily shifts the static structure of the magnetization out of equilibrium. The magnetization then relaxes back to the converged state within a few nanoseconds while generating a high-frequency small-angle precession dynamics [START_REF] Yan | Calculations of three-dimensional magnetic normal modes in mesoscopic permalloy prisms with vortex structure[END_REF]. The second option used in this work is based on a small misalignment of the effective Ąeld. In this method, the equilibrium state at a particular Ąeld is statically perturbed by applying a small Ąeld at a small angle with respect to the original Ąeld.

Once the equilibrium of the perturbed state is obtained, the additional small Ąeld is removed, so that the structure dynamically relaxes back to the converged state in the original Ąeld. Like in the case of the Gaussian Ąeld pulse, the relaxation dynamics of the magnetic ring-down process towards the equilibrium contains the relevant information on the high-frequency modes [START_REF] Baker | Proposal of a micromagnetic standard problem for ferromagnetic resonance simulations[END_REF].

In all cases, the magnitude of the perturbation was carefully chosen so that it does not cause any irreversible effect such as magnetization switching or any other major change in the static magnetic structure. The dynamics of the magnetic ring-down process of the excited state back to the energetic equilibrium is numerically recorded. This data is then processed by using our Fourier analysis tool, such as to extract the proĄles corresponding to different oscillation modes. The hysteresis loops are simulated like in the Ąrst case: the external Ąeld is varied in small steps, making sure that a new equilibrium state is reached for each step.

Static magnetic conĄgurations

The cylindrical nanowires were thin enough, so that at zero Ąeld they all tend to be axially magnetized. Although head-to-head and tail-to-tail domain walls are observed on larger dimensions, they usually only form as a transient, non-equilibrium state during hysteresis calculations. A micromagnetically interesting situation arises at the vertices, where three such nanowires meet and the magnetization of each wire has to

Static magnetic conĄgurations

adapt so as to accommodate the contributions from the neighbouring branches and, also to the change in geometry, see Fig. 6.2. This competing interaction at the vertices gives rise to locally frustrated magnetic conĄgurations. As seen on further discussion, these frustrated states affect the overall properties of the array. To explain the frustration of the vertices, it is convenient to represent the magnetization direction of each wire with an Ising-like dipole moment [START_REF] Castelnovo | Magnetic monopoles in spin ice[END_REF]. Since odd number of wires are meeting at each vertex, it is impossible to arrange the direction of each dipoles to satisfy all competing interactions. This situation inherently results in vertices with varying degrees of frustration. In general, we can classify the magnetic conĄgurations at the vertices into four types based on the relative orientation of these dipoles (Fig. 6.3) with respect to the vertex. Moreover, if we assign an index number to each of the 60 vertices we can prepare a small code that calculates the average value of ρ at each vertex. By plotting this computed value, we can easily identify and quantify the different kinds of charges are present in the system in an automated way, without visual inspection of the magnetic structure. An example of such a representation is shown in Fig. 6.5. One can clearly recognize the formation of different pleateaus, each corresponding to a speciĄc vertex conĄguration type.

Properties of triple-charges

In the context of artiĄcial spin ice lattices, the triple charges can be considered as defect structures with monopole-like properties [START_REF] Castelnovo | Magnetic monopoles in spin ice[END_REF][START_REF] Ladak | Direct observation of magnetic monopole defects in an artiĄcial spin-ice system[END_REF][START_REF] Ladak | Direct observation and control of magnetic monopole defects in an artiĄcial spin-ice material[END_REF][START_REF] Mengotti | Real-space observation of emergent magnetic monopoles and associated Dirac strings in artiĄcial kagome spin ice[END_REF][START_REF] Mól | Magnetic monopole and string excitations in two-dimensional spin ice[END_REF][START_REF] Loreto | Emergence and mobility of monopoles in a unidirectional arrangement of magnetic nanoislands[END_REF][START_REF] Rougemaille | Chiral nature of magnetic monopoles in artiĄcial spin ice[END_REF]. In single-charged vertices, since two of the wires have opposite moment orientation with respect to the vertex, the micromagnetic energy is minimized more efficiently [START_REF] Heyderman | ArtiĄcial ferroic systems novel functionality from structure, interactions and dynamics[END_REF][START_REF] Nisoli | Colloquium: ArtiĄcial spin ice: Designing and imaging magnetic frustration[END_REF][START_REF] Skjžrvø | Advances in artiĄcial spin ice[END_REF][START_REF] Qi | Direct observation of the ice rule in an artiĄcial kagome spin ice[END_REF]. The energetic considerations concern primarily magnetostatics, because in a single-charge state two out of the three charges are compensated, but also the exchange interaction; an aspect that is not of importance in disconnected ASIs.

Static magnetic conĄgurations

Here, since the nanowire structure is interconnected, the comparatively low degree of magnetic inhomogeneity developing at the single-charge vertices contributes to their structure being energetically favorable. In contrast, none of these energy minimization criteria are satisĄed in a triple-charged vertex conĄguration, which results in a comparatively higher energy density. Consequently, the triple charged vertices display a stronger local magnetic frustration. As a direct result of these two properties, the triple charged vertices are statistically less favourable. Simulations which were started from a randomized initial state usually yields relaxed zero-Ąeld states with two or three triple-charges out of the 60 vertices. A randomized initial state is an artiĄcial conĄguration in which the magnetization vectors at each descretization point are aligned in a random direction. Starting a simulation from such an initial state is useful to obtain results which are independent of memory effects of the material.

Despite their comparatively higher energy density, the triple-charged vertices are stable at zero Ąeld. In all the geometries we studied, it requires a Ąeld of at least 30 mT to annihilate a triple charge structure. The micromagnetic structure of the triple charged vertices shows a strong size dependence, in contrast to the single charged

Controlled generation and removal of triple-charges

The existence of defect charges is one among many driving forces for the research in ASI.

More often than not, the formation and various other behaviours of these defect charges are stochastic in nature, and hence difficult to control experimentally. In the case of buckyballs, the different three-dimensional orientations of the nanowires provides an easy method for the control of the formation and dissolution of these triple charges, namely by applying external magnetic Ąelds of suitable strength at an appropriate direction. An analogous behaviour for a controlled generation and removal of defects through a homogeneous external Ąeld is usually not possible in 2D-ASI systems. This possibility to manipulate magnetic defects is therefore an example of how the three-

Static magnetic conĄgurations

dimensional nature of the nanostructure leads to a qualitatively different and potentially signiĄcant behavior that is not found in the two-dimensional counterparts. The insertion and removal of triple-charge vertices can be demonstrated by simulating hysteresis curves (Fig. 6.7) of these structures. The Ąeld is swept from 500 mT to -500 mT in 10 mT steps, and the evolution of the magnetic structure is analyzed.

The Ąeld is applied along an axis which connects two diametrically opposite vertices.

The choice of these pair of vertices is arbitrary. Due to the spherical symmetry of the buckyball, any pair of vertices on opposite sides behaves identically.

At 500 mT, the Buckyball is in an onion-like (Fig. 6.8) state, where the two vertices along the axis at which the Ąeld is applied form the tip and the root of the onion. This state is characterized by the nanowires being magnetized at an angle with respect to the axial direction as the Zeeman energy contribution overpowers the shape anisotropy of the wires. When the applied Ąeld is reduced, the magnetization in the individual nanowires gradual aligns towards the local axis of the nanowires in a reversible way, thereby adapting to the geometry provided by the nanowire network. As the Ąeld is further reduced to zero, the magnetization structure progresses towards an ASI behaviour into a state characterized by an axial magnetization of the individual nanowires.

But the remanent structure is also inĆuenced by the initial onion-like state, which it partially preserves. Along with the alignment of the wires towards their axis, the tip This pair is gradually formed upon reduction of the external Ąeld remain stable even when a Ąeld in the opposite direction is applied. The zero-Ąeld remanent state hence contains a pair of triple charges at these two vertices. For convenience, we call this the triple-charge state.

Static magnetic conĄgurations

As the hysteresis loop is continued by applying a gradually increasing Ąeld in the opposite direction, the triple charges remains stable for until a particular value. On further increase of the Ąeld until the coercive Ąeld, the triple charges are dissolved by the formation of a domain wall emitted from the triple charge vertex, which results in the Ćipping of the magnetization in one of the bars. This Ćipping is carried forward through the connected vertices setting up a chain-reaction type switching leading to the reorientation of the magnetization in a number of neighboring nanowires and a corresponding rearrangement of the charge distribution in the associated vertices. As the applied (negative) Ąeld strength is increased, the Ąeld-induced dissolution of the triple charge vertices is followed by the reversal of individual nanowires at various switching Ąelds, depending on the orientation of the wires with respect to the applied Ąeld. These switching events are irreversible magnetization processes, and they can be observed as Barkhausen-type like steps in the hysteresis loop. The reversal of different nanowires occur at different Ąeld strengths, depending on the orientation of the wires with respect to the Ąeld direction (Fig. 6.9). The resulting conĄguration that forms

Static magnetic conĄgurations

as a result of this chain of events is free of any triple charges. It contains only icerule obeying single-charge vertices (+1, -1). The dissolution of triple charges by this process is also an irreversible process. Even if the Ąeld is reduced back to zero from the point of annihilation, the dissolved triple charges are not regenerated. Hence, by going to a zero-Ąeld state through a suitable minor loop, we can obtain a stable zero-Ąeld conĄguration which does not contain any triple charges. This state can be denoted as the single charge state. By simply applying a suitable sequence of external Ąelds, we can thus obtain two qualitatively different states at zero Ąeld: Either the triple charge state which contains a pair of +3, -3 defect charges, or the single-charged state which is free of such defect structures. The hysteresis curves were simulated for a total of ten geometries with wire lengths

Static magnetic conĄgurations

ranging from L = 25 nm to 250 nm. The results are analyzed and compared (see Fig. 6.10). The step-wise nature of the hysteresis curve, which is observed in all sizes, is related to the mechanism involving the reversal of individual nanowires. With an increase in size of the buckyballs, the steps tend to become smoother. It is also observed that the value of both coercivity and remanence decrease with increasing object size. This is a general tendency in micromagnetic, which is related to the reduced impact of exchange effects in larger samples. In this speciĄc case, the size-dependent changes in the hysteresis can partly be related to the evolution of the triple-charge vertices to a three-dimensional vortex-like state, which results in a more efficient Ćux-closure compared to the triple-headed domain wall structure developing in smaller geometries.

Magnonic spectrum of a buckyball: Switchable frequencies

The presence of triple charges has a strong impact on the magnonic spectrum of a magnetic buckyball structure. To demonstrate this, we simulate the small-angle precession modes of the equilibrium zero-Ąeld states of the buckyballs by the methods explained in section 6.3. The spectrum of the high-frequency response of a buckyball to a small perturbation at zero Ąeld in Fig. 6.11. It shows the case of a buckyball in single-charged state with L = 100 nm. We can clearly identify Ąve modes, with two lower frequency sharper peaks and three higher frequency broader peaks. These Ąve modes can be

Static magnetic conĄgurations

classiĄed into two types, based on their geometric origin. The Ąrst two peaks modes are oscillations localized at the vertices, in which the intensity of the activity is concentrated at the spheres connecting the nanowires. The remaining higher-frequency modes correspond to the oscillation of standing waves within the nanowires. The results of the discrete Fourier analysis provides information about the spatial proĄles of each mode, which are plotted in Figs. [START_REF] Yuasa | Giant roomtemperature magnetoresistance in single-crystal Fe/MgO/Fe magnetic tunnel junctions[END_REF].12 and 6.13. In these images, the average value of the modulus of the dynamic component of magnetization δm at each point over the entire time frame is plotted. The yellow regions indicate areas of high magnonic activity, while the purple regions are the inactive parts, or the nodes. The modes 1 and 2 at 9.07 GHz and 10.31 GHz, respectively, are caused by the activity of the single charged vertices. Almost half of the vertices oscillate at the Ąrst frequency and the rest oscillates at the second frequency as shown in Ągure 6.12. The modes 3 , 4 and 5 at 14 GHz, 17 GHz and 20 GHz, respectively, correspond to the Ąrst, second and third order oscillations of the nanowires 6.13.

In the next step, we repeat the same procedure for the triple-charged state and compare the results. A comparison of the frequency response of the triple-charge state with the single-charge state is given in Fig. 6.14. We can see that all the modes we discussed earlier remain essentially the same. However, now the spectrum also contains a pronounced low-frequency mode at 1.4 GHz, which was previously absent. The spatial proĄle of this mode is displayed in Fig. 6.15. This low-frequency mode is caused by the activity of the +3/ -3 triple-charge pair. This is an important observation with potential applications: Since the vertices oscillate at considerably lower frequency after In addition, the frequency of these triple charges can be controlled by varying the size of the buckyball and by means of an external Ąeld. The variation of the frequency of the vertices with changing size of the buckyball is shown in Fig. 6.16(a). Since the frequency of oscillation of the single charges are split into two groups, the frequency of the lowest mode in all sizes is used for the comparison. Both the single charge mode and the triple charge frequencies decreased linearly with an increase in the nanowire length: the frequency of oscillation of the single charged vertex decreased from 11.18 GHz in the smallest sample with nanowire length 50 nm to 1.8 GHz in the largest

Static magnetic conĄgurations

Hollow nanoarchitectures

the vertices or nearest neighbor interactions. We currently donŠt have a conclusive explanation for this splitting of the ensemble of nominally identical single-charge vertex oscillators into two populations of markedly different frequencies. The selection of the vertices belonging to one mode or the other does not appear to be based on details of the magnetic conĄguration. It might be the result of complex effects arising from a weak coupling between the oscillators, resulting in the partial synchronization of clusters of vertices. Future studies will be aimed at analyzing this effect in more detail.

The response of the frequency of oscillation of the triple-charged vertices of the buckyball with a side length L of 100 nm to an external Ąeld is shown in Fig. 6.17.

The Ąeld is applied along the diagonal connecting the two triple charged vertices in the positive direction. We can observe a sharp increase in the frequency of the mode with an increase in the Ąeld strength; from 1.4 GHz at zero Ąeld to a frequency of 2.1 GHz at 70 mT. Beyond this, the oscillation frequency remains constant despite an increase in the Ąeld strength. When the Ąeld was applied in the opposite direction, the frequency of oscillation was found to be decreasing.

Hollow nanoarchitectures

Apart from direct nano-patterning of ferromagnetic materials, three-dimensional structures similar to the buckyball structures discussed in theis chapter can also be fabricated by coating a non-magnetic structure with a ferromagnetic material. Buckyballs fabric- ated through this approach will be effectively hollow inside, at least from a magnetic perspective. They will be made of nanotubes interconnected by spherical shells. The magnetic properties of nanotubes has been extensively investigated in the recent past [START_REF] Landeros | Equilibrium states and vortex domain wall nucleation in ferromagnetic nanotubes[END_REF][START_REF] Escrig | Phase diagrams of magnetic nanotubes[END_REF][START_REF] Hertel | Ultrafast domain wall dynamics in magnetic nanotubes and nanowires[END_REF][START_REF] Yan | Fast domain wall dynamics in magnetic nanotubes: Suppression of Walker breakdown and Cherenkov-like spin wave emission[END_REF][START_REF] Sun | Magnetic properties of a long, thin-walled ferromagnetic nanotube[END_REF][START_REF] Landeros | Reversal modes in magnetic nanotubes[END_REF][START_REF] Usov | Nucleation Ąeld of a soft magnetic nanotube with uniaxial anisotropy[END_REF][START_REF] Wyss | Imaging magnetic vortex conĄgurations in ferromagnetic nanotubes[END_REF][START_REF] Skoric | Micromagnetic modeling of magnetic domain walls in curved cylindrical nanotubes and nanowires[END_REF][START_REF] Vasyukov | Imaging stray magnetic Ąeld of individual ferromagnetic nanotubes[END_REF], and it is already known that hollow tubes exhibit distinct magnetic properties that are different from those of solid nanowires.

Hollow nanoarchitectures

The hollow structure of nanotubes results in magnetic conĄgurations which can close the magnetic Ćux while avoiding magnetic singularities [START_REF] Landeros | Equilibrium states and vortex domain wall nucleation in ferromagnetic nanotubes[END_REF][START_REF] Landeros | Reversal modes in magnetic nanotubes[END_REF]. It was argued that this may result in faster magnetization switching under external Ąelds and thus offers better possibilities to manipulate magnetic conĄgurations compared to solid cylinders.

