Keywords:

étendu Résumé étendu

.a). Les skyrmions ont fait l'objet de nombreuses recherches depuis leur première observation en 2011. Ils sont généralement stabilisés grâce à une interaction chirale qui favorise la non-collinéarité des spins, connue sous le nom d'interaction de Dzyaloshinskii-Moriya (DM). Cette interaction se produit spontanément dans des structures ayant un fort couplage spin-orbite et une symétrie d'inversion brisée.

Après trois ans de travail acharné (et heureusement passionné), voici enfin ma thèse qui s'achève et une nouvelle page qui se tourne. La réussite (subjective) de ma thèse est évidemment liée aux différentes rencontres que j'ai eu la chance de faire durant ce parcours ainsi qu'au soutien inconditionnel de mes plus vieilles connaissances.

Il me paraît logique de commencer par remercier mes deux superviseurs, Vincent et Nicolas, qui m'ont permis de travailler dans un tel environnement sur un sujet qui me passionne. Malgré un contexte difficile de début de thèse avec le COVID, tout s'est miraculeusement bien déroulé durant ces trois années aussi bien au niveau des résultats que de mon épanouissement personnel. Merci donc Nicolas pour ton enthousiasme, pour ta curiosité qui mène toujours à de nouveaux axes de recherches et aussi pour ta patience au vu du nombre incalculable de fois où je suis venu te demander de l'aide... En plus, tu restes à jours sur les différentes manips ou sur les simus ce qui m'a été grandement appréciable et salutaire de nombreuses fois ! Ton sens du detail est aussi impressionnant et louable et il m'a bien souvent poussé à vérifier et à analyser de manière critique mes démarches et résultats ce qui a été très formateur. Merci Vincent pour ton entrain, ta constance et ta rigueur. Nos discussions m'ont souvent permis d'y voir plus clair et d'avoir de nouvelles idées grâce à tes connaissances impressionnantes de la spintronique. Malgré un agenda constamment surchargé, tu t'es toujours rendu disponible si nécessaire pour discuter et apporter ton point de vue éclairé sur mes résultats ou la suite de mon projet. Enfin merci pour ta réactivité et ton efficacité: malgré tes responsabilités, tu prenais toujours le temps d'apporter (d'innombrables) corrections à mes différents drafts pour les améliorer. Vous faîtes un travail très complémentaire d'encadrements et j'espère que vous ne changerez rien pour les prochains heureux qui trouveront leur place dans votre groupe ! Le seul reproche que je pourrais vous faire c'est d'avoir envoyé autant de fois un doctorant qui ne boit pas de café faire des beamtimes... À part cela, je vous remercie pour ces trois belles années à vos côtés ! Pour continuer, j'aimerais également remercier le jury pour leurs retours sur mon travail et les discussions intéressantes lors de la soutenance. Moreover, thank you Claire for your many inputs and your expertise on the laminography. Mon travail a également été porté par différentes collaborations qui m'ont permis d'avancer efficacement. Je pense notamment à nos collaborateurs de SOLEIL, Nicolas J. et Horia. Merci à eux pour leur pédagogie, leur gentillesse et leur patience alors que je faisais mes débuts dans le monde exclusif des synchrotrons. Leur expertise a permis d'avancer considérablement le projet ainsi que leur volonté de passer mes échantillons en in-house. Merci également à Cyril qui m'a tenu compagnie XII pendant le beamtime le plus fatigant de ma thèse à PSI ! I would also like to thank our German collaborators, particularly Riccardo, for his help and his everlasting enthusiasm. My best wishes to you for the completion of your PhD. J'ai également bénéficié de l'expérience et de l'entraide des autres membres du groupe skyrmion. First of all, thank you to Titiskha, Fernando, and Erick, who helped me get started at the beginning for the few months our paths crossed. Thank you Sougata for your good mood, your helpfulness, and your insights, it has been greatly appreciable to work alongside you. Thank you also for some of the best Indian food I've ever eaten! Thank you Sachin for your expertise, your funny streak, and your kindness. Finalement, un grand merci à Yanis qui fait preuve d'une complaisance sans pareille et qui m'a formé à mes débuts avec patience et bonne humeur ! Malheureusement pour toi, je crois que tu vas devoir me reformer à nouveau dans peu de temps... Heureusement, nous arrivons tout de même à se supporter en dehors du labo donc on devrait survivre à cette nouvelle épreuve ! J'aimerais également remercier les autres permanents du labo, notamment les ingénieurs qui m'ont permis d'utiliser efficacement les différentes manips du labo. Merci donc à Sophie, pour sa gentillesse et de m'avoir accordé beaucoup de temps pour les nombreuses lithos dont j'avais besoin. Merci également pour sa compréhension alors que je maltraitais la pulvé avec mes recettes à rallonge qui ont coûté une cible de platine en quelques mois et ont probablement achevé le bras avec le nombre important de rotation... Mais cela valait le coup pour faire croître les 541 échantillons de ma thèse ! Merci également à Florian pour la bonne ambiance qu'il sait mettre et son aide salvatrice avec la pulvé. Merci ensuite à Karim pour son humour acerbe mais qui prenait souvent le temps d'admirer mes magnifiques double contrastes au MFM. Merci également à Aymeric pour sa formation efficace sur le MFM qui m'a énormément servi durant la thèse. Je souhaite également remercier Anne, Julien, et Florence pour votre efficacité administrative au retour des missions. Un dernier remerciement pour Bruno, mon parrain au labo, qui a pris le temps de me suivre ces trois ans.

Finalement, passons aux non-permanents, avec d'abord les "vieux" doctorants. Merci Diane pour ta bonne humeur et les bons moments partagés notamment en conférence mais aussi autour de jeux ! Merci Laurette pour ton incroyable énergie et ta joie incomparable, pour les "chéris" qui résonnent dans le couloir ! Merci également à Pauline pour ton enthousiasme et ton aide à de bien trop nombreuses reprises sur les rayons X. Avec Yanis en plus, vous formiez vraiment un groupe accueillant, sympathique et toujours prêt à aider à tout moment. Merci aussi à Aurélien pour les excellentes soirées jeux grâce à notre passion commune pour les jeux en tous genres. Merci également à tous les autres que j'ai pu côtoyer au labo : Aya, Kévin, Diana, Sarah, Srijani, David... Il est difficile de citer tout le monde et de rester concis donc merci à tous. J'aimerais enfin remercier la vieille bande de CPI qui survit depuis maintenant quelques années et qui arrive toujours à me supporter ! Merci donc à Pierre, Lucas, Margaux, Lenaïg, Nolwenn et Angéline. Je n'avais malheureusement pas beaucoup de temps avec la thèse mais on trouvait toujours le moyen de tout de même se voir et ça changeait beaucoup les idées ! Un petit mot également pour les gens du tarot, mon père, Franck, Patrick, Nono, Gege, Sylvie et XIII j'en passe. Quand on joue plus de 16h par jour aux cartes, c'est assez efficace pour oublier la thèse (au détriment de mon sommeil) et en plus, c'est toujours dans la bonne ambiance et la rigolade ! Bon, je vais quand même te remercier, Gaétan, sinon tu ne vas jamais t'en remettre et tu es déjà assez embêtant comme cela... Merci pour toutes ces discussions, ces ragots, ces moments de rage et de désespoir sur la thèse (plus sur la tienne que la mienne), ces pauses baby ou autres jeux pour passer le temps. Par contre, essaie de t'entraîner à Smash Bros pour que ça soit un peu plus intéressant ! Enfin, des derniers remerciements à l'égard de mon cercle familial. D'abord, merci à Morgane et Jules pour leur sollicitude, leur soutien et les nombreux week-ends où ils nous ont accueillis et étaient aux petits soins. C'était grandement appréciable de se relaxer avec vous ! Et puis, on a chacun eu notre bébé de notre côté même s'il y en a un plus bruyant que l'autre... Merci donc de votre confiance aussi.

Merci également à ma famille pour leur soutien malgré leur incompréhension devant mon sujet (non ce ne sont pas juste des petites bulles). Sans eux et malheureusement les obstacles que nous avons dû affronter, je ne serais pas arrivé là où je suis aujourd'hui.

Pour finir, le remerciement le plus important (et court car tu sais ce que je pense) pour Marie qui, comme moi, a enduré l'épreuve qu'est la thèse. Merci pour ton soutien et ta présence dans ma vie.

Titre : Cocons skyrmioniques : nouvelles textures magnétiques 3D dans des multicouches magnétiques apériodiques Mots clés : Magnétisme, Spintronique, Skyrmion, Cocon skyrmionique, Multicouches magnétiques, Holographie, Tomographie

Résumé :

Le domaine du nanomagnétisme tri-dimensionnel (3D) connaît une expansion rapide au sein de la spintronique, englobant des nanostructures magnétiques et des textures de spin complexes qui s'étendent au-delà de deux dimensions effectives et varient selon la troisième, le long de la normale du film magnétique. Les multicouches magnétiques, composées de répétitions d'un élément ferromagnétique encapsulé entre des métaux, permettent un excellent contrôle des propriétés magnétiques et la stabilisation de différentes textures de spin, ce qui en fait d'excellentes plateformes pour le développement du nanomagnétisme 3D.

Cette thèse exploite le potentiel des multicouches magnétiques pour stabiliser de nouvelles textures magnétiques 3D, que nous avons dénommées cocons skyrmioniques. Ce sont des objets ellipsoïdaux occupant une fraction de l'épaisseur totale de la multicouche magnétique. Deux structures de multicouches distinctes ont été minutieusement étudiées qui sont caractérisées par une épaisseur variable de l'épaisseur de l'élément ferromagnétique. La première permet de stabiliser les cocons skyrmioniques dont l'extension peut être contrôlée à l'aide d'un champ magnétique externe ou d'ajustements structurels. La deuxième multicouche utilise des couches ferromagnétiques présentant une forte anisotropie magnétique perpendiculaire pour moduler les textures magnétiques le long de l'épaisseur. Il en résulte la coexistence de deux types de textures de spin : le skyrmion colonnaire 2D aux côtés du cocon skyrmionique 3D. Cette cohabitation est importante pour le développement de futurs dispositifs de spintronique, par exemple pour encoder plusieurs états de mémoire.

Cependant, l'observation et la mesure de l'aimantation en 3D continue d'être un défi, reposant souvent sur des techniques impliquant la lumière synchrotron. Au cours de cette thèse, diverses approches expérimentales ont été utilisées pour confirmer la nature en 3D des cocons skyrmioniques, notamment la microscopie à force magnétique et les mesures de magnétotransport, ainsi que des techniques avancées utilisant des rayons X comme l'holographie et la laminographie. Des simulations micromagnétiques ont également été réalisées, soutenant l'existence de textures de spin 3D et corroborant les résultats expérimentaux. Cette recherche ouvre la voie à une maîtrise plus poussée des multicouches pour exploiter la troisième dimension de l'aimantation, représentant ainsi une étape significative vers le développement de dispositifs skyrmioniques 3D. Title: Skyrmionic cocoons: novel 3D magnetic textures in aperiodic magnetic multilayers Keywords: Magnetism, Spintronic, Skyrmion, Skyrmionic cocoon, Magnetic multilayers, X-ray Holography, X-ray Tomography

Abstract:

The field of 3D nanomagnetism is rapidly expanding within spintronics, encompassing magnetic nanostructures and intricate spin textures that extend beyond two effective dimensions and vary along the third dimension, normal to the film's plane. Magnetic multilayers, composed of repetitions of a ferromagnetic element encapsulated between metals, enable excellent control over magnetic properties and the stabilization of various spin textures, making them excellent platforms for developing 3D nanomagnetism.

This thesis explores the tunability of magnetic multilayers to stabilize novel 3D magnetic textures, that we have baptized "skyrmionic cocoons". These are ellipsoidal objects which occupy only a fraction of the multilayer thickness. Two distinct multilayer structures have been thoroughly studied with varying ferromagnet thickness. The first allows for the stabilization of cocoons whose extension can be controlled with an external magnetic field or structural adjustments. The second multilayer utilizes ferromagnetic layers with strong perpendicular mag-netic anisotropy to modulate magnetic textures along the thickness. This results in the coexistence of two types of spin textures: the more common 2D columnar skyrmion alongside the 3D skyrmionic cocoon. This cohabitation is important for developing future spintronics devices, for example to encode several memory states.

However, observing and measuring magnetization in 3D remains a challenge, often relying on techniques involving synchrotron light. During this thesis, various experimental approaches have been employed to confirm the 3D nature of skyrmionic cocoons, including magnetic force microscopy and magnetotransport measurements as well as advanced X-rays techniques like holography and laminography. Micromagnetic simulations were also conducted, supporting the existence of 3D spin textures and corroborating the experimental findings. This research paves the way for further exploiting multilayer tunability to harness the third dimension of magnetization, representing a significant step towards the development of 3D skyrmionic devices. d) Exemples de structures magnétiques en 3D, de gauche à droite : bobber [1], toron [2] et hopfion [3].

Différents milieux sont propices à la stabilisation de ces objets, notamment les aimants chiraux et les multicouches magnétiques. Ces dernières sont constituées de couches ferromagnétiques alternées avec des métaux, souvent lourds, comme illustré dans la Figure 1.b). Pour rompre la symétrie, ces métaux sont généralement différents, ce qui favorise l'apparition de l'interaction de DM et donc la stabilisation de skyrmion. Dans ce cas, le skyrmion devient colonnaire, c'est-à-dire qu'il est présent VII dans toutes les couches ferromagnétiques de la pile. Bien qu'il puisse présenter quelques variations de chiralité et de diamètre, cet objet topologique reste relativement invariant en fonction de l'épaisseur. Afin de pleinement exploiter toutes les possibilités du système, il devient essentiel d'explorer également la troisième dimension, ce qui nous conduit vers des objets plus complexes variant dans toutes les directions de l'espace. Il s'agit donc du domaine du nanomagnétisme tridimensionnel (3D), qui offre davantage de perspectives pour d'éventuelles applications en raison du nombre accru de degrés de liberté disponibles. Deux axes de recherche principaux peuvent être mis en évidence. Tout d'abord, les nanostructures 3D consistent à sculpter des matériaux en trois dimensions pour obtenir des distributions complexes de spins soumises à des contraintes géométriques. Il est évident que l'accès à la troisième dimension offre plus de flexibilité que le nanomagnétisme 2D (Figure 1.c). L'autre aspect du nanomagnétisme 3D concerne les structures de spins en 3D, qui sont des objets variant dans toutes les directions de l'espace (contrairement aux skyrmions colonnaires). Quelques exemples d'objets, tels que le bobber, le toron ou le hopfion, sont présentés dans la Figure 1.d.

De tels nouveaux objets ouvrent la voie à l'expansion de la spintronique dans la troisième dimension, mais plusieurs défis doivent être relevés en premier lieu, principalement en ce qui concerne leur caractérisation et leur mise en oeuvre. En ce qui concerne cette dernière, la plupart de ces textures nouvellement observées ont été détectées dans des cristaux magnétiques chiraux ou dans des nanostructures, ce qui limite fortement leur fonctionnalité. En revanche, les multicouches magnétiques apparaissent comme une alternative prometteuse pour abriter des textures topologiques en 3D, car elles sont plus évolutives tout en offrant une plus grande ajustabilité de leurs propriétés magnétiques. Le défi le plus problématique demeure la caractérisation des textures magnétiques, car elle nécessite de sonder l'aimantation sur toute l'épaisseur. Cela est souvent réalisé en utilisant des techniques d'imagerie de pointe récentes qui reposent sur l'utilisation de rayons X mous ou d'électrons et nécessitent l'accès à des établissement spécialisés (synchrotron). Il est donc important de développer des méthodes plus facilement applicables afin de simplifier la caractérisation de ces milieux magnétiques.

Les travaux de ma thèse ont donc été dirigés sur la stabilisation ainsi que la caractérisation d'un nouveau type de textures de spin 3D, les cocons skyrmioniques. Afin d'atteindre cet objectif, nous avons tiré parti de la flexibilité offerte par les multicouches magnétiques. En effet, ces structures présentent un grand nombre de degrés de liberté, tels que les épaisseurs, les interfaces et les matériaux, qui permettent de contrôler de manière quantitative les interactions magnétiques qui influencent la distribution des spins. Parmi ces interactions, on peut citer l'échange d'Heisenberg, l'interaction DM, l'anisotropie uniaxiale et l'interaction dipolaire. Par exemple, en modifiant l'épaisseur de l'élément ferromagnétique, il est possible d'induire des modifications significatives dans l'anisotropie, passant ainsi d'une anisotropie hors du plan à une anisotropie dans le plan. La majorité des multicouches étudiées au cours de cette thèse étaient basées sur des répétitions de la tri-couche Pt/Co/Al. Afin d'atteindre la modulation des textures de spins sur l'épaisseur, notre première approche a consisté à utiliser différentes épaisseurs pour l'élément ferromagnétique afin d'induire une distribution inhomogène des interactions magnétiques. Cela a conduit à la création de la structure que nous appelons simple gradient (SG). Les épaisseurs utilisées étaient relativement épaisses, généralement supérieures à l'épaisseur de réorientation des spins (1.7 nm pour Pt/Co/Al), ce qui induisait une anisotropie dans le plan. Malgré cela, des textures de spins étaient stabilisées hors du plan grâce à l'interaction dipolaire.

Ces textures ont été soigneusement caractérisées à l'aide de différentes techniques, notamment la microscopie à force magnétique (MFM) et des mesures de magnetotransport, qui ont été corrélées avec des simulations micromagnétiques. Des exemples sont présentés dans la Figure 2, montrant l'évolution en fonction du champ magnétique.

Á rémanence, les objets magnétiques s'étendent sur toute l'épaisseur, ce qui en fait des skyrmions colonnaires. Cependant, l'augmentation du champ magnétique permet de les confiner selon l'épaisseur et de les détacher des interfaces les plus extrêmes. Ces nouveaux objets ellipsoïdaux ont été appelés cocons skyrmioniques en raison de leur forme et de leur nature 3D. Ce confinement vertical peut également être contrôlé en optimisant la multicouche, notamment en jouant sur les différences d'épaisseur entre les couches ferromagnétiques. En partant de la base des structures SG, des multicouches plus complexes ont été conçues pour permettre la coexistence de cocons skyrmioniques et de skyrmions colonnaires. L'idée fondamentale était d'incorporer des couches minces de Co, qui présentent une forte anisotropie magnétique perpendiculaire (PMA), afin de diviser les textures magnétiques en deux. La structure résultante, appelée double gradient (DG), se compose donc de deux SG séparées par des répétitions de couches minces de Co.

De manière similaire, des études combinant la microscopie à force magnétique (MFM) et des mesures de transport, associées à des simulations, ont permis de démontrer l'existence de deux types d'objets : ceux qui s'étendent sur toute l'épaisseur et ceux dont l'extension verticale est limitée (Figure 3). IX Cette coexistence est intéressante pour les applications car la présence de différents types d'objets permettrait d'encoder de manière stable différents états. The rise of 3D nanomagnetism I n many fields of physics, dimensionality governs the complexity of the system under consideration.

A simple one-dimensional (1D) model often provides an insight regarding the associated physics but might fail to capture more elusive effects or phenomena that hide into higher dimensionality.

The latter commonly gives access to a richer panel of properties which is of interest from both the fundamental and the technological standpoint, assuming one can harness the added intricacy. The spintronics and related nanomagnetism fields are no exceptions to this tendency. From [4].

Whereas electronics thrive on using the electron's charge, spintronics is based on the exploitation of its spin, namely its intrinsic magnetic moment, which can be coupled with the orbital moments to process or encode information. The spin is innately suitable for memory purposes as its two projections (up or down) can easily be identified with binary states (0 or 1). Systems resting on such a quantum number have been introduced as early as 1956 with the development of the hard disk drive by IBM.

Over the years, the development of leading-edge technologies (Figure 1.1) have been supported by various fundamental breakthroughs. Among those, the giant magnetoresistance (GMR) [5,6] and the tunneling magnetoresistance (TMR) [7] are key examples as they allow the creation of a spin current making use of a ferromagnet, explaining why they are still heavily used nowadays. Those discoveries marked the birth of the spintronic field and the research for spin-based nanoelectronic devices continued to prosper in the hope of fully utilizing the spin capabilities to obtain a reduced power consumption along with increased processing and memory functions [4]. This development is intrinsically linked to the magnetic properties down at the nanometer scale, i.e. nanomagnetism.

In the last few years, the spintronic field continues to thrive, for instance with the emergence of topological magnetic quasiparticles or textures. This is of the highest interest as modern electronics appear to reach its limit on a similar timescale, with the breaking down of Moore's law [START_REF] Waldrop | More than Moore[END_REF]. Indeed, semiconductors-based transistors, which are indispensable components for microprocessors, are getting harder to miniaturize due to various constrains such as Joule heating or more fundamental issues like quantum effects down at the atomic scale [START_REF] Kang | Skyrmion-electronics: An overview and outlook[END_REF]. Among several alternatives, topological magnetic textures thus appear promising, especially for their scalability and their non-volatility. Numerous candidates can be put forward such as vortices, monopoles or domain walls but one in particular has been retaining the community's attention since its discovery: the magnetic skyrmion.

A magnetic texture is defined by a representative space-dependent distribution of the local magnetization, i.e., by the arrangement of neighboring spins. Simply defined, a magnetic skyrmion corresponds to a circular two-dimensional (2D) whirling of the spins (Figure 1.2.a). More precisely, in the periphery of the skyrmion, the local magnetic moments are pointing upwards whereas they are pointing downwards in the core region (or reciprocally). The smooth spatial transition between those two regions displays a screw rotation of the spins which can take many forms, introducing the notion of chirality. description of the interfacial DMI emerging between a heavy metal (blue) and a ferromagnet (gray). c) Typical scheme of a magnetic multilayer: repetitions of a ferromagnet (Co) encapsulated between metals (Pt, X). d) Structure of a columnar skyrmion in a magnetic multilayer. Adapted from [START_REF] Fert | Skyrmions on the track[END_REF][START_REF] Moreau-Luchaire | Additive interfacial chiral interaction in multilayers for stabilization of small individual skyrmions at room temperature[END_REF].

Initially, skyrmions were first introduced in 1962 by Skyrme who thus formulated the first topological soliton to model a particle, in the field of nuclear physics. It was repurposed later in 1989 by Bogdanov and Yablonskii who predicted the existence of magnetic skyrmions [START_REF] Bogdanov | Thermodynamically stable "vortices" in magnetically ordered crystals. the mixed state of magnets[END_REF]. This theory was later extended in 1994 by including chiral interactions to the energy functional which favor the stabilization of enantiomeric non-collinear magnetic textures [START_REF] Bogdanov | Thermodynamically stable magnetic vortex states in magnetic crystals[END_REF]. The interaction in question corresponds the Dzyaloshinskii-Moriya interaction (DMI) [START_REF] Dzyaloshinsky | A thermodynamic theory of "weak" ferromagnetism of antiferromagnetics[END_REF][START_REF] Moriya | Anisotropic superexchange interaction and weak ferromagnetism[END_REF], an antisymmetric exchange of relativistic origin sustained by structures with large spin-orbit coupling (SOC) in presence of a broken inversion symmetry (Figure 1.2.b). In 2006, it was finally predicted that chiral skyrmions could be stabilized as ground states in B20 helimagnets [START_REF] Ulrich K Roessler | Spontaneous skyrmion ground states in magnetic metals[END_REF], helped by the DMI, which triggered the first experimental observation in 2009 by Mühlbauer in bulk MnSi [START_REF] Mühlbauer | Skyrmion lattice in a chiral magnet[END_REF] which is able to host 20 nm skyrmions [START_REF] Yu | Variation of skyrmion forms and their stability in MnSi thin plates[END_REF]. Since then, they have been characterized in multiple hosting materials such as B20 crystals or thin films but also in artificial heterostructures, called magnetic multilayers [START_REF] Kang | Skyrmion-electronics: An overview and outlook[END_REF][START_REF] Finocchio | Magnetic skyrmions: from fundamental to applications[END_REF][START_REF] Back | The 2020 skyrmionics roadmap[END_REF].

In this thesis, the work has been solely focused on magnetic multilayers, albeit structures more intricate than the usual ones. A typical multilayer is made of repetitions of a trilayer of one ferromagnetic element encapsulated between two different metals, often heavy elements, which thus possess a large SOC (Figure 1.2.c). Interestingly, a rigorous selection of the materials (and their thickness) allows to engineer and tune the DMI and the other magnetic interactions by controlling the various interfaces [START_REF] Ajejas | Element-selective modulation of interfacial Dzyaloshinskii-Moriya interaction in Pt|Co| metal based multilayers[END_REF]. This can facilitate the stabilization of topological textures as it rests on the interplay between the various magnetic interactions. For instance, by using heavy metals with opposite sign of the DMI on either side of the ferromagnet allows the chiral effect to be reinforced as the contributions from the two interfaces add up. In magnetic multilayers, the skyrmions are stabilized in all the magnetic layers, yielding a cylindrical profile of the magnetization (even though the magnetization is discontinuous along the vertical axis). In this case, they are called columnar skyrmion or skyrmion tube due to the invariant translation of the skyrmion along the vertical direction in the ferromagnetic layers1 (Figure 1.

2.d).

This columnar structure is important to increase their thermal stability up to room temperature and facilitate their characterization. Thus, magnetic multilayers are suitable alternatives to chiral magnetic crystals as they are more scalable while offering a broader tunability of their magnetic properties, an indisputable perk for potential applications.

Topological textures, including skyrmions, are said to be protected against small perturbations of the system which would make them reliable for technological purposes. Typically, this improved stability due to topological protection is characterized with a topological number N sk that cannot be changed by a continuous deformation of the spin configuration. In a two-dimensional case, it can be defined as [START_REF] Nagaosa | Topological properties and dynamics of magnetic skyrmions[END_REF]:

N sk = 1 4π ¢¢ m • ∂m ∂x × ∂m ∂y dx dy (1.1)
with m(x, y) the normalized magnetization at lattice site (x, y), a function of R 2 evolving on the surface of the unit sphere S 2 . It describes the spin distribution forming the magnetic arrangement.

The topological number N sk takes integer values which defines different homotopy groups: magnetic configurations with non-equal N sk thus have a different topology. Consequently, an uniform magnetization yields N sk = 0 whereas for a skyrmion N sk = ±1. However, the magnetization of physical systems is defined on a discrete lattice of localized atoms which violate the continuity hypothesis underlying the definition of the topological number and thus question the validity of the various homotopy classes from a mathematical standpoint. This breakdown explains why in real systems it is possible to create and stabilize magnetic skyrmions in the middle of a uniform ferromagnetic background, a transformation that would otherwise be forbidden.

To develop stable and efficient skyrmion-based devices, their nucleation, detection and motion must be fully controlled in order to write, read and process the information at room temperature.

The figureheads of such applications are the racetrack memory [START_REF] Stuart Sp Parkin | Magnetic domain-wall racetrack memory[END_REF] (Figure 1.3.a) or logic devices that would require the control of the aforementioned properties. For instance, the nucleation can be achieved via local heating of the system with electric current [START_REF] Sampaio | Nucleation, stability and current-induced motion of isolated magnetic skyrmions in nanostructures[END_REF][START_REF] Hrabec | Current-induced skyrmion generation and dynamics in symmetric bilayers[END_REF], by application of laser pulses [START_REF] Soong-Geun | Creation of magnetic skyrmion bubble lattices by ultrafast laser in ultrathin films[END_REF][START_REF] Gerlinger | Application concepts for ultrafast laser-induced skyrmion creation and annihilation[END_REF] or with the support of an external magnetic field. The precise localization of nucleation is often linked to the presence of local variations of the magnetic properties which can be induced by the growth itself or by selective irradiation [START_REF] Fallon | Controlled individual skyrmion nucleation at artificial defects formed by ion irradiation[END_REF][START_REF] Juge | Helium ions put magnetic skyrmions on the track[END_REF][START_REF] Hu | Precise tuning of skyrmion density in a controllable manner by ion irradiation[END_REF][START_REF] Kern | Deterministic generation and guided motion of magnetic skyrmions by focused He+-ion irradiation[END_REF] or by using geometrical deformation (such as notches in tracks) among other methods. Experimental observations of skyrmions are reported in Figure 1.3.b,c. Their motion is often supported by the use of a spin current, which generate torques on the magnetic objects [START_REF] Mark | Anatomy of spin-transfer torque[END_REF][START_REF] Manchon | Theory of spin torque due to spin-orbit coupling[END_REF][START_REF] Garello | Symmetry and magnitude of spin-orbit torques in ferromagnetic heterostructures[END_REF][START_REF] Av Khvalkovskiy | Matching domain-wall configuration and spin-orbit torques for efficient domain-wall motion[END_REF][START_REF] Legrand | Room-temperature current-induced generation and motion of sub-100 nm skyrmions[END_REF]. Finally, their detection can be envisioned with the use of electrical measurements based on magnetoresistance effects [START_REF] Neubauer | Topological Hall effect in the a phase of MnSi[END_REF][START_REF] Hanneken | Electrical detection of magnetic skyrmions by tunnelling non-collinear magnetoresistance[END_REF][START_REF] Maccariello | Electrical detection of single magnetic skyrmions in metallic multilayers at room temperature[END_REF].

Despite acquiring a third dimension within magnetic multilayers, skyrmions remain relatively invariant over the heterostructure thickness and are commonly regarded as 2D topological spin textures.