The presence of an additional internal curved surface in these hollow structures is also a point of interest. Hence, it is natural to extend the studies on buckyballs to hollow models, and to compare the results with those obtained for the solid structures.

The Ćexibility of the Ąnite element method permits the accurate modeling of magnetization in such complex geometries. We have simulated a range of hollow buckyballs from side length 100 nm to 200 nm with Ąxing the geometrical parameters in the following ratio, length of nanotube (L), external radius of nanotube (R e ), external radius of the spherical shell (S e ) internal radius of the nanotube (R i ), internal radius of the sphere (S i ) 50:6:8:4:6. The mesh length was chosen appropriately such as to ensure an accurate description of the geometry of the thin tubes.

Hollow nanoarchitectures

Comparison of nanowires and nanotubes

To understand the basic magnetization structure and magnonic response of these two types of buckyballs, we Ąrst simulated a miniature versions of their constituent units and compared the results. This involves

R i = 8 nm, R e = 12 nm
The magnetization structures of these units are shown in Figs. [START_REF] Yuasa | Giant roomtemperature magnetoresistance in single-crystal Fe/MgO/Fe magnetic tunnel junctions[END_REF].18 and 6.19. We can see that they have almost identical, axial magnetization conĄgurations. There are only minor differences in the ends depending on the presence of spheres or shells. The relaxed magnetic conĄgurations are excited with a Gaussian Ąeld pulse, and the frequency response is analyzed. In all these structures, the primary mode of oscillation is a coherent oscillation of the magnetization component at the two ends. But one can also see a considerable difference in the frequency of oscillation of this mode with a minor change in the geometrical parameters. SpeciĄcally, we observe a decrease in the frequency if we add a spherical structure at the end. Comparing the frequency of oscillation of all these structure, we can also see that the hollow tubes with closed ends have a very low frequency of only 610 MHz. Magnetic oscillations at such low frequencies are usually typical for the frequency of gyration of magnetic vortices. But in this case, there are no vortices present in this structure. Here, the low-frequency mode is caused by the slow rotation of the in-plane component of magnetization in the two nanodiscs which form the caps attached at the two ends of the nanotube. In comparison with the frequency of oscillation of other similar structures, this mode in the nanotube with closed end is more intense and sustained for a longer time period. These simple observations indicate that the magnonic properties of hollow structures can be signiĄcantly different from those of solid ones, and that they can depend sensitively on details of the geometry of the constituent units.

Hollow nanoarchitectures

Hollow nanoarchitectures

Hysteretic properties

The hollow structures exhibit a hysteretic behavior that is quite similar to that of the solid buckyballs. When the structure is saturated by applying a strong magnetic Ąeld of µ 0 H ext = 500 mT along an axis connecting two vertices locate on diametrically opposite positions, and if the Ąeld is then gradually reduced to zero, we obtain again a state which contains a pair of triple charges on the two vertices along the axis through which the Ąeld was applied. Applying a gradually increasing Ąeld in the opposite direction will result in the annihilation of the triple charges by the emission of a domain wall, and a subsequent relaxation back to zero Ąeld through a minor loop will again result in a single-charge state which is free of triple charges. The simulated hysteresis curve of the structure with a tube length of 100 nm and a comparison of the remanence ratio and coercive Ąeld strength of the samples with wire length is given in Fig. 6.24. We could observe that both the remanence ratio and the coercive Ąeld tends to decrease with size.

Hollow nanoarchitectures

High-frequency modes

A comparison of the high-frequency response of the magnetization to a short excitation in the case of the triple-charged state of the solid and hollow buckyball is shown in Fig. 6.25. We can identify three major modes which are marked as 1 , 2 and 3 . As in the case of the solid buckyballs, the Ąrst mode at 2.78 GHz is caused by the activity of the triple charged vertices, the second and third mode at 7.42 GHz and 9.10 GHz, respectively, are caused by oscillations of the single charged vertices. In comparison to

Chapter summary

the solid Buckyball, we can see that the characteristic frequency of the triple charges is shifted towards a higher value, while those of the single-charge vertex oscillations are shifted towards lower frequencies. The most striking difference in the magnonic spectrum of the hollow buckyball, compared with the solid case, is the absence of wire modes. The nanotubes connecting the vertices are not magnonically active. This is manifested as the absence of the three higher frequency modes, which we had seen in the previous case.

Chapter summary

In this chapter, we have investigated the fundamental properties of artiĄcial ferromagnetic buckyball nano-architectures by means of Ąnite-element micromagnetic simulations. The geometric Ćexibility of the Ąnite element approach allowed as to accurately model the cylindrical nanowires and the spherical vertex regions of these structures.

We simulated buckyballs of a wide range of size Ű from a side length of 25 nm to 250 nm.

In all the sizes we studied, the individual nanowires displayed a single-domain state at remanence, magnetized along the longitudinal direction. Based on the axial direction of the magnetization in each nanowire we could assign an Ising-like dipole moment to the nanowire. According to the relative orientation of these Ising moments, we 6.6 Chapter summary could identify different magnetic conĄgurations in the vertices with varying degree of magnetic frustration, and we distinguished between the ice-rule obeying single charges and the defect-type triple charges. Due to the three-dimensional spherical structure of the buckyballs, these triple changes can be generated and destroyed in a controlled manner by means of an external Ąeld. We studied the magnonic response of these structures and we could identify the vertex modes and the nanowire modes arsing from the corresponding geometric locations. We saw that the presence of triple charged vertices produced a signature with low-frequency peaks in the magnonic spectrum of these buckyballs. The lower frequency of oscillation of the triple charge vertices when compared to the single charged vertices is due to the absence of a strong pinning Ąeld in the triple charges due to the symmetrical arrangement of the magnetisation of the nanowires. The frequency of these peaks could be controlled by varying the geometry, and also by applying an external magnetic Ąeld. We compared these Ąndings with the results of a hollow buckyball and observed that they exhibit similar behavior to that of the corresponding solid geometries in the concerned length scales.

These structures can be considered as a model system to describe a transition from two to three dimensional artiĄcial spin ice systems. The spherical symmetry of these structure allows the to insert and remove triple-charge defects by means of external magnetic Ąelds. This opens a pathway for magnonic applications as, this control of the magnetic vertex conĄguration allows to manipulate the high-frequency spectrum of the nanostructures through the occurrence of a pronounced, sharp peak in the magnonic absorption spectrum in the case of triple-charge defects. We hope that the Ąndings of these simulation studies will inspire experimental groups to investigate these properties and to obtain an experimental veriĄcation.

In the previous chapter, we analyzed the micromagnetic properties of the static and dynamic magnetization in buckyball-type geometries. Although these structures are clearly three-dimensional, the nanowire network generating the buckyballs is, in itself, two-dimensional as it covers the surface of a sphere. While this combination of two-and three-dimensional features renders the buckyball geometry interesting to study effects emerging from the transition from two to three dimensions, other structures display a more genuine three-dimensional character. This is the case for periodic nanowire arrays with a periodic geometric arrangement of the vertex positions. The ability to fabricate three-dimensional structures of this type has recently been demonstrated [START_REF] Keller | Direct-write of free-form building blocks for artiĄcial magnetic 3D lattices[END_REF][START_REF] Sahoo | Observation of coherent spin waves in a three-dimensional artiĄcial spin ice structure[END_REF][START_REF] May | Realisation of a frustrated 3d magnetic nanowire lattice[END_REF]]. This is one example showing the important progress in nanofabrication and nanopatterning techniques of three-dimensional patterning techniques, such as FEBID [START_REF] Fernández-Pacheco | Three dimensional nanomagnetism[END_REF][START_REF] Fernández-Pacheco | Writing 3D Nanomagnets Using Focused Electron Beams[END_REF], which has made it possible to fabricate magnetic nanostructures of virtually any three-dimensional geometry. Using these techniques, it is possible to fabricate arrays consisting of periodic, three dimensional networks of interconnected nanowires, with vertex positions corresponding to those of atomic positions in a crystal lattice. Three-dimensional interconnected nanostructures of this type can be interpreted as an extension from two-dimensional artiĄcial spin-ice systems to three-dimensional ones. On the other hand, their particular geometry is reminiscent of artiĄcial structures which, in the case of optical or acoustic applications, are used for the design of metama-terials [START_REF] Kadic | 3D Metamaterials[END_REF]. One can therefore also see these three-dimensional nano-architectures as prototypes for a new form of magnetic metamaterials.

Metamaterials are artiĄcially prepared composite materials which exhibit properties that cannot be observed in the bulk state [START_REF] Rangu | Magnetic metamaterials: A comparative study of resonator geometry and metal conductivity[END_REF][START_REF] Liu | Metamaterials: a new frontier of science and technology[END_REF][START_REF] Kadic | 3D Metamaterials[END_REF], such as negative refractive index [START_REF] Smith | Metamaterials and negative refractive index[END_REF][START_REF] Valentine | Three-dimensional optical metamaterial with a negative refractive index[END_REF], unusual acoustic properties [START_REF] Cummer | Controlling sound with acoustic metamaterials[END_REF], exceptionally large optical susceptibilities, non-reciprocal behavior, sign reversal of thermal expansion constant etc. The emergence of such exotic properties in metamaterials is a result of their microscopic artiĄcial geometric structure, rather than a consequence of the intrinsic properties of the chemical constituents [START_REF] Liu | Metamaterials: a new frontier of science and technology[END_REF]. This particular property of the metamaterials opens up the potential to manipulate their exotic behaviors by Ąne-tuning their geometric parameters [START_REF] Louis | A tunable magnetic metamaterial based on the dipolar four-state potts model[END_REF]. An interesting sub-category of such metamaterials are artiĄcial crystals which are formed by a periodic arrangement of building blocks mimicking the behavior of natural crystals [START_REF] Burckel | Micrometer-scale cubic unit cell 3D metamaterial layers[END_REF][START_REF] Soukoulis | Past achievements and future challenges in the development of three-dimensional photonic metamaterials[END_REF]. If such structures are fabricated with magnetic materials, they may result in a new type of magnonic materials. ArtiĄcial magnonic crystals [START_REF] Gubbiotti | Magnetostatic interaction in arrays of nanometric permalloy wires: A magneto-optic Kerr effect and a Brillouin light scattering study[END_REF][START_REF] Krawczyk | Review and prospects of magnonic crystals and devices with reprogrammable band structure[END_REF][START_REF] Popov | Spin wave propagation in three-dimensional magnonic crystals and coupled structures[END_REF] (AMC) are largely an unexplored sub-category of artiĄcial magnetic nanostructures. In particular, little is known about the magnonic properties of threedimensional interconnected structures in the context of their potential application as magnetic metamaterials. Analogous to how the propagation of photons and phonons is manipulated in photonic and acoustic crystals, AMC have the potential to be tuned in such a way to control magnons -the fundamental excitation unit in magnetism.

In contrast to photonic an acoustic metamaterials, the ferroic character of AMCŠs introduces an additional degree of freedom, which can be exploited to generate a varying degree of disorder within the crystal. As discussed already on the example of the buckyball structures, it is possible to change the magnetic conĄguration at the vertex points, where two or more nanowires meet. Such manipulations of the magnetic structure could be used to modify the magnonic spectrum of the array, since the vertices can act as sources of magnetic frustration due to competing interactions [START_REF] Mistonov | Three-dimensional artiĄcial spin ice in nanostructured co on an inverse opal-like lattice[END_REF][START_REF] Koraltan | Tension-free Dirac strings and steered magnetic charges in 3D artiĄcial spin ice[END_REF][START_REF] May | Magnetic charge propagation upon a 3D artiĄcial spinice[END_REF], as already known from traditional artiĄcial spin ice (ASI) systems [START_REF] Heyderman | ArtiĄcial ferroic systems novel functionality from structure, interactions and dynamics[END_REF]. Because these network of interconnected nanowires can be interpreted as a three dimensional version of the familiar artiĄcial spin ice lattices, their multiple magnetization states could yield different and switchable magnonic properties. Conventionally, most of the research in artiĄcial spin ice structures was concentrated on two-dimensional disconnected grids of patterned single domain elements [START_REF] Heyderman | ArtiĄcial ferroic systems novel functionality from structure, interactions and dynamics[END_REF][START_REF] Skjžrvø | Advances in artiĄcial spin ice[END_REF]. However, recent studies have extended the

Methodology

investigation into interconnected two-dimensional networks [START_REF] Frotanpour | Vertex dependent dynamic response of a connected kagome artiĄcial spin ice[END_REF][START_REF] Shi | Kerr microscopy real-time imaging of the magnetization reversal process in kagome artiĄcial spin ice[END_REF][START_REF] Saavedra | Dynamic susceptibility of interconnected pentagonal spin ice lattices[END_REF] and threedimensional disconnected structures [START_REF] Fischer | Launching a new dimension with 3D magnetic nanostructures[END_REF][START_REF] Sahoo | Observation of coherent spin waves in a three-dimensional artiĄcial spin ice structure[END_REF][START_REF] Koraltan | Tension-free Dirac strings and steered magnetic charges in 3D artiĄcial spin ice[END_REF]. In this chapter, using Ąnite-element micromagnetic simulations, we investigate the frustrated states, Ąeld dependence and high-frequency properties of two types of artiĄcial magnonic crystals: the diamond lattice [START_REF] May | Realisation of a frustrated 3d magnetic nanowire lattice[END_REF] structure and the cubic lattice structure.

ArtiĄcial diamond lattice structure

An interesting sub-category of three-dimensional artiĄcial magnonic crystals are diamondtype nano-architectures, which are formed by a network of interconnected nanowires.

The nodes where these wires meet in space are located as per the lattice points of a natural Diamond crystal. Four nanowires meet at each node in a tetragonal arrangement and the lattice can be interpreted to be made of individual tetrapod units. When compared to the Buckyball structures discussed in the previous chapter which can be considered as a planar structure wound to form a spherical structure, the Diamond lattice is a full-Ćedged three-dimensional network.

Methodology

As in the case of the study on the buckyballs, we prepared Ąnite element meshes of these geometries by using Netgen and chose material parameters corresponding to those of FEBID-deposited Cobalt. The diamond-type lattice structure that we simulated had an overall size of 450 × 450 × 450 nm. The array was composed of nanowires with a length of 70.0 nm and a radius of 7.0 nm. The entire network contained 202 nanowires interconnected at 83 vertex node points. The Ąnite element model which we used had a maximum cell size of 4.0 units and contained 704,476 tetrahedral elements.

We performed micromagnetic simulations to investigate the static magnetization structure, different kinds of vertex conĄgurations, their hysteretic behaviour and also their magnetic high-frequency properties.

Static magnetization structure and vertex conĄgurations

Static magnetization structure and vertex conĄgurations

Similar to the case of buckyballs, in the geometries which we simulated, the wires are thin enough so that they are axially magnetized at equilibrium state. We can thus interpret the magnetization direction of each wire as that of an Ising-type macroscopic dipole [START_REF] Bramwell | Spin ice state in frustrated magnetic pyrochlore materials[END_REF] consisting of two equal and opposite magnetic charges (±q). In such a macroscopic dipole, the +q charge represents the end at the head of the magnetization and -q the tail. In the diamond-type geometry that we studied, each vertex is the intersection point of four nanowires. Accordingly, we can identify different types of vertex conĄgurations based on the relative orientation of the four dipoles meeting at each vertex, and assign a charge value to each conĄguration [START_REF] Montaigne | Size distribution of magnetic charge domains in thermally activated but out-of-equilibrium artiĄcial spin ice[END_REF][START_REF] Cheenikundil | Switchable magnetic frustration in buckyball nanoarchitectures[END_REF]. The concept of assuming a charge for each vertex conĄguration can be justiĄed as follows. Since each nanowire in the relaxed state is magnetized axially, depending on the orientation of magnetization with respect to the vertex, each nanowire either carries magnetic Ćux towards the vertex or away from it. Since there are four nanowires meeting at each vertex, a balanced state is established when the number of wires with incoming Ćux and outgoing Ćux are equal (two/two) and thus cancel out, resulting in a net zero charge.