Consequently, exploring the study of 3D objects that showcase significant variations along the third dimension becomes a natural progression, introducing an additional degree of freedom to the system.

This heightened dimensionality holds numerous advantages for future devices and applications, including enhanced storage density and memories with a greater range of available states. More possibilities also arise from the potential ability to displace the objects along the third dimension, further augmenting their potential.

Thus, following the interest for the development of more efficient and reliable devices, interest has also surged for 3D nanomagnetism [START_REF] Fernández-Pacheco | Three-dimensional nanomagnetism[END_REF][START_REF] Fischer | Launching a new dimension with 3D magnetic nanostructures[END_REF][START_REF] Göbel | Beyond skyrmions: Review and perspectives of alternative magnetic quasiparticles[END_REF][START_REF] Raftrey | The road to 3-dim nanomagnetism: Steep curves and architectured crosswalks[END_REF] which regroups magnetic nanostructures and complex spin textures that are no longer confined in a plane or invariant along the vertical direction (like the columnar skyrmion). For magnetic nanostructures, the third dimension offers more possibilities and complexity regarding the possible geometries (Figure 1.4.a) which broaden the effects and applications available. The recent progresses in that direction rest for example on the development of focused electron beam induced deposition (FEBID) [START_REF] De | Present and future applications of magnetic nanostructures grown by FEBID[END_REF][START_REF] Jm De | Review of magnetic nanostructures grown by focused electron beam induced deposition (FEBID)[END_REF][START_REF] Fernández-Pacheco | Writing 3D nanomagnets using focused electron beams[END_REF][START_REF] Magén | Focused-electron-beam engineering of 3D magnetic nanowires[END_REF] which allows growing complex 3D nanostructures out of a few chemical elements as illustrated with several examples in Figure 1. 4. For instance, one can deposit helices to carry information using domain walls or even interlace two of them to investigate the subsequent interactions. The third dimension also promotes new dynamics regimes with, for instance, the propagation of domain walls guided along cylindrical nanowires [START_REF] Yan | Chiral symmetry breaking and pair-creation mediated walker breakdown in magnetic nanotubes[END_REF] or even along 3D racetrack the data is encoded with binary states corresponding to the absence or presence of a skyrmion. b) Skyrmions imaging. Left: scanning transmission X-ray microscopy (STXM) performed on a Pt/Co/Ir heterostructure in different devices [START_REF] Moreau-Luchaire | Additive interfacial chiral interaction in multilayers for stabilization of small individual skyrmions at room temperature[END_REF]. Middle: photo-emission electron microscopy (PEEM) in Pt/Co/MgO multilayer [START_REF] Boulle | Room-temperature chiral magnetic skyrmions in ultrathin magnetic nanostructures[END_REF]. Right: STXM measurements on a Pt/Co/Ta heterostructure [START_REF] Woo | Observation of room-temperature magnetic skyrmions and their current-driven dynamics in ultrathin metallic ferromagnets[END_REF]. c) Successive STXM imaging of skyrmions in a track after the application of current pulses to highlight their motion [START_REF] Woo | Observation of room-temperature magnetic skyrmions and their current-driven dynamics in ultrathin metallic ferromagnets[END_REF]. [START_REF] Fernández | The racetrack breaks free from the substrate[END_REF][START_REF] Gu | Three-dimensional racetrack memory devices designed from freestanding magnetic heterostructures[END_REF]. For instance, complex networks of interconnected magnetic nanowires are investigated for the freedom they provide with their structure, topology and frustration [START_REF] May | Realisation of a frustrated 3D magnetic nanowire lattice[END_REF][START_REF] Edward C Burks | 3D nanomagnetism in low density interconnected nanowire networks[END_REF][START_REF] Cheenikundil | Magnetization dynamics in a three-dimensional interconnected nanowire array[END_REF]. Those studies also underline the necessity to understand the impact of curvature and other 3D geometrical deformations on the magnetization [START_REF] Streubel | Magnetism in curved geometries[END_REF][START_REF] Josten | Curvaturemediated spin textures in magnetic multi-layered nanotubes[END_REF][START_REF] Meng | Non-planar geometrical effects on the magnetoelectrical signal in a three-dimensional nanomagnetic circuit[END_REF].

Another aspect of 3D nanomagnetism, which is at the heart of this thesis, relates to the spin textures themselves, regardless of the geometrical properties of the hosting materials. Those correspond to magnetization distributions with significant variations along every direction and displaying topological properties. For instance, in bulk material, theoretical predictions, followed by experimental observation, describe magnetic bobbers [START_REF] Redies | Distinct magnetotransport and orbital fingerprints of chiral bobbers[END_REF]1] where one end of a skyrmion string goes to a surface while the other one ends with a point singularity inside the sample (Figure 1.5.a). The singularity, called a Bloch point, is associated with a vanishing magnetization and possesses topological properties. This pioneer example could be extremely important for the development of more stable racetrack memory for instance, as one could encode one state with bobbers and the other with columnar skyrmions. Another case of these skyrmions strings corresponds to dipole strings (Figure 1.5.c) [START_REF] Ao Leonov | Homogeneous and heterogeneous nucleation of skyrmions in thin layers of cubic helimagnets[END_REF]2], so named due to the two reversed Bloch points at their extremities, corresponding to two opposite topological charges, which render them topologically trivial. It is also predicted that antiferromagnetic bobbers could be coupled together, yielding a toron-like object with an inversion symmetry [START_REF] Bassirian | Breathing mode dynamics of coupled three-dimensional chiral bobbers[END_REF]. Finally, skyrmions strings can also be braided together in multiple ways in chiral magnets [START_REF] Zheng | Magnetic skyrmion braids[END_REF] (Figure 1.5.e) to obtain multiple states brought by the third dimension [START_REF] Fernández-Pacheco | Three-dimensional nanomagnetism[END_REF]. b) Artificial double cobalt helices grown using FEBID (left: simulated spin distribution, right: scanning electron microscope images) [START_REF] Sanz-Hernández | Artificial double-helix for geometrical control of magnetic chirality[END_REF]. c) Concept of using 3D connectors between two functional planes to carry information with the propagation of domain walls [START_REF] Donnelly | Complex free-space magnetic field textures induced by three-dimensional magnetic nanostructures[END_REF]. d) Example of 3D nanodevice for magneto-transport measurements to investigate the impact of the 3D geometry [START_REF] Meng | Non-planar geometrical effects on the magnetoelectrical signal in a three-dimensional nanomagnetic circuit[END_REF].

in a single system. Analogous 3D textures have been reported as well in magnetic multilayers with the recent observation of the elusive hopfion [START_REF] Kent | Creation and observation of hopfions in magnetic multilayer systems[END_REF][START_REF] Liu | Binding a hopfion in a chiral magnet nanodisk[END_REF], originally predicted to appear in chiral magnetic crystals [START_REF] Sutcliffe | Hopfions in chiral magnets[END_REF][START_REF] Filipp N Rybakov | Magnetic hopfions in solids[END_REF] which displays a complex torus-like distribution of the magnetization (Figure 1.5.d). It has been recently shown that, in hybrid ferromagnetic/ferrimagnetic multilayers, truncated skyrmions that end abruptly at a given interface can be stabilized [START_REF] Mandru | Coexistence of distinct skyrmion phases observed in hybrid ferromagnetic/ferrimagnetic multilayers[END_REF][START_REF] Oguz Yildirim | Tuning the coexistence regime of incomplete and tubular skyrmions in ferro/ferri/ferromagnetic trilayers[END_REF], as evidenced with a vertical cut of the magnetization shown in Figure 1.5.b. The zoology of 3D spin textures is thus ever expending and the research for new and more complex ones is ongoing.

Such new objects pave the way toward expending spintronics into the third dimension but several challenges must be overcome first, mostly their characterization as well as their implementation.

Regarding the latter, most of those newly observed textures have been detected in chiral magnetic crystals or in nano-structures, which strongly limits their functionality. In opposition, magnetic multilayers appear as a promising alternative to harbor 3D topological textures as they are more scalable while offering a broader tunability of their magnetic properties. The more problematic challenge remains the characterization of magnetic textures as it requires probing the magnetization over the thickness. This is often achieved by using recent high-end imaging techniques that rely on the use of soft X-rays or electrons. Transmission techniques such as scanning transmission X-ray microscopy (STXM), tube along with a proposal of racetrack memory based on them [1]. b) Magnetization profile of a truncated skyrmion tube stabilized in a magnetic multilayer [START_REF] Mandru | Coexistence of distinct skyrmion phases observed in hybrid ferromagnetic/ferrimagnetic multilayers[END_REF]: the skyrmion is annihilated in the middle layers (red: spin up, blue: spin down). c) Dipole strings or magnetic globules stabilized in chiral magnets [2]. d) Magnetization distribution of a hopfion [3]. e) Different skyrmion braids displaying interlaced skyrmion strings [START_REF] Zheng | Magnetic skyrmion braids[END_REF].

holography or photo-electron microscopy (PEEM) can be used to detect 3D spin textures [START_REF] Grelier | X-ray holography of skyrmionic cocoons in aperiodic magnetic multilayers[END_REF] or to probe the magnetization in 3D nanostructures such as nanowires [START_REF] Da Col | Observation of Bloch-point domain walls in cylindrical magnetic nanowires[END_REF]. Building on those transmission techniques, remarkable reconstructions of the magnetization distribution have been achieved using X-ray laminography for a permalloy disk [START_REF] Witte | From 2D STXM to 3D imaging: Soft x-ray laminography of thin specimens[END_REF], for 3D nanostructures [START_REF] Pip | X-ray imaging of the magnetic configuration of a three-dimensional artificial spin ice building block[END_REF] or for various textures with X-ray or electron tomography (see Figure 1.6) [START_REF] Donnelly | Three-dimensional magnetization structures revealed with X-ray vector nanotomography[END_REF][START_REF] Seki | Direct visualization of the three-dimensional shape of skyrmion strings in a noncentrosymmetric magnet[END_REF]. Recently, a magnetic skyrmion in a magnetic multilayer was also reconstructed [START_REF] Raftrey | Quantifying the topology of magnetic skyrmions in three dimensions[END_REF]. Most of those measurements are based on the X-ray magnetic circular dichroism (XMCD) [START_REF] Schütz | Absorption of circularly polarized x rays in iron[END_REF], which results in a significant difference of absorption between spins of opposite polarities at properly selected energies. As the aforementioned techniques allow to retrieve the full vectorial field of the magnetization over the complete space, they are powerful tools for the study of 3D textures.

My PhD thesis is focused on the stabilization and characterization of a novel type of 3D textures, the so-called skyrmionic cocoons.

Chapter 2 presents the theoretical framework that is essential for comprehending the subsequent discussions. It primarily focuses on magnetic interactions and their significance in stabilizing noncollinear spin textures. The fundamental concepts of topology are also introduced, laying the foundation for further exploration. Furthermore, this chapter refines the definition of the skyrmion, which will be compared to new 3D textures. Moreover, considering the reliance of numerous experiments on X-rays, the interaction between X-rays and magnetization in a ferromagnetic material is discussed.

An introduction to the growth and characterization of magnetic multilayers is given in Chapter 3.

The quality of the interfaces, a key property for engineering the magnetic interactions, and of the of the magnetization of a FeGe needle measured with holographic vector field electron tomography [START_REF] Wolf | Unveiling the three-dimensional magnetic texture of skyrmion tubes[END_REF]. X-ray tomography magnetic reconstructions of b) a GdCo 2 microscopic pillar displayed with isosurfaces [START_REF] Donnelly | Experimental observation of vortex rings in a bulk magnet[END_REF], c) a permanent Nd-Fe-B magnet along its hysteresis loop [START_REF] Takeuchi | Real picture of magnetic domain dynamics along the magnetic hysteresis curve inside an advanced permanent magnet[END_REF], d) a Bloch point observed in a boomerang shaped permalloy (NiFe) sample [START_REF] Hermosa | Topological magnetic dipoles and emergent field bundles in a ferromagnetic microstructure by X-ray magnetic vector tomography[END_REF] and e) the outer layers of a ferrimagnet trilayer [START_REF] Hermosa-Muñoz | 3D magnetic configuration of ferrimagnetic multilayers with competing interactions visualized by soft X-ray vector tomography[END_REF].

materials are investigated with various structural approaches. This allows to ensure that the intermixing remains negligible and that no consequent contamination (oxidation) is observed. In the second part of this chapter, the magnetic properties of the Pt|Co|Al trilayer are investigated as it serves as a basis for the majority of the multilayers in this thesis. Finally, the first aperiodic multilayers (with variable ferromagnetic thickness) are experimentally and numerically probed to verify the impact of the non-uniform anisotropy. This put forwards the so-called single gradient (SG) structure whose promising confinement capabilities are able to stabilize zero-field skyrmions.

In Chapter 4, the single-gradient structure is characterized in depth using magnetic force microscopy as well as micromagnetic simulations, notably regarding the field dependency and the impact of the structure parameter. A new spin texture, called skyrmionic cocoon, is observed. It is characterized by a typical ellipsoidal shape and a limited vertical extension as it does not extend from the bottom to the top magnetic layers. Its topology and its chirality are further discussed and the latter is probed with X-ray resonant magnetic scattering experiments.

The SG structure can also serve as a building block for more complex architecture. From it, the double gradient (DG) multilayer is engineered which allows for the coexistence of skyrmionic cocoons and more usual columnar skyrmions. This structure is characterized in depth in Chapter 5, similarly to the SG structure.

Chapter 1. The rise of 3D nanomagnetism

The complexity to characterize 3D spin textures led us to realize magnetotransport measurements in order to probe the average magnetization. In Chapter 6, those are successfully correlated with micromagnetic simulations, confirming the accuracy of the latter and thus of the coexistence of several spin textures with different vertical extensions. This also put forwards an easily implementable method to detect 3D textures. Additionally, their individual electrical contributions are studied and separated in micron-sized Hall bars. The ability to resolve their identity based on their electrical signature is an important asset for potential applications.

Finally, Chapter 7 is dedicated to synchrotron measurements using soft X-rays, allowing to directly retrieve information about the full magnetization texture. Different transmission experiments have been performed, such as holography or scanning transmission X-ray microscopy, that are sensitive to the average magnetization over the thickness. The DG structures are studied in depth at different magnetic fields and X-ray energies. To finish, an experimental reconstruction of the 3D magnetization distribution is discussed, obtained from laminography experiments. The results of this chapter unequivocally proves the 3D nature of skyrmionic cocoons as well as their coexistence with other spin textures. This particular property could be interesting for many 3D skyrmionic applications. The topology and the chirality of the different spin textures studied in this thesis is then discussed as this can strongly impact their behavior. Finally, the resonant X-rays interactions with magnetism are briefly explored as the imaging and the characterization of the 3D spin textures has often been performed using synchrotron based experiments.

Magnetization relaxation in magnetic multilayers

In this PhD, we rely on a micromagnetic model to investigate the magnetization behavior and metastable magnetic configurations in presence of an effective field. The micromagnetism approach serves as a bridge between quantum theory, which deals with atomic interactions, and Maxwell theory, which applies to macroscopic dimensions. It utilizes a continuous magnetization vector to describe spins, rather than considering individual atomic moments. While it does not capture distinct magnetic spins or quantum effects, micromagnetism provides a reliable theoretical understanding of magnetization processes down to a few nanometers scale. It assumes that the organizing forces (the magnetic interactions) are sufficiently strong to ensure that the magnetization varies continuously in space and time, without singularities or discontinuities [START_REF] Fidler | Micromagnetic modelling-the current state of the art[END_REF][START_REF] Abert | Micromagnetics and spintronics: models and numerical methods[END_REF].

The normalized magnetization is thus described with a spatially varying three-dimensional vector 1 field m(x, y, z) with |m| = 1. It corresponds to the local magnetization M normalized by the saturation magnetization M S , a system-dependent magnetic parameter (the magnetization strength). Note that in magnetic multilayers, the magnetization is discontinuous along the vertical direction z (along the normal of the layers) due to the non-magnetic spacers between two magnetic layers. Even though some coupling exists in between two adjacent layers, strong vertical variations can thus exist from one layer to another. Inside a given layer, the magnetization should not change significantly along z for thin films 2 so that the magnetization depends mostly on (x, y).

In multilayered systems, different interactions are at play which favors different spin configurations resulting in the formation of domain walls or complex spin textures for instance. After introducing the central equation of the magnetization dynamics, the Landau-Lifshitz-Gilbert equation, the magnetic interactions of interest for the stabilization of 3D textures in magnetic multilayers are discussed.

Landau-Lifshitz-Gilbert equation

The Landau-Lifshitz-Gilbert (LLG) equation is a fundamental equation in the field of magnetism that describes the dynamics of magnetization. It combines the Landau-Lifshitz equation, which describes the precession of magnetization, with the Gilbert damping term, which accounts for the dissipation of energy. The LLG equation can be written in the following form [START_REF] Landau | On the theory of the dispersion of magnetic permeability in ferromagnetic bodies[END_REF][START_REF] Thomas | A phenomenological theory of damping in ferromagnetic materials[END_REF]:

∂m ∂t = -γ µ 0 m × H eff + α m × ∂m ∂t (2.1)
with γ the gyromagnetic ratio, µ 0 the vacuum permeability, H eff the effective magnetic field and α the Gilbert damping constant. The first term on the right-hand side of Eq. 2.1 describes the precession of the magnetization vector around the local effective magnetic field. The second term represents is the Gilbert damping term that is linked to the dissipation of energy and causes the magnetization to relax towards the direction of the local effective field (see Figure 2.1). The parameter α determines the strength of the damping: for instance, α = 0.1 (resp. 0.01) indicates that the magnetization precesses about a 10 (resp. 100) times before relaxing.

The LLG equation is derived from the quantum mechanical description of the interaction between the magnetic moments of the atoms in the material and the external magnetic field. It is widely used in the study of magnetization dynamics in various magnetic systems, including thin films, magnetic nanoparticles, and magnetic multilayers. Solving the LLG equation allows to understand and predict 1 The vectorial quantities are written in bold symbols throughout this manuscript. 2 The ferromagnetic films investigated in the scope of this thesis typically have thickness lower than 3 nm. the behavior of magnetization in thin films, which is crucial for developing and optimizing magnetic devices and technologies such as magnetic memory devices, spintronics, and magnetic sensors. The equation provides a theoretical framework to investigate phenomena such as magnetization reversal, domain wall motion, and spin dynamics, and has been instrumental in advancing the understanding of magnetic materials and their applications. It also serves as basis for micromagnetic simulations solver such as mumax 3 [START_REF] Vansteenkiste | The design and verification of MuMax3[END_REF].

The contributions of the different magnetic interactions are contained inside the effective field H eff .

In ferromagnetic multilayers, we focus on five different interactions, namely the Heisenberg exchange, the magnetic anisotropy, the dipolar interaction, Dzyaloshinskii-Moriya interaction (DMI) and finally the Zeeman interaction. Using the same order, the effective field can thus be written as:

H eff ∝ H ex + H K + H demag + H DMI + H ext with H β = 1 M S ∂E β ∂m (2.2)
with β the previously mentioned interactions and E the corresponding energy. We do not include the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction [START_REF] Melvin | Indirect exchange coupling of nuclear magnetic moments by conduction electrons[END_REF], an antiferromagnetic coupling that can be obtained in magnetic multilayers as we only exploit ferromagnetic coupling in this thesis. Moreover, we predominantly relies on static measurements in this manuscript so that we do not consider spin-orbit torques or spin-transfer torques in the LLG equation. In the following, we discuss the interactions present in Eq. 2.2 independently.

Energy contributions

Symmetric exchange

The symmetric exchange (or Heisenberg exchange)in ferromagnets is a local effect that favors the collinear alignment of two neighboring spins, thus encouraging the formation of large domains of uniform magnetization. It arises from quantum mechanical considerations based on the Pauli principle and the Coulomb energy of a two-particle system (see [START_REF] Blundell | Magnetism in condensed matter[END_REF][START_REF] Cohen-Tannoudji | Quantum mechanics[END_REF][START_REF] David | Introduction to quantum mechanics[END_REF] for demonstrations). The exchange between two neighboring spins (S i and S j ) with an atomistic approach can simply be expressed as:

E ij ex = -JS i • S j (2.3)
with J the exchange integral expressed with the wave function of each particle. For a ferromagnetic coupling (J > 0), the energy is minimal (resp. maximal) for parallel (resp. antiparallel) spins.

Considering the continuous magnetization field m, assuming a cubic and isotropic lattice for simplicity, the exchange energy becomes:

E ex = ¢ A (∇m) 2 dV with A = J ⟨S 2 ⟩ n a (2.4)
with n the number of atoms per unit cell, a the lattice parameter, S the quantum spin number and V the volume of the magnetic system. The higher order of exchange are neglected as the significant contributions mostly arise from next neighbors interactions. Despite being initially derived for localized spins and isotropic lattice structures, this energy expression has been found to effectively describe a wide range of materials [START_REF] Huber | Magnetic domains: the analysis of magnetic microstructures[END_REF].

In our magnetic multilayer, the exchange is typically lower than 30 pJ/m for Co-based structures [START_REF] Ha Alperin | Observation of the dispersion relation for spin waves in hexagonal cobalt[END_REF][START_REF] Vaz | Magnetism in ultrathin film structures[END_REF] and is modulated by the thickness and materials under consideration [START_REF] Legrand | Spatial extent of the Dzyaloshinskii-Moriya interaction at metallic interfaces[END_REF][START_REF] Böttcher | Quantifying symmetric exchange in ultrathin ferromagnetic films with chirality[END_REF]. Indeed, some interfacial considerations arise for the thin film limits as not all atoms posses magnetic neighbors near the interfaces.

The exchange allow changes on a characteristic length larger than:

l ex = 2A µ 0 M 2 S (2.5)
Typically, for the 3d elements such as Co or Ni, it reaches 10 nm. In magnetic thin films with a smaller thickness than l ex , the magnetization can be considered invariant along the vertical direction.

Dipolar interaction

The dipolar interaction arises from the dipole-dipole interaction existing between each pair of spins.

For a single pair, it can be written as:

E = µ 0 4πr 3 µ 1 • µ 2 - 3 r 2 (µ 1 • r) (µ 2 • r) (2.6)
with r the distance between the two spins, µ i = -gµ B S i the magnetic moment associated to the spin momentum ℏS, g the Landé factor and µ B the Bohr magneton. It thus depends on their spatial separation as well as their degree of mutual alignment. In a magnetic material, all pairing between spins must be considered to compute the dipolar field, also called demagnetizing or stray field. In a

∇ • B = 0 (2.7) ∇ × H demag = 0 (2.8)
with the magnetic flux B = µ 0 (H demag + M ) with M the non-normalized magnetization. The first equation Eq. 2.7 indicates that whenever a divergence of the magnetization occurs a divergence of the stray field is also present. For instance near the edges where it stops abruptly, it is as if magnetic monopoles are left on the surface and serve as source of the dipolar field (Figure 2.2.a). From Eq. 2.8, the stray field is non-rotational and can thus be defined from a scalar potential U that ultimately respects Poisson's equation:

∆U = ∇ • M with H demag = -∇U (2.9)
It can be shown that the scalar potential can be expressed using Gauss's theorem with magnetic

volume charges ρ m (x) = -∇ • M (x) and surface charges σ m (x) = M (x)
• n located at the spin site (x), with n the vector along the surface normal [START_REF] Blundell | Magnetism in condensed matter[END_REF][START_REF] Abert | Micromagnetics and spintronics: models and numerical methods[END_REF][START_REF] Exl | Micromagnetism. Handbook of Magnetism and Magnetic Materials[END_REF]:

U = - 1 4π ¢ V ρ m |x -x ′ | dV + 1 4π S σ m |x -x ′ | dS (2.

10)

Except for a few magnetic arrangements, no analytical solutions exist for H demag but it can be numerically computed from the previous expression. The related energy can then be obtained as: The dipolar interaction favors the formation of domains of opposite magnetization direction (canceling the surface magnetic charges) to enable flux closure and thus reducing the energy. In

E demag = - µ 0 M S 2 ¢ V m • H demag dV (2.11)

Magnetic anisotropy

The presence of magnetic anisotropy in a material promotes spin alignment along specific axes (resp. planes) known as the easy axis (resp. plane). Since these axes are not oriented, the resulting energy is independent of the orientation of the magnetization projection on those axes. The origin of anisotropy can be attributed to the spin-orbit interaction and the partial quenching of angular momentum. Different types of anisotropy contributions can be distinguished, including shape anisotropy, magneto-crystalline anisotropy, and magneto-elastic anisotropy [START_REF] Mt Johnson | Magnetic anisotropy in metallic multilayers[END_REF]. Another term of interfacial origin is expected when a thin film is in contact with a heavy metal providing large SOC. In the context of our multilayered systems, the magneto-elastic anisotropy, which arises from structural strain such as lattice mismatch, is not considered significant and is not further discussed. Then, the anisotropy energy can be written as:

E K = - ¢ K u (m • z) 2 dV with K u = K u,s t + K V (2.12)
with K V the volume contribution and K u,s denoting the interface contribution for a ferromagnetic layer of thickness t. In our magnetic multilayers, where the thickness t remains smaller than 3 nm, the bulk anisotropy K V has been considered negligible (K u ∼ K u,s /t). Finally, the shape anisotropy arises due to the dipolar field created by the magnetization. It promotes magnetization alignment along the longest axis of the magnetic structure which reduce the magnetic charges present on the surface.

Specifically, in the case of a thin film, the magnetization is encouraged to lie in-plane to minimize the energy associated with the stray field. As a result, an additional term is introduced into the anisotropy expression, valid for a uniform magnetization:

K shape = - µ 0 M 2 S 2 (2.

13)

In a quasi uniform magnetization configuration, the total effective anisotropy is given by:

K eff = K u,s t - µ 0 M 2 S

(2.14)

To calculate the corresponding total anisotropy energy, this expression must substitute K u in Eq. 2.12. We make the assumption that both the volume anisotropy and higher-order terms of the uniaxial anisotropy can be neglected, which is often observed to be the case. When the effective anisotropy constant, K eff , is positive (resp. negative), it favors an out-of-plane (resp. in-plane) magnetization.

Dzyaloshinskii-Moriya interaction (DMI)

The Dzyaloshinskii-Moriya interaction (DMI) is a quantum mechanical effect in solid-state physics that influences the magnetic properties of materials. It was first proposed by Dzyaloshinskii [START_REF] Dzyaloshinsky | A thermodynamic theory of "weak" ferromagnetism of antiferromagnetics[END_REF] and Moriya [START_REF] Moriya | Anisotropic superexchange interaction and weak ferromagnetism[END_REF] independently in the late 1950s. The DMI arises due to spin-orbit coupling in structures with broken inversion symmetry. Unlike the usual symmetric exchange interactions, such as the Heisenberg exchange, which favors parallel alignment of neighboring spins, the DMI is an antisymmetric exchange that encourages the tilting of neighboring spins with a given chirality.

In a simple atomistic picture, the DMI energy can be expressed as:

E DMI = D • (S 1 × S 2 ) (2.

15)

with the D the DMI vector. The energy is thus minimal for an orthogonal arrangement of the spins.

The DMI is schematically represented in Figure 2.3.a at the interface between a ferromagnetic layer and a heavy metal [START_REF] Ar Fert | Magnetic and transport properties of metallic multilayers[END_REF][START_REF] Heide | Dzyaloshinskii-moriya interaction accounting for the orientation of magnetic domains in ultrathin films: Fe/W (110)[END_REF]. This interface breaks the inversion symmetry, while the heavy metal provides a strong spin-orbit coupling interaction, both of which are necessary for the emergence of the DMI. In this case, the DMI vector is perpendicular to the axis between the two spins. The exact formulation of the DMI term in the micromagnetic approach depends on the crystallographic structure under consideration. For interfacial effect in thin films, it corresponds to [START_REF] Ao Leonov | The properties of isolated chiral skyrmions in thin magnetic films[END_REF]: As previously mentioned, the DMI can be engineered at the interface of a ferromagnetic layer and a heavy metal. Magnetic multilayers, thanks to their high tunability, build on that property to control and enhance the DMI [START_REF] Ajejas | Interfacial potential gradient modulates Dzyaloshinskii-Moriya interaction in pt/co/metal multilayers[END_REF]. However, as illustrated in Figure 2.3.b, using the same heavy material at both interfaces of the ferromagnet drastically decrease the DMI3 . Indeed, as the stacking order is reversed, so is the DM vector, so that both interfaces have opposite contributions. To avoid the DMI cancellation, a second metal can be used as in Figure 2.3.c, with an opposite sign of D, effectively resulting in additive contributions. Thus, in magnetic multilayers, the DMI strength can be modulated by tuning the interfaces and materials.

E DMI = - ¢ D L (x) zx + L (y) zy dV = - ¢ D m z ∂m x ∂x -m x ∂m z ∂x + m z ∂m y ∂y -m y ∂m z ∂y dV (2.
In thin films of thickness t, the interfacial DMI can be described using a surface constant D s = D × t.

Typically, for Pt|Co|X with X a second metal, D s ranges between -2 and 0.5 pJ/m 2 [START_REF] Ajejas | Interfacial potential gradient modulates Dzyaloshinskii-Moriya interaction in pt/co/metal multilayers[END_REF][START_REF] Kuepferling | Measuring interfacial Dzyaloshinskii-Moriya interaction in ultrathin magnetic films[END_REF].