Any imbalance in the number of wires with in-and outĆowing Ćux results in a non-zero net magnetic charge at the corresponding vertex, as in the case of emergent magnetic monopoles in ASIs [START_REF] Mengotti | Real-space observation of emergent magnetic monopoles and associated Dirac strings in artiĄcial kagome spin ice[END_REF][START_REF] Hügli | ArtiĄcial kagome spin ice: dimensional reduction, avalanche control and emergent magnetic monopoles[END_REF][START_REF] Gilbert | Emergent ice rule and magnetic charge screening from vertex frustration in artiĄcial spin ice[END_REF]. Thus, in the diamond lattice or in the case of any other such structure with a coordination number of four, there can exist Ąve different vertex conĄgurations as listed below. such as, 1. Two-in / two-out (0) 2. Three-in / one-out (+2q) 3. One-in / three-out (-2q) 4. Four out (+4q)

Four in (-4q)

Out of these Ąve conĄgurations, the Ąrst one with a net charge of zero is the ice-rule obeying structure, while the rest can be classiĄed as different types of magnetic defect . charges. As our simulations show, the theoretically proposed conĄgurations number 4 and 5 are energetically unstable and do not develop in the parameter range that we studied. In none of our simulations could we observe these states as stable conĄgurations. However, the states 2 and 3 with a net charge of +2q and -2q respectively are obtained and are found to be stable. For convenience, these states will be referred to as the Şdouble-chargeŤ states henceforth in this thesis. In addition to the Ąve stats listed above, another type of charges can be distinguished if one also considers the magnetization state of the naowires at the surface of the artiĄcial crystal. The dangling free ends of the surface nanowires represent a situation that is different from the vertex structures within the volume. We can assign a charge value of +1 or -1 to these ends, based on the direction of magnetization at the ends pointing inwards or outwards Owing to the symmetric, three-dimensional arrangement of the nanowires at each vertex, there is only one ice-rule obeying state. This is in contrast to the two different possible ice-rule obeying state known from two dimensional square ASI, evan though planar square ASIs also have vertices with a coordination number of four [START_REF] Gilbert | Frustration by design[END_REF]. In the case of a 2D square ASI, not all the conĄguration of three nanowires meeting at a vertex are equivalent with respect to a fourth wire. We can differentiate between the magnetic structure in wire on the opposite side and that in the two orthogonal wires.

Static magnetization structure and vertex conĄgurations

Because of this difference, there exist two Ćavors of two-in/two-out state in a 2D square ASI depending on the relative orientation of the incoming/outgoing wires, as indicated in Fig. 7.6. In the diamond lattice, however, all three neighboring wires meeting at a vertex are equivalent with respect to the fourth wire. The angle between any of the wires in the tetragonal arrangement is the same, and therefore the distinction between wires on opposite or neighboring directions does not apply. Consequently only a single Ćavour of two-in/two-out state exists in this case. As a result of this equivalence of the relative orientation of any pair of nanowires at the vertices, each charge value for the tetrapod-type vertex has only one possible magnetic conĄguration associated with it. 7.8. It is to be noted that this disordered state is only one realization out of a quasi-continuum of many possible conĄgurations [START_REF] Nisoli | Ground state lost but degeneracy found the effective thermodynamics of artiĄcial spin ice[END_REF][START_REF] Morgan | Thermal ground-state ordering and elementary excitations in artiĄcial magnetic square ice[END_REF][START_REF] Möller | Magnetic multipole analysis of kagome and artiĄcial spin-ice dipolar arrays[END_REF].

Static magnetization structure and vertex conĄgurations

Hysteretic properties of the crystal: zero charged state

A simulated hysteresis loop of the network is shown in Fig. 7.9, and a visualization of the evolution of the magnetization of the structure at various stages of the hysteresis is displayed on the next page. For convenience and better visualization, the magnetization of a single tetrapod unit within the bulk of the lattice is visualized, instead of displaying the whole network. The external Ąeld is applied along the x direction, so that it is not parallel to any of the nanowires in the network. At the starting Ąeld of 800 mT, the network is in an technically saturated state in which the nanowires are magnetized parallel to the Ąeld direction rather than along the axial direction. As the Ąeld is lowered, the magnetization in the wires rotates smoothly towards the axial direction. On further lowering the Ąeld to zero, we obtain a state where all the individual nanowires are magnetized along their axis and the network has a net remanent magnetization along the x direction. This remanent state obtained after saturation along the x direction is highly ordered and contains only zero-charged vertices. Upon increasing the Ąeld in the opposite direction, the magnetization along the wire initially remains axial in the case of low Ąelds. However, once the Ąeld exceeds about -240 mT, the magnetization Ćips irreversibly towards the Ąeld direction resulting, in a sharp jump in the M -H curve. After this point, increasing the Ąeld further results in the reversible rotation of the magnetization towards the external Ąeld. This gradual rotation appears as an almost linear increase of M ♣♣ /M s with increasing external Ąeld, until saturation is reached.

Magnonic excitations: Zero-charged state

Magnonic excitations: Zero-charged state

As we have seen already in the case of the buckyball structures, the formation of ice-rule-violating defect charges can have a strong effect on the magnonic response of the artiĄcial spin ices. To demonstrate this, we simulate the small-angle precession modes and extract the frequency modes with a Fourier analysis, as explained in the previous chapters, and compare the magnonic signature of the zero-charge state and the disordered defect state. The magnonic spectrum of the zero-charge state is shown in Fig. 7.10. We can identify Ąve major modes with the frequencies 9.9 GHz, 14.1 GHz, 16.6 GHz, 19.0 GHz, and 28.3 GHz, respectively. Each of these prominent modes can be attributed to distinct magnetic oscillation developing at various geometric locations within the lattice: the dangling wire ends at the surface, the vertices, and the cylindrical nanowires. Each mode is marked with a number from 1 to 5 , which will be used in the following discussion to refer to the individual modes. We can classify the Ąve modes into two categories: the low frequency modes which are caused by the activity of the ±1 and 0 charged vertices (and free ends) and the higher frequency modes which are caused by different orders of oscillations of the magnetization in the nanowires. The mode 1 at 9.9 GHz arises from the activity of the ±1 charges at the dangling free ends. The vertices in the bulk and the nanowires are inactive at this frequency (Fig. 7.11). The relative height of the peak 1 with respect to the other peaks depends on the geometry of the crystal geometry, which determines the surface-to-volume ratio; that is the ratio of ±1 charges to 0 charges. The oscillation of the dangling free ends appears to be decoupled, meaning that we could not observe any phase correlation between the oscillations at different sites. It can be observed that there is no distinction between the oscillation at +1 charges with that of the -1 charges. The spatial proĄle of the mode is shown in Fig. 7.12. We can observe a clear localization of the mode at the zero-charged vertices. Contrary to the oscillation of the free ends, we can visually identify a phase correlation in the oscillation of the zero charged vertices. This is discussed in more detail in a later subsection.

Magnonic excitations: Zero-charged state

Magnonic excitations: Zero-charged state

The modes 3 4 and 5 at 16 GHz, 19.0 GHz and 28.3 GHz, respectively are the wire modes. These peaks correspond to the development of standing waves within the nanowires. The spatial proĄle of the three standing waves are displayed in Fig. 7.13. For convenience, the wires are graphically isolated from the rest of the lattice, although in the simulation they are structurally embedded in the lattice. In the case of mode 3 and 5 the vertices remain inactive and act as nodes, while for mode 4 the magnetization also oscillates at the vertices. A clear macroscopic correlation of the oscillations at different sites of the crystal can only be observed for the mode 4 , which is the only one in which the vertices takes part in the activity. This suggests that long-range dynamic correlation results from the dynamic interaction between the vertices. According to the orientation of the nanowires, there are six different distinct directions in the lattice.

However, the magnetic modes developed in the nanowires in the defect free state do not appear to be affected by the orientation of the wires.

Magnonic response of defect charges

at 14.2 GHz only the zero charged vertices. In spite of this clear separation based on the magnetic conĄguration, there exists a considerable difference in the intensity of the activity even within the vertices belonging to the same type. We attribute these differences to inhomogeneities in the local magnetic Ąelds arising due to the disordered charge distribution. This difference only concerns the zero charges; we could not observe any difference in the activity between +2q and -2q double charges. Even though the oscillating frequency and the mode proĄle of the zero-charged vertices remains the same as in the case of the defect free state, a clear phase correlation between different vertices, which was observed in the previous case, was absent in the discorded state.

The number of zero-charged vertices in the defect state ( 48) is almost 40 % less than that of in the defect free state [START_REF] Kronmüller | Micromagnetism and the Microstructure of Ferromagnetic Solids[END_REF]. When we compare the intensity of the peaks, we can observe a proportional decrease in the amplitude of the disordered state, which is in accordance with the decrease in the number of zero-charged vertices. 1. type 0/0 wires located within the bulk and connecting two zero-charged vertices 2. type +1/0 wires at the surface, connecting a +1 charged end to a zero-charged vertex 3. type -1/0 wires connecting a -1 charged end to a zero-charged vertex It is to be noted that wires connecting -1 to +1 charges cannot exist, as the ±1 single charges are always formed on the single, dangling free ends. They are always connected to a vertex within the volume, and never with each other. In the ice-rule obeying zero-charge state, there are 83 zero-charge vertices, 36 vertices with +1 charge, and 36 with a charge of -1. As a result, there are 130 wires of 0/0 type, 36 wires of +1/0 type, and 36 wires of -1/0 type. To obtain a deeper understanding about the dependence of the presence different types of nanowire conĄgurations on the magnonic response of the whole structure, we carried out localized Fourier analysis for each nanowires. In this analysis process, each descretization point corresponding to a particular nanowire is identiĄed, isolated and the magnetization dynamics in this region is Fourier-analyzed such as to yield the individual magnonic response of each nanowire.

It is observed that nanowires belonging to the same category display similar magnonic properties. We further found that the peculiar magnonic signature of each type of wire is charge-symmetric: that is, a set of wires connecting charges x and y exhibits a basically identical magnonic spectrum as that of a set of wires connecting -x and -y type charges. This can be explained as follows. In the absence of an external Ąeld, the magnetic conĄguration in a nanowire oscillates under the effect of the dipolar Ąeld generated by the magnetic conĄgurations at the vertices. The magnetic volume charge densities developing in a -2q vertex and a +2q vertex act as a sources and sinks for the magnetic Ąeld H. As a result, a longitudinal magnetic Ąeld develops in wires representing a Dirac string [START_REF] Mengotti | Real-space observation of emergent magnetic monopoles and associated Dirac strings in artiĄcial kagome spin ice[END_REF][START_REF] Gliga | Spectral analysis of topological defects in an artiĄcial spin-ice lattice[END_REF] connecting +2q a charge to a -2q charge (-2q → +2q).

The presence of such a magnetic Ąeld can shift the frequency of the standing wave with respect to the same oscillation at zero Ąeld [START_REF] Jorzick | Brillouin light scattering from quantized spin waves in micronsize magnetic wires[END_REF]. Thus, all the wires having same kinds of charges attached to their ends are exposed to similar dipolar Ąelds, resulting in similar magnonic properties. This also explains the charge symmetry effect. Due to time-inversion symmetry, a +2 charge and a -2 charge are equivalent, which in turn results in dipolar Ąelds of equal strength. Since there are three types of wires in the reference state, we can have three types of interactions. A comparison of the magnonic response of all the 0/0 type nanowires is given in Fig. 7. [START_REF] Parkin | Magnetic domain-wall racetrack memory[END_REF]. One can see that all the individual wires in this group exhibit a similar magnonic spectrum. The minute variations in the power spectrum can be attributed to asymmetries in the next-nearest neighbour interactions, and to variations in the orientation direction with respect to the direction of the Ąeld pulse by which the oscillation is triggered. Comparing the magnonic features of the nanowires with that of the whole structure, we can see that the mode 1 at 9.9 GHz, which is caused by the activity of ±1 charges, is absent in the 0/0 type. A detailed comparison of the 0/0, +1/0 and -1/0 type nanowire groups is shown in Fig. 7.17. We can clearly see that the +1/0 and -1/0 groups have a similar magnonic response to the Ąueld-pulse exitation, and that their spectrum is signiĄcantly different from that of the 0/0 group, as expected.

Different types of nanowires: InĆuence of local Ąelds on the magnonic spectrum

In the case of the disordered double-charge state, which contains ±2 charged vertices along with zero-charges vertices, more types of nanowires can be observed as explained in the following table. In such a disordered conĄguration, we can distinguish twelve different types of nanowires, with the 0/0 type being the most common one. A detailed graph depicting the magnonic response of all twelve types of nanowires is presented in Fig. 7.18. We can see that all the nanowires in the same group exhibit a similar magnonic spectrum, especially in the lower-frequency modes below 15 GHz. Above that frequency, the modes tend to broaden and to become less coherent. The charge- symmetric behaviour explained in the previous sub-section can also be observed.

Tuning the mode frequencies with an external Ąeld

Tuning the mode frequencies with an external Ąeld

To investigate the Ąeld dependence of the frequency of the modes, small-angle precession modes of the lattice are simulated in the presence of an external Ąeld. The resulting changes of the frequency can be displayed in a heatmap representation of the modes, as shown in Fig. 7.19. Since the equilibrium magnetization state was obtained by relaxing to zero Ąeld from saturation along a particular direction, the structure retains

Effect of a geometric defect in the crystal lattice

The strong impact on the magnonic spectrum of the diamond-type artiĄcial lattice by the formation of defect charges is comparable with that of Nitrogen vacancy centers (NV centers) on the optical properties in the case of a Nitrogen-doped real diamond crystal. To further investigate the potential for manipulating the properties by defects in the artiĄcial lattice, we conducted the same studies as described before, but on a structure with geometrical defects. The individual defects in the AMC are introduced by removing single nanowires from the bulk of the lattice. Such a removal results in the creation of two vertices with odd coordination in the bulk of the lattice, namely with a coordination number 3. Similar to how we assigned a charge value to each vertex conĄguration in section 7.2, we can now assign a charge value to these structural defect vertices as well. Since an odd number of wires is meeting at these kinds of vertices, they

Effect of a geometric defect in the crystal lattice

cannot adopt a zero-charge conĄguration. Similar to the case of the ice-rule obeying states in the buckyball structure, only two types of conĄgurations can be found in these vertices:

1. Two-in/one-out state (+1.5q)

2. One-in/two-out state (-1.5q) Theoretically, one could imagine also to Ąnd three-in and three-out conĄgurations at these vertices. But in our simulations we never observed a situation like this, in which all the wires of a defect vertex are magnetized towards or away from the vertex. To be consistent with the previous discussion of the charged vertices in the buckyball lattice, one could have assigned charges of +1 and -1 to these two magnetic states forming at the tripods of the defect vertices, respectively. But it is necessary to differentiate between these charges and those from the ±1 charges occurring at the surface. The distinction is not only required because the magnetic conĄguration is entirely different in those two case, but also because the numerically computed average value of ∇ • m at the geometrical defect points is signiĄcantly different from that at the free ends. It lies between that of the single charges at the surface and the value of the double-charged vertices. We therefore attribute a value of ±1.5q to the charges at these structural defects; more for practical than for formal reasons.

The magnonic spectrum of the ice-rule obeying zero-charged state of the structure The weakly developed peak 2 can be directly ascribed to the defect in the lattice. with the geometrical defect is shown in Fig. 7.21. We can identify two strong peaks at 9.9 GHz and 13.7 GHz, respectively. As already seen, these peaks correspond to the activity of the ±1 charged vertices at the free ends and to the zero-charged ice-rule obeying vertices, respectively. In between these peaks we can observe a low-intensity mode (labelled Ş2Ť) at 11.7 GHz. The proĄle of this mode is localized at the ±1.5 charged defect vertices. It has a low intensity because the signal originates from just two defect vertices out of a total of more than 100 surface and volume vertices in the AMC.

Besides the appearance of a this mode, the introduction of geometrical defects results in the shifting of the wire modes 3 , 4 and 5 towards lower frequencies. Adding more structural defects exacerbates this effect and leads to further changes in the spectrum.

Macroscopic spin wave in a diamond-type AMC

Upon increasing the density of structural defects, the extent to which the spectrum changes quickly reaches a degree at which it becomes impossible to consider the defects as mere perturbations or modiĄcations of the defect-free state. An illustration of the spatial proĄle of the mode at the structural defect is shown in Fig. 7.22.