Zeeman interaction

The Zeeman interaction results from an applied external magnetic field. As an external contribution, its amplitude and direction can be adjusted to modify the effective field acting on the magnetization.

It can thus help to control and modulate the magnetization distribution. The corresponding energy contribution is written as:

E Z = -µ 0 M S ¢ V m • H ext dV (2.17)
with H ext the external magnetic field. The energy is minimal when the spins are aligned with the magnetic field.

Magnetic interactions

The equilibrium of the magnetization thus relies on the interplay of all the significant magnetic interactions in magnetic multilayers (see

Skyrmion chirality

The interplay of the different magnetic interactions also play an important role in defining the effective chirality of non-collinear spin textures which can ultimately lead to variations over the thickness. Before going further in this statement, let us introduce the four peculiar chiralities available for a 2D skyrmion (see Figure 2.4).

The chirality can be defined using the helicity ϕ h : for Néel type skyrmion, ϕ h = nπ with n ∈ N and for Bloch, ϕ h = nπ + π/2. For a Néel skyrmion (resp. Bloch), taking a radial cut of the magnetization yields a smooth rotation in the plane of the cut (resp. in the plane orthogonal to the cut). For each chirality, two senses of rotation can also be defined, depending on the evolution of the spins along the radial cut: either clockwise (CW) or counterclockwise (CCW). This also depends on the polarity P of the core, i.e. if the spin at the center is up or down. For instance, for P = 1, ϕ h = 0 (resp. π) has a CCW Néel chirality (resp. CW) and ϕ h = π/2 (resp. 3π/2) has a CCW Bloch chirality (resp.

CW chirality). Generally, the interfacial DMI existing in magnetic multilayers promotes a given Néel chirality while the bulk DMI favors a Bloch one. Moreover, depending on its sign, the DMI encourages one given sense of rotation. In a standard skyrmionic magnetic multilayer, the skyrmions are stabilized in all magnetic layers, with a slightly varying radius and chirality. This results in a barrel-shaped skyrmion, where the size is larger in the middle layers compared to the outer layers [START_REF] Legrand | Hybrid chiral domain walls and skyrmions in magnetic multilayers[END_REF].

Columnar skyrmion

In this thesis, we compare the conventional skyrmions with new three-dimensional textures.

Therefore, it is crucial to establish a precise definition of what we will refer to as a skyrmion. In the following, a columnar skyrmion, also interchangeably known as a skyrmion tube, refers to a spin texture that spans across the entire thickness of a magnetic multilayer, from the bottom to the top interface. Consequently, the core of the skyrmion remains relatively invariant along the thickness, even though its diameter and chirality may exhibit variations in the vertical direction. Furthermore, an ideal skyrmion would possess a single spin direction (either up or down) at its center, but in experimental observations, they tend to be more extended and do not exhibit this characteristic. In this context, we will consider larger circular objects that possess a columnar structure as skyrmions, even though they may not strictly conform to the theoretical definition of a solitonic solution (which exists in 2D only). This extended definition was also put forward recently as Bogdanov and Panagopoulos suggested referring to the skyrmion as a skyrmion-bubble hybrid [START_REF] Bogdanov | Physical foundations and basic properties of magnetic skyrmions[END_REF].

A practical introduction to topology

A practical introduction to topology

This thesis expands on the current understanding of magnetic skyrmions and explores newly discovered 3D textures that are stabilized in magnetic systems. It is important to distinguish these textures from the well-established spin textures, especially skyrmions. Chapter 1 serves as an introduction, providing an overview of skyrmionic concepts and presenting pertinent experimental findings. The subsequent discussion delves into the investigation of the topological and chiral properties of spin textures, both in the general case and specifically focusing on skyrmions. Since our research involves novel magnetic objects in magnetic multilayers, comprehending their topological properties as well becomes crucial for assessing their stability.

Topology basics

This section makes the relation between the mathematical approach of topology and the magnetic order parameter m. The following discussion is heavily based on the reviews [START_REF] Braun | Topological effects in nanomagnetism: from superparamagnetism to chiral quantum solitons[END_REF] and [START_REF] Zang | Topology in magnetism[END_REF]. For simplicity, it is assumed that the magnetic medium is continuous, thus ignoring the discrete nature of the lattice4 . In real system, due to its finite and discrete nature, topological objects may deform and disappear but topology is still a powerful concept to study their stability with those approximations.

It is important to make the distinction between singular topological defects where the magnetization is forced to vanish and smooth topological features for which the magnetization vary smoothly between each site. Many magnetic configurations of interest from the latter kind, such as domain walls, vortices or skyrmions, are stable topological features whose stability arises from topological arguments as we will demonstrate in the following. Other properties, such as the emergent magnetic field of the skyrmions can also be explained from topological considerations in real space [START_REF] Nagaosa | Topological properties and dynamics of magnetic skyrmions[END_REF][START_REF] Zang | Topology in magnetism[END_REF].

Homotopy groups

Let d ∈ J1, 3K be the physical space dimension and n = d -1 the associated contour dimensionality5 , p ∈ J1, 3K the number of magnetization components and r = p -1 the corresponding contour dimensionality (see Table 2.2). The magnetization dimension is related to the Ising model (limited to spin up or down) for n = 1, XY model for n = 2 and Heisenberg model for n = 3. In most physical systems, the inherent nature is three-dimensional, represented by three-dimensional Heisenberg spins that can be effectively mapped onto the surface of a sphere (r = 2) that we just call sphere later on.

However, in the case of thin films or nanowires, the anisotropy is so high that the other dimensions can also be effectively realized, allowing for a more generalized discussion. The n-th homotopy group π n (S r ) summarizes all the ways in which an n-dimensional sphere in the base manifold can be mapped onto an r-dimensional one in the target manifold. For instance, π 0 (S 0 ) corresponds to the homotopy group describing Ising spins (m = ±m z ) in a one-dimensional chain while π 2 (S 2 ) is related to a three-dimensional system with Heisenberg spins. To allow for the existence of topological defects, π n (S r ) must be non-zero, otherwise it is simply the identity (all contours can be contracted to a point), i.e. a uniform magnetization. For non-identity homotopy, the projection of the magnetization onto a sphere of dimension r cannot be contracted to a point: at some moment the magnetization orientation will be undefined and thus corresponds to a topological defect. It can be furthermore be shown that, for n ≥ 1:

π n (S n ) = Z (2.

18)

For n = 0, the defect is either here or not, yielding π 0 S 0 = Z 2 . Thus the different elements of π n (S n ) can be uniquely characterized by an integer number which is often referred to as the winding number. Two magnetization configurations are thus topologically (or homotopically) equivalent if they have the same winding number: they can be deformed continuously into each other without having to surpass an infinite energy barrier. On the contrary, topological defects or textures exist when the mapping of the magnetization m cannot be deformed to the uniform ferromagnetic state, i.e. they are not topologically equivalent (or in the same homotopy group) and have a non-zero winding number.

Note that if n < r, the homotopy group π n (S r ) is trivial. It corresponds for instance to mapping a circle onto a sphere: it can easily be reduced continuously to a point. Finally, for n > r, more complex mapping can occur such as the Hopf fibration.

Winding number

To understand schematically how the winding number w is linked to the topology, several examples are given in Figure 2.6 for mappings S 1 → S 1 . The first case corresponds to w S 1 = 0 with spins that are pointing in a uniform direction or spins that only maps one half of the target manifold. For w S 1 = 1, with skyrmion-like domain wall (either Néel or Bloch), the magnetization wraps one time around the circle. Note that reversing the magnetization rotation in the domain wall, from clockwise to counterclockwise, does not impact the topological number. As the contour is oriented, if the rotation direction changes, the winding number also reverses and w S 1 = -1. Finally, it is possible to obtain higher winding number if the magnetization rotates as to wrap more than once the target manifold. ) counting the number of times the circle is wrapped around itself. Adapted from [START_REF] Zang | Topology in magnetism[END_REF].

For π n (S m ), i.e. mappings f : S n → S m , the general expression of the winding number is expressed as:

deg f = ¡ M f * Ω ¡ N Ω = 1 8π ¢ M ϵ abc m a ∂ i m b ∂ j m c dx i ∧ dx j (2.19)
where Ω is a differential n-form defined on the target manifold N and f * Ω denotes its pullback onto the base manifold M . Moreover, ϵ abc is the Levi-Civita tensor, summed over the three components of m, and ∂ i m a = ∂m b /∂x i . With this background, we now focus on the two types of topological objects with non-zero winding number: the singular defects or the smooth topological textures.

Bloch points and other singular topological defects

Singular topological defects have a vanishing magnetization of dimension d ′ in the d dimensional real space. The dimension d ′ of the defect is related to the magnetic contour dimensionality r or the number of components of the magnetization p = r + 1 by [START_REF] Toulouse | Principles of a classification of defects in ordered media[END_REF]:

d = d ′ + r + 1 = d ′ + p (2.

20)

The different possibilities are summed up in Figure 2.7. From the formula, we see that topological defects cannot exist when the spins have a larger dimensionality than the real effective space. This corresponds to π n (S r ) = 0 with n < r. For instance, we can think of the case of a one-dimensional chain with a domain wall (d = 1). If it were an Ising model (p = 1), a point singularity (d ′ = 0) would exist at the transition but for p = 2 or 3 (XY or Heisenberg model), the spins can rotate smoothly to avoid it. This is the so-called "escape in the higher dimension". Even though it is a singularity, BPs have a finite energy cost that scales with the exchange [START_REF] Thiaville | Magnetism and topology[END_REF]. In magnetic thin films, for which the characteristic exchange length (Eq. 2.5) is larger than the thickness, the system is effectively two-dimensional (d = 2) with three-dimensional spins, thus preventing the emergence of BPs.

Topology of the magnetic skyrmions

The magnetic skyrmion pertains to the second category of topological objects for which the magnetization has no singular point but instead varies smoothly from one site to another. Other examples of this kind include π or 2π domain wall, merons or even hopfions 6 . In this case, the magnetization far enough from the object has a uniform configuration. Importantly, this means that the direct space R d can be directly projected onto a sphere of dimensionality d using, for instance in 3D, the stereographic projection from the north pole on a plane tangent to the south pole. Then, topological defects arise for d = r.

The magnetic skyrmion in a 2D plane with 3D spins, i.e. d = r = 2 with the corresponding mapping S 2 → S 2 , belongs to the homotopy group π 2 (S 2 ) = Z. Thus, they have a non-zero winding number N sk whose expression can be derived from Eq. 2.19 [START_REF] Zang | Topology in magnetism[END_REF]:

N sk = 1 4π ¢¢ m • ∂m ∂x × ∂m ∂y dx dy (2.

21)

Different configurations, topologically different, are presented in Figure 2.9 to illustrate the different topological classes. For an anti-skyrmion, the magnetization maps the sphere one time but partially pointing inside, thus yielding N sk = -1. For an in-plane magnetization taking a uniform value around the core of the texture (b), the topological number is zero as the spin only partially cover the sphere. bubble with a uniform in-plane magnetization around its core. c) Néel skyrmion. d) Higher order skyrmion. Adapted from [START_REF] Thiaville | Magnetism and topology[END_REF].

For a Néel skyrmion (c), the topological number is 1. Note that reversing the rotation or the chirality would also yield N sk = 1 as they are topologically equivalent. Higher order skyrmion can also be stabilized for instance with N sk = 2. Those four objects are thus topologically different and cannot be continuously deformed from one to the other. Apart for the trivial case (b), they cannot be deformed to a uniform magnetization as well. By definition, the topological number depends on the magnetization of the core so that if the core switches polarity, so does N sk 7 .

The methodology employed in this study focuses on 2D skyrmions to adhere to the principles of topology based on continuity approximation. However, in the context of magnetic multilayers, skyrmions possess a vertical dimension as they are stabilized across all magnetic layers. Due to the significant discontinuity between the magnetic multilayers, which is much larger than the ignored distance between two lattice sites, it becomes challenging to define topology accurately in threedimensional space. Consequently, at present, the topological properties of these skyrmions can only be analyzed using a mathematically valid 2D approach within each individual magnetic layer.

Interaction with X-rays

In order to investigate three-dimensional systems, X-ray techniques prove to be valuable, as they allow to probe the magnetization through the path of light, typically averaging it along the thickness of the sample. The resulting magnetic information on the magnetization of the spins far below the surface would otherwise be challenging to access. This thesis includes measurements utilizing X-ray techniques, emphasizing the importance of understanding how X-rays interact with magnetization to extract meaningful information.

X-ray absorption

Entangling the different contributions X-rays interaction with matter leads to a decrease of the initial intensity I 0 due to different scattering processes. In its most general form, its attenuation through a thin layer of thickness z can be described with the Beer-Lambert law:

dI dz = -ρσ tot I =⇒ I = I 0 e -ρ σtot z (2.22)
with ρ the density of scattering centers and σ tot the total cross sections for uniform σ tot . The final intensity is a exponentially decreasing function of the material thickness. The exact details of the physical phenomena at play lie within the σ tot term that includes many contributions: coherent scattering (Rayleigh), incoherent scattering (Compton), photo-absorption processes, pair or triplet productions (positron).

σ tot = σ coh + σ incoh + σ ph + σ pair + σ triplet (2.

23)

However, the relative strength of each contribution depends on the material and the energy of the X-rays. In Figure 2.10, the different cross sections are identified for Co over a large range of energy.

In the soft X-rays range (100 to 3000 eV), highlighted by the red background, the photo-absorption process dominate the others by two to three orders of magnitude, making them negligible. This is also verified for a vast majority of the elements [START_REF] D E Cullen | Tables and graphs of photon-interaction cross sections from 10 eV to 100[END_REF].

The photo-absorption process corresponds to the transfer of energy ℏω from the incident X-ray to an electron, promoting it to an excited state. For certain energies, called absorption edges, the transition rate is increased as the energy corresponds to the difference between a core level to an unoccupied state. The L and K-edges are identified on Figure 2.10 for the Co, respectively at about 780 eV and 7700 eV.

Fermi's golden rule

The Fermi's golden rule is a fundamental principle in quantum mechanics that relates to the transition rate between quantum states. The rule provides a method to calculate the probability of a quantum system undergoing a transition from an initial state to a final state in the presence of a perturbation or interaction. It is particularly useful in systems where the perturbation is weak and the transitions can be treated as a first-order process. In its general form, it is written as:

Γ i,f = 2π ℏ ⟨ψ i | Ĥ|ψ f ⟩ 2 ρ(E f ) (2.

24)

with Γ i,f the transition rate between the initial state ψ i and the final state ψ f of energy E f , H the Hamiltonian of the system and ρ(E F ) the density of states at the final energy. In this case, we consider the Hamiltonian of an electron in an electromagnetic field with vector potential A and electric potential V (r):

H = - 1 2m [ p -eA(r, t)] 2 + V (r) (2.

25)

with p the momentum operator of the electron. Neglecting the term in A 2 , the perturbation part of the Hamiltonian (in the Coulomb gauge) is:

H ′ = - e 2m ( p • A + A • p) = - eℏ im A • ∇ (2.

26)

For an incoming X-ray wave, the vector potential takes the form of an electromagnetic plane wave:

A = A 0 e ik•r e -iωt ê (2.27)
with ê the polarization vector of the light, k the wave vector of the X-rays and r the position of the electron. For the absorption of X-ray, the dipole approximation is used to simplify the computation.

It assumes that the electromagnetic field varies slowly over the spatial extent of the system under consideration, i.e., that k • r ≪ 1. In this case, only the linear term in the interaction with the beam is kept:

A ∼ A 0 e -iωt ê (2.28)

Thus, within the dipole approximation, the contribution to the X-ray absorption from a core level ψ i is given by Fermi's golden rule as [START_REF] Van Bokhoven | X-ray absorption and X-ray emission spectroscopy: theory and applications[END_REF]:

σ ∝ f |⟨ψ i |ê • p|ψ f ⟩| 2 δ(ℏω + E i -E f ) (2.

29)

with ℏω = E f -E i the energy of the photon equal to the different between the initial state energy E i and the final one E f . The sum is performed over the final unoccupied states. In the end, the dipole approximation also sets selection rules for the different transitions that are allowed [START_REF] Richard | Introductory quantum mechanics[END_REF]: ∆l = ±1, ∆j = 0, ±1, ∆m j = 0, ±1, ∆n unrestricted (2.30)

with l the azimuthal quantum number, j the total angular momentum quantum number (j = |l ± s|), m j the projection of j and n the principal quantum number. For instance, at the L 2,3 -edge (n = 2), considering p orbitals (l = 1), the transitions are only allowed for a final state with a s (l = 0) or d (l = 2) symmetry in any shells, the latter being more probable [START_REF] Ebert | L-edge x-ray absorption in fcc and bcc cu metal: Comparison of experimental and first-principles theoretical results[END_REF]. Finally, note that no spin operator is present in the perturbation Hamiltonian so that the spin number is unchanged by the transition.

X-ray Magnetic Circular Dichroism (XMCD)

X-ray magnetic circular dichroism (XMCD) is a technique used in condensed matter physics and materials science to probe the magnetic properties of selected materials. XMCD provides valuable information about the magnetic moments and their orientations in a material, as well as the interactions between magnetic atoms or ions. Circular dichroism refers to the differential absorption of left-and right-circularly (CL and CR) polarized light by a material.

The first theoretical studies on XMCD were performed on Ni in 1975 [START_REF] James | Calculation of the m 23 magneto-optical absorption spectrum of ferromagnetic nickel[END_REF] but the first experimental realization were only realized in 1987 in Fe [START_REF] Schütz | Absorption of circularly polarized x rays in iron[END_REF] thanks to the development of synchrotron radiation.

The two-steps model [START_REF] Schütz | Absorption of circularly polarized x rays in iron[END_REF][START_REF] Stöhr | Exploring the microscopic origin of magnetic anisotropies with x-ray magnetic circular dichroism (xmcd) spectroscopy[END_REF][START_REF] Van Bokhoven | X-ray absorption and X-ray emission spectroscopy: theory and applications[END_REF] can be used to illustrate how the XMCD works (see Figure 2.11). In particular, for ferromagnetic elements (Fe, Co and Ni), it focuses on excitations of core electrons from the L 2 or L 3 edges, i.e. from p states split by spin-orbit coupling, to unfilled 3d shells 8 . The spin-up and down are defined relatively to the X-ray propagation direction (either parallel or antiparell). Adapted from [START_REF] Stöhr | Exploring the microscopic origin of magnetic anisotropies with x-ray magnetic circular dichroism (xmcd) spectroscopy[END_REF].

In the first step, the polarized X-rays (CL or CR) transfer their angular momentum (±ℏ) to the excited photo-electron. In the case of levels split by spin-orbit, the angular momentum can be partially transferred to the spin. The two helicities transfer opposite momentum so that the corresponding photo-electrons are excited with opposite spins. The L 2 and L 3 edges (p 1/2 and p 3/2 ) have opposite spin-orbit coupling, so that the spin polarization will be opposite at the two edges for a given helicity. It is equivalent to change the helicity or reverse the magnetization.

Magnetization is taken into account in the second step. In a ferromagnet, the density of states near the Fermi level depends on the spin orientation (see Figure 2.11). This acts like a detector for the spin of the excited photo-electron with a quantization axis given by the direction of the magnetization which, for maximum contrast, should be parallel to the X-rays propagation direction. For instance, if there are only spin-down holes above the Fermi level, the detector will only be sensitive to spin-down electrons, maximizing the dichroism.

In this thesis, XMCD serves as the underlying effect of many advanced X-ray techniques such as X-ray holography or scanning transmission X-ray microscopy that will be used for imaging magnetic textures. XMCD probes the component of magnetization aligned with the wave vector of the incident X-rays. When X-rays are incident at normal angles, it provides sensitivity primarily to the out-of-plane component (m z ), while tilting the sample enables the detection of in-plane components. It is worth noting that XMCD measurements can also be utilized to estimate the spin and orbital moments by utilizing the signal from the L 2 and L 3 edges, applying associated sum rules [START_REF] Bt Thole | X-ray circular dichroism as a probe of orbital magnetization[END_REF].

To ensure sufficient flux and brilliance for the experiments, X-ray experiments were conducted at various synchrotron facilities. Numerous books and reviews are available that discuss the generation and application of X-rays in such facilities [START_REF] Duke | Synchrotron radiation: production and properties[END_REF][START_REF] Beaurepaire | Magnetism: A synchrotron radiation approach[END_REF].

T

o investigate the existence of 3D spin textures in magnetic multilayers, it is important to rely on a deposition technique that allows for reproducible growths with preserved interface quality and limited roughness. For the samples studied in this thesis, DC magnetron sputtering has been used because of its ease of use, reliability and the high tunability it provides for growing complex structures. This chapter aims to put forward the structural properties obtained in film grown with this technique as they dictate the underlying magnetic interactions. Finally, after introducing the standard magnetic multilayers generally used to stabilize skyrmion tubes, more complex multilayered systems with variations of the uniaxial anisotropy are numerically and experimentally investigated.

Precise and controlled growths via magnetron sputtering

This section provides an introduction to magnetron sputtering and the advantages that makes it the preferred choice for growing magnetic multilayers. Using regular calibrations, it allows for reliable and reproducible growths of various thin films and magnetic multilayers on large areas.

Principle of magnetron sputtering

As thin films occupy a predominant place in modern technologies, magnetron sputtering has become a widely used growth technique for the production of high quality coatings on an industrial scale [START_REF] Peter | Magnetron sputtering: a review of recent developments and applications[END_REF][START_REF] Bräuer | Magnetron sputtering-milestones of 30 years[END_REF]. It possesses several advantages, notably the possibility to deposit on large areas (up to several square meters) with minor variability, as well as a large choice of materials whose deposition can be finely optimized, giving broadly tunable film properties. Thanks to its scalability, this deposition tool is also suitable for research purposes activity and, in our case, for the development and optimization of magnetic heterostructures made of stacks of ultrathin magnetic and non-magnetic films. The first observation of the phenomena of sputtering turns out to be quite old as it was first put forward in 1852 by Grove in his ion etching experiments [START_REF] Robert | On the electro-chemical polarity of gases[END_REF]. In the absence of magnetron, the basic sputtering process relies on the bombardment of a target plate by energetic ions, which generates a plasma in the vicinity of the target. The extracted atoms can then deposit themselves on a nearby substrate, creating the desired thin film as illustrated on Figure 3.1.a. During the bombardment, secondary electrons are created which have the ability to sustain the plasma thanks to electron-atoms collisions but, since they are not confined, the corresponding rate is quite small. Thus, this approach is notably limited to low deposition rates and a low ionization efficiency of the plasma. To overcome those drawbacks, the magnetron was introduced in the 1970s, which coupled with the progress in vacuum equipment, launched the sputtering as an industrial process. Different types of magnetron exist, but the most common remains the planar magnetron which consists of a magnet centered below the target Magnetron sputtering is also a versatile technique that allows to deposit many elements, using a DC power for conducting materials and RF power for insulating ones. The deposition rates scale linearly with the power and do not depend on the target temperature, allowing for a fine tuning of the sputter yield. Depending on the pressure and the power, the rates are typically between 0.1 nm/s and 10 nm/s, which can be used to reach a precision smaller than an atomic layer. Moreover, to ensure the quality of thin films, the deposition needs to be realized at the lowest possible pressure to limit the contamination or the oxidation of the deposited elements.

Our growth conditions

In our setup, the sputtering chamber reaches a pressure close to 10 -8 mbar, for an operating pressure of Argon around 2.5 × 10 -3 mbar and DC power lower than 100 W. It hosts up to seven 3-inches targets of different materials, enabling the growth of multilayers within the same vacuum. The targets are located 75 mm away from the chosen substrate which, for 2-inches wafers, yields a uniform deposition with less than 2% thickness variations from the center to the edge. Unless mentioned otherwise, the deposition is performed at room temperature on commercially available wafers of silicon capped by 280 nm of thermally oxidized silicon oxide, i.e.

an amorphous substrate.

Calibration with thin films

To fully exploit the potential of magnetron sputtering, it is primordial to properly assess the deposition rates of each material for given operating pressure and DC power. To that end, calibrations are performed using thin films deposited on silicon oxide which are then characterized using X-ray reflectivity (XRR) [START_REF] Yasaka | X-ray thin-film measurement techniques[END_REF]. XRR is a non-destructive approach, fitting for crystalline and amorphous materials, that helps retrieve many structural properties such as the film thickness, the density and the roughness. The measurements are performed by measuring the reflection of X-rays at K α Cu wavelength while sweeping the grazing incidence angle θ. Below a critical angle θ C (usually between 0.2 • and 0.5 • ), whose value scales with the density, a phenomenon of total reflection is observed. When the incidence angle surpasses this value, the X-rays are able to penetrate the material which results in interferences upon their reflections at various interfaces. The quality of the interface, i.e. its roughness, can also be responsible for scattering the X-rays, contributing to the decay of the signal at higher incidence angles.

XRR is also suitable to study multilayers, as the penetration length can reach several micrometers. However, increasing the number of variables consequently makes more complex the extraction of the structural parameters.

In Figure 3.2, a typical reflectivity curve for a thin film of Pt is presented, displaying the characteristic oscillations arising from the interferences. As their frequency is related to the thickness of the film, the position of the main peak of the Fourier Transform allows to obtain the thickness and thus to calibrate the deposition rate. Given the width of the peak, an incertitude of a few percents can be expected on the actual rate. The deposition rates for the materials used in this PhD thesis along with the average operating DC power are indicated in Table 3 Table 3.1 -Typical deposition rates and the associated DC power with intensity (at 2.5 × 10 -3 mbar).

Structural characterization of periodic multilayers

In this section, we investigate the structural properties of magnetic multilayers to evaluate the growth quality. As more layers are deposited on top of each other, the probability to have defects increases and some intermixing can occur between two adjacent layers [START_REF] Zhou | Atomic scale structure of sputtered metal multilayers[END_REF][START_REF] Biswas | Defect growth in multilayer chromium nitride/niobium nitride coatings produced by combined high power impulse magnetron sputtering and unbalance magnetron sputtering technique[END_REF]. Such problems impact the magnetic properties, notably the saturation magnetization, the anisotropy and the DMI, which strongly rely on the quality of the interfaces. It is thus important to verify the quality of the stacking and the reproducibility of the growths.

Multilayer structure notation

In the following, the structure of a given multilayer is written from the substrate to the surface with each element and the associated thickness in nanometers. For instance, a multilayer grown on silicon oxide with a seed of Ta (5 nm) made of five repetitions of aluminum (2 nm), cobalt (1 nm) and platinum (3 nm), corresponds to: SiO 2 ||Ta5|(Al2|Co1|Pt3) 5 . As most of the samples are grown on silicon oxide, it is now omitted in the notation.

In most cases, a chemically inert and amorphous seed of 5 nm of Ta is used to facilitate the growth and adhesion of the multilayer on the substrate. It also helps to increase the lifetime of Pt films at high temperature [START_REF] Samara L Firebaugh | Investigation of high-temperature degradation of platinum thin films with an in situ resistance measurement apparatus[END_REF] and can serve as a diffusion barrier [START_REF] Holloway | Tantalum as a diffusion barrier between copper and silicon: Failure mechanism and effect of nitrogen additions[END_REF]. Similarly, in most multilayers, the capping is often made of Pt, at least 3 nm thick, in order to prevent the oxidation of the remaining layers. This thickness was determined after an XPS (X-Ray photoelectron spectrometry) study focusing on oxygen contamination.

Reproducibility of the growths

As a first approach to probe the structural properties of a given multilayer, similarly to thin films, a reflectivity curve can be measured. Its fit yields information regarding the thicknesses, that can be compared to the nominal ones, and the roughness of the interfaces. Typically, the latter is in the range 0.2 to 0.5 nm. In Figure 3.3.a, an example of reflectivity curve is provided along with the corresponding fit for the multilayer Pt5|(Pt3|Co1.7|Ru1.4) 5 |Pt3. The oscillations display more variations than for a single thin film but there is still a periodicity that arises from the Pt|Co|Ru trilayer repetition.

Typically, the fit shows a 5% variation between the nominal and fitted thicknesses.

This evidences some deviations arising from the incertitude on the deposition rates but as long as those biases are constant, the growths should be reproducible. To verify this, two identical samples at the exception of the capping layer were grown on the same day and their corresponding XRR are shown in Figure 3.3.b. From the associated fits, the various thicknesses and roughnesses are presented in Table 3.2.

The fitted thicknesses for both samples are identical within error margins (±0.2 nm), thus demonstrating the reproducibility of DC sputtering. Moreover, for all materials, the roughness is smaller than 0.5 nm. This highlights the reliability of magnetic sputtering for the study of magnetic multilayers. b) Measurements and fits on similar multilayers to check reproducibility of the growths. The curves have been shifted for clarity. Extracted from [START_REF] Yanis | Magnetic skyrmions in multilayers: control of their current induced nucleation and motion[END_REF].