Macroscopic spin wave in a diamond-type AMC

As indicated in the previous section, we could observe a clear long-range phase correlation in the oscillation of the zero-charged vertices in the defect free state. The oscillation appears to propagate from the core of the lattice towards the surface in the form of a macroscopic spin wave with a wavelength of approximately twice the wire length. To verify this observation, we modelled an elongated AMC network with a length of 20 times the estimated wavelength of the macroscopic spin wave. The relaxed magnetization structure was obtained by saturating the sample along the longitudinal direction and then relaxing the system to a zero Ąeld state. This resulted in a well-deĄned defect-free magnetization state in which the whole structure has a net magnetization along the longitudinal direction and where all vertices within the crystal are in a two-in/two-out zero-charged conĄguration. To investigate the possible existence and propagation of macroscopic spin waves in the form of correlated oscillations of the zero-charged vertices within the artiĄcial lattice, the elongated structure was excited with a spatially homogeneous Gaussian pulse. The magnetic Ąeld pulse was applied perpendicular to the longitudinal direction. The subsequent global response of the vertex modes is analyzed with the same methods we used before. In order to verify whether such macroscopic vertex modes can propagate through the lattice, we also studied the case in which the elongated structure was excited only at one of its ends by means of a spatially inhomogeneous pulse. In both cases, the magnetic ring-down dynamics triggered by the Ąeld pulse was analyzed.

The high-frequency spectrum of the elongated structure, obtained from a homogeneous Ąeld pulse excitation, is shown in Fig. 7.24. Unsurprisingly, the structure exhibits high-frequency response that is similar to the one of the other, smaller diamond-type lattice structure. The extracted oscillations of the vertex modes are displayed in Fig. 7.25.

Contrary to what we expected, we could not observe any longitudinal variation in the phase of the oscillation. All the vertices appear to be oscillating at the same phase along the entire lattice. However, we could detect a propagation along the radial direction of the lattice. We can visually identify a phase delay in the oscillation of the vertices near the surface from that of the vertices in the bulk, in particular thos close to the central axis. This effect is probably caused by small differences in the properties of the vertices depending on their nearest-neighbor interactions. The vertices lying closer to the surface of the lattice are connected to two other bulk vertices while the two other branches are dangling free ends. Contrary to this, in the case of vertices within the bulk, all the four nanowires originating from them are connected to four other vertices within the volume. This difference in nearest-neighbor interaction could be responsible for the appearance of variations of the properties of the vertices within the lattice depending on their location, which in turn could explain a phase delay between their oscillation.

To test whether a localized perturbation can propagate along the lattice in the form of a macroscopic spin wave, the elongated lattice was excited at only one of its ends with

Cubic lattice structure

In the previous section, we investigated the formation of different vertex conĄgurations in a diamond AMC and their effect on the magnonic signature of the network. To demonstrate the robustness of these observations, we expand our investigation into a different kind of AMC: the cubic lattice. The cubic lattice structure is formed by the interconnection of nanowires in a simple cubic arrangement, so that six wires meet at each vertex. Our model structure was made of cylindrical nanowires of radius 7.0 nm and length 70.0 nm, arranged in a 7 × 6 × 5 lattice, see Fig. 7.27.

Cubic lattice structure

Vertex conĄguration

In contrast to the four nanowires meeting at the vertices of the diamond structure, there are six nanowires at each vertex in the case of a cubic lattice. Adopting the same principles as before, we assign an Ising type magnetization to each nanowire. Based on the total number of Ising type moments coming in or leaving out at each vertex site, we can have seven different types of vertex conĄgurations in the cubic lattice: Out of the seven theoretically possible vertex conĄgurations listed above, the Ąrst and the last states with sixfold charges could not be observed in any of the simulations we carried out, while all the other vertex states can be stable at zero Ąeld.

In addition to these vertex conĄgurations, the dangling free ends at the surface of the network can also be assigned a charge of ±1, where the sign depends on the direction of magnetization. We note that the total charge of the system, given by the sum of volume charges and surface charges is zero. Moreover, the charge distribution at the surface is related to the charge distribution in the volume. According to GaussŠs divergence theorem, the total magnetostatic volume charge density can be expressed as D ∇ • M dV = S M • dA. Accordingly, one can conclude that the total volume charge within the bulk depends on total number of surface vertices with magnetization pointing inwards or outwards.

There are a few qualitative differences between the diamond AMC and the cubic version of an artiĄcial crystal. In addition to having different types of vertex charges, Out of all these possible vertex conĄguration,s the zero-charged vertices are the only ice-rule obeying states. Both, the ±2q and ±4q type vertices represent defect charges.

By saturating the AMC in different directions by means of external Ąelds and then relaxing to a zero Ąeld state, we could generate three qualitatively different ground states.

• Reference state: Contains only head-tail type zero charge vertices. This state can be produced by saturating the entire structure by applying a strong external Ąeld along the [1, 1, 1] direction and then relaxing to zero Ąeld.

• ±2q state: contains head-head type zero charged vertices and ±2q charged vertices. Can be prepared by saturating the structure along [1,0,0] direction and relaxing to zero.

• ±4q state: contains ±4q charged vertices along with other types listed above. Relaxing to zero Ąeld from a random-initial conĄguration results in this magnetic state.

Cubic lattice structure

A statistics of the number of different types of vertex conĄgurations in each type is given in table 7.31. The existence of more number of charge types and multiple Ćavors of each charge type in the cubic lattice opens a wider potential for manipulation. To investigate these effects, we simulated these three states and compared their highfrequency properties. As already seen in the case of buckyballs and in the diamond AMC, the introduction of defect charges in the crystal will activate additional frequency modes which are absent in the defect-free reference state. This is demonstrated by comparing the frequency response to a short-pulse excitation of the two defect states with that of the reference state. The Ş+2q stateŤ contains in total three charge types, compared to just one type in the reference state: (i) head-head type zero-charged vertices and (ii) +2q/ -2q defect charges along with the (iii) head-tail type zero-charged vertices already seen in the reference state. The presence of these two results in two additional peaks: mode 6 at 8.8 GHz corresponds to the activity at head-head type zero charges and mode 7 at 9.4 GHz corresponds to the activity of ±2q-charged vertices. It is to be noted that despite having same magnetic charge densities, the head-tail and head-head type zero charges oscillate at signiĄcantly different frequencies, because of their differences in the micromagnetic structure and consequently the difference in the effective exchange The third magnetic state, the Ş+4q stateŤ contains a pair of +4q/ -4q charges in addition to the second state. Mode 8 , which is present only in the third state, is generated by these quadruple charges. Despite there being only two quadruple charges, mode 8 is as intense as the contribution arising from the 60 zero-charged vertices in the Ąrst state. This difference in intensity is a result of a signiĄcantly higher magnonic activity of the quadruple charges. In the third state, the intensity corresponding to the oscillation of the +2q and the head-head zero-charges is increased in proportion to an increase in their number in the AMC consequently resulting in a drop of the intensity of the mode corresponding to the head-tail zero charged to almost 1/7 th . The introduction of the quadruple charges and an increase in the number of the double charges in the cubic lattice combined has a strong impact on the wire modes. In this case, the modes are further broadened and intertwined, becoming basically indistinguishable from one another.

Comparing the frequency of oscillation of different types of charged vertices, we can see that, in general, all vertices with the same charge type oscillate at the same

Chapter summary and outlook

To summarize, we have conducted detailed Ąnite-element micromagnetic simulation studies on three-dimensional diamond and cubic-type artiĄcial magnonic crystals. We looked into the details of the magnetization structures developing at the vertices of these structures. All our simulations yielded axially magnetized nanowires at equilibrium states, resulting in a spin-ice behavior in these three-dimensional arrays. Based on this observation and a corresponding Ising-type spin assignment to the indivdual nanowires, we could classify different vertex conĄgurations with varying levels of magnetic frustration and energetics. We could identify and automatize the recognition of these vertex conĄgurations based on the magnetostatic volume charge densities they produce. The quasistatic magnetic properties of the diamond AMC in an external 7.11 Chapter summary and outlook magnetic Ąeld were studied by simulating hysteresis loops. We found that the magnetic switching in these structures involved the abrupt reversal of individual nanowires, resulting in Barkhausen-type jumps in the simulated hysteresis curves. With such hysteretic studies, we also demonstrated how different magnetic conĄgurations Ű including in particular magnetic defect structures Ű can be selectively generated in the lattice by saturating the structure at various crystallographic directions and then relaxing to a zero Ąeld state. In addition to creating magnetic defects by external Ąelds, we could also inject frustrated states by manipulating the geometry of the lattice through the insertion of a structural defect. We simulated the small-angle precessional modes within these AMC and, by means of inverse Fourier transform, extracted and analyzed their magnonic origins. In general, the modes can be classiĄed into two types: the fundamental modes including the surface, vertex, and wire modes which are related to corresponding geometric constituents, and the defect modes which originate from the magnetic defects within the AMC. We could see that the formation of magnetic defects gives rise to distinct additional modes and yields unmistakable Ąngerprints in the magnonic spectrum. We veriĄed the robustness of these results by extending a similar investigation into the cubic AMC and found that the general observations are valid there as well. The existence of a larger number of vertex conĄgurations and even multiple Ćavors of the same charge type in the cubic lattice offers additional methods to control the frequency properties.

The phenomena of magnetic defects creating additional frequency peaks in the spectra of these AMC are analogous to the changes in the optical spectra of natural diamonds by the presence of chemical defects [START_REF] Mäki | Properties of optically active vacancy clusters in type IIa diamond[END_REF][START_REF] Aharonovich | Diamond nanophotonics[END_REF]. The ability to control the formation of these defects by Ąeld and geometry manipulation opens up an entire new Ąeld for spin manipulation using these artiĄcial magnonic crystals. The re-conĄgurability offered by the artiĄcial spin ices, when combined with the advantages of magnonic crystals, makes these new architectures an promising tool for magnonic applications.

Moreover, this behavior of the magnetic charges can be exploited as an indirect means to detect the presence or absence of these defects in an AMC network. This type of investigations using Ąnite-element simulations on the high-frequency properties of interconnected three dimensional networks has not been reported extensively in the literature; and though these results are rather convincing and promising, they require experimental conĄrmation. We anticipate that our Ąndings, which demonstrate that

Chapter summary and outlook

high-frequency modes can be controlled through both the 3D geometry and the magnetic structure of an AMC, will inspire experimental groups to study the magnonic properties of these structures and will further promote three-dimensional interconnected nanowire arrays as a new category of magnonic metamaterials with promising potential for applications in devices for storage, magnonic operations [START_REF] Burks | 3D Nanomagnetism in low density interconnected nanowire networks[END_REF] [251], and neuromorphic computations [START_REF] Grollier | Neuromorphic spintronics[END_REF][START_REF] Burks | 3D Nanomagnetism in low density interconnected nanowire networks[END_REF].

Three-dimensional Sierpinski structures

In this thesis, we conducted extensive micromagnetic Ąnite-element simulations to study the magnetization structures and the resulting high-frequency properties of different types of three-dimensional nano-architectures. The initial part of this research was concerned with the development of a python-based postprocessing software; a Fourier analysis tool which was necessary to carry out the frequency analysis of these studies.

The working principles and the usage of this tool is explained in Chapter 2. This Fourier analysis software was helpful in identifying and isolating the high-frequency modes developed in the structures which we simulated. The Ćexibility of python permitted to modify the code effortlessly based on the concerned problem and to carry out advanced analysis on the results.

We investigated primarily four different types of geometries, 1. Three-dimensional fractal Sierpinski structures 2. Three-dimensional buckyballs 3. Three-dimensional diamond AMC 4. Three-dimensional cubic AMC

Three-dimensional Sierpinski structures

A Sierpinski tetrahedron is formed by four tetrahedrons arranged on the vertices of a larger tetrahedron. Higher iterations of the fractals are generated by dividing the individual tetrahedrons into smaller units. We kept the overall size of the fractal constant at 512 nm and simulated Ąve generations of the fractal. The individual tetrahedron units in the Ąrst two stages of the fractal were large enough so that their equilibrium magnetization structure at zero Ąeld consisted of the formation of a three-dimensional vortex structures while for the higher stages the individual units existed in a single domain state and the Ćux closure was achieved by the formation of of vortex-like structure were the magnetization of the individual units curl together in a ring. The magnetization reversal of these structures was investigated by simulating their hysteresis loops.

The high-frequency magnonic response of the relaxed magnetic conĄgurations of these fractals to a small perturbation was simulated by exciting these states with a Gaussian pulse, recording the magnetic ring-down for an extended period of time, followed by a discretized Fourier analysis. We could see that the fractal nature of these geometries

Buckyball nano-architectures

had a considerable impact on their magnonic spectrum. Especially, in the third and fourth stages we could see the appearance of a wide-band frequency response stretching across several GHz instead of individual isolated peaks. This property of these fractals has potential applications in a wide range of Ąelds, such as microwave absorption for telecommunication antennas and for the development of radar absorbing materials for military applications.

Buckyball nano-architectures

The Buckyball nano-architectures are made of 90 cylindrical nanowires, interconnected at 60 spherical vertices so that three nanowires meet at each vertex to form a spherical network of interconnected hexagons and pentagons. The Buckyball can be regarded as a model system which depicts a transition from two to three dimensional spin-ice systems. To restrict the parameter space, we Ąxed the ratio between the length of the nanowire L, the radius of the nanowire R and the radius of the spheres S at the vertices to ratio 25:3:4 and simulated a wide range of Buckyballs by varying the side length L from 25 nm to 250 nm. We investigated their relaxed magnetic states, different types of vertex conĄgurations, their hysteretic properties and high-frequency magnonic excitation modes. In the size ranges we studied, the individual nanowires were magnetized in single domain state, i.e., they are magnetized along the axis. A situation of interest, from a micromagnetic perspective, arises at the vertex points where three such nanowires meet. Depending on the direction of magnetization of each nanowire and the Ising-like dipole moment of each nanowire, we could identify different vertex conĄgurations. We noted that these structures exhibit artiĄcial spin ice behavior, including the existence of different degenerate vertex conĄgurations, the appearance of monopole-like magnetic defects, and a quasi-continuum of nearly degenerate states with different spatial distributions of defect structures. Based on the direction of the magnetization of the three nanowires meeting at a vertex, we distinguished two different types of vertex conĄgurations: the ice-rule obeying single-charged vertices and the defect-type triple charges with monopole-like properties. Owing to the three-dimensional nature of the buckyballs structures, the defect-type triple charges could be generated and removed in a controlled way by means of a suitable sequence of external magnetic Ąelds. We could thus selectively generate two qualitatively different magnetic states in such a buckyball: a defect-free state which only contained ice-rule obeying single charged vertices and a

Three-dimensional diamond AMC

defect-state which contained a pair of triple charge vertices along with single-charge vertices.

We simulated the small-angle precession modes of the relaxed magnetic conĄgurations and compared the results. There exists two different magnetic modes in these buckyballs based on their geometric origin: the vertex modes which are localized at the vertices and the higher-frequency wire modes which can be attributed to the formation of standing waves in the nanowires. We could see that the presence of triple-charged vertices results in the appearance of a characteristic, intense low-frequency mode in the magnonic spectrum. This is a potentially important observation, as this feature in combination with the possibility to insert and remove the triple charges reposnible for this peak through external Ąelds opens up a pathway for re-conĄgurable magnonic applications. This effect can also be exploited as an indirect tool to probe the magnetic conĄgurations in such nanoarchitectures. The robustness of the results was veriĄed by simulating hollow buckyballs made of nanotubes and spherical shells. In the simulation of these geometries we observed that, in the concerned length-scales, the hollow structures exhibited a quite similar behavior to that of the solid structures.

Three-dimensional diamond AMC

The diamond lattice was formed by a three-dimensional network of cylindrical nanowires so that four nanowires meet at each vertex point in a tetragonal orientation. The vertex points where four such wires meet are distributed in space corresponding to the atomic locations of a natural diamond lattice -hence the name. The network we simulated was made of nanowires of length 70 nm and thickness 14 nm. The extended network Investigation into the magnonic excitations of the lattice revealed the existence of surface modes, vertex modes and wire modes based on the geometric origin of the active regions. The presence of defect charges created additional low frequency modes which were absent in the reference state. We could also create additional modes in the spectrum by introducing geometric defects in the lattice. Since the defects can be used to manipulate the magnonic spectrum and their formation could be controlled, these types of periodic AMC networks offers immense potential for magnonic applications.

Cubic AMC

The cubic lattice is formed by the interconnected network of cylindrical nanowires so that six wires meet at each vertex in an orthogonal way. The wires had a similar dimension to that of the diamond AMC and they exhibited similar single domain be-

Summary

To summarize, we carried out fundamental investigations on the static and magnonic properties of several three-dimensional nanoarchitectures. The geometric Ćexibility of our Ąnite-element approach allowed an accurate modelling of their complex geometries, especially the curved surfaces and the interior surfaces of the hollow structures. We

  ln view of the foregoing, I hereby certify that the work described in my thesis manuscript is original work and that / have not resorted to plagiarism or any other form of fraud.

Numériquement, lŠévolution temporelle

  du vecteur dŠaimantation peut être obtenue en intégrant lŠéquation LLG dans le temps. Toutes les simulations abordées dans cette th' ese sont réalisées avec le logiciel de simulation micromagnétique par éléments Ąnis tetmag, qui a été développé ' a lŠIPCMS par v Riccardo Hertel. Le logiciel tetmag prend en entrée le mod' ele ' a éléments Ąnis de la géométrie, les param' etres du matériau ferromagnétique et les conditions externes telles que les champs ou les courants appliqués, et produit en sortie le champ vectoriel de magnétisation ' a chaque point en fonction du temps. Le logiciel tetmag utilise des techniques mathématiques avancées telles que les schémas de compression matricielle hiérarchique, qui permettent de simuler de grandes géométries tout en utilisant des ressources numériques modestes. Il met également en İuvre un calcul massivement parall' ele en utilisant lŠaccélération GPU, ce qui permet dŠatteindre des vitesses de simulation rapides, en particulier dans le cas de probl' emes ' a grande échelle impliquant des millions de cellules de discrétisation. Une structure dŠaimantation dŠéquilibre statique dans la géométrie concernée est obtenue en simulant lŠévolution dissipative de lŠaimantation dans le temps jusquŠ' a ce quŠun minimum énergétique soit atteint. Pour étudier les effets dynamiques tels que les modes de précession ' a petit angle et la propagation des ondes de spin, les structures dŠaimantation détendues sont excitées avec une perturbation appropriée telle quŠune impulsion de champ gaussienne et la relaxation oscillatoire de lŠaimantation (un phénom' ene appelé ŞringdownŤ magnétique) depuis cet état excité jusquŠau minimum énergétique est enregistrée sous forme de série chronologique. La dynamique de lŠaimantation dans un tel processus de ringdown contient généralement plusieurs signaux haute fréquence différents superposés, et il est généralement impossible dŠidentiĄer les composantes de fréquence par de simples inspections visuelles. La partie initiale du doctorat a consisté ' a développer un code de transformation de Fourier en Python qui peut identiĄer et extraire les modes de fréquence individuels dŠune telle superposition. Avant de les utiliser pour étudier la dynamique haute fréquence de nanoarchitectures tridimensionnelles, ces techniques ont été appliquées sur un syst' eme plus traditionnel : un ensemble de nanodisques cylindriques. Cette étude a été réalisée en collaboration avec un groupe expérimental de lŠUniversité Versailles. Les coll' egues expérimentateurs ont fabriqué un réseau bidimensionnel de nanodisques en cobalt et ont mesuré les modes haute fréquence de la structure en fonction du champ appliqué au moyen dŠun dispositif vi VNA-FMR. Dans les données expérimentales, on a pu identiĄer deux modes de fréquence proéminents. En effectuant des simulations par éléments Ąnis, nous avons pu reproduire cette oscillation ' a deux fréquences proches et iden-tiĄer le proĄl spatial relatif ' a ces modes. LŠexcitation de fréquence plus basse est un mode localisé sur le bord, dans lequel lŠoscillation de lŠaimantation aux deux extrémités opposées du nanocylindre est synchronisée. Le deuxi' eme mode, de fréquence plus élevée, a pu être identiĄé comme une oscillation plus complexe de lŠaimantation au centre du nanocylindre. En effectuant des simulations de ce type ' a différentes intensités de champ, nous avons pu reproduire numériquement les fréquences des deux modes en fonction du champ obtenues dans les expériences de FMR. La fréquence des deux modes augmente presque linéairement avec lŠintensité du champ externe appliqué. En raison des limitations des techniques de mesure, il y avait une incertitude concernant la valeur des param' etres du matériau, comme lŠaimantation ' a saturation. En effectuant un grand nombre de simulations et en faisant correspondre la fréquence de ces modes aux données expérimentales, nous avons pu obtenir indirectement ces param' etres matériels ' a partir de nos simulations micromagnétiques. Les premi' eres structures artiĄcielles tridimensionnelles que nous avons étudiées sont des géométries 3D fractales de tétra' edres du type Sierpinksi. Ces structures se composent de quatre tétra' edres disposés de mani' ere récursive aux sommets dŠun tétra' edre plus grand. Des itérations ou étapes supérieures des fractales ont été générées en divisant les tétra' edres individuels en unités plus petites. Ces structures fractales peuvent être considérées comme une version tridimensionnelle des triangles de Sierpinski bidimensionnels plus connus. Dans des études précédentes rapportées dans la littérature, de telles nanostructures fractales ferromagnétiques bidimensionnelles ont été étudiées expérimentalement et ' a lŠaide de simulations micromagnétiques aux différences Ąnies. Il a été démontré que leur structure auto-similaire et lŠexistence de différentes échelles de longueur ont un effet considérable sur leurs propriétés, notamment en ce qui concerne leur réponse en fréquence.

Figure 1 :

 1 Figure 1: La structure de lŠaimantation du troisi' eme stade de la fractale : Tous les tétra' edres individuels sont aimantés dans une structure semblable ' a une forme ŞΛŤ. (c) Vue agrandie de lŠamas formant une structure tourbillonnaire déconnectée. (d) Une perspective différente montrant la structure en forme de ŞAŤ de lŠamas.

Figure 2 :Figure 3 :

 23 Figure 2: Spectres de Fourier des différentes étapes ' a champ zéro.

Figure 4 :

 4 Figure 4: Spectre dŠexcitation magnétique ' a haute fréquence de lŠétat de charge simple et triple, avec le proĄl spatial correspondant des modes.

Figure 5 :

 5 Figure 5: (a) Mod' ele par éléments Ąnis du réseau de diamants utilisé pour la simulation (b).

Figure 2 . 1 :

 21 Figure 2.1: Schematic description of the magnetization dynamics described by the LLG equation. The sketch represents the damped precessional motion of the magnetization vector M (r, t) around the effective Ąeld H. The precessional term is shown in red and the damping term in blue.

( 2 .

 2 [START_REF] Maruo | Three-dimensional microfabrication with two-photon-absorbed photopolymerization[END_REF] on the one hand and byGilbert (2.28) on the other hand can be reconciled and interpreted as merely two different representations of the same equation. If the Gilbert form is reformulated in an explicit form, taking in to account that M • dM /dt = 0, one obtains the so-called Landau-Lifshitz-Gilbert equation, given by

Figure 2 . 2 :

 22 Figure 2.2: Comparison between a Bloch and a Néel wall. In Bloch wall the magnetization is rotated 180 degree around the normal of the domain wall while in the case of a Néel wall the magnetization is rotated in the plane of the Ąlm.[101]

Figure 2 . 3 :

 23 Figure 2.3: Micromagnetic simulation of a junction of four Permalloy nanowires of thickness 14 nm. The opposite wires are magnetized in anti-parallel direction and a combination of a head-to-head and tail-to-tail domain wall is formed at the interface junction

Figure 2 . 4 :

 24 Figure 2.4: Micromagnetic simulation of a three-dimensional vortex formed in a Permalloy nanosphere of radius 100 nm

Figure 2 . 5 :

 25 Figure 2.5: Shape anisotropy of an elongated structure (a) magnetization along the easy direction resulting in a comparatively lower surface charge formation (b) the energetically unfavorable conĄguration, resulting in a higher demagnetising energy due to the formation of higher amount of surface charges.

Figure 2 . 6 :

 26 Figure 2.6: Crystalline structure of water-ice. Each Oxygen atom is surrounded by four Hydrogen atom: two are bound by covalent bond and the remaining two by hydrogen bonds with the neighboring H 2O molecule

Figure 2 . 7 :

 27 Figure 2.7: The two familiar artiĄcial spine ice lattices square ASI and Kagome ASI along with the 16 theoretically possible vertex conĄgurations in a square ASI[START_REF] Skjžrvø | Advances in artiĄcial spin ice[END_REF]

Figure 3 . 1 :

 31 Figure 3.1: Comparison of a Ąnite-element mesh of a sphere (a) with a uniform grid approximation of the same shape. The geometric Ćexibility of the Ąnite element method method allows

Figure 3 . 2 :

 32 Figure 3.2: Example of a shape function in 1D. (a) A continuous function (blue) is approximated by a piece-wise linear function (red) (b) The function is represented by means of a basis function (blue) and their linear combination (red)

13 )

 13 integrals over the basis functions. Using the representation of a function u(r) according to eq. (3.1), the integral of u(r) over a tetrahedral element e of volume V e can be written as The second part of the integral is simple: N e i dV = V e /4. This denotes the fraction of the volume of the tetrahedron shared by each node i. Thus, eq. (3.13) becomes Ve ũ

. 28 )

 28 It is worth noting that the integral on the right-hand sideVc ∇N e,i ∇N e,j dV = K e i,j =x,y,z α c e iα c e jα V e(3.29)

  Where ρ = -∇ • M is the magnetostatic charge density. The normal vomponent of the magnetostatic Ąeld is discuntinuous at the boundary surface ∂(Ω) of the volume containing the magnetic material (Ω), which yields the condition∂U in (r) ∂n -∂U out (r) ∂n = M • n (3.37)for the normal derivatives of the potential at the urface, where the superscripts ŞinŤ and ŞoutŤ represent the inner and outer limits of the derivative at the boundary surface ∂Ω, respectively, and n is the normal vector oriented towards the outside. To solve for (3.36) inside the magnetic volume Ω, we follow the ansatz by Fredkin and Koehler to split U (r) into two parts U (r) = U 1 (r) + U 2 (r), each with speciĄc properties. The Ąrst part U 1 is the solution of the Poisson equation inside the magnetic volume Ω

41 )

 41 This technique of splitting U into two parts allows us to calculate U by only having to solve inside the region Ω, which is otherwise not possible as for the original problem the boundary condition at the surface ∂Ω is unknown. The Ąrst part U 1 is calculated by solving the Poisson-Neumann problem (3.38) by converting it to the weak form, as discussed before in the case of the exchange Ąeld V ψ∆U 1 dV = V ψ∇M dV (3.42)

. 46 )

 46 Even though equation(3.46) is valid everywhere inside the body, its computation at all discretization point is computationally expensive. Hence, it is integrated only at the boundary surface to obtain the necessary Dirichlet boundary condition to solve 3.45. This results in the following integral connecting the values of U 1 with those of U 2 at the surface[START_REF] Koehler | Finite element methods for micromagnetics[END_REF]:

3. 5

 5 Simulation method: working of tetmag relaxation simulations where a large time-step can be used. The second (CVODE) solver is more stable but slightly slower. It is mostt suitable for the accurate simulation of time-dependent dynamics with a low value of the Gilbert damping. Both solvers can exploit acceleration through massive parallelization, both thread-based as well as GPU based.

53 )

 53 The function F Ű the Fourier transform function described in equation (3.52) Ű describes a transformation from the time domain to the frequency domain. If f (t) is a periodic function of time, the corresponding Fourier transform g(ω) contains information about f (t) that can be decomposed into individual sinusoidal signals of different amplitude. This transformation is widely used in digital signal processing (DSP) to extract individual frequency components from a mixed signal. The discrete Fourier transform (DFT) is the discrete equivalent of the continuous Fourier transform, deĄned as X (ω k ) ≜ N n=0 x (t n ) e iω k tn , k = 0, 1, 2, . . . , N -1 (3.54)

Figure 3 . 3 :

 33 Figure 3.3: (a) Snapshot of the oscillation of m(r, t), deĄned at each node of the Ąnite-element mesh (b). The time-series of m i (r, t) of the magnetization at a node i is a superposition of several oscillations (c,d). The oscillation at the i th node can be Fourier-transformed to a power spectrum, from which three modes can be identiĄed. (f) A windowed power-spectrum is generated by Ąltering out the Ąrst mode and suppressing the higher modes. From this windowed Fourier spectrum, an inverse Fourier transform (g) yields the isolated dynamic proĄle of the desired frequency at this node. Repeating this process for all the N nodes will give the overall spatial proĄle of the desired frequency mode (h).

Figure 4 . 1 :

 41 Figure 4.1: VNA-FMR data of Co/Ag nanodot arrays, with Ąeld applied along the plane of the nanodots [183]

Figure 4 . 2 :

 42 Figure 4.2: Finite element mesh of the nano-disc with thickness 20 nm and radius 100 nm. The unstructured grid contains 10,750 nodes and 59,780 tetrahedrons

4 . 3 .

 43 Relaxing the Ąeld of 200 mT to 200 mT from a uniformly saturated state results in a ŞonionŤ-type [187] magnetic state, see Fig. 4.3(c), in which the center part of the disc is magnetized along the Ąeld direction, while the magnetization at the edges curves along the surface to form a structure resembling the cross-section of an onion. As the Ąeld is further reduced, the outermost layer that is curved along the surfaces grow in size. Below a Ąeld of 2 mT a vortex state develops, which stays stable until a negative Ąeld of about 130 mT is applied. Simulated magnetization structures of the disc at various Ąelds are displayed in Ągure 4.4. As a next step, we conducted dynamic simulations in an attempt to reproduce the experimental FMR results. Using the perturbation methods described above, small-angle precession modes of the disc were stimulated at various Ąelds.

Figure 4 . 3 :

 43 Figure 4.3: Simulated hysteresis loop of the Cobalt nanodot thickness 50 nm and radius 100 nm and a saturation magnetization of 1.1 T

Figure 4 . 5 :

 45 Figure 4.5: Power spectrum showing the typical frequency response of the disc at 400 mT at a damping of 0.1 and a simulation time period of 10 nanoseconds.

Figure 4 . 6 :

 46 Figure 4.6: Extracted spatial proĄle corresponding to the oscillations at the two main frequencies. Yellow colored region are the anti-nodes, where the oscillation amplitude is maximal and the blue colored areas are the inactive regions (arbitrary units are used). (a) Edge mode oscillation of the lower-frequency mode (b) Oscillation of the higher-frequency mode.

Figure 4 . 7 :

 47 Figure 4.7: The variation of the frequency of the modes with change in saturation magnetization

Figure 5 . 1 :

 51 Figure 5.1: (a) A two-dimensional Sierpinski triangle (b) Stage 1 of the 3D fractal composed of four tetrahedrons of side length 256 nm (c,d,e,f) Stage 2,3,4 and 5 of the fractal structure respectively.

Figure 5 . 2 :

 52 Figure 5.2: Change in the volume and surface/ volume ratio with change in stage

Figure 5 . 3 :

 53 Figure 5.3: Relaxed magnetization structure at zero Ąeld of the stage one fractal (a) M x component (b) Arrows represents the direction of the reduced magnetization m at each discretization point, vortex structures devolved at the faces perpendicular to the direction of saturation can be identiĄed. (c) Perspective view on the ŞAŤ-like structure.

Figure 5 . 4 :

 54 Figure 5.4: Magnetization structure of the second stage (a) M x component (b) arrow representation. Formation of a disconnected vortex structure can be observed on the central cluster.

Figure 5 . 5 :

 55 Figure 5.5: Magnetization structure at the third stage: All the individual tetrahedrons exist in an ŠΛŠ like structure. (c) Zoomed-in view of the cluster forming a disconnected vortex structure. (d) A different perspective showing the ŞAŤŠ-shaped structure of the cluster.

Figure 5 . 6 :

 56 Figure 5.6: Magnetic structure of the fractal at the fourth stage: the local clusters forming the disconnected vortices are sub-divided into smaller units, however the structure retains its shape.