Imaging the atomic structure

STEM

To study more precisely its atomic structure on a smaller scale, especially to account for the interfaces quality, some scanning transmission electron microscopy (STEM) measurements have been performed, in collaboration with T. Denneulin from Ernst Ruska-Centre for Microscopy and Spectroscopy with Electrons, Jülich, Germany. STEM relies on the transmission of an electron beam, focused down to roughly a 1 Å spot, through a thin sample [START_REF] Av Crewe | A scanning microscope with 5 å resolution[END_REF][START_REF] David | Structure and bonding at the atomic scale by scanning transmission electron microscopy[END_REF]. By recording the signals from the scattered electrons and the ionized atoms at various spatial positions, a two-dimensional map can be obtained with chemical and bonding information. It is nowadays used conjointly with deep-learning algorithms in order to understand more in details the chemistry and physics of more complex materials [START_REF] Ziatdinov | Deep learning of atomically resolved scanning transmission electron microscopy images: chemical identification and tracking local transformations[END_REF][START_REF] Sergei V Kalinin | Machine learning in scanning transmission electron microscopy[END_REF].

For imaging, STEM typically relies on the technique of high-angle annular dark-field (HAADF) [START_REF] Max T Otten | High-angle annular dark-field imaging on a tem/stem system[END_REF] for which the contrast is highly sensitive to the atomic number. Moreover, to obtain more quantitative chemical information, energy-dispersive X-ray (EDX) spectroscopy mapping can be realized on a STEM-system with a lesser spatial resolution than HAADF. Indeed, the incoming electrons can eject inner shell ones, creating X-rays with given energies that depend on the specific elements and transitions.

Such measurements were performed on the multilayer Ta5|Pt8|Co0. To study more closely the compositional changes at the interfaces, EDX mapping have been recorded for the elements of the multilayers as well as for the oxygen to probe for possible oxidation.

In Figure 3.4.b, specific elements profiles have been selected along the normal of the surface. The corresponding colored maps are shown below in Figure 3.4.c. Each colored region corresponds to a given element and their width correspond quite well with the expected thicknesses. The remaining data on the right side of the plot thus represents the noise level for each of the elements. Regarding the oxidation, the level of oxygen stays below the noise of 35% in the multilayer itself, indicating that the platinum capping of 3 nm is efficient at preventing it. For the intermixing, some overlapping of the curves can be seen, especially at the Co interfaces, but they still appear to be well-defined. Moreover, this phenomenon can be accentuated by the roughness along the propagation of the beam as it averages in this direction.

Atomic probe tomography

Atomic probe tomography can be used to reconstruct the 3D composition of a multilayer [START_REF] Thomas | Atom probe tomography[END_REF].

Such measurements were realized in collaboration with M. Descoins and K. Hoummada from the IM2NP, CNRS/Aix-Marseille Université, Marseille, France, through a METSA project deposited by the laboratory. This destructive technique allows to fully reconstruct the structural and chemical properties of a given material, with the same sensibility to all kind of atoms, even the lighter ones. To that end, the multilayer is first protected with a layer of Ni then patterned into a tip using Focused Ion Beam (FIB) etching. An example is shown in Figure 3.5.a. evidencing the periodicity in both structures. Note the clearly defined interfaces for both samples, confirming that the intermixing is negligible. Importantly, no oxygen or hydrogen was measured in the magnetic layers, demonstrating the high-quality of the growth.

These combined structural studies have demonstrated the reliability of the growths via magnetron sputtering, in terms of reproducibility and quality. It is especially important to have such clean interfaces and low intermixing as it plays a primordial role to engineer and control the magnetic interactions, notably the anisotropy and the DMI. In the following, we overview some properties of the magnetic multilayers hosting skyrmion tubes since this knowledge serves as basis for more complex spin textures.

Breaking the periodicity to tune the magnetic interactions

After investigating the structural properties of the multilayers, we focus on the magnetic properties and how we can control them in order to develop new 3D magnetic orders.

The Pt|Co|Al trilayer

The main advantage of magnetic multilayers compared with bulk chiral magnets is the versatility they present when it comes to controlling the magnetic interactions. Indeed, by carefully tuning the various interfaces and materials, whose quality has been accessed in the previous section, it is possible to impact given interactions such as the anisotropy or the DMI. In our laboratory, many stackings have been investigated, often based on the Pt|Co bilayer with another element on top to disymmetrize it, such as Ru, Cu, Ir and Al. In the recent years, our group has however investigated in more depth the Pt|Co|Al stacking, acquiring important insights regarding its properties. Fernando

Ajejas, former postdoc in our group, demonstrated the existence of a large DMI in this structure [START_REF] Ajejas | Interfacial potential gradient modulates Dzyaloshinskii-Moriya interaction in pt/co/metal multilayers[END_REF].

Moreover, Sachin Krishnia, another former postdoc, investigated the different torques strength in different stackings [START_REF] Krishnia | Large interfacial rashba interaction and giant spin-orbit torques in atomically thin metallic heterostructures[END_REF] and showed that they can be optimized for the Pt|Co|Al interfaces. Finally, this trilayer allows to tune the anisotropy, either with the Pt or with the Co thickness, a key property for our endeavor with 3D textures. We now discuss each interaction separately in the Pt|Co|Al trilayer.

Pt|Co|Al Trilayer

To limit the degrees of freedom and to use the existing knowledge on skyrmionic multilayers, the Pt|Co|Al trilayer has been chosen to try studying 3D spin textures and serves as the basic block for the growths. Noticeable, for low Co thickness (≈ 0.2 nm), the saturation magnetization is close to zero. This is often referred to as the dead layer phenomenon which displays an apparent absence of magnetization [START_REF] Soo | Magnetic dead layer in amorphous CoFeB layers with various top and bottom structures[END_REF][START_REF] Sharma | Study of ultrathin magnetic cobalt films on MgO (001)[END_REF]. However, the magnetic material is still present so that this effect is likely linked to interfacial considerations. Even in thicker layers, the magnetic thickness contributing fully to the magnetization is thus slightly lower than the nominal one due to interfacial effects.

Saturation magnetization

For more accuracy, the saturation magnetization can be measured using superconducting quantum interference device (SQUID) magnetometry. Note that some uncertainty arises from geometrical considerations [START_REF] Sawicki | Sensitive SQUID magnetometry for studying nanomagnetism[END_REF]. In our typical multilayers made of Pt|Co|Al, the saturation magnetization has been measured as:

M S = 1.22 ± 0.02 MA/m (3.1)
This value assumes that all the Co atomic layers share the same properties which is not necessarily verified near interfaces. It is slightly lower than the value obtained with AGFM as here we take into account the full thickness, neglecting the interfacial effects that might lower the magnetization near them.

Uniaxial anisotropy

Sputtered Pt|Co allows to grow the ferromagnet with a (111) texture and a low roughness. It has been studied extensively because of the strong interface anisotropy that sustains a strong perpendicular magnetic anisotropy (PMA) for small thicknesses of Co [START_REF] Bertero | Structure-property correlations in Pt/Co multilayers for magneto-optic recording[END_REF][START_REF] Ga Bertero | Interface structure and perpendicular magnetic anisotropy in Pt/Co multilayers[END_REF][START_REF] Kisielewski | Magnetic anisotropy and magnetization reversal processes in Pt/Co/Pt films[END_REF], a property interesting for data storage purposes. Upon the increase of the ferromagnet thickness, a competition arises between the interfacial and the dipolar interactions, the latter trying to pull the magnetization in-plane. Thus, the PMA disappears for thicknesses large enough, whose threshold depends on the system under consideration.

In the average m z . For t Co = 1.5 nm, the transition happens near remanence, evidencing that the easy axis lies along the field, i.e. out-of-plane, so that the anisotropy is effectively out-of-plane. On the contrary for t Co = 1.9 nm, the field starts to pull the magnetization out-of-plane at high field (around 250 mT) because of its in-plane effective anisotropy. Those observations already indicate that the spin reorientation thickness, for which K eff is null, is in between those two cases. To determine it precisely from those measurements, several approaches can be envisioned. In the first one, we recover the saturation field H sat along the hard axis which allows to estimate K eff as:

K eff = µ 0 H sat M S 2 (3.2)
Using the M S measured with SQUID (Eq. 3.1). This method suffers from a certain difficulty to sometimes pinpoint precisely the saturation field, especially if the external field is slightly misaligned.

To compare, we also implement a second computation, based on the area between the out-of-plane (OOP) and the in-plane (IP) curves (see Figure 3.7.b) which is directly equal to K eff , assuming that the field is expressed in T and the measured magnetization in A/m. Using both methods to retrieve the effective anisotropy for different thicknesses, we plot K eff t Co in Figure 3.7.c. The effective anisotropy K eff is related to the surfacial uniaxial anisotropy K u,s by:

K eff t = K u,s - µ 0 M 2 S 2 t (3.3)
A linear trend is thus expected which slope depends on the saturation magnetization. Using the M S measured by SQUID (1.22 ± 0.02 MA/m) to set the slope, a linear fit is performed for each data set.

The value of K u,s then simply corresponds to the intersection at the origin: K u,s = 1.58 ± 0.10 mJ/m 2 using H sat and K u,s = 1.55 ± 0.10 mJ/m 2 using the integral method. Moreover, the spin reorientation thicknesses are close to 1.7 nm in both cases. This thickness corresponds to a zero effective anisotropy and thus, below (resp. above) this value, the effective anisotropy is out-of-plane (resp. in-plane) as it is dictated by the interfacial (resp. dipolar) interaction.

The anisotropy can also be obtained from magnetotransport measurements as an alternative to the AGFM methods. Indeed, it can be extracted by fitting the variation of the magnetoresistance at saturation as a function of the angle of the external field. This method will be applied in subsequent chapters.

DMI

To obtain a non-zero DMI, the inversion symmetry needs to be broken, hence the need to have two different materials surrounding the Co. In fact, in the Pt|Co|X stacks, the DMI comes mainly from the bottom interface [START_REF] Ma | Interfacial Dzyaloshinskii-Moriya interaction: effect of 5 d band filling and correlation with spin mixing conductance[END_REF][START_REF] Ajejas | Interfacial potential gradient modulates Dzyaloshinskii-Moriya interaction in pt/co/metal multilayers[END_REF] thanks to its strong spin-orbit coupling. To measure the strength of the DMI constant, many approaches have been proposed, sometimes yielding divergent values. They can be divided into three main categories: the domain wall, the spin-orbit torque or the spin-waves methods (see Ref. [START_REF] Kuepferling | Measuring interfacial Dzyaloshinskii-Moriya interaction in ultrathin magnetic films[END_REF] for a comprehensive review). The latter method uses the non-reciprocity introduced by the DMI that leads to a non-reciprocal propagation of the spin waves [START_REF] Cortés | Influence of the Dzyaloshinskii-Moriya interaction on the spin-wave spectra of thin films[END_REF][START_REF] Di | Direct observation of the Dzyaloshinskii-Moriya interaction in a Pt/Co/Ni film[END_REF][START_REF] Belmeguenai | Interfacial Dzyaloshinskii-Moriya interaction in perpendicularly magnetized Pt/Co/AlO x ultrathin films measured by Brillouin light spectroscopy[END_REF]. In practice, this is measured with the Brillouin light scattering (BLS) technique [START_REF] Carlotti | Brillouin scattering and magnetic excitations in layered structures[END_REF]. It relies on the inelastic scattering of photons by spin waves which are either created (Stokes process) or annihilated (anti-Stokes process) in a sample saturated in-plane. Concretely, the scattering of a monochromatic focused beam on the surface is measured within a solid angle. To probe different wave vectors, the incidence angle simply needs to be changed.

The presence of DMI in the system shifts the frequency of the Stokes peak f S (resp. anti-Stokes peak f AS ) towards negative frequency for negative DMI and inversely for positive DMI:

f A,AS = f 0 + sign(D) × f DMI (3.4)
with f 0 the frequency of the peak without DMI and f DMI the shift arising from the antisymmetric

D = f DM I k sw πM S γ (3.5)
with γ the absolute value of the electron gyromagnetic ratio and k sw = 4π sin θ/λ the wave vector of the spin waves with θ the incidence angle and λ the wavelength. Such measurements were performed by Sachin Krishnia, former postdoc in our group, in collaboration with André Thiaville (Laboratoire de Physique des Solides, Orsay, France) on the multilayer Ta5|Pt8|Co0.9|Al1|Pt3. In

Heisenberg exchange

Similarly to the DMI, the symmetric exchange can be measured using BLS but this time considering the position of the central peak f 0 in absence of DMI [START_REF] Liu | Exchange stiffness, magnetization, and spin waves in cubic and hexagonal phases of cobalt[END_REF][START_REF] Belmeguenai | Interfacial Dzyaloshinskii-Moriya interaction in perpendicularly magnetized Pt/Co/AlO x ultrathin films measured by Brillouin light spectroscopy[END_REF][START_REF] Böttcher | Heisenberg exchange and Dzyaloshinskii-Moriya interaction in ultrathin Pt (W)/CoFeB single and multilayers[END_REF][START_REF] Legrand | Spatial extent of the Dzyaloshinskii-Moriya interaction at metallic interfaces[END_REF]. Other methods can also be applied such as the temperature dependence of the magnetization, studying the domain periodicity, with neutron scattering or even using ab-initio computations [START_REF] Vaz | Magnetism in ultrathin film structures[END_REF][START_REF] Böttcher | Quantifying symmetric exchange in ultrathin ferromagnetic films with chirality[END_REF].

In Co layers, the exchange is expected to be thickness independent with a bulk value close to 30 pJ/m [START_REF] Ha Alperin | Observation of the dispersion relation for spin waves in hexagonal cobalt[END_REF][START_REF] Vaz | Magnetism in ultrathin film structures[END_REF]. However, in thin films, it displays some variations depending on the thickness and the surrounding materials [START_REF] Legrand | Spatial extent of the Dzyaloshinskii-Moriya interaction at metallic interfaces[END_REF][START_REF] Böttcher | Quantifying symmetric exchange in ultrathin ferromagnetic films with chirality[END_REF]. For instance, BLS measurements performed by Fernando Ajejas, former postdoc of our group, in collaboration with André Thiaville (Laboratoire de Physique des Solides, Orsay, France) on stacking of the type Pt|Co t Co |Al yield different exchange value as a function of the Co thickness t Co . For t Co = 0.5 nm, no quantifiable symmetric exchange was measured while for t Co = 1 nm and 1.4 nm, A = 6 ± 3 pJ/m and 14.6 ± 3 pJ/m respectively. It thus seems to increase with the Co thickness and should saturate once it reaches the bulk value.

Pt|Co|Al magnetic parameters

The different magnetic properties of the Pt|Co|Al stacking have been investigated with different approaches. The resulting parameters are summed up in Table 3.3.

Parameter M S (MA/m) K u,s (mJ/m 2 ) D s (pJ/m) A (pJ/m) Technique SQUID AGFM BLS BLS Value 1.22 ± 0.02 1.58 ± 0.1 -0.8 ± 0.05 14.6 ± 3 Table 3.3 -Magnetic parameters measured in Pt|Co|Al multilayers.
Given the quality of the interfaces that were observed with STEM or atomic probe, the number of repetitions should not induce variations in the magnetic parameters. Note that A and M S are thickness dependent so that their value might vary depending on the precise multilayer structure.

Micromagnetic parameters

The experimental parameters can be used as input to properly define the materials properties in micromagnetic simulations. The numerical parameters that have been used in this thesis, unless mentioned otherwise, are shown in Table 3.4. They are relatively close to the experimental values, apart for D s . The corresponding BLS study was led simultaneously with the work presented in this thesis as it is a recent technique. At the beginning, we set D s to this higher value as a first guess.

However, in further chapters we show that the dipolar interaction is dominant in the multilayers that we consider so that this inaccuracy does not impact significantly the stabilization of spin textures. This is discussed in more details in Appendix B, in which similar simulations with different D s values are presented.

Additionally, for more precise simulations, one would need to introduce a thickness dependent exchange. Here, the value was fixed slightly higher than the experimental one that was measured for t Co = 1.4 nm as we consider larger thicknesses.

Parameter M S (MA/m) K u,s (mJ/m 2 ) D s (pJ/m) A (pJ/m) Value 1.2 1.62 -2.34 18
Table 3.4 -Micromagnetic parameters of Pt|Co|Al. Note that A is for t = 1.4 nm.

Though not perfectly accurate, this set of micromagnetic parameters serves as a starting point for understanding our magnetic multilayers.

Importance of the Pt thickness

In the multilayers under consideration, the Pt thickness plays an important role in controlling the magnetic interactions. However, to tune it, a trade-off needs to be considered between the following considerations:

• The dipolar interaction decreases with the thickness of the spacer.

• Platinum is a heavy element with thus a high x-ray absorption which, as some multilayers are characterized in synchrotron facilities, might hinder the signal and spatial resolution.

• The larger the thickness, the easier it is to obtain PMA as the [START_REF] Zang | Topology in magnetism[END_REF] orientation is preserved.

• The minimal thickness of Pt to reach the highest efficiency for spin-orbit torque corresponds to about two times the diffusion length which is close to 3 nm for Pt|Co|Al [START_REF] Zhang | Role of transparency of platinum-ferromagnet interfaces in determining the intrinsic magnitude of the spin Hall effect[END_REF][START_REF] Krishnia | Large interfacial rashba interaction and giant spin-orbit torques in atomically thin metallic heterostructures[END_REF]. This allows for better motion efficiency, a key factor for potential application.

To try respecting those various constrains, the platinum thickness is typically set to 3 nm and more rarely to 2 nm (for synchrotron measurements).

Standard skyrmionic magnetic multilayers

Columnar skyrmions are typically considered as two-dimensional (2D) textures, exhibiting minimal variation along the vertical axis except for chirality and small diameter differences over the thickness.

However, our objective is to study and stabilize new objects that exhibit variations along the third dimension. It thus becomes crucial to distinguish them from the conventional columnar skyrmions. In general, those can be stabilized in various multilayers through interface engineering, which enhances the interfacial DMI. These multilayers typically consist of Co, CoFeB, or Ni layers of uniform thickness sandwiched between non-magnetic elements like Pt, Ir, Ru, Al, and others. The size of columnar skyrmions can range from 10 nm [START_REF] Meyer | Isolated zero field sub-10 nm skyrmions in ultrathin Co films[END_REF] to several hundreds.

To detect skyrmions and other spin textures, magnetic force microscopy (MFM) measurements can be performed while applying an external out-of-plane magnetic field. MFM is sensitive to the derivative of the stray field emanating from the sample's magnetization, providing insights into the actual magnetization distribution.

MFM specifications

Magnetic force microscopy (MFM) is extensively used in this thesis to characterize the magnetic textures stabilized in multilayers. As it yields a signal related to the stray field of the spin textures, it is a convenient technique to identify them even if it remains mostly sensitive to the top layers. To measure the magnetic signal, a magnetic tip is used which can disturb the magnetization and sometimes nucleate or annihilate textures in the observed multilayer. It is thus important to select the appropriate tip as to ensure a good signal to noise ratio while minimizing its interaction with the sample under consideration. Unless mentioned otherwise, the MFM images have been acquired using cantilevers from TeamNanotech (TN) with nominal stiffness of 3 N/m, with its tip capped by 7 nm of magnetic (proprietary) material and 10 nm of Pt. Moreover, we use a tapping mode with a lift height of 10 nm, at 75% of the drive amplitude with double-passing and at room temperature. Note that the lift height is defined relatively to the first pass height recording the topography, implying that the effective height of the probe is larger than 10 nm, typically of the order of 50 nm. For field measurements, two field modules are available: the first one allows to scan from -150 mT to +150 mT while the second one adds a permanent magnet, thus accessing the range 150 mT to 300 mT.

As MFM is used extensively throughout this thesis, we quickly analyze the origin of the magnetic contrast. For this, we consider the force F exerted on the magnetization M of the tip by the stray field of the sample H (or inversely) [START_REF] Hartmann | Theory of magnetic force microscopy[END_REF][START_REF] Feng | Magnetic force microscopy contrast formation and field sensitivity[END_REF]:

F ∝ ¢ ∇ (M tip • H sample ) dV = ¢ ∇ (M sample • H tip ) dV (3.7)
This very general expression can be simplified considering that the magnetic tips are often perpendicularly magnetized and thus probe only the vertical component of the magnetization:

F z ∝ ¢ dH z dz dV (3.8)
This assumes that the tip is a perfect magnetic dipole. Practically, the shift of the cantilever's resonance frequency ∆f , i.e. the phase, is measured instead of the deflection of the cantilever by the magnetic force. It corresponds to:

∆f ∝ ∂F z ∂z ∝ ∂ 2 H z ∂z 2 (3.9)
The numerous MFM images presented in this thesis corresponds to this frequency shift and are thus called phase maps.

In Figure 3.9, the field dependency of a typical skyrmionic multilayer, Ta5|Pt8|(Co1.9|Al1.4|Pt3) 5 |

Pt3, is investigated. The AGFM loop measured with an out-of-plane field shows that the easy axis lies in-plane due to its relatively high saturation field. This is expected from the Co thickness above the spin-transition thickness. Noticeably, the loop opens at high field, near saturation. In Figure 3.9.b, MFM phase maps are presented, evidencing the evolution of the magnetization as the field is increased. At remanence, a complex distribution of worms is stabilized with no favored alignment direction. At 100 mT, many have split and reduced to circular shapes which allows to identify them as skyrmions. This transition coincides with the opening observed in the AGFM loop.

At 113 mT, all textures are now skyrmions and they remain stable at 143 mT even though they have reduced in size because of the Zeeman effect. The use of an external magnetic field is a useful method to stabilize skyrmions and they are easily observed using MFM.

First designs of intricate magnetic multilayers

To facilitate the investigation of 3D magnetic objects, we focus on studying the impact of a single magnetic parameter. The logical choice that arises is to play with the magnetic anisotropy as the transition from out-of-plane to in-plane magnetization could help to create vertical variations. Indeed, creating abrupt variations in the anisotropy, for instance from strongly out-of-plane to strongly in-plane, might cut an existing spin texture at the corresponding interface. However, there is a delicate balance between all interactions so that the modifications of the anisotropy, either with the Pt or the Co thickness, also impact the other parameters.

To build on this idea, different structures have been grown with DC sputtering using the Pt|Co|Al trilayer. However, contrarily to the usual skyrmionic layer, the thickness of Co is variable depending on the layer position, thus breaking the periodicity in order to induce inhomogeneities in the magnetic interactions.

Gradient thickness notation

Most of the multilayers grown in this thesis present a variable thickness for the ferromagnetic material. To mark this evolution, we use the following shortcut: [X 1 :S:X 2 ] represents successive layers of ferromagnet from the initial thickness X 1 to the final one X 2 with steps S. For example, (Pt3|Co[1.2:0.1:1.4]|Al1.4) corresponds to Pt3|Co1.2|Al1.4|Pt3|Co1.3|Al1.4|Pt3|Co1.4|Al1.4.

In Figure 3.10, several multilayers with drastically different Co evolutions have been investigated

with MFM. They are built from the same pattern: Ta5|Pt3|(Co X i |Al1.4|Pt3) N where X i is the thickness of the i th layer and N the number of repetitions. The samples were saturated with an OOP field before being measured without any magnetic field. It is important to remark that the vast majority of the thicknesses used are superior to the spin-reorientation thickness (∼1.7 nm), thus having an in-plane effective anisotropy. This range of thickness was selected for a first try to have an in-plane effective anisotropy which might help to modulate out-of-plane spin textures along the vertical direction. Moreover, as shown on all the MFM images, it does not impede the formation of vertical domains or isolated textures. As the anisotropy encourages the magnetization to lie in-plane, this observation evidences the importance of the dipolar interaction. Additionally, the spacer between two magnetic layers should be sufficiently thick to make the interlayer exchange negligible so that it is purely a dipolar interaction.

Let us now discuss each multilayer architecture separately. The first attempts that were made This structure is now called a single gradient (SG) multilayer. The corresponding second magnetic state (IV) displays isolated circular textures, i.e. skyrmions, stable at remanence. With an overall effective in-plane anisotropy, it highlights again the importance of the dipolar interaction in such thick layers.

Stabilizing skyrmions at remanence is already an accomplishment so that the single-gradient structure (state IV) sets itself as an interesting candidate for building more complex multilayers. A few first tries are presented with the two last states. An extended SG was first considered by repeating Those representative trends allow to span a variety of behaviors based on the variations of magnetic parameters along the vertical axis. It put forwards the many possibilities brought by the tunability of the magnetic multilayers. The most promising structure appears to be the single gradient as it stabilizes skyrmions at remanence. Until now, such observations were only made possible by the use of bias layer [START_REF] Gaurav | Room-temperature skyrmions at zero field in exchange-biased ultrathin films[END_REF][START_REF] Ajejas | Densely packed skyrmions stabilized at zero magnetic field by indirect exchange coupling in multilayers[END_REF], in antiferromagnetic systems [START_REF] Yu | Room-temperature skyrmions in an antiferromagnet-based heterostructure[END_REF][START_REF] Legrand | Room-temperature stabilization of antiferromagnetic skyrmions in synthetic antiferromagnets[END_REF] or through advanced engineering using X-rays [START_REF] Guang | Creating zero-field skyrmions in exchange-biased multilayers through x-ray illumination[END_REF]. This provides a new and less restrictive way of obtaining isolated textures at remanence.

Thus, the SG structure has been studied in depth to understand in more depth its intriguing properties.

Single gradient multilayer

The single gradient (SG) multilayer displays promising properties as it allows to stabilize skyrmions at remanence. Its structure is characterized by a discrete variation of the ferromagnetic thickness:

starting from the bottom layer, it increases up to the middle one before decreasing symmetrically.

The effective anisotropy is thus maximal in the outer layers. The SG multilayer is extensively characterized in the following chapter.

Controlling the vertical extension with an external magnetic field

To demonstrate the impact of the anisotropy and understand how it can help to stabilize 3D objects, a few representative examples of micromagnetic simulations run with mumax 3 [START_REF] Vansteenkiste | The design and verification of MuMax3[END_REF] are shown in Figure 3.11, using the parameters from Table 3.4 (see Appendix A for more details on the simulations).

They are based on the previous multilayer architectures but under an external field of 250 mT to investigate how it transforms the spin textures. The simulations are displayed using isosurfaces to better apprehend the shape and dimension of the isolated textures: in red, we have m z = -0.8 while the white isosurface delimits the domain walls (m z = 0). The first three cases have roughly the same average uniaxial anisotropy but with different trends: uniform, with a minimum or a maximum near the middle magnetic layer as illustrated with the anisotropy plots. Thanks to the vertical cuts of m z on the right, the effect of those anisotropy variations are clear. Indeed, for the uniform case ( This simple numerical study confirms the possibility brought forth by the manipulation of the magnetic anisotropy, i.e. the thickness of the ferromagnet, for the stabilization of 3D spin textures which hints at their experimental realization.

Chapter 3 Summary

The structural quality of our magnetic multilayers and the reproducibility of their growth is established using reflectivity measurements along with high quality reconstruction performed with STEM or atomic probe tomography. Following this, we focus on systems based on the Pt|Co|Al to try stabilizing 3D textures as its magnetic properties have been extensively characterized. Finally, tentative growths are realized with structures displaying a variable thickness for the Co, allowing to tune spatially (including along the vertical axis) the magnetic interactions. Even though those aperiodic multilayers are new architectures, based on our extensive characterization of similar multilayer with fixed thickness, their structural properties should be of the same high level quality. The micromagnetic simulations of such structures demonstrate the impact of such thickness variations and show how it could be used to engineer an artificial confinement of spin textures at given position, notably in the single-gradient structure. Their experimental realization already put forward interesting properties as isolated objects can be stabilized at remanence after an out-of-plane saturation.

N

on-periodic magnetic multilayers can be used to stabilize spin textures with a vertical dependency, as illustrated with the micromagnetic simulations in Chapter 3. In this chapter, we focus on the characterization of the so-called single gradient (SG) multilayers using complementary experimental and numerical approaches. This thorough study allows us to reveal the existence of a new type of magnetic object, the skyrmionic cocoon, that differs from the usual columnar skyrmion as it presents a limited vertical extension.

First approach of skyrmionic cocoons

To probe the signature of the spin textures inside the SG multilayers, MFM measurements are performed with field along with micromagnetic simulations to correlate with the experimental observations. Unlike typical skyrmionic multilayers, the ferromagnetic layers in this structure exhibit an effective in-plane anisotropy but still stabilize out-of-plane textures through a relatively strong interlayer dipolar interaction. Furthermore, we investigate how the varying thickness of the materials in the multilayer affects the stabilization of spin textures.

Optimization of the single gradient multilayers

To stabilize 3D magnetic textures, we utilize the magnetic multilayers' tunability by creating a complex sample architecture with varying thicknesses of the ferromagnetic films within the multilayer.

This introduces an uneven distribution of magnetic interactions along the vertical dimension. This first sample heterostructure, named the single gradient (SG), is thus designed with a specific evolution for the ferromagnetic thickness as rapidly introduced in the previous chapter.

Single gradient (SG)

The cobalt thickness X i transitions from the initial value X 1 to X center with a fixed thickness step and then comes back to the initial thickness, following the same steps in reverse order. In other words, it can be recursively defined as:

X i+1 =        X i + S for i ≤ N/2 X i -S for i > N/2 , ( 4.1)
with S the thickness step between two consecutive layers, N the total number of layers and X 1 the bottom layer thickness. To have a symmetric structure, N requires to be odd. Typically, we will use S = 0.1 nm and N = 13 repetitions. The thickness of Pt is written t N Pt and the aluminium thickness is fixed at 1.4 nm.