Figure 5 . 7 : 5 . 5 Figure 5 . 8 :

 575558 Figure 5.7: Stage 5, the individual tetrahedrons are small enough to form a single domain structure (c) Zoomed-in view showing the local-clusters retaining the vortex structure (d) and the lateral ŞAŤ-shaped structure.

Figure 5 . 9 :

 59 Figure 5.9: Fourier spectra of the different stages at zero Ąeld.

Figure 5 . 10 :

 510 Figure 5.10: Vortex core gyration of the stage one fractal at 700 MHz. The color code represents the change in the z component of magnetization in time.

Figure 5 . 11 :

 511 Figure 5.11: Simulated heatmaps showing the Ąeld dependence of stage 2,4 and 5.

6. 1

 1 Geometry and Ąnite element modelling vertex to form a three-dimensional Y-type junction with planar angles 108°and 120°.

Figure 6 . 1 :Fig. 6 . 1 .

 6161 Figure 6.1: (a) a Ąnite element model (FEM) of the Buckyball (b) Zoomed-in view of the mesh demonstrating the accurate modelling of the geometry and its curved surfaces.

Figure 6 . 2 :

 62 Figure 6.2: (a) Magnetization structure at zero Ąeld of a Buckyball of wire length 100 nm and thickness 24 nm (b) Zoomed in view showing the axial magnetization of the wires and the magnetization at the vertices, yellow arrows represent the total magnetization direction of each nanowire

Figure 6 . 3 :

 63 Figure 6.3: Four types of magnetic conĄgurations at the vertices of a buckyball structure: (a) one-in / two-out (b) two-in / one-out (c) three-in (d) three-out.

Figure 6 . 4 :

 64 Figure 6.4: Comparison of different visualization methods: (a) The magnetization vector at each discretization point is represented by arrows, (b) the y component of the magnetization is plotted with a color code, (c) The value of ∇ • m at each point is plotted. By plotting the divergence of the magnetization, the different types of vertex conĄgurations can be easily identiĄed by means of the color the magnitude of the charge is given by the intensity of the color. Bright purple and green spots indicate +3, -3 charges respectively and mild spots indicate +1/ -1 charges.

Figure 6 . 5 :

 65 Figure 6.5: Plot of the average value of ρ with vertex indices, we can see that there exists four types of vertices and the number of vertices in each type.

6. 4

 4 Static magnetic conĄgurations states, whose structure remains almost the same in all size ranges. The magnetization structure of triple-charge vertices shows a gradual progression from a triple head-tohead domain wall structure in the smallest buckyballs to a three-dimensional vortex conĄguration in the largest structure, see Fig. 6.6.

Figure 6 . 6 :

 66 Figure 6.6: Progression of the triple-charge with size of the sphere (a) 16 nm (b) 32 nm (c) 48 nm (d) 64 nm (e) 80 nm (f) Comparison of the evolution of the average value of the magnetostatic volume charge density at the spheres for the triple charge vertices with an increase in sphere radius

Figure 6 . 7 :

 67 Figure 6.7: Simulated hysteresis loop, including a minor loop, of a buckyball with L : R : S = 100 nm : 12 nm : 16 nm.

Figure 6 . 8 :

 68 Figure 6.8: (a) Onion state at 500 mT (b) Zoomed-in view showing a canted magnetization in the wires.

Figure 6 . 9 :

 69 Figure 6.9: Different steps in the annihilation of the triple charges. Initially, the triple charge vertex at 0 mT is locate on the upper left (a). The yellow arrows represent the magnetization directions of each wire. (b) Transformation to a single-charge vertex via the emission of a domain wall at -50 mT. (c) Final state free of triple-charges, with rearranged charge distribution. Branches that are Ćipped are highlighted with red arrows. (d) Single charged state at 0 mT obtained through a minor loop.

Figure 6 .

 6 Figure 6.10: (a) Comparison of hysteresis curves of Buckyballs of different sizes. The plots show the reduced magnetization component M ∥ /M s along the Ąeld direction. (b) With increasing size, both the remanence and the coercive Ąeld strength tend to decrease.

Figure 6 . 11 :

 611 Figure 6.11: Power spectrum of the Fourier-transformed magnetic oscillations in single-charged state at zero Ąeld of a buckyball with L = 100 nm.

Figure 6 . 12 :

 612 Figure 6.12: Oscillation of the vertices (a) Mode 1 (b) Mode 2 in single-charge state. (light green areas indicate higher magnonic activity).

Figure 6 . 13 :

 613 Figure 6.13: Oscillation of the nanowires (a) Mode 3 (b) Mode 4 and (c) Mode 5 in Singlecharge states

Figure 6 .

 6 Figure 6.16: (a) Variation of the frequency of the oscillation of the single and triple charge vertices with size of the buckyballs. (b) The oscillation frequency of the single-charged vertices splits into two modes.

Figure 6 . 17 :

 617 Figure 6.17: Variation of the frequency of the triple charges in the presence of an external Ąeld applied along the axis connecting the two triple-charge vertices.

1 .

 1 cylindrical solid nanowire of length L = 100 nm, radius R = 12 nm 2. cylindrical open tube of length L = 100 nm, R i = 8 nm, R e = 12 nm 3. hollow nanotube with closed ends L = 100 nm, R i = 8 nm, R e = 12 nm 4. solid nanotube with spheres attached on both ends L = 100 nm, R = 12 nm, S = 16 nm 5. hollow nanotube with hollow spherical shells attached on both sides, L = 100 nm,

Figure 6 . 18 :

 618 Figure 6.18: Relaxed magnetization structure at zero Ąeld of a solid cylindrical nanowire of length L = 100 nm, radius R = 12 nm (a) and of a solid nanowire with two spheres attached at the end S = 16 nm(b).

Figure 6 .

 6 Figure 6.19: (a) Hollow open nanotube (L = 100 nm, R i = 8 nm, R e = 12 nm), (b) hollow nanotube with two spherical shells attached at the ends (L = 100 nm, R i = 8 nm, R e = 12 nm), and (c) hollow nanotube with closed ends capped by disks (L = 100 nm, R i = 8 nm, R e = 12 nm).

Figure 6 . 20 :

 620 Figure 6.20: Comparison of the frequency of he primary modes of various hollow and sloid structures with different geometries.

Figure 6 . 22 :

 622 Figure 6.22: Relaxed magnetization structure of a hollow buckyball with side length, L = 100 nm (a) the charges at vertices are visualized by plotting the volume charge density ∇ • m (b) The local magnetization is plotted by means of arrows (c) Zoomed-in view of one of the triple charged vertices (d) Zoomed-in view of a single charge vertex structure.

Figure 6 . 23 :

 623 Figure 6.23: Evolution of the triple charge stucture, from a triple-head domain wall structure to a vortex-like state. Stable triple charges at different geometries (a) L = 100 nm, (b) L = 125 nm (c) L = 150 nm, (d) L = 175 nm.

Figure 6 .

 6 Figure 6.24: (a) Simulated hysteresis curve of the hollow Buckyball with a tube length of L = 100 nm (b) Comparison of the variation in remanence ratio and the coercive Ąeld strength with size

Figure 6 . 25 :

 625 Figure 6.25: Comparison of the frequency response of the triple-charges states of the solid and hollow Buckyballs with side length L = 100 nm.

Figure 7 . 1 :

 71 Figure 7.1: (a) Finite element model of a diamond-type structure. (b) Enlarged view of an individual tetrapod unit representing the building block of the artiĄcial diamond structure.

Figure 7 . 2 :

 72 Figure 7.2: (a) Two-in/two-out state (b) One-in/three-out state (c) Three-in/one-out state

7. 2

 2 Figure 7.4: (a) The micromagnetic structure of the one-in/three-out state (b) Zoomed-in view highlighting the tail-tail domain wall structure at the vertex

Figure 7 . 6 :

 76 Figure 7.6: Micromagnetic structure of two different Ćavors of the two-in/two-out state in a 2D square ASI: (a) head-tail state, (b) head-head state

Figure 7 . 7 :

 77 Figure 7.7: Automatized detection of the charge type, number and position of each type of vertex conĄguration based on the value of ∇•m at each vertex. Different types of conĄgurations can be uniquely identiĄed by clustering the vertices based on the average value of the local divergence (volume charge density) at each vertex.

7. 3

 3 Hysteretic properties of the crystal: zero charged state state which only contains double charges: All the vertices within the crystal display the magnetic structure of ice-rule violating states, resulting in a periodic distribution of alternate layers of +2q and -2q vertices. Relaxing from a random initial conĄguration results in a disordered state, which contains both zero charged vertices and double charged vertices distributed randomly within the AMC. The disordered state which we simulated and used for all the upcoming discussions contained 48 zero-charged vertices and 35 double-charged vertices, out of which 17 are of +2q and 18 are of -2q type. A detailed description of the number of each charge type in each lattice type is given in table

Figure 7 . 9 :

 79 Figure 7.9: Simulated hysteresis loop of the diamond AMC lattice

Figure 7 . 10 :

 710 Figure 7.10: Frequency spectrum of the zero-charge state. Different prominent modes are marked from 1 to 5.

Figure 7 .

 7 Figure 7.11: (a) Mode 1: 9.9 GHz, Oscillation of the +1 and -1 charges at the free ends of the lattice. (b) Zoomed-in view showing the activity at the ends.

Figure 7 .

 7 Figure 7.12: (a) Mode 2: 14.1 GHz, Oscillation of the ice-rule obeying zero-charged vertices (b) Zoomed in view showing the activity of the vertices

Figure 7 . 14 :

 714 Figure 7.14: Comparison of the magnonic spectrum of the zero-charge and double-charge state

Figure 7 . 15 : 7 . 6 7 . 6

 7157676 Figure 7.15: Spatial proĄle of the vertex modes in the disordered state (a) Activity of the zerocharge vertices. (b) Activity of the double-charge vertices. It can be visually observed that the two modes are complementary, meaning that the regions active at one frequency are inactive at the other.

Figure 7 .

 7 Figure 7.16: comparison of the magnonic response of all the 0/0 type nanowires, with the magnonic response of the whole structure overlayed on top (green) for reference. Each thin, blue line corresponds to a different 0/0 type nanowire.

Figure 7 . 17 :

 717 Figure 7.17: Comparison of the magnonic spectrum of 0/0, +1/0 and -1/0 type nanowire groups.

7. 7 Figure 7 . 18 : 7 . 8

 771878 Figure 7.18: Comparison of the magnonic response of all twelve types of wires

Figure 7 . 19 :

 719 Figure 7.19: Field response of the frequency of the modes in the case of (a): an external Ąeld applied parallel to the direction of effective magnetization of the lattice and (b) a Ąeld applied perpendicular to the direction of effective magnetization.

Figure 7 .

 7 Figure 7.20: (a) Diamond lattice structure with a geometrical defect (yellow circle). (b) Zoomed-in view highlighting the defect vertices with coordination number 3.

7. 8 Figure 7 . 21 :

 8721 Figure 7.21: Magnonic spectrum of a diamond-type AMC lattice with two geometrical defects.

Figure 7 . 22 :

 722 Figure 7.22: Spatial proĄle of various modes in the AMC lattice with structural defect. (a) Oscillation of the ±1 charges at the ends, (b) oscillation of the ±1.5 charges at the defect vertices (the inset shows a zoomed view on the defect vertices), and (c) oscillation of the zero-charged regular vertices, with the inset showing a zoomed view on the defect vertices.

7. 9 Figure 7 . 23 :

 9723 Figure 7.23: Time evolution of the vertex mode in an elongated diamond-type AMC. The oscillation appears to be propagating from the core towards the surface.

Figure 7 . 24 :

 724 Figure 7.24: Frequency response of the elongated lattice upon excitation by a Ąeld pulse. The vertex modes which are of interest are marked.

7. 10

 10 Cubic lattice structure a position-dependent Gaussian Ąeld pulse. As usual, the following dynamical response was recorded and the oscillation of the vertices was extracted. The time evolution of the oscillation of the vertices is displayed in Fig.7.[START_REF] Fernández-Pacheco | Three dimensional nanomagnetism[END_REF]. We can see that the oscillations of the vertices do not propagate through the lattice. This suggests that the magneto-dipolar coupling between the vertices is not strong enough for an efficient energy transfer and for a related spin-wave propagation.

Figure 7 . 25 :

 725 Figure 7.25: Time evolution of the vertex modes of the elongated lattice with the corresponding lateral view. Even though there is no propagation along the length of the lattice, we can see a radially propagating behaviour from the centre of the lattice towards the exterior region.

Figure 7 . 26 :

 726 Figure 7.26: Oscillation of the vertices after being excited locally. We can see the development of a standing wave, but the wave does not propagate through the lattice.

1 .Figure 7 . 27 :

 1727 Figure 7.27: Geometry and Ąnite element mesh of a cubic lattice structure

7. 10 Figure 7 . 28 : 123 7. 10

 1072812310 Figure 7.28: Two different types of zero state conĄguration (a) Head-head state containing a pair of head-head/tail-tail domain wall, (b) Head-tail state with a magnetization structure describing a continuous Ćow across all three branches.

Figure 7 . 29 :

 729 Figure 7.29: Defect-type double-charge vertex conĄgurations: (a) +2q charge formed by a twoin/four-out conĄguration. The two incoming wires are orthogonal to each other, the other possible state in which the two incoming wires are mutually opposite is not observed in the simulations. (b) -2q charge formed by the inverse, four-in/two-out conĄguration.

Figure 7 . 30 :

 730 Figure 7.30: Defect quadruple charge conĄgurations: (a) +4q formed by the Ąve-out/one-in state (b) -4q charge pattern formed by the Ąve-in/one-out state. There is no multiplicity for quadruple charges.

Figure 7 . 31 :

 731 Figure 7.31: Overview of the number of different types of charged vertices in cubic lattices with different macroscopic magnetization state, as described in the main text.

7.10 Cubic lattice structure 7 . 10 . 2

 7102 Magnonic excitationsHigh-frequency small-angle precession modes of the magnetization are simulated by exciting the various relaxed states with a small perturbation and recording the ringdown process of the magnetization. The frequency response to this perturbation is studied for different zero-Ąeld remanent states and the effect of the various vertex conĄguration on the magnonic properties are compared. We can identify three different types of modes in the lattice differing by their geometric localization: vertex modes -in which the activity is concentrated at the vertex points, end modes -which are caused due to the oscillation of the dangling free ends, and wire-modes which are caused by the formation of standing modes in the nanowires.

Figure 7 . 32 :

 732 Figure 7.32: Frequency response of the reference state. Modes are marked from 1 to 5 .

Figure 7 . 33 :

 733 Figure 7.33: Mode proĄles of the reference state. (a) Edge modes corresponding to mode 1 (b) Vertex modes corresponding to mode 2 (c) Wire modes corresponding to modes 3 , 4 and 5

7. 10 Figure 7 . 34 :

 10734 Figure 7.34: Comparison of the frequency response of the three different magnetic states. The additional peaks observed in the +2q charged state ( 6 , 7 ) and +4q charged state ( 8 ) are marked .

7. 11

 11 Figure 7.35: Charge-dependent frequency of different vertex types in a cubic AMC lattice.

  contained 202 such nanowires, interconnected at 83 vertex points. The whole structure has a dimension of 450 × 450 × 450 nm Similar to the case of buckyballs, the individual nanowires in the network were magnetized axially at zero Ąelds and their interaction at the vertex points leads to frustrated vertex conĄgurations. We could identify two types of vertex conĄgurations: the ice-rule obeying two-in/two-out zero charged vertices and the defect charges with three in / one out arrangement. Based on the distribution of these different types of vertices we could develop three qualitatively different states in such a diamond lattice: a reference state which only contained ice-rule obeying zero charged vertices a defect state which contained only defect vertices which were periodically distributed in the lattice and a disordered state which contained both types of 8.4 Cubic AMC vertices which were all randomly distributed.

  havior. The increased coordination number in the cubic structure[START_REF] Yuasa | Giant roomtemperature magnetoresistance in single-crystal Fe/MgO/Fe magnetic tunnel junctions[END_REF] compared to the diamond (4) and the orthogonal arrangement of wires results in the emergence of additional possible vertex conĄgurations and even different Ćavors of the same type of conĄguration. Along with the vertex conĄgurations seen in the diamond lattice, there exist two additional vertex types in the cubic lattice; two Ćavors of the zero charged vertices and ±4q quadruple charges. As expected, the presence of double charges and quadruple charges introduces characteristic Ąngerprint modes in the magnonic spectrum of the lattice. Despite having same number of moments coming in and leaving out, and having moreover almost identical magnetic volume charge density, the two Ćavors of two-in/two-out vertex conĄgurations exhibit completely different dynamic properties as a consequence of the difference in their local micromagnetic structure. The availability of a larger number of defect-type vertex conĄgurations in the cubic lattice offers additional pathways for manipulating the magnonic properties of the lattice.

Contents 1 Introduction 2 Fundamental aspects of micromagnetism 2

  .1 Theory of micromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Exchange energy . . . . . . . . . . . . . . . . . . . . . . . . . . .

	Les propriétés magnoniques du réseau ont été étudiées en excitant les différents terne par rapport ' a la direction de saturation du treillis. Lorsque le champ identiĄer une corrélation de phase ' a longue portée. Cela sugg' ere lŠexistence 2. Quatre entrées / deux sorties (+2q) type de sommet dans le réseau entraîne lŠapparition dŠun nouveau pic ca-met. Nous avons identiĄé trois conĄgurations de sommets stables dans
	états relaxés et en enregistrant le ŞringdownŤ magnétique qui sŠensuit. est appliqué parall' element ' a la direction de saturation, la fréquence de tous dŠune onde de spin macroscopique se propageant du cİur du réseau vers le 3. Deux entrées / quatre sorties (-2q) ractéristique. En général, les sommets appartenant au même type de charge la structure du diamant, dont les charges nulles obéissant ' a la r' egle de
	Le spectre haute fréquence de lŠétat sans défaut contient trois types de modes. Premi' erement, les modes de surface, qui sont formés par lŠactivité les modes augmente linéairement avec lŠintensité du champ. En revanche, lorsque le champ est appliqué dans la direction opposée, la fréquence des centre, par le biais dŠune corrélation spatiale et temporelle de lŠoscillation des modes des sommets. Comme nous lŠavons déj' a dit, les propriétés magno-oscillent ' a la même fréquence, et la fréquence dŠoscillation diminue lorsque la glace et deux types de charges doubles. Différents microétats peuvent 4. Cinq entrées / une sortie (+4q) la magnitude de la charge augmente. Cette observation est vraie pour tous être générés dans un tel treillis en saturant le treillis dans différentes direc-
	modes diminue. Lorsque le champ externe est appliqué perpendiculairement niques de ces structures peuvent être contrôlées en manipulant leur struc-5. Une entrée / cinq sorties (-4q) les types de charge, sauf pour les sommets ' a charge nulle. Les sommets ' a tions. Le spectre de fréquence du réseau contient trois types de modes : les des extrémités libres avec des charges +1q et -1q ' a la surface du treillis. Deuxi' emement, les modes de sommet qui sont dus ' a lŠoscillation des som-' a la direction de saturation, les modes de surface et des nanoĄls se divisent ture magnétique et aussi en introduisant des défauts structurels. Si une 6. Six entrées (+6q) charge nulle dans le treillis cubique oscillent ' a deux fréquences différentes, modes de surface, les modes de sommet et les modes de nanoĄls. Dans les
	mets ' a charge nulle ' a lŠintérieur du réseau, et enĄn les modes de nanoĄls en deux et quatre sous-branches, en fonction de leur orientation par rapport onde de spin macroscopique se propageant peut être réalisée dans une telle 7. Six-sorties (-6q) selon leur type ou leur variant. modes de sommet, les sommets défectueux oscillent ' a une fréquence beau-
	' a plus haute fréquence, qui sont causés par des oscillations de type onde au champ externe. structure, cela pourrait ouvrir de nouvelles perspectives pour des applica-Pour résumer, nous avons réalisé des simulations micromagnétiques par coup plus basse que celle des sommets ' a charge nulle. Ces modes peuvent
	éléments Ąnis sur la dynamique de lŠaimantation ' a haute fréquence dans être contrôlés plus précisément ' a lŠaide de champs externes. LŠintroduction
	les nanocylindres de cobalt et dans plusieurs types de nanoarchitectures de défauts géométriques dans le treillis a entraîné lŠapparition de modes de
	tridimensionnelles : Tétra' edres de type Sierpinski, structures Buckyballs, fréquence supplémentaires. Pour vériĄer la validité générale de ces résultats,
	structures ' a réseau de type diamant et structures ' a réseau cubique. La Ćex-un autre cristal magnonique artiĄciel a également été étudié, ' a savoir le
	ibilité de la méthode dŠéléments Ąnis sŠest avérée cruciale pour modéliser treillis cubique. Ses sommets sont formés par lŠintersection orthogonale
	avec précision la géométrie compliquée et les surfaces courbes de ces struc-de six nanoĄls. En raison de lŠaugmentation du nombre de coordination,
	tures. Nous avons identiĄé les différentes conĄgurations dŠéquilibre dans les davantage de types de conĄgurations de sommets sont présents dans le treil-2.1.2 Crystalline anisotropy energy . . . . . . . . . . . . . . . . . . . .
	tétra' edres de Sierpinski, y compris un état de vortex déconnecté, et étudié lis cubique. De plus, en raison des effets géométriques, différents variants 2.1.3 Zeeman energy . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	comment ces états affectent les propriétés magnoniques. Nous avons ob-de conĄgurations de sommets avec la même charge ont été observées. Le 2.1.4 Demagnetizing energy . . . . . . . . . . . . . . . . . . . . . . . .
	servé une réponse en fréquence ' a large bande dans les étages supérieurs de spectre de fréquence du réseau cubique présente en grande partie le même 2.1.5 Total energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	la fractale. Dans les structures Buckyball, nous avons pu identiĄer quatre comportement que celui du réseau de type diamant, en ce sens quŠils con-2.2 Magnetization dynamics: The Landau-Lifshitz-Gilbert equation . . . . .
	conĄgurations de sommets différentes, dont deux états de charge simple tiennent tous deux des modes qui peuvent être clairement attribués aux 2.3 Basic micromagnetic structures and effects . . . . . . . . . . . . . . . . .
	obéissant ' a la r' egle de la glace et deux conĄgurations de charge triple de type oscillations de la surface, des sommets et des nanoĄls, et que les sommets 2.3.1 Magnetic domain walls . . . . . . . . . . . . . . . . . . . . . . . .
	défaut, en fonction de la conĄguration magnétique des nanoĄls. La form-avec différents types de conĄguration oscillent ' a des fréquences différentes. 2.3.2 Magnetic vortices . . . . . . . . . . . . . . . . . . . . . . . . . . .
	ation de ces charges de défaut peut être contrôlée en utilisant des champs 2.3.3 Shape anisotropy: axial magnetization of elongated structures . .
	1. deux entrées / deux sorties (0q) magnétiques externes. La présence de ces défauts a un fort impact sur les 2.4 ArtiĄcial spin ices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	2. trois entrées / une sortie (+2q) propriétés haute fréquence de ces structures. Cette capacité ' a contrôler le
	spectre magnonique ouvre de nouvelles possibilités dŠapplications potenti-3. une entrée / trois sorties (-2q) elles de ces structures en magnonique. De plus, cet effet peut être exploité
	4. quatre entrées (+4q) comme un outil indirect pour détecter expérimentalement ces défauts par
	5. quatre sorties (-4q) façon aléatoire. Le renversement de lŠaimantation dans ces structures en et comparé. Le réseau cubique présente des caractéristiques magnoniques des mesures de fréquence. LŠétude a été étendue aux Buckyballs creuses, qui
	fonction du champ est étudiée en simulant leur boucle dŠhystér' ese. Nous LŠévolution du spectre de fréquence du réseau en fonction dŠun champ similaires ' a celles du réseau diamant : L' a aussi, les modes dans lŠétat de présentent globalement un comportement similaire ' a celui de leurs homo-
	Parmi ces quatre arrangements, le deuxi' eme et le troisi' eme, ainsi que le observons que le renveresement de lŠaimantation implique une commutation magnétique externe a été étudiée en appliquant le champ dans deux dir-référence sans défaut peuvent être classés en modes de surface, de som-logues solides. Deux types de cristaux magnoniques artiĄciels ont également
	quatri' eme et le cinqui' eme sont respectivement équivalents en raison de magnétique abrupte des nanoĄls, ce qui entraîne des sauts brusques dans la ections, puis en observant les changements résultants. Nous avons observé met et de nanoĄl. Les différents types de conĄguration des sommets oscil-été étudiés. Le réseau en forme de diamant est constitué dŠun réseau
	courbe M-H. que la réponse ' a un champ externe dépend de lŠorientation du champ ex-lent ' a des fréquences différentes, cŠest pourquoi lŠintroduction dŠun nouveau périodique de nanoĄls avec quatre nanoĄls se rencontrant ' a chaque som-
	xviii xix xx xxi xxii xxiii

Nous avons étudié leurs structures dŠaimantation statique, leurs propriétés hystérétiques et les modes dŠoscillation haute fréquence apparaissant dans différents micro-états. Comme dans le cas des Buckyballs, les nanoĄls sont suffisamment Ąns pour quŠ' a champ nul, les branches individuelles aient tendance ' a être aimantées le long de leur axe. Par conséquent, comme dans le cas précédent, nous pouvons attribuer un moment dipolaire magnétique de type Ising ' a chaque nanoĄl. Puisque quatre nanoĄls se rencontrent ' a chaque sommet, il existe cinq dispositions possibles des sommets, ' a savoir : la symétrie dŠinversion temporelle. LŠétat énergétiquement optimal, qui obéit ' a la r' egle de la glace (Şice ruleŤ), est le premier arrangement dans xvii lequel le Ćux magnétique vers et ' a lŠopposé du sommet est parfaitement équilibré, ce qui conduit ' a une conĄguration dans laquelle la charge volumique nette au sommet est nulle. Les deuxi' eme et troisi' eme arrangements sont les charges doubles de type défaut avec des propriétés de type monopole. Contrairement aux sommets ' a charge nulle, ces types de sommets développent une charge magnétostatique nette non nulle. Étant donné que les charges magnétiques peuvent agir comme des sources et des puits du champ magnétique H, la présence de telles charges de défaut donne ' a ces structures des propriétés supplémentaires. Les conĄgurations 4 et 5 nŠont pas été observées dans un état stable dans aucune de nos simulations, car ces arrangements semblent être énergétiquement instables. En plus des sommets, nous pouvons également attribuer une charge de +1q ou -1q aux extrémités libres ' a la surface du treillis. Les états dŠaimantation relaxés formés dans ces structures sont sensibles ' a leur histoire magnétique. Ainsi, nous pouvons générer divers états qualitativement différents en saturant le réseau le long de différentes directions, puis en relaxant jusquŠau champ zéro. LŠapplication dŠun champ suffisamment fort pour saturer lŠéchantillon dans la direction [1,0,0], de sorte que le champ ne soit pas parall' ele ' a lŠun des nanoĄls, puis la relaxation jusquŠau champ zéro, donne un état ordonné ' a charge nulle qui ne contient que des sommets obéissant ' a la r' egle de la glace avec une charge nette nulle. En revanche, en saturant le réseau dans la direction [1,1,1] de façon ' a ce que le champ soit aligné parall' element ' a lŠune des directions dŠorientation des nanoĄls, et en relaxant le syst' eme jusquŠau champ zéro, on obtient un état de charge défectueuse qui ne contient que des sommets chargés de type +2q et -2q. En relaxant la structure ' a un état stable ' a champ nul ' a partir dŠune conĄguration initiale aléatoire, on obtient un état désordonné, dans lequel les sommets ' a charge nulle et les sommets ' a charge double sont répartis de stationnaire au sein des nanoĄls. Pour étudier lŠeffet de la formation des charges de défaut sur le spectre magnonique, les modes haute fréquence de lŠétat désordonné (qui contient des charges +2q et -2q, ainsi que des charges nulles) sont comparés ' a ceux de lŠétat ' a charge nulle. Nous avons observé que tous les modes précédents sont restés presque les mêmes, tandis quŠun mode basse fréquence supplémentaire et intense est apparu, qui est localisé aux sommets de la double charge. Il est intéressant de noter que les sommets ' a double charge oscillent ' a une fréquence considérablement plus basse que les sommets ' a charge nulle, bien que lŠoscillation se produise ' a des emplacements géométriques similaires. Nous avons en outre constaté quŠil nŠy a pas de différence de fréquence ou dŠintensité dŠoscillation entre les sommets chargés +2q et -2q. Nous avons également observé une baisse de lŠintensité de lŠoscillation des sommets ' a charge nulle par rapport ' a lŠétat de référence. Cela est dû ' a la diminution du nombre de sommets ' a charge nulle dans lŠétat désordonné, car certains sommets acqui' erent une conĄguration ' a double charge. Alors que les modes du nanoĄl dans lŠétat de référence apparaissent comme trois pics bien déĄnis, dans lŠétat désordonné, ils sont mélangés et essentiellement indiscernables les uns des autres. Cela peut être attribué ' a lŠeffet des charges +2q et -2q, qui agissent comme des sources de champs magnétiques locaux qui modiĄent la fréquence dŠoscillation des nanoĄls correspondants. Apr' es avoir établi comment la présence de défauts magnétiques affecte le spectre magnonique, nous avons étudié comment les propriétés magnoniques peuvent être manipulées par la formation de défauts géométriques. Pour cela, nous avons introduit un défaut structurel dans le réseau en retirant un nanoĄl individuel ' a lŠintérieur du réseau, créant ainsi une paire de sommets avec un numéro de coordination de trois. Comme dans le cas des états obéissant ' a la r' egle de la glace dans la structure buckyball, y résultent deux types de conĄgurations stables dans ces sommets défectueux : 1. deux entrées / une sortie 2. une entrée / deux sorties Pour comprendre lŠeffet de la formation de défauts géométriques sur les propriétés magnoniques, les excitations haute fréquence qui se développent dans un réseau avec un défaut géométrique sont comparées ' a celles dŠun réseau sans défaut en état de charge nulle. La présence dŠun défaut structurel introduit un pic supplémentaire dans le spectre de puissance, localisé uniquement sur les sites des défauts. LŠintensité de ce nouveau pic est comparativement faible, car il provient de seulement deux sommets sur les 83 sommets totaux. LŠobservation dŠun pic dŠabsorption supplémentaire est comparable aux phénom' enes de modiĄcation des spectres optiques des cristaux naturels par la présence dŠimpuretés et de défauts. JusquŠ' a présent, nous avons étudié les modes statiques dans le sens o' u les oscillations sont localisées ' a des positions géométriques spéciĄques, sans évoluer dans le temps. Cependant, en analysant lŠoscillation Ąltrée par une analyse de Fourier des sommets dans lŠétat de charge zéro, nous avons pu tions potentielles en magnonique. Pour étudier cette possibilité, nous avons modélisé une version allongée du réseau et, au lieu dŠexciter la structure enti' ere, nous avons appliqué une impulsion de champ localisée ' a lŠune des extrémités de la structure et nous avons analysé les oscillations magnétiques qui en résultent. Contrairement ' a ce que nous attendions, nous nŠavons pas pu observer de propagation signiĄcative dans le sens longitudinal lors de nos premi' eres tentatives. Cela sugg' ere que le couplage magnéto-dipolaire entre les sommets nŠest pas assez fort pour un transfert dŠénergie efficace et pour une propagation macroscopique des ondes de spin qui en résulte.

Cependant, il existe un large éventail de param' etres ' a faire varier, tels que la longueur et lŠépaisseur des nanoĄls, les propriétés du matériau etc., qui pourraient être abordés dans de futures études.

Dans la section précédente, nous avons étudié la formation de différentes conĄgurations de sommets dans un cristal magnonique artiĄciel (AMC) de type diamant ainsi que leur effet sur la signature magnonique du réseau.

Pour démontrer la robustesse de ces observations, nous étendons notre enquête ' a un autre type dŠAMC avec une structure de réseau cubique. La structure de treillis cubique est formée par lŠinterconnexion de nanoĄls dans une disposition cubique simple, de sorte que six Ąls se rencontrent ' a chaque sommet. Notre structure mod' ele est composée de nanoĄls cylindriques de 7,0 nm de rayon et 70,0 nm de longueur, disposés en un treillis de 7 × 6 × 5.