In Figure 4.1.a, an example of the thickness evolution as a function of the layer index i is shown for a case where the outer layers are the thinnest ones (S > 0), yielding a typical arrow shaped-trend. It corresponds to the SG structure described by S = 0.1 nm, X 1 = 1.7 nm, N = 13 layers and t N Pt = 3 nm. The magnetic hysteresis measured with AGFM for an out-of-plane (OOP) and an in-plane (IP) field are shown in Figure 4.1.b. The in-plane effective anisotropy of most layers implies that the easy axis of the overall anisotropy lies within the plane. This is supported by the AGFM loops, which exhibit an easier transition with an in-plane field, with a lower saturation field (approximately 200 mT) compared to the out-of-plane measurements (approximately 300 mT).

To optimize SG structures, various growths were performed while changing different parameters, notably S and X 1 . The magnetic states obtained at zero field are considered as a first hint on how the structure parameters impact the stabilized textures. The samples' magnetization has been saturated with an out-of-plane field of 1.8 T before being measured with MFM at remanence. We start by considering the variations of the thickness of the bottom layer X 1 between 1.6 nm and 2.1 nm with the other parameters fixed (Figure 4.2.a). For thicker X 1 , i.e. at least 2 nm, we mostly observe elongated objects and stripes. Moreover, if we increase the thickness step S while conserving the minimum thickness X 1 = 1.8 nm and the maximum one X center = 2.4 nm (Figure 4.2.b), similar observations can be made as the textures expands more and more. The optimal value thus appears to be S = 0.1 or 0.15 nm when comparing samples. In Figure 4.2.c, the average characteristic length, density and circularity are plotted as a function of X 1 to corroborate the previous affirmation. The characteristic length corresponds to the long axis of the ellipse used to fit each object after binarizing the phase map with the appropriate threshold. The circularity is computed as 4 × A × π/P 2 with A the area of the texture and P its perimeter. For a circle, it should thus be equal to 1 while for a square, it is about 0.78. We observe a decrease of the characteristic length and an increase in the circularity and density for a bottom thickness of 1.8 nm, in agreement with the previous qualitative observations. The impact of the number of ferromagnetic layers N was not investigated to limit the degrees of freedom.

Additionally, for each structure presented in Figure 4.2, an MFM image is simulated using micromagnetic simulations (see bottom left corner of each image). Qualitatively, the trend a similar to what is observed experimentally: the stabilized objects become more elongated as X 1 and S increase.

Having established the optimal structure for stabilizing isolated circular spin textures at remanence in the SG structure (S = 0.1 nm, X 1 = 1.7 or 1.8 nm, N = 13 layers, t N Pt = 3 nm), the next logical step is to investigate how these textures evolve under an external magnetic field.

Revealing skyrmionic cocoons with an out-of-plane magnetic field

Out-of-plane field dependency

Thus, we investigate in more details the field dependency of the following SG structure Ta5|Pt3| (Co[1.7:0.1:2.3]|Al1.4|Pt3)|(Co[2.2:-0.1:1.7]|Al1.4|Pt3), using MFM to probe for the evolution of the isolated textures. In Figure 4.3.a, some phase maps are presented at different out-of-plane magnetic field along with the simulated MFM images obtained with mumax 3 . At remanence, the magnetic state corresponds to the previously observed configuration of tightly arranged isolated objects. Increasing the magnetic field up to 165 mT homogenizes their shapes and sizes due to the Zeeman effect gaining in strength: they appear as circular skyrmion tubes with a negligible dispersion of the magnetic contrast.

For µ 0 H ⊥ = 225 mT, a loss of contrast can be observed along with a reduction of the objects size.

Finally, near saturation, at 262 mT, the density and the contrast decrease further. The behavior is in qualitative agreement with the corresponding simulated MFM images (see insets) which thus appear to appropriately model the density, the shape and the size of the observed objects1 . The spin textures distribution properties are summed up in Table 4.1. They were obtained by binarizing the phase maps with an appropriate threshold before extracting the properties of the isolated pixels groups. In agreement with the previous qualitative observations, the average apparent diameter decreases as the field increases, going from 186 nm at remanence down to 86 nm near saturation. On the contrary, the circularity increases and stays close to 1, a value representative of quasi-circular shapes. The density increases slightly at first, as the small worms divides into several magnetic objects and then decreases as µ 0 H increases due to annihilation events. As MFM is sensitive to the stray field of a magnetic object, the resulting apparent diameter may differ slightly from the real one but it still provides a reasonable estimation of the size. As all the images have been recorded with the same experimental parameters, the reduction of signal with increasing field indicates a change in the configuration of the spin textures2 . To gain a more detailed understanding, micromagnetic simulations with an external field to mimic the experimental conditions can be considered as shown in Figure 4.3.b (using the parameters of Table 3.4). For the simulations, we assume that the DMI and the anisotropy arises from interfacial contributions only: [START_REF] Jiang | Skyrmions in magnetic multilayers[END_REF][START_REF] Zhang | Creation of a thermally assisted skyrmion lattice in Pt/Co/Ta multilayer films[END_REF][START_REF] Zázvorka | Skyrmion lattice phases in thin film multilayer[END_REF]. They possess a typical barrel shape that is wider in the middle layers, due to the dipolar interaction [START_REF] Legrand | Modeling the shape of axisymmetric skyrmions in magnetic multilayers[END_REF]. For a field of 225 mT, the objects become more and more ellipsoidal along the vertical direction and start to vanish from the top magnetic layer. Then at 275 mT, they become vertically confined and remain in a fraction of the layers only, acquiring a characteristic deformed ellipsoid shape (evidenced in Figure 4.4). This confinement in the bottom layers3 causes a diminution of the stray field near the surface, hence the observed drop in the MFM signal.

K u = K u,s /X i and D = D s /X i (

Layer-resolved magnetization

To take a closer look at the shape evolution of the spin textures, we present in

Skyrmionic cocoons

The micromagnetic simulations and their agreement with the MFM dependency put forward the stabilization of a new type of spin textures in the aperiodic multilayers. With the application of an external magnetic field, the more common tubular skyrmions evolve into vertically confined objects that do not span the entire thickness of the multilayer, thereby exhibiting evolution along the third dimension. Due to this significant difference and their typical ellipsoidal shape, we have named them skyrmionic cocoons. These objects differ from similarly shaped structures like magnetic torons or globules [2,[START_REF] Ao Leonov | Homogeneous and heterogeneous nucleation of skyrmions in thin layers of cubic helimagnets[END_REF] due to their discontinuous nature which precludes the emergence of Bloch points (a zone of vanishing magnetization). the equilibrium between the DMI, which promotes CCW Néel chirality for this particular stacking order (Pt|Co|Al), and the dipolar interaction, which stabilizes CW Néel caps to achieve flux closure [START_REF] Legrand | Hybrid chiral domain walls and skyrmions in magnetic multilayers[END_REF]. In the remaining middle layers, the skyrmion walls display a magnetization pointing uniformly in the same in-plane direction as the interlayer dipolar interaction and the DMI nearly cancel each other out. Anew, this evidences the importance of the interlayer dipolar interaction in those heterostructures that is strong enough to stabilize out-of-plane objects in presence of an in-plane effective anisotropy. Similar observations can be made for skyrmionic cocoons at higher fields with a chirality evolving from CCW Néel to non-chiral to CW Néel from bottom to top. However, the dipolar strength diminish with their size, leading to a weaker dipolar interaction. With the DMI unchanged, there should be a reduced presence of CW Néel chirality in this scenario, as confirmed by simulations.

This numerical and experimental study in presence of an external magnetic field has demonstrated the possibility to control the vertical confinement of spin textures, transitioning from columnar skyrmion to skyrmionic cocoons.

Energy distribution

To gain a more detailed understanding of the stabilization of isolated textures in these aperiodic multilayers, we can extract energy maps from the simulations and analyze their contribution. In the different layers at remanence and at 275 mT in all the simulation space. Both plot evidence the importance of the dipolar interaction (red points) and the anisotropy (pink points) while the exchange, both symmetric (blue points) and antisymmetric (green points), appear negligible in comparison, especially at high field. This is however representative of the full simulations spaces, averaging many skyrmions at remanence or skyrmionic cocoons at high field. To focus on the profile of a given object, energy maps are presented in Figure 4.6.b for a layer with a CCW Néel chirality (left) and a non-chiral one (right). For the former, the domain wall is costly in terms of exchange and anisotropy but it is favored by the demagnetization and the DMI. For the layer with no well-defined chirality in the domain wall (the spins point uniformly in a single direction), the DMI effect is effectively zero as two nodes of opposite sign cancels each other. Additionally, the exchange is less costly as the spins are more aligned in the domain wall. The dipolar and the anisotropy interaction remains similar as in the other layer. This energy study also demonstrates why the DMI constant we used in the simulations (D s = 2.34 pJ/m), inaccurate with respect to BLS measurements (Subsubsection 3.3.1.3), has not a drastic impact on the stabilization of the resulting textures. Indeed, the dipolar interaction and the anisotropy are at least an order of magnitude stronger which limits the DMI impact, at least on the stabilization of cocoons. Zooming in on given object (Figure 4.6.c) however shows that, in the domain wall itself, the different energy terms have similar amplitude. Thus, the DMI might play a role for the configuration of the domain wall. We find that the chirality in the domain wall is not significantly modified when using smaller and more accurate D s . However, the vertical position of the cocoons at high field is altered: they are vertically centered, disconnected from both the bottom and the top interfaces of the multilayer (see Appendix B).

Impact of the thickness gradient on the vertical confinement

Numerical confinement study

To further investigate the vertical evolution of the magnetic solitons existing in the SG structure, we then focus on the thickness parameters of the multilayer itself. The magnetic multilayers properties can be accurately tuned to answer specific requirements which is deeply interesting for the engineering of three-dimensional textures. Until now, we have chosen to focus on SG structures with a step thickness S of 0.1 nm but a thorough numerical study was done on other possible architectures. To investigate the magnetic textures that could be hosted in such structures, micromagnetic simulations were performed using mumax 3 for different thickness steps values, at various magnetic fields and X 1 , while N = 13 layers was fixed. For each set of parameters, the magnetization was initially set out-of-plane with additional random noise and then relaxed at each amplitude of the increasing OOP magnetic field (see Appendix A for more details on the simulations). Then, the average heights of all the textures present in the simulation space has been computed as well as their average circularity C to differentiate the stripes states (C < 0.9) from the states presenting more circular and isolated objects (C ≥ 0.9). The heights have been obtained by computing the vertical extension of each individual spin textures, i.e. the number of magnetic layers it is stabilized in.

In Figure 4.7.a,b, three representative cases are displayed: S = 0 nm, which corresponds to the uniform case, S = 0.1 and 0.2 nm with the larger layer in the middle. The average heights of the magnetic textures for different thicknesses and magnetic fields are presented. Those graphs evidence the impact of the thickness gradient: the larger S is, the more confined the textures can be at equal field and mean Co thickness. Indeed, with S = 0.2 nm, with fields stronger than 200 mT and an average thickness below 2 nm, some textures that are only present in half the overall thickness (less than 7 layers). Similar conclusions can be drawn on the intermediate case, S = 0.1 nm, for which at high field more confined textures are present. On the contrary, for S = 0 nm the vertical confinement is very limited even if, for large average thicknesses and magnetic field, the textures can be confined down to 10 layers. In the general case however, the textures remain almost columnar. Moreover, in our simulations, the range of stability in thickness is broadened by the presence of the gradients as it is possible to stabilize textures with lower thicknesses than the uniform case. Finally, by introducing a variable thickness into the structure, it is possible to obtain states with isolated textures at lower fields and thicknesses rather than stripes or worms as shown by the circularity.

To support the importance of the gradients, vertical cuts of the magnetization of a selected axisymmetric object are displayed below their respective graph, for a magnetic field of 250 mT and an average thickness close to 2 nm. We observe that for S = 0 nm, it corresponds to a columnar skyrmion ( We also consider what we call Reversed Single Gradient (RSG) with the inverse thickness evolution, i.e S < 0, so that the middle layer has the thinnest thickness. For S = -0.1 nm, the impact of the gradient on the vertical confinement is minimal as the height distribution is close to the uniform one.

However, the magnetization cut shows an object with a higher diameter and smaller extremities at equivalent field and average thickness. A stronger effect is noticeable when using a larger step but it is restricted to a small range of thicknesses and the majority of the relaxed magnetic states corresponds to stripe states and not to isolated objects, thus limiting the interest of this other possible structure.

Note that we observe a magnetic object split in half in the vertical cut, with the scission occurring in the layer with the stronger OOP anisotropy (of thickness 1.2 nm in that case). It shows the importance of strong PMA layers in the manipulation of the 3D magnetization distribution. Overall, for negative S the objects are less confined than with positive S and the bottom and top layers display a more complex behavior. MFM images of RSG multilayers are shown in Appendix C. These simulations enable the exploration of a wide range of parameter space, encompassing both field strength and average thickness. They reveal the significant influence of the thickness step S on the SG architecture, wherein larger steps result in more confined magnetic textures.

Probing the height with MFM

To build on the previous numerical study, MFM measurements have been performed on three SG samples, with different thickness steps but a similar average thickness close to 2.1 nm. To enable the comparison between them, identical experimental conditions were used for the lift height, the drive amplitude and the magnetic tip. In Figure 4.8, the phase maps are shown for different OOP magnetic fields along with line cuts to evidence the evolution of the signal. Due to experimental constraints, we cannot record a full hysteresis using our MFM 5 . In this particular study, we have focused on the nucleation of the textures, thus starting from saturation at 300 mT and lowering the magnetic field.

The evolution of the contrast with the field might give insight into the vertical extension of the objects.

Since the nucleation happens at different magnetic fields depending on the samples, different field ranges have been probed.

• S = 0 nm with Ta5|Pt3|(Co2.07|Al1. The contrast seems similar for all of the textures and does not evolve significantly with the magnetic field (only a factor 1.25 between the line cut at 243 mT and the one at 172 mT). In Figure 4.8.b we can see that the line cut at the highest field (268 mT) has a much smaller signal than the rests but it corresponds to an isolated object that nucleated at a much higher field than the rest and is thus not representative of the trend. Finally, the skyrmions appear to be quite large, with an apparent diameter close to 300 nm at the lowest measured field. In 221 mT. Two contrasts (≈ -0.2 • and -0.12 • ) are distinguishable and appear uncorrelated with the apparent diameter of the corresponding textures, defined as the full width at half maximum (FWHM). Indeed, both populations have a similar apparent diameter (about 170 nm), indicating that the signal variations is not due to size considerations (as the resolution is high enough to resolve the spin textures). This disparity likely indicates a height distribution that is not uniform. Moreover, as the field is decreased, the signal increases as shown with the line cuts in Figure 4.8.b, showing a factor 4 in the evolution of the signal from highest to lowest field. For those cuts, a linear trend adequately fits the evolution of their maximum amplitude with the field, yielding a slope of -5 mdeg/mT (for a R-square coefficient of 0.98). This evolution evidences the height evolution of the objects as they expend in all direction upon the decrease of the field, a result coherent with the numerical expectations (Figure 4.7). Once, the cocoons have fully extended upon lowering the field further, the amplitude of the peaks should stay constant. We can also notice that the skyrmionic cocoons are much smaller than the skyrmions with an apparent diameter of around 100 nm.

• S = 0. The experimental evolution of the MFM contrast puts forward a vertical extension for the cocoons in the aperiodic magnetic multilayers that is not seen in their periodic counterparts. It thus highlights the importance of the ferromagnetic thickness gradients in addition to provide insight into the 3 rd dimension of the cocoons.

Resilience to an in-plane magnetic field

To complete the experimental characterization of skyrmionic cocoons with MFM, the field dependency has been measured using an in-plane magnetic field. With our current setup, it can continuously reach 400 mT starting from remanence. We have thus studied the same SG as in At remanence, we observe the usual distribution of isolated objects but only the domain walls are resolved (both up and down domains have the same contrast). This likely originates from a switching of the magnetic tip due to the strong stray field of the multilayer. It is usually solved by applying a small bias OOP field which is not possible in the IP setup. However, the phase map that is obtained is still representative of the magnetic state. Increasing the field up to 90 mT, we can observe that the textures are shrinking and tend to align with the direction of the magnetic field. At 110 mT, the features get blurry as the magnetization is tilting more and more in plane. This effect worsens as the field increases until the textures merge together at 150 mT, near saturation. Going down from the saturated state at 400 mT, the remanent state shows an ensemble of stripes aligned with the magnetic field. The field dependency of this SG demonstrates the resiliency of the spin textures to an in-plane field.

Topology in a discontinuous medium

After the introduction of the skyrmionic cocoon and subsequent exploration through MFM and correlated simulations, we delve into is its topological properties. To this end, we rely on numerical studies as it is not accessible experimentally (by MFM). However, since cocoons are stabilized in a magnetic multilayer, it results that the magnetization is discretized along the vertical direction which violates the continuity principle that topology is based on. We will thus concentrate on a 2D approach, mathematically correct, at first before considering the 3D case with a thought experiment.

Two-dimensional lattice-based topological number

To study the topology of the skyrmionic cocoons, we consider the same approach used for skyrmions in thin films and magnetic multilayers (see Section 2.2). A 2D topological number N sk in each magnetic layer can be numerically computed using a lattice-based approach [START_REF] Kim | On quantifying the topological charge in micromagnetics using a lattice-based approach[END_REF] at different fields. In the continuous approach, N sk is defined as:

N sk = 1 4π ¢¢ m • ∂m ∂x × ∂m ∂y dx dy (4.2)
The previous formula is valid under a hypothesis of continuity in the plane which is not rigorously valid as the spins are localized on the atoms of a lattice. However, assuming that the angles vary slowly from one spin to its neighbors, we can assume that the medium is continuous. Typically, for a magnetic skyrmion, the topological number is ±1 depending on the polarity of the core. This indicates that the magnetization can be mapped around a unity sphere S 2 . In mathematical term, it thus pertains to the homotopy group π 2 S 2 [START_REF] Thiaville | Magnetism and topology[END_REF].

More precisely, in the simulations, we sweep the magnetic field from positive saturation towards negative saturation and compute the topological number in each layer every 25 mT, mimicking half of a hysteresis cycle. The results are presented with the map shown in This simple approach respects the mathematical premisses of the integral definition but it is not really representative of a three-dimensional object.

Three-dimensional approach

To further probe the topology of skyrmionic cocoons, we can consider the 3D generalization of the winding number [START_REF] Im | Dynamics of the Bloch point in an asymmetric permalloy disk[END_REF][START_REF] Oleksandr V Pylypovskyi | Bloch point structure in a magnetic nanosphere[END_REF] (adapted from Eq. 2.19):

q = 1 4π 4 S ϵ ijk m • (∂ j m × ∂ k m ) dA i (4.3)
where the integral is taken over a surface S, enclosing the structure(s) and with ϵ ijk the Levi-Civita tensor, summed over three components x, y and z. For a single enclosed Bloch point, q takes the value ±1 (see Section 2.2), and q can be zero for even numbers of Bloch points, like for the magnetic torons.

In our case, when the cocoon is fully buried into the multilayer, the magnetization on a contour is quasi uniform with no curvature, and therefore leads to q = 0 (see Figure 4.12.a). Alternatively, if the cocoon "emerges" at the surface of the multilayer, with a skyrmion in the topmost (or bottommost) magnetic layer, then q is entirely determined by the 2D skyrmion number of this last layer (Figure 4.12.b) and is thus non-zero. However, we stress that the lack of continuity of the magnetization6 , especially along the vertical direction, breaks the premises of the underlying mathematical foundations: in particular, q ̸ = 0 is not associated with the presence of Bloch points, which are "hidden" between the magnetic layers. As illustrated in Figure 4.12.c, when the symmetric exchange is low enough and the layers only coupling through dipolar exchange, there is no more continuity of the magnetization texture along the out-of-plane direction. As the symmetric exchanged is reduced, keeping only the dipolar coupling between the layers, no more continuity of the magnetization is expected between layers, and the Bloch points can be truly avoided. Therefore, the name skyrmionic cocoons was introduced to indicate this fundamental difference with the continuous textures called torons [START_REF] Ao Leonov | Homogeneous and heterogeneous nucleation of skyrmions in thin layers of cubic helimagnets[END_REF]2]. This thought experiment demonstrates the complexity to define the topology in a discontinuous three-dimensional medium. The same conclusion could be drawn for tubular skyrmions in magnetic multilayers. Thus, with a 2D approach, the cocoons demonstrate a well-defined topology but in 3D we are missing the correct approach to study them. However, the possibility to sustain a reversal of the magnetization along the vertical axis without any BP is an important feature (smaller energy cost), made possible thanks to the discontinuity of the magnetization.

Investigating the effective chirality

Thanks to the previous MFM and numerical study, we have introduced the skyrmionic cocoon, a 3D texture stabilized in aperiodic multilayers that displays some variations, either by tuning the external magnetic field or by playing with the materials properties. For those discussions, we have mostly relied on the micromagnetic simulations to discuss the chirality of the domain walls and the topology of the objects. In the following, X-ray resonant magnetic scattering (XRMS) measurements are presented as a probe of the chirality in the SG multilayers. Due to the complexity of the scattering events occurring in the multilayer, only an effective chirality, i.e., a weighted average of the thickness dependent chirality, is obtained.

XRMS principle

Interactions between soft X-rays and the magnetization provide with powerful techniques to investigate the spin distribution. In standard X-ray diffraction theory, the role of the magnetic moment has been neglected for long since its non-resonant impact in ferromagnets is much weaker than the Thomson scattering contribution. Indeed, the magnetic scattering is reduced by a factor ℏω/mc 2 compared to the Thomson one [START_REF] Beaurepaire | Magnetism: A synchrotron radiation approach[END_REF][START_REF] Blume | Magnetic scattering of x rays[END_REF][START_REF] Bykova | High-resolution X-ray ptychography for magnetic imaging[END_REF] with m the mass of the electrons, ω the X-ray energy and c the light celerity. For instance, in the soft X-rays range (at the Co L 3 edge), this factor is about 10 -3 . Thanks to the energy tunability and the high energy of synchrotron sources, it is possible to use resonant scattering close to the absorption edges of magnetic elements which increases by orders of magnitude the magnetic contribution.

The first demonstration of XRMS to determine the direction of the flux closure of magnetic domains was performed on FePd films by Dürr et al. in 1999 [START_REF] Ha Durr | Chiral magnetic domain structures in ultrathin FePd films[END_REF]. Since then, it also has been shown that information about the effective chirality, i.e. the magnetic winding and the DMI sign, can also be extracted from the reflection pattern [START_REF] Chauleau | Chirality in magnetic multilayers probed by the symmetry and the amplitude of dichroism in x-ray resonant magnetic scattering[END_REF][START_REF] Legrand | Hybrid chiral domain walls and skyrmions in magnetic multilayers[END_REF]. Moreover, it is a non-perturbative technique that can be applied to non-patterned materials with a probing depth that typically extends over about a hundred nanometers beneath the surface.

The experimental configuration is shown in Figure 4.13: for a given helicity of the X-ray, the beam is reflected on the sample with an incidence angle θ and the scattered signal is captured with a CCD camera. To protect it from the intense signal coming from the specular signal (in the low frequencies region), a beamstop is used, which thus creates missing pixels in the recorded pattern. Then, by considering the difference of the two circular polarizations of the light, it yields a typical azimuthal signal that depends on the effective chirality of the probed layers. The sensitivity of the measurements can also be tuned by varying the incidence angle or the energy to change the penetration depth of the X-rays.

Analytical approach

In the following, we synthesize the main steps of the dichroism demonstration with circularly polarized X-rays at resonance to understand its origin. Within the Born approximation (in kinematical scattering), the scattered intensity can be expressed as:

I(q, ω) ∝ n f n (ω) exp (iq • r n ) 2 (4.4)
with q = k f -k i the scattering vector, k i,f the initial and final wave vector. Moreover, f n is the resonant scattering amplitude of the atomic site located at r n which contains the magnetic dependency. Indeed, in the electric dipole approximation, Hannon et al. showed that it could be written as [START_REF] Jp Hannon | X-ray resonance exchange scattering[END_REF]:

f n (ω) = ê′ • êf (0) n -i (ê ′ × ê) • mf (1) n + (ê ′ × m) (ê × m) f (2) n (4.5)
with ê′ and ê the polarization vectors of the incident and scattered beam, m the reduced magnetization.

Respectively, f

(0) n , f (1)
n and f

(2

)
n correspond to the monopole, magnetic dipole and quadrupole parts of the scattering signal. The first term, f (0) n , refers to the resonant anomalous charge scattering and is thus independent from the magnetic moments. For structures with smooth interfaces, it mostly contributes to the specular signal and is added to the Thomson scattering. The term f (1) n is linear with the magnetization and, near the appropriate absorption edges, can generate a substantial XMCD effect. It will cause the apparition of magnetic peaks at wave vectors ±2π/p, with p the periodicity of the magnetic domains. Similarly, the term f (2) n is quadratic in m and thus gives rise to a magnetic contribution. Although it is often negligible, it can provoke the apparition of a second order signal with peaks at twice the periodicity in the reciprocal space. Each of those scattering terms can be expressed as a real dispersion part coupled to an imaginary absorption part by the Kramers-Kronig relation.

Finally, using F n (ω), the Fourier transform of f n (ω), and µ the density matrix of the incident X-ray beam, the intensity can be expressed as [START_REF] Jp Hannon | X-ray resonance exchange scattering[END_REF]:

I(q, ω) = Tr F n (ω)µ F n † (ω) ≡ I c (q, ω) + I m (q, ω) + I i (q, ω) (4.6)
It thus comprises a charge (I c ), a magnetic (I m ) and an interference term (I i ). The density matrix depends on the polarization of the X-rays, which will be circular in our case. Using the Stoke-Poincaré representation, it yields [START_REF] Van Der Laan | Soft x-ray resonant magnetic scattering of magnetic nanostructures[END_REF]:

µ CL =   1 -i +i 1   , µ CR =   1 +i -i 1   (4.7)
After development, the charge term in the density matrix relies solely on the diagonal coefficients and is thus independent of the helicity in the case of circular polarization. On the other hand, the magnetic and interference terms rely on the off-diagonal coefficients, which differ between the two helicities. This explains why it is essential to consider the dichroic signal, which is the difference between the two polarizations. Doing so enables the separation of the charge and magnetic scattering in reciprocal space since the latter reverses when the polarization changes. The full proof for each term is available in [START_REF] Van Der Laan | Soft x-ray resonant magnetic scattering of magnetic nanostructures[END_REF].

Setup specifications (SEXTANTS)

On the SEXTANTS beamline at SOLEIL, the 2048 × 2048 pixels Peltier cooled CCD camera is located at 26 cm from the sample, covering an angular range of 6.1 degrees, with a pixel size of 13.5 µm. It records the signal coming from the RESOXS diffractometer [START_REF] Jaouen | An apparatus for temperature-dependent soft x-ray resonant magnetic scattering[END_REF] in high vacuum conditions with a beam of 50 µm diameter, allowing to average over a large area. Thanks to the undulators, a polarization efficiency for circular or linear polarization superior to 99% can be reached. The range of energies that can be reached span from 70 eV to 1200 eV. For SG multilayers, we need to be sensitive to the Co and thus will place the photons energy near the L 3 Co edge (at 778.5 eV). Note that the nominal energy depends on the calibration of the beamline so that the exact value used here is not absolute and might vary by a few eV on other beamlines. An energy scan is shown in Figure 4.14.a.

As only an in-plane field can be applied for this setup, a permanent magnet is used to provide an out-of-plane component, with the sample placed on top.

Understanding the dichroic signal

The circularly polarized light will be denoted CL (resp. CR) for circular left (resp. right). As previously explained, the magnetic contribution can be studied by considering the dichroic signal, i.e. Extracted from [START_REF] Chauleau | Chirality in magnetic multilayers probed by the symmetry and the amplitude of dichroism in x-ray resonant magnetic scattering[END_REF].

I

XRMS chirality

By convention, the zero of the azimuthal angle ψ is set at the northern end of the scattering pattern, as illustrated on Figure 4.15 with a clockwise rotation. As shown analytically before, the normalized difference exhibits evidence of a magnetic signal which, for periodic system, appears as positive or negative nodes around the specular signal. The relative position of the node with respect to the azimuthal angle is directly linked to the effective chirality probed by the X-rays [START_REF] Chauleau | Chirality in magnetic multilayers probed by the symmetry and the amplitude of dichroism in x-ray resonant magnetic scattering[END_REF]. In Figure 4.14.b, the different cases are put forward. For simplicity, in

Extracting the periodicity

Before considering more complex interpretations, an easy analysis can be performed for any given incidence angle: the study of the periodicity. Indeed, the scattering ratio should display magnetic peaks at wave vectors ±2π/p according to the expression of the scattering amplitude (Eq. 4.5). To that end, the real coordinates have to be mapped to the reciprocal space which depends on the geometry of the setup. Using the spatial referential defined in Figure 4.13.a where the beam travels in the (yz) plane, they can be expressed as:

q j = 2π λ × [sin θ + sin (θ + dθ)] with dθ = arctan jL D (4.8)
with j = x, y the pixels index, D the distance to the detector (26 cm), L the pixel size (13.5 µm)

and λ = ℏc/ω the wavelength of the X-rays (1.59 nm -1 at the L 3 Co edge). With this updated coordinate system, it becomes possible to extract the characteristic size of the scattering pattern.