Comme six nanoĄls se rencontrent ' a chaque sommet, au lieu des quatre du réseau de type diamant, un plus grand nombre de conĄgurations de sommets sont possibles, comme indiqué ci-dessous :

1. Trois entrées / trois sorties (0q) ( État respectant la r' egle de la glace) Parmi ces sept conĄgurations possibles, les deux derni' eres nŠont pas été observées en tant que structures stables, ce qui sugg' ere quŠelles sont énergétiquement instables. Dans le cas du réseau de type diamant, puisque les quatre nano-Ąls sont disposés de mani' ere tétragonale, les trois voisins dŠun nanoĄl sont équivalents. En revanche, les cinq voisins du réseau cubique ne sont pas équivalents. Quatre des cinq voisins sont disposés normalement au nano-Ąl, et le cinqui' eme Ąl est disposé ' a lŠopposé. ' A cause de cette disposition géométrique particuli' ere, il existe différents ŞvariantsŤ de conĄgurations de sommets ayant la même charge. Par exemple, il existe deux types différents dŠétats Şdeux entrants/deux sortantsŤ, avec une structure micromagnétique différente et, par conséquent, des propriétés dynamiques différentes. Comme dans le cas précédent, il est possible de générer des états qualitativement différents en saturant la structure le long de différentes directions, et ensuite relaxant le syst' eme au champ zéro. Nous avons pu générer trois états différents : un état de référence qui ne contient que des sommets ' a charge nulle de type un obéissant ' a la r' egle de la glace et un autre état qui contient des sommets ' a charge double ainsi que les deux types de charges nulles et un troisi' eme état qui contient des sommets ' a charge quadruple, ainsi que des sommets ' a charge double et nulle. Le spectre haute fréquence des différents états statiques a été simulé

3 Finite element modeling, simulation and post-processing

	CONTENTS
	3.4 Time integration of the LLG equation . . . . . . . . . . . . . . . . . . .
	3.5 Simulation method: working of tetmag . . . . . . . . . . . . . . . . . .
	3.6 Post processing: Extraction of frequency modes with a windowed inverse
	Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	3.2 Finite element formulation . . . . . . . . . . . . . . . . . . . . . . . . . .
	3.2.1 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	3.2.2 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	3.3 Computation of the effective Ąeld . . . . . . . . . . . . . . . . . . . . .
	3.3.1 Exchange Ąeld: weak formulation . . . . . . . . . . . . . . . . . .
	3.3.2 Anisotropy Ąeld . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	3.3.3 Demagnetizing Ąeld: FEM/BEM method . . . . . . . . . . . . .
	xxv

4 Micromagnetic simulations of high-frequency modes in Cobalt nan- odots

  Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	4.1 Finite element modeling and simulation . . . . . . . . . . . . . . . . . .
	4.2 Magnetization structure . . . . . . . . . . . . . . . . . . . . . . . . . . .
	4.3 Magnetization dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . .
	4.4

5 Three-dimensional Sierpinski fractal structures 5

  .1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Magnetization structure at zero Ąeld . . . . . . . . . . . . . . . . . . . . 5.4 Hysteretic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnonic excitations . . . . . . . . . . . . . . . . . . . . . . . . . 7.11 Chapter summary and outlook . . . . . . . . . . . . . . . . . . . . . . .

	CONTENTS
	7 ArtiĄcial magnonic crystals
	7.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	7.2 Static magnetization structure and vertex conĄgurations . . . . . . . . .
	7.3 Hysteretic properties of the crystal: zero charged state . . . . . . . . . .
	7.4 Magnonic excitations: Zero-charged state . . . . . . . . . . . . . . . . .
	7.5 Magnonic response of defect charges . . . . . . . . . . . . . . . . . . . .
	7.6 Different types of nanowires: InĆuence of local Ąelds on the magnonic
	spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	7.7 Tuning the mode frequencies with an external Ąeld . . . . . . . . . . . .
	7.8 Effect of a geometric defect in the crystal lattice . . . . . . . . . . . . .
	7.9 Macroscopic spin wave in a diamond-type AMC . . . . . . . . . . . . . .
	7.10 Cubic lattice structure . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	7.10.1 Vertex conĄguration . . . . . . . . . . . . . . . . . . . . . . . . .
	7.10.2 8 Conclusion
	5.5 Magnonic excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Three-dimensional Sierpinski structures . . . . . . . . . . . . . . . . . . 5.6 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Buckyball nano-architectures . . . . . . . . . . . . . . . . . . . . . . . .
	6 ArtiĄcial Buckyball nano-architectures 8.3 Three-dimensional diamond AMC . . . . . . . . . . . . . . . . . . . . .
	6.1 Geometry and Ąnite element modelling . . . . . . . . . . . . . . . . . . . 8.4 Cubic AMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	6.2 Material parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	6.3 Simulation method and post-processing . . . . . . . . . . . . . . . . . .
	6.4 Static magnetic conĄgurations . . . . . . . . . . . . . . . . . . . . . . . .
	6.4.1 Properties of triple-charges . . . . . . . . . . . . . . . . . . . . .
	6.4.2 Controlled generation and removal of triple-charges . . . . . . . .
	6.4.3 Magnonic spectrum of a buckyball: Switchable frequencies . . . .
	6.5 Hollow nanoarchitectures . . . . . . . . . . . . . . . . . . . . . . . . . .
	6.5.1 Comparison of nanowires and nanotubes . . . . . . . . . . . . .
	6.5.2 Hollow Buckyballs: magnetization structure . . . . . . . . . . . .
	6.5.3 Hysteretic properties . . . . . . . . . . . . . . . . . . . . . . . . .
	6.5.4 High-frequency modes . . . . . . . . . . . . . . . . . . . . . . . .
	6.6 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	xxvi

  section 2.1.5 we discussed the different energy contributions in micromagnetism. A stable equilibrium magnetization state can be obtained by minimizing the total energy of the system. If we are instead interested in obtaining the time dependent evolution of magnetization M (r, t), we must numerically integrate (ŞsolveŤ) the discretized LLG equation. The magnetic state M (r, t + δt) at time t + δt is obtained by the time integration of the LLG equation based on the magnetic state M (r, t) at the time t. For this, the LLG equation is discretized to 3N set of partial differential equations

This can also be explained using the magnetostatic reciprocity theorem[START_REF] Brown | Magnetostatic Principles in Ferromagnetism[END_REF][START_REF] Aharoni | International series of monographs on physics[END_REF]: The energy of a magnetic moment µA in the dipolar Ąeld of another moment µB is identical to the energy of the other moment µB in the dipolar Ąeld of the magnetic moment µA. Obviously, this energy should only be counted once for a pair of magnetic moments.

We point out that the visualization represents the divergence of the magnetization, while the magnetostatic volume charges are deĄned with an opposite sign, ρm = -∇m. For the sake of simplicity, we donŠt always adhere to this sign convention in the discussion of the charges at the vertices. To highlight the properties of the vertex conĄgurations, the presence of regions with divergence of different sign and magnitude is more important than the sign convention.

Acknowledgements

The three years of my life in which I did this PhD is an important part of my life and I would like to express my heartfelt gratitude to all the people who

Acknowledgement

The C++ code which was used for the generation of the Ąnite element models were based on a previous code written by Swapneel Amit Patak for the generation of two dimensional Sierpinski triangles. I acknowledge and express my gratitude for Swapneel for this contribution.

Funding

This work was funded by the LabEx NIE (ANR-11-LABX-0058NIE) in the framework of the Inter-disciplinary Thematic Institute QMat (ANR-17-EURE-0024), as part of the ITI 2021-2028 program supportedby the IdEx Unistra (ANR-10-IDEX-0002-002) and SFRISTRATUS (ANR-20-SFRI-0012) through the FrenchProgramme dŠInvestissement dŠAvenir. The authors ac-knowledge the High Performance Computing center ofthe University of Strasbourg for supporting this work byproviding access to computing resources.

Chapter 3

Finite element modeling, simulation and post-processing Chapter 4 Micromagnetic simulations of high-frequency modes in Cobalt nanodots

Magnetization dynamics

Magnetization dynamics

The magnonic response of the disc at 400 mT is shown in Fig. 4.5. There exist two principal modes. The lower frequency mode at around 15.7 GHz is the edge mode, caused by symmetric oscillations of the head/tail part of the onion structure. The higher-frequency mode at around 20.3 GHz is caused by the oscillation involving both the central part and the edges of the disc. These individual frequency peaks are identi-Ąed and isolated using the Fourier extraction tool as explained in the previous chapter.

The extracted spatial proĄle of the two frequency modes are displayed in Fig. 4.6.

These results qualitatively agree with the observation of two major frequency peaks in the FMR measurements. Various parameters such as the thickness, radius, the saturation magnetization, and the strength of the anisotropic Ąeld were varied to match the simulated frequency with the observed values.

Vibrating sample magnetometry measurements of the Co dot array yielded a saturation magnetization of 343 mT, a value that is much lower than the one of bulk Cobalt, which is 1700 mT. Such a reduction of the saturation magnetization M s may be caused by a combination of aging and inter-phase diffusion processes in the sample. To investigate the impact of the value of the saturation magnetization on the high-frequency oscillations, we performed simulations in which we varied M s from 343 mT to 1700 mT.

According to information obtained from the colleagues, we also included a crystalline anisotropy corresponding to a Q-factor of 0.01. The resulting frequency response was compared with the experimental data 4.7. It was observed that the frequency of the In all the cases we simulated -except for the smallest (L = 25 nm ), the frequency of oscillation of the single charged vertices always splits into two groups 6.16(b). This splitting into two lower and higher frequency modes is not based on the charge of 6.5 Hollow nanoarchitectures

Conclusion

Hollow Buckyballs: magnetization structure

In the length ranges concerned, similar to the case of solid nanowires, the nanotubes tend to adopt an axially magnetized state at zero Ąelds, as would be expected from the shape anisotropy effect. As a result, despite differences in the properties of the constituent nanowires and nanotubes, the solid and hollow Buckyballs exhibit similar magnetic structures at the vertices. Based on the direction of magnetization of each nanotube meeting at a vertex we can apply the same classiĄcation of vertex conĄgurations as in the case of the solid buckyball be distinguishing between single and triple charges. A detailed visualization of the relaxed magnetic conĄguration of a hollow buckyball structure with side L = 100 nm is displayed in Fig. 6.22. The simulations show that the triple charge vertices have the same triple head-to-head domain wall structure, as seen in the solid model. As the size of the structure is increased, the triple-charge vertex structures evolves towards a three-dimensional vortex-like conĄguration, in essentially the same way as previously discussed in the case of the solid buckyball.

Chapter 7

Artificial magnonic crystals

Static magnetization structure and vertex conĄgurations

The ice-rule obeying zero-charged vertices are energetically optimal states for the lattice. Their conĄguration results in a minimum possible magnetostatic and exchange energy, while the charged vertex conĄgurations yields both a net magnetic charge, thereby increasing the magnetostatic energy, and a higher degree of inhomogeneity, and thus higher exchange energy. The formation of defect charges within the AMC results in a higher magnetic disorder, which can be quantiĄed by the density of charged vertices within a lattice. Despite their energetically unfavorable structure compared to the ice rue obeying states, the defect charges are stable once they are formed. In general, the removal of a defect charge from the system requires an external Ąeld strong enough to magnetically saturate the sample. Based on the presence or absence of defect charges, we can deĄne two qualitatively different ordered magnetic states within the lattice: A defect-free state, which only contains ice-rule obeying zero-charged vertices, and a defect state which contains several +2q/ -2q charged double charge vertices. The two different magnetic states can be simulated in a controlled way by changing the initial conĄguration of the simulation.

Relaxing the structure to a zero-Ąeld state when starting from a saturated state along the [1, 0, 0] direction so that none of the wires is aligned parallel the applied Ąeld results in a defect-free, zero-charged state. On the other hand, if the initial point is a saturated state with the Ąeld applied along the along the [1, 1, 1] direction of the lattice, a subset of the nanowires (one out of six) in the lattice is magnetized parallel the Ąeld direction. This initial direction of the magnetization in the subset of nanowires will be preserved if the Ąeld is subsequently lowered. Relaxing to zero Ąeld from this state results in a

Magnonic response of defect charges

Magnonic response of defect charges

So far, we have investigated the high-frequency modes arising in the defect-free lattice.

We could identify Ąve modes, each corresponding to a speciĄc magnonic activity. In the next step, we repeat the same procedure for the magnetic conĄguration which contains ice-rule violating defect charges; the disordered state or the double-charge state. A comparison of the magnonic response of this double-charge state with the zero-charge state is shown in Fig. 7.14. On Ąrst glance, we can see that the Ąve modes we discussed in the previous section remain almost unchanged. The most striking difference is the appearance of an additional, intense low-frequency mode at 7.9 GHz. This mode describes the oscillation of the double-charged vertices, which were absent in the previous state. The appearance of this peak, caused by the introduction of magnetic defects, is a manifestation of the dependence of the high-frequency properties on the magnetic conĄguration of the AMC. The zero-charged vertices present in the disordered state oscillate at the same frequency discussed previously, which is quite different from that of the double charges, even though the oscillations occur at the same geometrical location.

The spatial proĄle of the oscillation of the zero-charged and double charged vertices is shown in Ągure 7.15. The mode proĄles of the zero-and double-charge vertices are complementary, that is, at 7.9 GHz only the double charged vertices are active, and Chapter 8

Conclusion

Summary

demonstrated how the magnonic properties of artiĄcial spin ice lattices can be manipulated and re-conĄgured by means of defect charges. Investigations on three-dimensional magnetic nanostructures and nanoarchitectures is still at an early stage, especially as far as their high-frequency dynamics is concerned. The studies conducted in this thesis have been performed with the goal of achieving a better understanding of the properties of these novel three-dimensional structures. One can assume that many of their properties and potential applications are yet to be discovered. We hope the Ąndings of this thesis will aide the experimentalists to gain a deeper insight into the physics of these structures and also inspire future studies to follow.

Summary

Rajgowrav CHEENIKUNDIL

Simulations micromagnétiques de nano-architectures tridimensionnelles

Micromagnetic simulation of three-dimensional nanoarchitectures

Résumé Cette thèse traite de simulations micromagnétiques de la dynamique hyperfréquence de l'aimantation dans des nanoarchitectures tridimensionnelles (3D) constituées de réseaux de nanofils interconnectés. Les propriétés magnétiques de tels nanomatériaux artificiels sont fortement influencées par leur géométrie. En étudiant la structure magnétique statique de ces systèmes, nous montrons des configurations correspondant à des réseaux de glace de spin artificiels 3D, avec des interactions frustrées et des structures de défaut monopolaires aux points d'intersection. Nos simulations révèlent une activité élevée de ces sites dans les excitations magnétiques de haute fréquence. Nous étudions diverses nanoarchitectures 3D et montrons que leur géométrie et leur structure magnétique donnent des signatures hyperfréquences caractéristiques. Le contrôle de ces caractéristiques pourrait ouvrir de nouvelles voies pour la recherche magnonique et dans le développement de métamatériaux magnétiques reprogrammables.

Mots clé : Simulations micromagnétiques, nanostructures tridimensionnelles, dynamique de l'aimantation, hyperfréquences

Résumé en anglais

The thesis discusses micromagnetic simulation studies on the high-frequency magnetic dynamics in three-dimensional (3D) nanoarchitectures made of interconnected magnetic nanowire networks. Such artificial magnetic materials with nanoscale features have recently emerged as a vivid topic of research, as their geometry has a decisive impact on their magnetic properties. By studying their static magnetization structure, we find that these systems display a behavior analogous to that of 3D artificial spin ice lattices, with frustrated interactions and the emergence of monopole-like defect structure at the wires' intersection points. Our simulations reveal a high activity of these defect sites in the magnonic high-frequency spectrum. We study various 3D nanoarchitectures and show that their geometry and magnetization state results in characteristic high-frequency signatures. Controlling these features could open new pathways for magnonics research and reprogrammable magnetic metamaterials.

Keywords: Micromagnetic simulations, three-dimensional magnetic nanostructures, high-frequency magnetization dynamics