In

Reflectivity curves

To quickly probe for XMCD effects, reflectivity curves can be measured by varying the angle of incidence θ of the X-rays, recording the intense specular signal with a diode this time. Such measurements are shown in Figure 4.17 for the previous SG on a permanent magnet for two different energies and the two helicities. Due to the absence of real periodicity in the structure because of 7 The beamstop is used to protect the CCD camera from the more intense specular reflection.

the thickness gradient of the ferromagnet, it is expected to have a complex evolution of the angular signal. For both energies, Bragg peaks that correspond to optimal scattering conditions (constructive interferences) are clearly visible as evidenced by the black arrows. Off resonance, at 770 eV, in between the various Bragg peaks, multiple oscillations are present contrarily to the curves at 778 eV. Reversing the helicity do not create any significant changes in the signal: there is no evidence of dichroic signal when considering those reflectivity curves. However, they still yield data regarding the structural properties of our SG multilayer. Following this first measurement, let's now consider the evolution of the asymmetric ratio at different incidence angles to probe the evolution of the effective chirality.

Effective chirality from angular maps

Qualitative evolution of the normalized difference

To study the depth evolution of the chirality, the energy was set to 778.5 eV while the incidence angle was changed. The probing depth should increase with the incidence angle but its dependency can be quite complex due to the interferences arising from the multiples interfaces and their respective roughness as well as due to the resonant character of the absorption. In lobe is located at ψ = 270 • (resp. 90 • ). Thus, the effective chirality of the magnetic textures in the top layers of the SG structure is CW Néel. Interestingly, a second order is also visible at twice the wave vector which thus arises from the f

(2)

n term discussed previously. This effect disappears at higher incidence angle even though the chirality remains the same for θ = 24.1 and 39 • . A significant change occurs at high incidence angle, for θ = 43 • , as the two nodes switch position resulting in an inversion of the dichroic signal. This evidences that the effective chirality of the spin textures in the bottom layers is CCW Néel. To facilitate the analysis of the many images recorded at different angles, one can perform a radial integral over the azimuthal angle ψ. An example of the process is shown in Figure 4.18.b. First, the image is binarized with an adjusted threshold which allows to fit the dichroic signal with a circle.

Based on this, a ring mask is applied to the initial image that restricts the signal to its magnetic contribution. Then, the integral is performed radially over the ring as a function of the azimuthal angle. This method has been applied to the previously discussed images and the resulting signals are

SG effective chirality

An inversion of the dichroism in the XMCD ratio is observed between the scattering signal at low and high incidence angles. This corresponds to the presence of CW Néel domain walls in the top layers and CCW Néel below. This distribution of the effective chirality stems from the interplay between the DMI and the dipolar interaction. Indeed, the DMI arising from the stacking order (Pt|Co|Al) promotes CCW Néel domain walls while the dipolar interaction favors CW Néel caps in the top layers to reduce the energy of the system via flux closure [START_REF] Legrand | Hybrid chiral domain walls and skyrmions in magnetic multilayers[END_REF].

Radial integration of the signal

The azimuthal dependency has been studied for the remaining incidence angles (in total 67 different ones, not linearly spaced). The results are shown in Figure 4.19.a as a map of the asymmetric ratio in function of (θ, ψ). The beamstop is again visible at ψ = 270 • with the drop in the signal. Overall, this detailed evolution appears more complex than what previously stated.

• For low incidence angles (θ < 29 • ): the effective chirality is CW Néel. Those Néel caps are imposed by the strong dipolar interaction to reduce the energy of the system. The maximum intensity for the scattering are located near θ = 10 and 19 • , which are slightly at higher angle than the Bragg peaks shown in Figure 4.17.

• For high incidence angles (θ > 42 • ): an inversion of the dichroism has occurred, leading to an effective chirality that is CCW Néel. Due to the scattering nature of the experiment, the signal originating from layers near the surface has a stronger contribution compared to the underlying layers as there are fewer interferences and less absorption. In this case, the top layers display a CW Néel chirality. However, at higher incidence angles, a CCW Néel chirality is detected, suggesting its presence in multiple layers, potentially exceeding the number of layers exhibiting CW Néel chirality. This is also predicted in the simulations, with the 6 (resp. 4) bottom (resp. top) layers showing CCW (resp. CW) Néel winding.

• In the medium case, it is harder to draw qualitative conclusions. For 29 < θ < 32 • , the dichroism disappears: the XMCD ratio remains relatively constant with respect to ψ. As the scattering is sensitive to all the layers probed by the X-rays, with weights decreasing as they penetrate more, it can average the signal of CW and CCW Néel walls, thus canceling the dichroism. For θ = 32 • , the dichroic signal appears again but shifted by 90 • , the maximum is at ψ = 180 • , indicating a CW Bloch chirality. However, for 34 < θ < 38 • , it reverses to a CW Néel ordering and 38 < θ < 42 • the dichroism vanishes.

In Figure 4.19.b,c, the relaxed state of a micromagnetic simulation at 150 mT is displayed with horizontal and vertical cuts of the magnetization. Note that at this field, in the simulation, the columnar skyrmion only starts to get vertically confined in this structure. The micromagnetic simulations are in agreement with the trend obtained from the XRMS measurements, i.e. to have CW (resp. CCW) Néel in the top (resp. bottom) layers, allowing to reduce the energy via flux closure. However, it is complex to be more quantitative than that due to the complexity of the multilayer under consideration. As the X-rays probe various layers with different weights because of the interferences, it is extremely difficult to draw conclusions regarding the chirality. This is especially the case as the incidence angle increasing, mixing the signals coming from various layers with different weights. In the middle of the angular range, a complex evolution of the dichroic signal is seen, showing drastic changes even with 1 • step. To be more quantitative, numerical support is needed to obtain the precise distribution of the domain wall chirality. This can be done by modeling the magnetic state and using it to simulate the associated scattering signal [START_REF] Flewett | General treatment of off-specular resonant soft x-ray magnetic scattering using the distorted-wave born approximation: numerical algorithm and experimental studies with hybrid chiral domain structures[END_REF].

Chapter 4 Summary

The optimization of aperiodic multilayers, the single gradient (SG) structures, allowed to observe and characterize a new type of 3D texture, the skyrmionic cocoon. Its stabilization relies mostly on the dipolar interaction, as the effective anisotropy is in-plane. Using concomitant MFM and micromagnetic simulations, its vertical extension has been demonstrated through the variation of contrast. This vertical confinement can be controlled, either thanks to an out-of-plane magnetic field or through the parameters of the multilayer itself, notably the thickness step S between two consecutive ferromagnetic layers. This ability to disappear from the outer layers is vastly different from the properties of the usual skyrmions that are mostly invariant over the thickness. The skyrmionic cocoons hosted by the SG structure are also quite resilient to the application of an in-plane field, needing more than a 100 mT to be annihilated.

The topology of such objects was briefly broached as the study of 3D objects in a discontinuous medium such as magnetic multilayers remains intricate. However, using a 2D approach, their topology can be studied with a similar approach than for the typical two-dimensional skyrmion. Importantly, the cocoons do not display any Bloch points at their extremities, a feature allowed by the non-magnetic space that breaks the continuity of the magnetization along the vertical direction.

Following this study, the SG multilayers were probed with synchrotron light for resonant magnetic scattering (XRMS). Although it is not yet possible to be fully quantitative, the observations of CW Néel in the top layers and CCW Néel in the bottom ones are in agreement with the corresponding simulations. This configuration of the domain walls results from the equilibrium of the negative DMI sign, which favors CCW Néel, with the dipolar interaction that encourage flux closure with CW Néel caps.

Thus, the cocoons have been stabilized and characterized in those aperiodic multilayers, displaying variations over the 3 rd dimension. To build upon that knowledge, the SG will now serve as a building block for more complex architectures to obtain the coexistence of various magnetic textures, an interesting feature for potential applications.

S

kyrmionic cocoons can be stabilized in the single gradient (SG) structure in which the thickness of the Co layers depends on their vertical position, as described in Chapter 4. Those objects display a vertical confinement that can be controlled using an external magnetic field or by carefully engineering the properties of the multilayers. Using the knowledge gained from this study, the SG multilayers have become the building blocks of more intricate structures, later on labeled as the double gradient (DG) architectures. Those can accommodate the coexistence of distinct magnetic textures, namely 3D columnar skyrmions and 3D skyrmionic cocoons. Beyond the fundamental interest, for example to study the transition processes between topologically different magnetic textures, the purpose is also to investigate how these cocoons can be intentionally localized at different z-positions, that in the perspective of applications, would be a first step in the design of 3D skyrmionic platforms.

Optimization of the double gradient structure

Firstly, let's delve into the architecture of the double gradient (DG) multilayer, which incorporates the utilization of strong perpendicular magnetic anisotropy (PMA) layers to effectively modulate the textures. By integrating these PMA layers within the multilayer structure, the DG architecture demonstrates enhanced control and manipulation of various magnetic textures.

First approach of the DG multilayer

To modulate the textures along the vertical direction, we have put forward the role of the anisotropy (see Subsection 3.3.4), which can be finely tuned with the thickness of the ferromagnet in our Pt|Co|Al based heterostructures. In this chapter, we attempt to complement the optimized SG structure that allows to stabilize skyrmionic cocoons with the insertion, in between several SG blocks, of layers with strong perpendicular magnetic anisotropy (PMA). The latter have the potential to modulate isolated magnetic objects along the thickness thanks to the drastic changes in the anisotropy. To that end, we developed the double gradient (DG) multilayers by using two SG separated by repetitions of trilayers with strong PMA (see schematic in Figure 5.1.a).

Double gradient (DG)

The double gradient structure is made of two single gradients (defined by the thickness step S, the number of magnetic layers N , the bottom thickness X 1 and the Pt thickness t N Pt ) separated by M repetitions of Pt|Co|Al with a thinner Co of thickness Y to obtain a stronger perpendicular magnetic anisotropy (PMA). The aluminum remains fixed at 1.4 nm. For the platinum, the trade-off discussed in Subsubsection 3.3.1.6 between its role for the anisotropy, dipolar interaction, torques and its absorption of X-rays is still valid. Its thickness in the SG parts will be written as t N Pt and in the middle layers as t M Pt . Note that the capping layers will always be 3 nm of platinum. The typical evolution of the ferromagnetic thickness is shown in Figure 5 For simplicity, we will refer to the top and bottom parts of the multilayer as SG or gradient blocks, while the middle layers will be referred to as strong PMA layers.

Hysteresis

First, the effect of the strong PMA layers can be observed in the magnetic hysteresis presented in likely permitted by the stray field of the gradients parts as the strong PMA layers on their own would likely have a uniform remanent state.

In the gradient parts, the average thickness is 2.3 nm, which corresponds to an uniaxial anisotropy of 686 kJ/m 3 and thus K eff = -248 kJ/m 3 (using the experimental surface anisotropy of Table 3.4).

For comparison, in the central PMA layers with 1 nm Co, we obtain K u = 1580 kJ/m 3 for an effective anisotropy of 644 kJ/m 3 .

Unveiling the coexistence of magnetic textures

The DG magnetic state strongly depends on its magnetic preparation and evolves drastically with the application of a magnetic field, whether out-of-plane or in-plane. The structure under study in this section corresponds to the one presented in Figure 5.1, described by S = 0.1 nm, X 1 = 2.0 nm, N = 13 layers, M = 15 repetitions, Y = 1 nm, t N Pt = 2 nm and t M Pt = 3 nm. For simulations, voxels of 2 × 2 × 2.13 nm 3 are used for a simulation space of 1.024 × 1.024 µm 2 . The numerical spacer is set to 3.4 nm between two magnetic layers as the voxels dimensions are fixed.

Out-of-plane field: toward saturation

In Figure 5.2.a, we present the MFM images at different OOP magnetic fields. The sample was demagnetized using a decreasing magnetic field tilted by 30 • away from the surface's normal. This procedure was used to neither favor the in-plane or the out-of-plane anisotropy. At remanence, two contrasts are visible in the phase maps: the bright distorted red stripes along with weakly contrasted white dots in the blue domains. Some triangular distortion is also detected in the red domains with very weak white dots. To further probe this phenomenon, the magnetic field is increased which significantly modifies the magnetic states. For µ 0 H = 230 mT, the stripes are straighter and more white dots have nucleated in the blue domains. At higher field, at 275 mT, the stripes have shrunk to strong contrasted dots amidst many weaker white dots. Near saturation, at 300 mT, this coexistence stops, leaving only the smaller objects. first the ones that penetrate through all the layers, the stripes, and second the skyrmionic cocoons present only in the bottom SG or top SG block of the multilayer. For simplicity, we will refer to the former as 3D stripes (or 3D worms) in the following. The strong contrasts in Figure 5.2.a therefore correspond to the columnar objects whereas the cocoons are associated with a weaker white signal. For a larger OOP magnetic field (230 mT), the red 3D stripes become narrower and the density of cocoons increases, yielding a configuration showing alternatively stripe domains and chains of cocoons. This contraction of the red stripes (opposed to the field) likely annihilates the cocoons within (aligned with the field), killing the triangular nodes. For a slightly higher field, at 350 mT (resp. 275 mT) in the simulations (resp. experiments), the elongated side of the 3D stripes shrinks, causing them to transition into skyrmion tubes (large red dots) whereas the cocoons (white small dots) appear more resilient to the field. We can also notice that the columnar skyrmions appear significantly larger than the cocoons. This can be understood by considering the domain wall energy, scaling as 4 √ AK u -πD, that is more costly in the strong PMA regions. To counterbalance this loss, the dipolar needs to be stronger and thus the skyrmions expand horizontally.

Finally, when the field is high enough, at 400 mT, the layers with strong PMA point uniformly in its direction, thus transforming the remaining skyrmion tubes into two cocoons, one in each SG block.

We hence identify a field range in which we can stabilize cocoons in separated blocks that could be piled up to create more complex architectures. The exact nature of the transitions at play and their associated mechanisms remain to be elucidated.

Furthermore, the simulations indicate that in most scenarios, there are two cocoons present, with one located in each SG block, arranged in a stacked manner. As the MFM signal comes mainly from the stray field of the top layers, this cannot be confirmed experimentally using only MFM. This will be addressed using transmission techniques such as holography or STXM (see Chapter 7). However, with MFM, we can still distinguish the signal arising from the top layers cocoons and the 3D stripes.

Already, this simple field dependency measured with MFM demonstrates the interest of DG multilayers.

The use of strong PMA layers allows for the stabilization of cocoons at different vertical levels as well as their coexistence with more conventional entities like columnar skyrmions. Note that, similarly to the SG structure, the vertical extension of the cocoons in DG multilayer evolves with the magnetic field.

Out-of-plane field: from saturation

Similar measurements have been performed with MFM on the other branch of the hysteresis (coming down from saturation) to investigate the sharp jump in the OOP hysteresis loop (Figure 5.1.b).

Due to experimental constraints, the maximum field that can be applied in the setup allowing to measure close to remanence is about 150 mT. Thus, to reach the saturation of the magnetization, an external magnet (of 500 mT) was brought close to the sample before restarting the measurement. The clear change in the magnetization (see state (I) and (II) in Figure 5.3) confirms the success of this process. Indeed, before the application of a saturation field (I), there are both 3D worms and cocoons at 142 mT but the former have disappeared in (II) at the same field after saturation.

The field steps are evidenced on the magnetization curve measured with AGFM. From this state, the field is lowered before the step visible in the magnetization (see Figure 5.1.b), yielding a magnetic configuration (III) with only cocoons. In this case, skyrmionic cocoons appear slightly elongated in the horizontal plane (at 63 mT), a shape later on called partial worms as they reside in each SG block.

Those partials worms are being studied in more details by some of our collaborators, namely Riccardo Battistelli, a PhD student from the Helmholtz-Zentrum Berlin für Materialien und Energie in Germany.

Going down to 50 mT, on the other side of the transition, worms of negative m z (red) have nucleated (IV), disturbing the cocoons distribution 1 . The step in the magnetization curve near 60 mT is thus linked to the formation of domains in the PMA layers that propagate over the 1 The tip also interacts with the magnetization as it appears to cut off the red worms near the center. full thickness (3D worms). Interestingly, cocoons appear to nucleate inside the newly formed red domains, thereby having a reversed polarity from the ones in the blue domains to reduce the stray field. Continuing toward remanence, the worms extend more and more in the plane but many cocoons remains stable (V, VI).

Coexistence of skyrmions and cocoons

The presence of two contrasts in the previous MFM study (Figure 5.2, Figure 5.3) is directly linked to the stabilization of two kinds of objects with different stray field. With the support of micromagnetic simulations, they can be identified as:

• Domains vertically extending through all the magnetic layers (strong contrast) that we will call 3D worms or stripes. With increasing field, they evolve into columnar skyrmions.

• Skyrmionic cocoons with a limited vertical extension (low contrast).

Moreover, decreasing the field from saturation shows that the nucleation of domains in the strong PMA layers, i.e. 3D worms, occurs at the jump visible in the hysteresis loops. This informs about the magnetic phase diagram.

Importance of the magnetic history

The DG structure ability to host different magnetic textures at remanence strongly depends on its magnetic history. In

In-plane field

Lastly, the response to an in-plane field has been also investigated to check the resiliency of both magnetic textures as shown on Figure 5.5. At remanence, we can observe that the distribution of up and down cocoons is not at equilibrium as they are more numerous in the darker domains. At 100 mT, the domains are still defined along with the cocoons. However, at 200 mT, the density of cocoons has dropped even though there are still distinguishable. The 3D worms also start to rotate in-plane as evidenced by the domains walls which lose their sharp boundaries. Note that the tip magnetization might also rotate in-plane which could contribute to this smoothing effect. At higher fields, 300 mT and 400 mT, no clear pattern is visible even though some features are still present. Note that 400 mT is not enough to fully saturate the magnetization in-plane (see hysteresis in Figure 5.1.b). Lowering the magnetic field from this field value allows to (re)nucleate skyrmionic cocoons around 190 mT with a faint contrast. At remanence, the usual mixed state of 3D worms and cocoons is stabilized but with fewer cocoons than after an OOP saturation.

Thickness optimization of the DG structure

After having confirmed the existence of various spin textures in our intricate aperiodic multilayers, it becomes crucial to investigate how changes in structural parameters affect their stability and resilience under an external magnetic field. Due to the vast number of degrees of freedom (X 1 , S, Y , N , M , t N Pt and t M Pt ), it is impractical to examine the entire parameter space. However, we aim to gain a qualitative understanding of the influence of each parameter. To this end, we study the field dependency of different structures using MFM imaging.

Range of thickness

Let us first consider the impact of the thickness parameter X 1 , the thickness of the bottom layer in the SG blocks, which was already probed in SG structure (Subsection 4.1.2). The parameters are fixed according to the following table, changing only the thickness parameter. This strongly impacts the anisotropy of the magnetic layers as well as the dipolar interaction and the DMI which leads to different categories of behaviors [START_REF] Legrand | Spatial extent of the Dzyaloshinskii-Moriya interaction at metallic interfaces[END_REF]. In Figure 5.6, MFM images near remanence are presented for X 1 ∈ [1.4, 2.2] nm after an OOP saturation. Apart for 1.4 nm, they all display double contrasts that correspond to 3D worms and skyrmionic cocoons. However, for 1.6 nm, the density is sensibly lower than for higher thicknesses.

X 1 (nm) N (repetitions) S (nm) Y (nm) M (repetitions) t N Pt (nm) t M Pt (nm) 1.4 → 2.2 13
Additionally, increasing the thickness parameter up to 2.2 nm leads to an elongation of the skyrmionic cocoons as some of them appear to extend horizontally along the 3D worms direction. Moreover, in this case, the populations of up and down cocoons seem at equilibrium which is not the case for lower X 1 (more white dots in the blue domains).

As expected from the widely different remanent states, the field evolution displays different behaviors depending on the thickness parameter X 1 . For X 1 = 1.4 nm (not shown), some double contrasts can appear with the field but no clear range of stability emerges for the cocoons. In Figure 5.7, the field evolution is presented for the remaining cases. Note that the measurements require a change of field module so that the magnetic states might not be recognizable from one state to the other. For 1.8 and 2.0 nm, it follows the same trend described in Subsection 5.1.2: with increasing magnetic field, the 3D worms contracts to columnar skyrmions before splitting into two cocoons at high field. Then, for X 1 = 2.2 nm, the two cocoons populations are still stable are relatively high field, for instance near 64 mT. Increasing the field makes the 3D worms shrink before disappearing (at lower field than for its thinner counterparts), leaving only cocoons at 170 mT. After saturation and upon the decrease of the field, cocoons start to nucleate first with a weaker contrast, indicating a stronger vertical confinement.

For X 1 = 1.6 nm, the cocoons are the first to disappear: at 140 mT, mostly 3D worms are remaining. At 240 mT, it appears that a dominant population of columnar skyrmions is stabilized, without coexistence with cocoons. After saturation, going down to 170 mT, smaller objects with a slightly lower contrast are nucleated, likely to be skyrmionic cocoons. Thus, except at low field, the coexistence of columnar skyrmions with skyrmionic cocoons is not observed.

To ensure that the textures nucleated after saturation are indeed cocoons for X 1 = 1.6 nm, a minor loop can be investigated. This also allows to stabilize columnar skyrmions along with skyrmionic cocoons which makes this measurement an effective method to probe the capacity of the multilayer to allow their coexistence. The results are shown in Figure 5.8 with MFM images along with the corresponding AGFM measurements illustrating the field evolution and its impact on the magnetization.

At 230 mT (state II), only columnar skyrmions are remaining. By lowering the field at 170 mT from this state, smaller objects nucleate next to them with a weaker contrast, evidencing the coexistence of two populations. At 200 mT (state IV), the cocoons are still stable in this minor loop but they disappear when coming back to 230 mT. Note that we can identify the same columnar textures from one frame to another. This minor loop allows to obtain cocoons at higher field and simultaneously with columnar skyrmions. The MFM images are recorded at different magnetic fields, as indicated on the zoom of the hysteresis recorded with AGFM.

Following the study of those different samples, it evidences that skyrmionic cocoons can be stabilized over a large range of thickness even though the field dependency might vary.

Impact of t N P t

The platinum thickness plays a dominant role in controlling many of the magnetic properties that are of interest to nucleate and move the spin textures. Indeed, it can impact the anisotropy, the DMI, the torque efficiency and indirectly the dipolar interaction between two separate magnetic layers. In the DG structures, different thicknesses have been probed, between 2 and 4 nm, to check the impact on the magnetic properties. The rest of the parameters are fixed according to the following table.

Note that X 1 = 1.7 nm which is relatively low to obtain coexistence of cocoons and skyrmions (for t N Pt = 3 nm) as evidenced in the previous discussion.

X 1 (nm) N (repetitions) S (nm) Y (nm) M (repetitions) t N Pt (nm) t M Pt (nm) 1.7 13 0.1 0.9 15 2 → 4 3
In Figure 5.9, the field dependency measured for three DG with different t N P t is presented with representative magnetic states. For 3 or 4 nm, the trend is similar with cocoons stable near remanence with 3D stripes but the former disappear first when increasing the magnetic field. Similarly, the coexistence can be obtained at higher field by entering the minor loop as shown with the last right image. For 2 nm of platinum, the cocoons are more stable and coexist with columnar skyrmion with a monotonous increase of the magnetic field. When decreasing the magnetic field from saturation, only cocoons nucleate at first. For the range of Pt thicknesses explored (above 2 nm), the effective anisotropy should not vary significantly [START_REF] Ajejas | Densely packed skyrmions stabilized at zero magnetic field by indirect exchange coupling in multilayers[END_REF] nor the DMI and the saturation magnetization. Thus, only the dipolar interaction is significantly impacted, as it varies with the Pt thickness since the distance between the magnetic layers is changed. With a thinner spacer, the strong dipolar interaction helps to stabilize skyrmionic cocoons on larger range of magnetic field. The platinum role is thus complementary to the X 1 cobalt thickness, allowing us to keep one parameter fixed while modifying the others to achieve the desired properties.

Number of strong PMA layers

Continuing with the same structure (see table below), the number of strong PMA layers M can be modified to identify on which range cocoons can be stabilized.

X 1 (nm) N (repetitions) S (nm) Y (nm) M (repetitions) t N Pt (nm) t M Pt (nm) 1.7 13 0.1 0.9 5 → 15 3 3
In Figure 5.10, the field dependency for M = 5, 10, 15 is studied. Noticeably, for the two highest M number of repetitions, cocoons can be stabilized at remanence and after a saturation they nucleate first at high field. However, for M = 5, no cocoons are detected even after saturation. Additionally, the stabilized magnetic textures appear to be larger than for higher M values. Hence, a minimum number of strong PMA layers is required to facilitate the stabilization of cocoons, and it is strongly influenced by the remaining parameters. It is also expected that there is an equilibrium to respect between the number in the SG blocks N and M .

Changing the thickness step

In the previous chapter, the thickness step S was highlighted for its role in the vertical confinement of the cocoons. It thus follows to verify its impact on the stabilized spin textures in the DG structure. down still appear to be stable. However, at 180 mT, it seems that only the columnar structures are stable even though some weaker white contrast is still visible that could be attributed to confined cocoons. On the second line of Figure 5.11, the case S = 0 nm, i.e. for uniform thickness of the Co (2.3 nm here), is displayed. It shows a field dependency that resembles the typical DG structure. The thickness gradient is thus not essential to obtain double contrast but still provides with a stronger confinement (based on the SG study presented in Subsubsection 4.1.3.1) and thus less hybrid chirality. Finally, for S = -0.1 nm, a few cocoons are observed but the minor loop process is not enough to stabilize them efficiently which limits the interest of this configuration.

X 1 (nm) N (repetitions) S (nm) Y (nm) M (repetitions) t N Pt (nm) t M Pt (nm) 1.7 → 2.
Thus, similar results can be obtained for S = 0, 0.1, 0.2 nm but the vertical confinement may vary in consequence.

Reducing the number of layers

As already evidenced with the previous studies, many structures can be envisioned based on the parameters available. To study the dynamics and for future applications, it is however interesting to reduce the overall number of layers. This would make growths more reliable and it might provide with better current efficiency for moving spin textures. Based on the previous conclusions, many architectures can be envisioned to stabilized cocoons and skyrmions with fewer layers. In Figure 5.12, a few examples are given that successfully reach these aims (see table for their structure). As a first step in the decrease of the total number of layers, the architecture I with 30 magnetic layers allows to obtain double contrasts at remanence. As X 1 is relatively low, the cocoons disappear first with the field. However, they can still be stabilized after saturation and lowering the field at 185 mT for instance. The second sample (II) with only 17 repetitions also yields satisfying results

Number X 1 (nm) N (repetitions) S (nm) Y (nm) M (repetitions) t N Pt (nm) t M Pt (nm) I 1.7 9
as the two types of objects can be observed, even though not at the same magnetic field. Finally, the third sample with the most drastic decrease of layers, only 15 layers, and with a zero thickness step, is promising. Indeed, at remanence, 3D worms with cocoons are observed and at 220 mT, a mixed state with skyrmions and cocoons in high density is obtained. Going down from saturation, at 180 mT, columnar skyrmions seems to have nucleated which is a different behavior than the thicker samples. Those examples evidence the possibility to stabilize multiple textures even with a reduced number of layers (by more than a factor 2). It also put forward the many different magnetic properties that can be obtained by changing the structure parameters.

DG optimization

The tunability of the DG structure offers many different behaviors and renders complex the exploration of the parameter space. Varying one parameter at a time yields qualitative understanding of their impact and put forward large parameters ranges that allow to stabilize skyrmionic cocoons along with columnar spin textures.

• The thickness parameter X 1 is an extremely important parameter that determine the uniaxial anisotropy and dipolar interection which play a dominant role in the stabilization of the cocoons and skyrmions. An in-plane anisotropy seems to favor the stability of skyrmionic cocoons at remanence and under an external magnetic field.

• The platinum thickness in the gradient parts t N P t is complementary to the role of X 1 regarding the anisotropy. However, its role in the torque efficiency, the dipolar interaction and the X-ray absorption needs to be taken into account to chose its thickness. We will thus often keep it fixed at 2 or 3 nm.

• The number of strong PMA layers M determines the ability to obtain textures at different vertical levels. Intuitively and experimentally, we find that there is a lower threshold to split the structure in different magnetic areas which strongly depends on the overall number of layers.

• Those conclusions were drawn for a fixed number of magnetic layers in the gradients (N = 13).

However, they can be generalized to thinner structures, making possible to stabilize skyrmionic cocoons and columnar skyrmions with a reduced number of layers (down to N = 5).

Exploring other multilayer architectures

For the sake of completeness, other structures making use of abrupt changes in the magnetic anisotropy have also been investigated. They also allow the stabilization of different spin textures.

Encapsulated gradient

Split gradient

Finally, we present the structure called the split gradient that is made of a single gradient cut in half by M strong PMA layers. A schematic of the multilayer is shown in Figure 5.14 along with MFM images corresponding to different M . For each M values (between 5 and 15), the stable states display two kinds of contrasts that corresponds to 3D worms and skyrmionic cocoons. It is reminiscent of the DG structure with fewer layers even though the thickness distribution is different. The OOP hysteresis Those two structures were not studied in more depth, but they also demonstrate the ability to stabilize different magnetic textures and they show the malleability of our approach.

Probing 3D textures with XRMS

To investigate qualitatively the depth profile of the magnetic states stabilized in DG multilayers, X-ray resonant magnetic scattering experiments have also been performed. The technique and its interpretation have already been broached in (Section 4.3). First, we discuss the expected chirality obtained from the simulations depending on the magnetic history before presenting the complex signals obtained experimentally.

Chirality from micromagnetic simulations

The structure under consideration corresponds to the parameters: S = 0.1 nm, X 1 = 2.0 nm, N = 13 layers, M = 15 repetitions, Y = 1 nm, t N Pt = 2 nm and t M Pt = 3 nm. The numerical spacer is again set to 3.4 nm, the sum of t N Pt = 2 nm and t Al = 1.4 nm.

After IP saturation

As evidenced in Figure 5.4, the magnetic states at remanence are strongly influenced by the magnetic preparation of the sample. After an in-plane saturation, only 3D stripes are stabilized throughout all the magnetic layers (see plane and thus evidence the winding in each of the layers. In the gradient parts, the observations are similar to the SG case: the chirality is CCW Néel (resp. CW Néel) in the bottom (resp. top) layers. In between, a few layers present non-chiral domain walls (spins pointing uniformly in the same direction).

In the strong PMA layers, only CCW Néel domain walls are stabilized. This facilitates the flux closure with the top gradient but not the bottom one. With the changes of chirality between the bottom SG block and the strong PMA layers, spins pointing in opposite directions located in two adjacent layers can be stabilized (see green circle). This is allowed due to the discontinuity of the magnetization and because the layers are only coupled by a dipolar interaction.

Note also that the domain wall width parameter ∆ in the strong PMA layers appears much thinner than in the gradient parts. Assuming a hyperbolic tangent wall profile, the width π∆ is proportional to3 A/K u [START_REF] Thiaville | Domain wall dynamics in nanowires[END_REF]. As the uniaxial anisotropy is expected to be much larger in the strong PMA layers, it follows to have thinner domain walls. Numerically, the domain wall width obtained from hyperbolic tangent fits is close to 13 nm in the middle layers while it is larger than 30 nm in the rest. Using A = 18 pJ/m and K u = 1580 kJ/m 3 , the resulting domain wall width is 10 nm for the strong PMA layers. For the SG blocks, using K u = 686 kJ/m 3 (considering an average thickness of 2.3 nm), yields a larger width of 16 nm. The discrepancy for the latter value might be attributed to the simplicity of this 2D model, not representative of our complex aperiodic multilayer. Because of those thin domain walls, the magnetization can rotate over relatively short lengths, especially in the middle layers. However, the maximum rotations between neighboring voxels remain below 30 • , a reasonable threshold for accurate simulations [START_REF] Vansteenkiste | The design and verification of MuMax3[END_REF]. Moreover, the simulations have been checked by using voxels with smaller lateral dimension (1 × 1 nm 2 ), which did not yield any significant difference (see Appendix A).

After OOP saturation

A similar study is performed in a DG after an OOP saturation. In this case, cocoons are stabilized along with 3D stripes (Figure 5.

Field evolution of the topological number

To complement this evolution of the chirality, the topological number N sk has been computed in all the layers at different OOP magnetic fields from micromagnetic simulations (similarly to Figure 4.11 for the SG structure). The resulting map and representative magnetic states are presented in Figure 5.17.

Starting from positive saturation, skyrmionic cocoons start to nucleate at 325 mT while the strong PMA layers are still saturated. As the chirality is well-defined in the bottom and top layers of each cocoon, it yields a non-zero topological number. The layers in between, for instance 7 to 9, which have non-chiral domain walls (spins pointing uniformly in the same direction), are topologically trivial. This trend is valid as the field is decreased, causing a vertical extension of the cocoons in the top layers of each gradient. At 75 mT, a single domain nucleates in the middle layers and since it has a CCW Néel chirality, N sk = -1 (area in dotted black square). At negative field, the sign of the topological number switches due to the core inversion but the conclusions are the same.

Investigating a stripe state

The DG multilayer under consideration (S = 0.1 nm, X 1 = 2.0 nm, N = 13 layers, M = 15 repetitions, Y = 1 nm, t N Pt = 2 nm and t M Pt = 3 nm) has been demagnetized in-plane before the measurements. This stabilized 3D worms that extend over the full thickness. In Appendix F, results are presented for a second multilayer measured after a similar preparation.

Setup specifications (BOREAS)

The following measurements have been performed at the BOREAS beamline located at the ALBA synchrotron (Spain). There, the 2048 × 2048 pixels cooled CCD camera is located at 40.2 cm from the sample, covering an angular range of 2.1 degrees, with a pixel size of 13.5 µm. It records the signal coming from the MARES diffractometer [START_REF] Barla | Design and performance of BOREAS, the beamline for resonant x-ray absorption and scattering experiments at the ALBA synchrotron light source[END_REF] in ultra-high vacuum conditions with a beam of 50 µm diameter. Thanks to the undulators, a polarization efficiency for circular or linear polarization superior to 99% can be reached. For DG multilayers, we need to be sensitive to the Co and thus will place the photons energy near the L 3 Co edge (at 777 eV). Note that the energy calibration is not exactly equivalent to the SEXTANTS beamline at SOLEIL (up to several eV shift).

Reflectivity curves

Several reflectivity curves have been acquired at different photons energy (Figure 5.18). For both energies, the curves do not display any periodicity (no periodic oscillations) as expected from the complex structure of the multilayer. Far from the resonance, oscillations are however visible up to high incidence angles. Near resonance, at 650 eV, the curves are smoother and no clear XMCD effect is visible between the two helicities.

When changing the energy, even by a slight amount, the penetration depth can change significantly and thus the X-ray can probe entirely different layers [START_REF] Flewett | General treatment of off-specular resonant soft x-ray magnetic scattering using the distorted-wave born approximation: numerical algorithm and experimental studies with hybrid chiral domain structures[END_REF].

Multiple dichroic inversions

XRMS chirality

Note that the referential at the BOREAS beamline is rotated with respect to the one of the SEXTANTS beamline. Thus, to respect the previous convention, the zero of the azimuthal angle ψ is set on the right end of the scattering pattern, as illustrated on Figure 5.19 with a clockwise rotation. Thus, all patterns are rotated by 90 • (see below). The sample was oriented to have the stripes parallel to the beam. Therefore, in Fourier space, only two points are expected at wave vector ±2πp, with p the stripes periodicity. Following the reflectivity measurements, different images were recorded at 777 eV at various incidence angles as shown in A strong evolution of the background can also be noticed, showing vertical stripe-like variations at θ = 18.5 • for instance. This cannot easily be explained without further numerical simulations.

To quantitatively analyze the evolution of the signal, the radial integrals have been performed along a ring-shaped mask ( The majority of the previous scattering patterns display either a well-defined chirality (either CW or CCW Néel) for low and intermediate incidence angle or no dichroism at high incidence angle. However, more unusual patterns arise for some incidence angles such as θ = 7.75 • (or 14.5 • ). Indeed, in this case the signal exhibits four nodes with alternating sign (see Figure 5.22.a) that do not correspond to any known pattern for the chirality. For a 7.75 • incidence angle, the two rights nodes also appear more contrasted than the left ones. To verify that this signal is not an artefact, the sample was rotated at ±30 • which yielded non-dichroic signal (either fully blue or red).

To understand this evolution, we can consider the simple and ideal example presented in Figure 5.22.b.

It illustrates three different organizations of magnetic states for which the domain wall chirality is assumed to be CW Néel. For a disorganized state, i.e. a pattern with randomly propagating worms, the scattering pattern yields the expected circular dichroic signal. There are sinusoidal modulations of the intensity of the ratio with ψ (see Figure 4 Let's consider that the magnetization presents stripes that are parallel to the wave vector k. This situation is equivalent to taking a cut along the vertical dotted line of the first pattern: it yields two dichroic dots with a maximum intensity. If the sample is rotated by 90 • (stripes ⊥ k), the cut is now horizontal and located along the minima of the signal due to the sinusoidal modulations. Thus, even though the domain walls are chiral, the resulting intensity will be close to zero. In a way, changing the stripes direction probes the scattering pattern of the disorganized states at various ψ angles. For the measurement at θ = 7.75 • , we can extrapolate the signal of the worm state using the previous considerations as the magnetic state corresponds to stripes. Based on the variations of the asymmetric ratio, a schematic four quadrants signal is presented in Figure 5.22.c. The vertical cut corresponds to the stripes aligned with the wave vector, yielding the four nodes observed experimentally.

Taking a cut after rotating the sample by 30 • (equivalent to rotating ψ by the same angle), we obtain the same two blue nodes signal. The schematic signal thus appears to fit with the experimental data.

However, there is no simple explanation to understand its origin. Indeed, it does not correspond to any expected chirality and also displays an asymmetric distribution of the intensity (between the left and right hemisphere). It most likely arises from the complex distribution of the magnetization over the vertical axis (different chirality, domain wall widths...). Numerical support would be needed to understand in more depth this phenomenon which is why a collaboration is starting with Samuel Flewett (DESY synchrotron, Hamburg, Germany). Additional measurements in energy at this incidence angle are provided in Appendix F.

Cocoons and 3D worms: a multinodal signal

The same DG structure has been studied after a different magnetic history. In the previous section, only stripes were stabilized thanks to an IP demagnetization, yielding multiple inversion of the dichroism with the incidence angle. Following this, the DG was demagnetized with an OOP field to obtain a mixed state of 3D worms and cocoons (see inset of Figure 5.23.a). Similarly, it was investigated at different incidence angles, for ℏω = 777 eV. Some scattering patterns are displayed in Figure 5.23.a, evidencing the complexity of the signal. At low incidence angles, θ = 7.65 • , the multinodal signal discussed previously appears again, with less asymmetry in the intensity (between the right and left hemisphere). Increasing the incidence angle yields a circular pattern that corresponds to CW Néel chirality. At 18 • , the signal shows multiple variations that could be seen as the superposition of two scattering patterns with different chirality. This might arise from the two different magnetic populations stabilized at remanence. However, from the MFM image, a smaller periodicity (and thus a larger ring) would be expected for the cocoons than for the 3D worms. Once again, it is thus not straightforward to understand. At slightly larger incidence angle (18.3 • ), the ring has disappeared, leaving two nodes of opposite sign with CCW Néel chirality.

The full signal map is shown in Figure 5.23.b. At low angles, below 21 • , the quadrupolar signal is measured regularly while it disappears at higher incidence angles. Note that important variations can be obtained with change in θ as small as 0.1 • so that we cannot expect to probe the full spectrum.

However, qualitatively, the map still displays several inversions of the chirality (following the dotted lines at ψ = 90, 270 • ). Further numerical studies are needed to quantitatively understand these scattering measurements. The energy dependency of the signal is discussed in Appendix F. Additional data are also provided, notably XRMS measurements on DG structures with S = 0.2 nm that demonstrate an even more complex signal (up to 6 nodes visible).

Chapter 5 Summary

The double gradient (DG) structure offers the capability to stabilize together skyrmionic cocoons as well as textures that span the entire thickness of the multilayer. These observations are supported by the correlation between MFM imaging and corresponding micromagnetic simulations for a specific structure. The resilience of cocoons is demonstrated through the investigation of field dependencies, both in-plane and out-of-plane. Furthermore, the magnetic history of DG structures is found to be crucial in stabilizing different magnetic textures. A comprehensive experimental study of the structure parameters (thicknesses, number of layers, etc.) provides a qualitative understanding of their impact.

It reveals two main behaviors: the first behavior allows cocoons to be stabilized at remanence, and they prove to be more stable than 3D worms and skyrmions when increasing the magnetic field. In the second behavior, the opposite trend is observed, but cocoons can still be stabilized at high field by entering a minor loop. Importantly, the experimental study emphasizes the significance of having a strong in-plane anisotropy and dipolar field for cocoon stabilization, while also demonstrating that it can be achieved across a wide range of parameters.

Building upon this preliminary study, XRMS measurements have been conducted on the DG structures to address the limitations of MFM, which primarily captures information from the top layers with no information about the chirality. A specific DG structure has been examined after two different magnetic preparations, both of which exhibited multiple chirality inversions, as predicted by micromagnetic simulations. However, many recorded patterns are too intricate to be simply understood as they display more than two nodes. Further numerical analysis will be necessary to comprehend the origin of this multinodal signal.

In summary, this chapter indicates that DG multilayers have the ability to stabilize two types of magnetic textures with different vertical extensions, although resolution along the vertical direction is still lacking. This coexistence could prove advantageous for future applications such as the racetrack, where each bit value can be encoded by a specific magnetic entity. the rapidly decaying stray field. To complement and probe further the magnetization over the total thickness of the multilayers, we perform magnetotransport measurements, allowing to probe all the magnetic layers. The experimental curves of longitudinal and transverse magnetoresistances are in excellent agreement with the ones that can be calculated using the micromagnetic simulations as input parameters for the magnetic configuration inside the multilayers. These results corroborate the existence of 3D confined cocoons. Moreover, in devices of reduced dimensions, the magnetotransport approach shall permit to record the signal corresponding to a single quasi-particle, enabling hence their electrical detection

Magnetotransport measurements

Probing the average magnetization

To investigate the existence and structure of cocoons, in addition to the MFM imaging, we perform magnetotransport measurements in Hall bar devices patterned in the previously studied multilayers.

We measure both the longitudinal resistance R xx and the transverse resistance R xy while sweeping the magnetic field along different orientations. In order to fit these electrical measurements, a simple model taking into considerations several observed contributions to the magnetoresistive effects has been considered [START_REF] Maccariello | Electrical signature of noncollinear magnetic textures in synthetic antiferromagnets[END_REF]. The current is injected along x, the film normal is along z, θ corresponds to the angle between z and the field while the angle ϕ is between x and y (see the schematic in Figure 6.1.a).

For example, θ = 0 • corresponds to an out-of-plane field while for θ = 90 • and ϕ = 0 • the field is lying in-plane in the direction of the track x. A first component to the magnetoresistive signal is related to the presence of heavy metallic layers in contact with the magnetic films. They induce a spin Hall effect (SHE) in the heavy metal that gives rise to the spin Hall magnetoresistance (SMR) [START_REF] Sinova | Spin Hall effects[END_REF][START_REF] Dang | Anomalous Hall effect in 3 d/5 d multilayers mediated by interface scattering and nonlocal spin conductivity[END_REF], which depends on the angle between the magnetization m of the ferromagnet and the spin polarization due to the SHE [START_REF] Chen | Theory of spin Hall magnetoresistance[END_REF], [START_REF] Kobs | Anisotropic interface magnetoresistance in Pt/Co/Pt sandwiches[END_REF]. The change in R xx due to the corresponding SMR is thus proportional to m 2 y . The second contribution is associated to the anisotropic magnetoresistance (AMR) [START_REF] Mcguire | Anisotropic magnetoresistance in ferromagnetic 3d alloys[END_REF], that is proportional to m 2

x , i.e. when the magnetization is parallel to the current. Finally, the evolution of the transverse resistance is governed by the anomalous Hall effect (AHE) [START_REF] Dang | Anomalous Hall effect in 3 d/5 d multilayers mediated by interface scattering and nonlocal spin conductivity[END_REF], [START_REF] Nagaosa | Anomalous Hall effect[END_REF], that is proportional to the out-of-plane component of the magnetization: R xy ∝ m z . From those considerations, R xx and R xy can be written as:

R xx = R 0 xx + R SMR m 2 y + R AMR m 2 x (6.1) R xy = R 0 xy + R AHE m z (6.2)
with R 0 xx the longitudinal resistance for |m z | = 1 and R 0 xy an experimental offset. The coefficients R AMR , R SMR and R AHE are the proportionality coefficients of the associated effects whose are experimentally determined in the following. Note that the contributions related to giant magnetoresistance (GMR) and the planar Hall effect (PHE) are negligible and thus not considered. Moreover, as we found in previous work [START_REF] Maccariello | Electrical detection of single magnetic skyrmions in metallic multilayers at room temperature[END_REF] that the topological Hall effect (THE) contribution in standard skyrmion multilayers was negligible compared to AHE, we do not include this contribution either. Finally, we found that the Lorentz magnetoresistance contributing to R xx and the normal Hall effect contributing to R xy were withing the noise of the measurements (about 5 mΩ) so that they were also neglected (see Appendix E).

Transport simulations

To correlate with the experiments, we have performed micromagnetic simulations using the same field cycle, with increments of 25 mT. The micromagnetic parameters specified in Table 3.4 were used in a simulation space of 1.024 × 1.024 µm 2 (512 × 512 cells) with periodic boundary conditions. In order to facilitate the formation of cocoons, the simulation at for the first field branch was conducted at a temperature of 500 K for a duration of 2 ns, using a Gilbert damping parameter α = 0.01. For the other field branch, since the magnetic objects are already stabilized, the magnetization is simply allowed to relax. For the DG, domain nucleation in the strong PMA layers could not always be achieved by solely relying on the temperature or the magnetic noise.

It was thus artificially induced by implementing one skyrmion (radius of 15 nm) in the middle of the simulation space in the strong PMA layers and letting relax at different fields near the experimental transition.

To compare with R xy , the vertical component of the magnetization for each magnetic layer i was weighted by its thickness t i relative to the total thickness t tot . This weighting is necessary since the AHE scales linearly with the volume of the ferromagnet in a first approximation (the Pt layers can contribute due to the spin polarization induced by SHE upon the injection of current [START_REF] Dang | Anomalous Hall effect in 3 d/5 d multilayers mediated by interface scattering and nonlocal spin conductivity[END_REF]). Therefore, we calculate the following expression:

R Sim xy = 1 t tot N i=1 ⟨m z ⟩ i × t i (6.3)
with ⟨m z ⟩ i the average of m z over all cells in layer i. Likewise, for R xx , we compute the average of the squared in-plane components over the thickness. However, for the SMR effect, we do not employ weighted contributions as this contribution is of interfacial origin and therefore should not rely on the layer thickness. Hence, the expression we obtain is as follows:

R Sim xx = R 0 xx + R SMR ⟨m 2 y ⟩ + R AMR 1 t tot N i=1 ⟨m 2 x ⟩ i × t i (6.4) with ⟨m 2 x ⟩ i the average of m 2
x over all the cells in layer i, ⟨m 2 y ⟩ the average of m 2 y over all simulation space.

Fabrication of the tracks

In order to measure the magnetoresistance in our aperiodic magnetic multilayers, Hall bar devices have been patterned using UV lithography by projection. After exposition and development of the resist, the unprotected materials of the magnetic multilayer are etched via Ion Beam Etching (500 V, 100 mA, 5 • C, 5 • incidence) coupled with Secondary Ion Mass Spectrometry (SIMS) to stop at the appropriate layer. An example of devices is shown in Figure 6.1.b. The voltage probes used to measure the resistance are typically 5 µm wide. Aluminum contacts are made directly on the multilayer itself.

To limit the effect of edge effects and shape anisotropy, the tracks are typically large, at least 20 µm wide.

In the following, the SG multilayer under study corresponds to X 1 = 1.7 nm, N = 13 layers, S = 0.1 nm and t N Pt = 3 nm. It has been patterned as a track of 30 × 150 µm 2 . For the DG, the structure is defined by S = 0.1 nm, X 1 = 2.0 nm, N = 13 layers, M = 15 repetitions, Y = 1 nm, t N Pt = 2 nm and t M Pt = 3 nm. In this case, the track is 20 × 100 µm 2 . Typically, a current of 100 µA was used.

Measuring the resistance coefficients

In the R xx model introduced previously in Eq. 6.1, two coefficients appear that are associated with various magnetoresistance effects, namely R SMR and R AMR , as well as an experimental offset R 0 xx . To determine them experimentally, the longitudinal resistance can be measured while varying θ with a magnetic field high enough to ensure saturation of the magnetization. In order to isolate each effect, the magnetic field rotates either in the (yz) plane or in the (xz) one which keeps one of the in-plane component close to zero and thus nullify a term. For instance, let's consider the (yz) plane (ϕ = 90 • ). When the field is pointing out-of-plane (θ = 0 • ), the magnetization is only along z, hence m 2

x = m 2 y = 0. Similarly, when the field is in-plane (θ = 90 • ), the magnetization points along y so that m 2

x = m 2 z = 0. The difference ∆R xx (ϕ = 90 • ) between those two values thus yields R SMR , as follows:

   R xx (θ = 0 • , ϕ = 90 • ) = R 0 xx R xx (θ = 90 • , ϕ = 90 • ) = R 0 xx + R SMR =⇒ ∆R xx (ϕ = 90 • ) = R SMR (6.5)
A similar reasoning can be held in the (xz) plane which isolates the R AMR constant. In Figure 6.2, R xx (θ) are presented in both planes for the SG (a) and the DG (b) under consideration. The arrows evidence how the constants are determined. Note that the scale for each curve is different. In both cases, the signal is maximum for an OOP field (θ = 0 • ) and decreases when some in-plane components appear. Thus, both the AMR and the SMR effects have negative contributions.

The constants extracted from the previous measurements are summed up in Table 6.1. It evidences that the SMR effect is larger than the AMR in both structures, by a factor 10 for the SG and 6 for the DG. For the resistance R 0 xx , some variations are observed depending on the measurements as they were performed on two different tracks on the same sample. For R xy , the determination of the coefficients is more straightforward as only the AHE contribute to its evolution. Thus, measuring a field cycle with an out-of-plane field (θ = 0 • ) allows to determine R AHE as half the total amplitude and R 0 xy as the offset. The experimental determination of those coefficients allows to predict magnetoresistances values from micromagnetic simulations and to compare them with the measurements.

Structure R SMR (mΩ) R AMR (mΩ) R 0 xx (Ω) SG -252 -25 ≈ 36 DG -108 -18 ≈ 12.93

In a single gradient multilayer

We first focus on the study of the single gradient (X 1 = 1.7 nm, N = 13 layers and S = 0.1 nm).

To this end, R xx and R xy are measured while sweeping the magnetic field at fixed (θ, ϕ) and compared to the output of micromagnetic simulations.

SG transversal resistance R xy

The transversal resistance R xy (H) is measured in the (yz) plane every 30 • . Note that as it is only proportional to m z , it does not depend on the selected ϕ (90 • here). The experimental curves are presented in Figure 6 For instance, for an out-of-plane field, it ranges between ±1 as expected. For θ = 60 • , at saturation, m = sin 60y + cos 60z so that the component along z reaches ±0.5, similarly to the corresponding blue curve.

SG longitudinal resistance R xx

Complementary measurements can be performed for R xx . As it is sensitive to the in-plane components of the magnetization, the component of the field along x or y plays an important role which is why we measured R xx for ϕ = 0 • and 90 • . The results are presented in Figure 6.4 along Using a field with in-plane components impacts the magnetization and thus the resistance with two different mechanisms: the field can uniformly tilt the magnetization or it can transform the pre-existing spin textures. The former effect will be most visible at high field with a saturated magnetization while the latter affects the magnetization once nucleation events have occurred. At large field, the magnetization lies partially in-plane along the magnetic field orientation. For θ = 30 • , when the field is lowered, the resistance decreases smoothly as m uniformly tilt towards y, thus increasing the SMR contribution. This is verified as ⟨m 2 y ⟩ increases while ⟨m 2 z ⟩ decreases as shown in Figure 6.5. For θ = 60 and 90 • , ⟨m 2 y ⟩ remains fairly constant with the field which does not impact significantly the SMR, hence the plateau near saturation. The transitions near 150 mT correspond to the nucleation of up and down textures, either cocoons for θ = 30 • or stripes for the other θ angles (see relaxed states in Appendix E). This provokes a loss in the in-plane components, notably in m y , and thus an increase in the resistance. The maximum resistance is reached at remanence when the only-in plane components are in the DWs. It is interesting to notice that the three curves end up with the same resistance at zero field, different than θ = 0 • , attesting that another magnetic state is stabilized (stripes or worms for θ > 0 • and 3D skyrmionic or 3D worms textures for an out-of-plane field).

• ϕ = 0 • and θ > 0 •
The trend in the (xz) plane has a reduced amplitude compare to the (yz) plane. Its evolution is also strikingly different with a sharp minimum at remanence. When the nucleation happens, a drop in the resistance is measured. Similarly to the previous case, this causes a drop of the m x component1 (that is not compensated by the DW contribution). This should cause an increase of the resistance since the AMR has a negative contribution. However, the domain walls also provoke the apparition of m y components which are linked to the SMR effect with an amplitude ∼10 times larger than the AMR (see Table 6.1). This thus overcomes the positive gain arising from the loss of m x and decreases the overall resistance. This effect is again maximized at remanence. Interestingly, in Figure 6.5, for θ > 0 • , we can notice that the average ⟨m 2 y ⟩ in the layers 7 to 11 is non-negligible and higher than the ⟨m 2

x ⟩ contribution. This corresponds to the layers for which the domains wall are usually non-chiral, but in this case the in-plane field orients the domain walls in the y direction, thus disymmetrizing the in-plane contributions.

Additional transport measurements are presented in Appendix E. Those magnetotransport measurements coupled with the corresponding micromagnetic simulations allow us to probe all the magnetic layers and to understand the evolution of the magnetic textures with the field.

In a double gradient multilayer

A similar study has been led on the DG structure (S = 0.1 nm, X 1 = 2.0 nm, N = 13 layers, M = 15 repetitions, Y = 1 nm, t N Pt = 2 nm and t M Pt = 3 nm) to get a better understanding of the field dependency.

DG transversal resistance R xy

In Figure 6 Starting from high positive field, the behavior is the same as the one introduced in Chapter 5.

First, cocoons nucleate in both the gradient parts and they extend in all directions upon lowering the magnetic field. Near 75 mT, domains nucleate in the middle layers which lower the average m z , hence the sharp jump in R xy . It corresponds to the 3D worm visible in the simulations at remanence that extend over the full thickness. Going to negative field, the cocoons remain stable while the 3D worms shrink and transition to cocoons. Indeed, the blue 3D domain stable at -200 mT first transitions to columnar skyrmions before splitting into pairs of cocoons at -375 mT.

DG longitudinal resistance R xx

The same measurements of R xx (H) for θ ∈ [0 • , 90 • ] and ϕ = 0, 90 • are performed, as shown in Figure 6.7. The trends appear relatively similar to SG ones although the amplitudes are different due to weaker SMR and AMR effects (see Table 6.1). In the (xz) plane, the scale is reduced but the accuracy is similar as in the (yz) plane, with a maximal error again less than 10 mΩ. We first consider the case ϕ = 90 • and θ = 0 • (out-of-plane field) for which only minor variations are expected since the in-plane components should remain small throughout the field sweep. Starting from high positive field to negative field, the magnetization is first saturated along the z direction until the first skyrmionic cocoons nucleate in both SG near 325 mT (state I). Their apparition create in-plane components which yield, due to the sign of the AMR and SMR, a negative contribution to R xx .

As they expend in all directions upon lowering µ 0 H, it continues to decrease, reaching its minimum at remanence. According to the simulations, it corresponds to a state in which 3D worms coexist with the skyrmionic cocoons. The symmetric trend is then observed for negative magnetic field and it is similar to the one previously described with the MFM study: the spin textures shrink and the 3D worms end up disappearing, leaving only cocoons (state III) until the sample saturates.

The nucleation of domains in the strong PMA layers is barely noticeable at 50 mT (state II) but is evidenced within the inset in which a drop of a few mΩ is visible. Since the domains walls are very narrow in the middle layers (due to the high PMA), their in-plane contributions are small, hence the small jump. The numerical resistance, although slightly shifted in field, reproduces well the trend and the amplitude of this drop. This transition is discussed in more details in Appendix E, especially regarding the peak shape.

To further validate our approach, the field was tilted at different angles (30, 60, 90 • ) in order to introduce more in-plane components. This leads to a stronger field response that challenges the sensitivity of the micromagnetic simulations. Since the magnetic field lies partially (or totally) inplane, the nucleation events provoke significant positive jumps in the resistance linked to a loss of IP components. It is also noticeably harder to saturate the sample along those directions due to the strong PMA layers, leaving variations at higher fields. The application of a tilted magnetic field leads to complex spin textures that are displayed in Appendix E. At zero field however, we retrieve a state with 3D stripes and skyrmionic cocoons for θ = 30, 60 • (although different from θ = 0 • ) whereas θ = 90 • yields only 3D stripes. Regardless of the angle of the field, the resistances extracted from the simulations fit remarkably well the experimental ones. The micromagnetic simulations and the magnetic states they described thus match well with our DG structure, supporting the coexistence of skyrmionic cocoons and 3D worms.

For ϕ = 0 • , a similar reasoning than for the SG can be held, taking into consideration that the AMR is smaller than the SMR effect. For θ > 0 • , the nucleation events provoke a loss of m x and a gain in m y (in DW) which, as R SMR > R AMR , reduces the overall resistance. Due to the smaller scale, the discrepancy between the simulations and the experiments appears larger. However, the difference between the two is typically of a few mΩ which can originate from the simplicity of our model and its neglected contributions as well as from some experimental angular inaccuracy (less than 0.5 • for ϕ).

Moreover, the measurement for R AMR was more noisy than for R SMR (see Figure 6.2.b).

At higher fields, in the presence of a negligible normal Hall effect, the resistances reaches a plateau for all tilting of the magnetic field (see Appendix E).

Planar Hall Effect

So far, in the model in consideration, we have neglected the planar Hall Effect (PHE) contribution.

To check that it is indeed negligible, R xy (ϕ) have been measured at the maximum field for θ = 90 • .

In such magnetic multilayers, the PHE displays a sinusoidal behavior when rotating the magnetic field in plane. Its maxima are found at ϕ = π/4 + nπ/2 (n ∈ Z) and importantly, the PHE cancels along ±x and ±y. The corresponding measurement in the double gradient structure is shown in Figure 6.8.a along with the raw data of R xy (H, θ = 0 • ) to facilitate comparison. First, note that the amplitude of the PHE (18 mΩ) is about one fourth of the total amplitude of R xy (81 mΩ). Moreover, this measurement corresponds to a saturated state, meaning that all the spins contribute equally to the PHE. However, when considering the R xy measurement with an OOP field, only those present in the domain walls (the in-plane component) of the various textures contribute, and this greatly reduces its amplitude. Additionally, working under the assumption that the chirality and symmetry of the objects are well-defined based on our simulations, the contribution for a Néel (or Bloch) closed domain wall (DW) should cancel out (see Figure 6.8.b). We could argue that this circular symmetry is not always preserved in the layers with strong PMA but due to their strong anisotropy, the associated DW width remains small which again correlates to a very low PHE contribution. Moreover, this corresponds to an ideal case as the physical objects are not axisymmetric. As a sanity check, the average m x and m y have been extracted from the simulations over the corresponding field range and the larger value is found to be lower than 5 × 10 -3 over the whole simulation space, implying that the PHE should indeed be negligible. We can easily generalize the previous reasoning to magnetic fields with a finite in-plane component (θ ̸ = 0 • ). Indeed, as it lies in the (yz) plane, locally, the spins are more tilted along the y axis but their PHE contributions still cancel each other so that the PHE remains negligible. Even at saturation, the magnetization lies along a direction for which the PHE is null. Note that this conclusion would not stand for ϕ ̸ = nπ/2.

Sensitivity of the measurements

Due to the vertical variations of the cocoons with field, it is important to estimate the sensitivity of the electronic transport measurements to determine if this confinement is measured accurately. To that end, we consider the longitudinal resistance of an SG structure in presence of an out-of-plane decreasing magnetic field (Figure 6.9.a) and compare it to different states extracted from the micromagnetic simulations. We focus on the state at 200 mT because it displays a significant evolution over the thickness and modify it to investigate four different states: the unchanged one (A) and some in which layers have been set uniformly up (B, C, D). This modification is performed over the whole simulations space. Between the most different states A and D, a difference of 3 mΩ is obtained which is superior to the characteristic noise of our measurements. Between the closest states A and B, this difference drops to 0.6 mΩ which would be harder to resolve experimentally and both states are close to the experimental value. On the contrary, the more confined states C and D are further away from the experimental curve. Therefore, the transport measurement shows sensitivity to the 3D profile with an accuracy of one magnetic layer along the normal direction. A similar procedure has been followed on the DG measurements at 325 mT. In this case, the cocoons have been artificially removed from the top gradient to yield the state F. This yields a R xx variation close to 4 mΩ with the unchanged state and the corresponding data point is far away from the experimental curve. Moreover, it also corresponds to a difference of 0.1 in the average m z which should also be noticeable in R xy . This simple check demonstrates that cocoons must be easily detected in both gradients.

Establishing a magnetic phase diagram

From the previous analysis, we have a solid foundation for interpreting the signal in R xx and R xy measurements. To further leverage these measurements, we can closely examine the distinct transitions and changes in slope observed in the magnetoresistance curves. In Figure 6.11, R xy and R xx are shown for an out-of-plane magnetic field (θ = 0 • ), along with their derivative indicated by the dotted maroon curve. In the upper portion of the graphs, the background colors correspond to the magnetic states predicted by the simulations: uniform (U, shown in red), states with cocoons (C, shown in blue), and states with cocoons and 3D worms (C+W, shown in green). State I (C) and State II (C+W) serve as typical examples for the latter two cases. Similarly, in the lower portion of the graphs, the same colors are employed, but in this case they delineate the inflection points observed in the magnetoresistance derivative which correspond to the magnetic transitions. The predicted transitions happen at identical fields for both R xx and R xy (the extrema of their derivative are found at similar fields). The experimental results can be compared to the numerical phase diagram (top colors), and the agreement is generally excellent, except for a slight shift near negative field saturation. It is worth noting that the simulations have been computed with a relatively large step size of 25 mT, increasing the likelihood of such a shift. However, this still supports the correlation between the different magnetic phases and the variations in the derivatives. Thus, the derivative of the experimental signal can be utilized to determine the magnetic phases with accuracy, benefiting from the combined use of both R xx and R xy measurements. In Appendix E, the phase diagram as a function of temperature is discussed as well as the one for a thinner DG multilayer (N = 5, M = 5).

Exploiting the transport measurements

The investigation of R xx (H) and R xy (H) through transport experiments, combined with their sensitivity to distinct magnetization components and their strong correlation with micromagnetic simulations, facilitates a more comprehensive understanding the evolution in field of the different magnetic textures. Together with the MFM examinations, these findings provide evidence for the coexistence of diverse spin textures, including skyrmionic cocoons and columnar skyrmions, within the double gradient structure. Furthermore, by analyzing the transitions in the experimental signal (and its derivative) as a function of the magnetic field, a precise phase diagram can be derived.

Detecting skyrmionic cocoons

In practical applications, the ability to detect the existence of cocoons or skyrmions without imaging and isolate their signals is one of the important challenges that will have to be achieved. To this aim, we rely on the anomalous Hall effect and measure the Hall resistance (R xy ) in tracks of reduced dimensions, thereby enhancing the influence of individual magnetic textures. For such measurements, the longitudinal resistance R xx is not employed as it captures a signal proportional to a larger area which complicates the extraction of individual texture contributions. In the subsequent discussion, we present preliminary findings that, although promising, still require additional optimization efforts.

The results presented in this section have been obtained with a DG structure with parameters: S = 0.1 nm, X 1 = 1.6 nm, N = 13 layers, M = 15 repetitions, Y = 0.9 nm, t N Pt = 3 nm and t M Pt = 3 nm. It has been chosen to have a saturation field low enough to be able to saturate it in the MFM setup (less than 300 mT). Moreover, with this X 1 value, it will be easier to isolate the contributions of each magnetic species as they do not coexist (see Subsection 5.1.3). Indeed, only skyrmions are stabilized towards saturation and, while decreasing the field from there, only cocoons are nucleating.

Resistance contribution of a single object

The nucleation (or annihilation) of spin textures impacts the transversal resistance due to AHE. For a single magnetic entity i, the resistance is modified by:

∆R i = ±2R AHE V i V tot (6.6)
with V i the magnetic volume of the magnetic texture under consideration and V tot the total magnetic volume. The sign depends on the nature of the transition, either a nucleation or an annihilation, and on the direction of the field sweep. The area of the magnetic texture in each layer might fluctuates so we express its volume with its average area and its height:

V i = ⟨A i ⟩ × h i .
For the total magnetic volume, it is approximately taken as the active area2 (zone between the voltage probes) multiplied by the total ferromagnetic thickness of the multilayer A tot × t tot . Thus, the previous equation can be written as:

∆R i = ±2R AHE ⟨A i ⟩ A tot h i t tot (6.7)
Based on this equation, the contribution of a skyrmionic cocoon is expected to be lower than for a skyrmion as the cocoons have smaller dimensions. The transitions in R xy should thus allow to quantity and differentiate between the two textures. In the following, to compare between several devices with different geometries, ∆R i is multiplied by the area:

∆R i A tot = ±2R AHE ⟨A i ⟩ h i t tot (6.8)
We multiply by the active area only as the total magnetic thickness is the same between devices prepared on the same multilayer. The R AHE amplitude should also not vary significantly between devices.

Simultaneous imaging and electrical measurements

Using Ta contacts

For increasing the signal due to the nucleation of single entities, the tracks width needs to be reduced as the AHE scales with the average m z in the Hall bar area. Moreover, it helps to use contacts that are not directly made of the magnetic multilayer material, otherwise it increases the active area and broadens the current lines, making more complex the interpretation of the signal. To this end, we first pattern narrow (linear) tracks with width around 1 µm using UV lithography. A second step is then done to deposit transverse tantalum contacts (30 nm thick) to serve as Hall voltage probe. Ta has been chosen because of its high resistivity which should not disturb the current flowing through the main track while still allowing to measure the voltage. The contacts are however very fragile and prone to break easily. Examples of such devices are shown in Figure 6.12 with AFM (atomic force microscopy). It also evidences the consequences of the IBE (ion beam etching) process which is responsible for the redeposition near the edges of the track. Those tracks correspond to the one measured in the following.

Each resistance data point corresponds to the average of 200 measurements (inverting the polarity between each measurement), yielding a standard deviation typically close to 10 nV.

Skyrmion contributions

We first focus on the evolution of the signal towards saturation which should mostly present 3D

worms and columnar skyrmion. In Figure 6 Thanks to the MFM phase maps, the changes in R xy can be explained as we directly observe the modifications of the magnetization. For instance, the state 2 measured after a jump shows that a red worm has been shortened (see green ellipse) when compared to state 1, hence the decrease in m z .

Similarly, between 6 and 7, the worm on the right disappears. In frame 8, a worm is annihilated, leaving behind a columnar skyrmion. Those measurements also show that not only the magnetization located strictly between the Ta contacts is probed, as annihilations happening further away still contribute to the signal with a lesser intensity. Because of this, we cannot be quantitative as the worms extend over the length of the track and thus their signal cannot be isolated. However, the magnetic state can still be related to the evolution of R xy . Finally, the last state presented ( 8) is interesting as it shows two columnar skyrmions in between the Ta contacts.

We thus focus on the field range in which isolated objects are obtained, i.e., below -260 mT. In To delve deeper into the topic, we discuss the red curve and its more densely populated magnetic states. Initially (frame 1), we identify eight distinct columnar skyrmions, with half of them appropriately positioned between the contacts. The first transition occurs through tip interaction, transforming a columnar skyrmion into a cocoon (states 2 and 3), which could exist in both gradients. To comprehend the associated jump, let's consider a simplified model. As the Hall resistance R xy is proportional to the average out-of-plane magnetization m z , it scales with the magnetic thickness. In this particular structure, the strong PMA layers account for 24% of the total magnetic layers. Therefore, assuming that a columnar skyrmion contributes approximately 100 µΩ, if it splits into two cocoons, the resulting increase in m z should correspond to a jump of 24 µΩ for R xy . Experimental observations indicate that the jump before state 2 is close to 32 µΩ, deviating from our approximation by a few µΩ. Thus, it is likely that two cocoons, one in each gradient, have formed as a result.

Between states 2 and 3, no change is visibly detected with MFM. However, two small jumps (approximately 40 µΩ), corresponding to the blue values, are measured, which are likely associated with the annihilation of skyrmions located further away or cocoons in the bottom gradient. In the last three frames, we observe the sequential annihilation of three columnar skyrmions situated in the active area, accompanied by three jumps in R xy , each close to 100 µΩ. These very close values (±3%)

suggests that they occupy a similar magnetic volume.

In conclusion, our analysis, combining MFM imaging and Hall measurements, successfully demonstrates that columnar skyrmions contribute approximately 106 ± 11 µΩ to R xy , noted ∆R sk . Additionally, smaller discernible jumps hint at the presence and annihilation of cocoons or other magnetic textures that are not visually observable with MFM.

Cocoons contributions

To follow on the skyrmion contribution, we focus on the nucleation of skyrmionic cocoons in track II (same multilayer structure but different batch). To this end, we measure R xy lowering µ 0 H from saturation which stabilizes cocoons first in the range of field under study (less than -160 mT) as the strong PMA layers only form domains around -50 mT in this DG structure. In Figure 6.15.a, consecutive measurements of the same field cycle are shown. An MFM image is taken every mT (indicated by black arrows) while R xy is measured with steps ten times smaller. Note that the resistance is measured before and after each image at the same field as the tip might disturb the magnetization and consequently induce transitions. The curves all display a similar behavior with multiple recurring steps of similar amplitude. The range of field is relatively small (close to 10 mT) to focus on the nucleation of a low number of magnetic objects.

To better identify and quantify those transitions, three typical curves have been isolated and plotted along with the difference between consecutive points dR xy in Figure 6. To understand the origin of the transitions at play, we use the support of the MFM phase maps.

In Figure 6.16, the same three curves are presented along with the corresponding sequence of images.

The slope at high field is subtracted to facilitate the identification of the steps. In the different graphs, the transitions are identified with gray boxes (with Roman numerals) linked to nucleation events in the MFM images (dotted black circles). Let us now analyze each of the measurements consecutively (top to bottom).

• Starting from the saturated state (1), a cocoon nucleates near the top Ta contact for a jump of 22 µΩ (I). Then in the 4 th frame, three cocoons appears but farther away from the contacts, yielding in total a transition close to 24 µΩ. In frame 6, a cocoon is stabilized on the far right of the track which do not appear to be correlated with the resistance due to its distance from the contacts. It thus define a lower spatial limit to be able to detect them. In the next frame, a double transition (III) occurs: the first cocoon widens and becomes more contrasted while a similar texture nucleates close to it. Because of the size evolution and increased contrast, it is likely that those are double cocoons, i.e., one in each gradients on top of each other. In frame 8, the transition IV occurs with the apparition of a single cocoon in the top gradient (smaller and less contrasted), causing a 19 µΩ jump. Already, this establishes that the contribution of a single cocoon nucleation is around 20 µΩ. Interestingly, between the two last frames, two unidentified transitions as indicated by the red boxes on the curve (16 and 22 µΩ) happen. Given their heights, the most likely corresponds to single cocoons nucleating in the bottom gradient, invisible with MFM. Finally, in the last frame (9), a double cocoon appear (V) yielding a jump of 54 µΩ. • The first visible transition (I) is the same as the previous sequence with a single cocoon appearing near the contacts, for a slightly higher contribution of 26 µΩ. In frame 4, two cocoons nucleate on the left, far away from the contacts, with thus a reduced impact on R xy . The transition III (frame 6) is linked to three distinct changes: a single and a pair of cocoons nucleate while the first cocoon transitions to a double one. An isolated jump of 20 µΩ is detected, likely associated to the single cocoon nucleation, followed quickly by a larger step of 71 µΩ. This is roughly 3 times the contribution of a cocoon which is coherent with the MFM observation (a cocoon and a pair of cocoons). The transition IV corresponds to the transformation of a single to a pair with a contribution of 36 µΩ, similarly to the next one (V) with a reduced contribution (11 µΩ) as it happens far from the contacts.

Finally, between the last two images, no change is visible even though the resistance shows two jumps (22 and 17 µΩ). Again, according to their height, it is likely to correspond to cocoons nucleating in the bottom gradient.

• In the third sequence, the cocoon nucleating first between the contacts is not visible in R xy although it is picked up by MFM. It might be due to its small size that would represent a contribution lost in the characteristic noise of the measurements. The first transition (I) in frame 3 thus corresponds to the nucleation on the left of the track for 16 µΩ. Then, another cocoon nucleates (II) and the first one sensibly increase in size for 21 µΩ. The third transition is exactly the same as in the first curve,

showing the reproducibility of the measurements. Transitions IV and V respectively corresponds to the nucleation of a cocoon and a transition from single to double for 17 and 18 µΩ. Finally, two pairs of cocoons appear in the last frame causing a jump of 62 µΩ.

Taking into account this analysis, we can estimate that a single cocoon, depending on its position and size, contributes by 20 ± 5 µΩ, called ∆R c . This is consistent with the nucleation of a double cocoon observed in the middle curve (transition IV, frame 7), which correspond to 36 µΩ, roughly twice ∆R c . For transitions with an even higher number of cocoons involved, often 3, the signal is typically around 55-60 µΩ, i.e., a multiple of ∆R c . It is however impossible to be definitive about the contribution of multiple cocoons as nucleation could occur simultaneously in the bottom gradient which cannot be verified with MFM. Additionally, in the previous curves, many steps of height near ∆R c were observed but not correlated with MFM. Those transitions can thus indicate the nucleation of single skyrmionic cocoons in the bottom gradients, too far away from the surface to be detected by the tip. This thus helps to pick up the signal coming from magnetic objects not detected by MFM imaging.

Interestingly those measurements also picked up single cocoons which has not been the case in MFM images so far. Several reasons can be put forth to explain this. The first one is that the DG field dependency was measured here with accuracy (small field steps), allowing to observe the evolution of isolated textures. Moreover, the magnetic properties are also modified due to the shape anisotropy of the track which plays a role in the stabilization of single cocoons. It however shows the evolution of the contrast, not only from a single to double, but also for a single cocoon expending (see the first one to nucleate near the contacts in each loop).

The systematic measurement of the same field cycle additionally puts forward that the nucleation events appear at similar places on the track. This indicates that some pinning point exists, for which the energy barrier is smaller.

Skyrmions versus cocoons

For comparison's sake, let's consider the previous results measured in track I for the columnar skyrmion assuming a simplified model. To compare between the two tracks, the signal needs to be multiplied by the area between the two contacts (Eq. 6.8) which, because of the roughness of the edges and the contacts, can be inaccurate. Track I (resp. II) has an active area A I between the Ta contacts of roughly 2.2 ± 0.3 µm 2 (resp. A II = 3.4 ± 0.3 µm 2 ). Then the contribution of a skyrmion becomes ∆R I sk × A I = 233 ± 53 µΩ µm 2 by error propagation. The uncertainty is high due to inaccuracy on the area. Each gradient accounts for 38% of the magnetic material so that we obtain ∆R I c = 89 ± 21 µΩ µm 2 for a single cocoon. In track II, the contribution of a cocoon is ∆R II c × A II = 68 ± 27 µΩ µm 2 so that the two intervals coincide. The two series of measurements are thus coherent but would benefit to be performed on a single track for easier comparison. This study still shows that we can isolate the electric contributions of both cocoons and columnar skyrmions thanks to the correlation between imaging and magnetoresistance measurements.

Statistical data

After establishing that it was possible to identify the skyrmions and cocoons, within some limits, thanks to the electrical signal, we perform similar measurements without imaging on another setup in order to acquire statistical data. Unfortunately, the previous tracks were destroyed due to the fragility of the Ta contacts so that 3 new tracks have been measured to account for variability in between devices. Their various characteristics are presented in Table 6.2 as well as the measurements that were performed on them. For each track, at least 800 transitions were detected, allowing to draw some statistical conclusions. The following measurements focus on the nucleation of skyrmionic cocoons only, thus starting from saturation and lowering the field.

Step detection

To Following this process, the difference between consecutive data points dR xy is computed (see Figure 6.17.b) to identify the steps. A primary threshold is defined as ⟨dR xy ⟩ -0.5σ(dR xy ), i.e. the difference between the average and the standard deviation, to probe for points above the noise level (red dots). Then, in the vicinity of those points, a second threshold ⟨dR xy ⟩ -0.375σ(dR xy ) is considered to add points (green dots) in order to estimate the total resistance step of a given transition. This method seems relatively robust between measurements and thus allows for a comparative study between different field cycles.

Transition distributions

The process of step detection is applied to all measurements performed in the different tracks, yielding the distribution of transitions in R xy . To be able to compare between tracks, those are multiplied by the area between the Ta contacts of each track (W Track × W Ta ), given in Table 6.2. Note that the incertitude is relatively high as it was measured with optical microscopy images.

The full hysteresis for each track is shown in Figure 6.18.a. They display a similar amplitude, close to 70 mΩ, although the trends are slightly different. The first two tracks (III and IV) show a similar behavior while the last track (V) has a lower nucleation field and more numerous sharp steps. This is likely due to their width difference as track V is thinner which strengthen the edge effects and the shape anisotropy. This also increases the contributions of each nucleated object and make it easier to isolate as fewer textures nucleate, making track V the most suitable track for measurements. In Figure 6.18.b, the histograms of the transitions extracted from the multiple field sweeps are presented for the three tracks. Note that the field range under study is different in each case: for track III, it is between ±200 and ±-175 mT, for track IV between 200 and 180 mT while for track V we work at lower field for observing nucleation, between -160 to -115 mT (highlighted on the corresponding hysteresis). Typically, the histograms are bell-shaped with various maxima and vanishing tails. Because of the noise intrinsic to the measurements and the fact that R xy picks up objects away from the contacts with a signal decreasing with the distance, the distribution is a continuum and not peaks centered around multiple of the contribution of skyrmionic cocoons. However, we can still identify some maxima in the distributions as indicated by the Gaussian outlines. For track III for instance, there seems to be one at 157 µΩ µm 2 and one near 330 µΩ µm 2 , nearing a factor 2. Similarly, in the track IV, the same trend can be detected. In both cases, the larger transitions vanish quickly as the field range is still relatively high so that no consequent nucleation happens. In the last track (V), the distinct features are even more isolated with a first maxima near 165 µΩ µm 2 and the second near 307 µΩ µm 2 while a clear drop is seen in between. As the field range is larger, larger transitions are measured that most likely corresponds to the nucleation of many objects at the same time at various distance from the contacts, thus the quasi-continuum.

Although the uncertainty is large regarding the area of each track, all of them, especially the last one, appear to display extrema positioned near similar transition value. This would indicate that the nucleation of a skyrmionic cocoons contribute by ∆R c = 166 ± 20 µΩ.µm 2 while a double cocoon corresponds to ∆R c = 311 ± 20 µΩ µm 2 , i.e. 1.87∆R c .

Following the previous discussion, it appears that the last track (V) offers the most well-defined peaks. This originates from it reduced width (around 1.4 µm) which correlates with less numerous nucleation events. Thus, the contribution of each object is effectively larger and there is less nucleation away from the contacts which reduces the dilution effect (decrease of the resistance variations with the distance). However, the field range is still relatively large in this case which makes direct comparison with the MFM study more complex. To solve this, we focus on a reduced range of magnetic field (between -163 and -150 mT) and perform the same analysis, allowing to focus on the first nucleation events only. The resulting histogram is presented in Figure 6.19.a. Similar observations can be made, along with the emergence of 3 rd maxima near 650 µΩ µm 2 (∼ 4∆R c ). Moreover, the first two peaks, respectively corresponding to the nucleation of one or two cocoons, are also sharper.

In The Hall resistance measurement, when performed without simultaneous imaging, reveals a distribution characterized by peaks occurring at consistent intervals. Remarkably, these peaks exhibit comparable values across various devices, reinforcing the hypothesis that the first peak signifies the initiation of a singular cocoon formation, whereas multiple peaks indicate multiple nucleation events.

However, the presence of a continuous range of transitions arising from nucleation events occurring away from the contacts affects this observation. To address this, it would be advantageous to explore alternative device designs and employ smaller contacts, potentially optimizing the measurement process.

Comparison with simulations

Contrarily to MFM, the full hysteresis loop is recorded which allows to obtain the amplitude of the AHE. Considering the more reliable track (V), we have an amplitude 2R V AHE = 66 ± 2 mΩ after subtracting the high field slope (in Figure 6.18). In the simulations, coming from saturation, the nucleation of cocoons occurs first at 325 mT (see Figure 6.6). In this case, a number n c = 28 of cocoons nucleate in the simulation space of area A sim = 1.0486 µm 2 . Their apparition provokes a drop dm z = 0.15 in the average ⟨m z ⟩, i.e. a variation of dm z /2 × 100 = 7.6%. Thus, per object, this yields a final contribution of:

∆R sim c = 2R V AHE A sim n c dm z 2 ∼ 187 ± 5 µΩ µm 2 (6.9)
The numerical value is of the same order of magnitude as the experimental one (∆R c = 166 ± 20 µΩ.µm 2 ), although slightly higher (13% relative deviation). This difference can arise from two considerations. First, the simulations corresponds to the case X 1 = 2.0 nm so that there may be small size variations for X 1 = 1.6 nm. Second, in the simulations we use 25 mT steps and encourages the nucleation by running the system with temperature for 1 ns. In this case, it occurs at 325 mT but with smaller steps it might happen at slightly higher field (below 350 mT) which would nucleate more confined cocoons and thus would yield a lower ∆R sim c . Taking into account those considerations, the 6.2. Detecting skyrmionic cocoons correlation with the experimental value is thus satisfying.

Size estimation

Using Eq. 6.8, we can estimate the size of the nucleated cocoons with their estimated contribution to the Hall resistance, ∆R c = 166 ± 20 µΩ µm 2 . This corresponds to:

⟨A i ⟩ = ∆R c t tot 2R AHE h i = 6400 ± 800 nm 2 =⇒ ⟨r c ⟩ = ⟨A i ⟩ π = 45 ± 3 nm (6.10)
with r c the radius of a cocoon and using h i = 24.4 nm and t tot = 62.3 nm. We approximate that the cocoon extends over the full SG block which might not be the case, especially for nucleation at high field. This thus yields a diameter of about 90 nm, coherent with the MFM observations made at high field in a SG structure which yielded 86 nm diameter at 262 mT (see Table 4.1).

Electrical detection

Our analysis of transport measurements presented in this chapter has shown the possibility to electrically detect columnar skyrmions as well as single cocoons, conjointly with imaging. They also showed transitions of heights corresponding to single cocoons that could not be correlated with any nucleation in the MFM images. Thus, they likely correspond to nucleation in the bottom gradients. Without imaging, the transition distribution shows maxima around multiple of ∆R c for three different devices with comparable values. Those measurements need to be optimized further (smaller tracks and contacts) but they work as a proof of concept: it is possible to discern columnar skyrmion from cocoons with Hall measurements.

Chapter 6 Summary

The transport measurements in single or double gradient multilayers allowed to deepen the understanding of the behavior of the corresponding magnetic textures. By first studying large tracks, the average magnetization was probed over the full thickness while sweeping the magnetic field. By worms which then annihilate at lower field than cocoons.

Following this successful understanding of the average magnetization, the dimensions of the tracks have been reduced and Ta contacts were used to not disturb the current lines as well as the magnetization. Using smaller devices increase the contribution to R xy of single entities whether they appear or disappear. In a DG multilayer with no coexistence of skyrmions and cocoons in the major loop, the contribution of skyrmions and cocoons were successfully isolated by systematic comparison with simultaneous MFM imaging in two different tracks. Interestingly, some steps in R xy could not be correlated with nucleation in the phase maps which thus corresponds to skyrmionic cocoons appearing in the bottom gradient. The extracted values are also coherent as the contribution of the skyrmions is larger than for the cocoon by a factor 3. Finally, statistical data were acquired without imaging to study the nucleation of skyrmionic cocoons, yielding distributions with peaks located at regular interval.

This observation was particularly evident for the smallest track measured (amongst three) for which fewer objects nucleate and their contributions are larger. It is likely that the first peak corresponds to the nucleation of a cocoon and the higher order are multiple nucleation events. The next step would thus be to perform similar measurements in a DG showing coexistence to ensure that is it possible to distinguish the two contributions but this preliminary study works as a proof of concept.

We thus have a far better understanding the magnetization in those aperiodic multilayers and visualize the objects at play thanks to the micromagnetic simulations. However, we are still lacking imaging along the third dimension which is why we turn to synchrotron light in the following chapter.

7

C h a p t e r

Probing the 3rd dimension with soft X-rays T he investigation of three-dimensional spin textures is particularly complex as acquiring data about the actual magnetic configuration in the vertical direction of a multilayer is still a hardship with many conventional methods. To circumvent this, high-end techniques, often relying on soft X-rays, can be used in synchrotron facilities. Indeed, thanks to its high flux and brilliance, the synchrotron light is able to transmit across thick magnetic multilayers, allowing to recover information about the actual shape of complex magnetic textures. Moreover, the light energy can be tuned to different absorption edges to enhance the sensitivity to specific transitions of the chosen element. In this chapter, we present different approaches that we have used to characterize aperiodic multilayers, starting with transmission techniques based on the circular dichroism of X-ray holography or STXM (scanning transmission X-ray microscopy) yielding 2D maps of the magnetization averaged over the thickness. To finish, we discuss an experimental reconstruction of the 3D vectorial field of the magnetization obtain with the recently developed X-ray laminography technique.

7.1

Measuring in transmission: X-ray holography

FTH and HERALDO

Two different holography techniques have been used to study aperiodic magnetic multilayers, namely X-ray Fourier transform holography (FTH) experiments [START_REF] Eisebitt | Lensless imaging of magnetic nanostructures by X-ray spectro-holography[END_REF] along with X-ray holography with extended reference by autocorrelation linear differential operator (HERALDO) [START_REF] Guizar | Direct image reconstruction from a fourier intensity pattern using HERALDO[END_REF]. HERALDO and FTH are lensless high-resolution transmission techniques that record the hologram resulting from the interferences between an object hole and a reference aperture, a slit for the former and a hole for the latter. Those create a reference wave that encodes the phase so that a simple Fourier transform directly allows to recover a real-space image of the actual magnetic configuration. The spatial resolution is typically limited by the size of the reference aperture as well as by the numerical aperture of the setup and the size of the hologram (defining the highest recorded frequencies). To improve the contrast and the resolution, one can additionally make use of phase retrieval algorithms [START_REF] Marchesini | Invited article: A unified evaluation of iterative projection algorithms for phase retrieval[END_REF] for the image reconstruction.

Holography principle is illustrated in

Introduction to iterative phase retrieval algorithm

Phase retrieval algorithms are computational techniques used to recover the phase information of a signal or wavefront from its measured intensity data alone. In various scientific and engineering fields, such as optics, X-ray crystallography, and imaging, the phase of a wave is often inaccessible or lost during measurement processes. However, the phase carries vital information about the structure and properties of the signal, making its retrieval important for accurate analysis and reconstruction.

Phase retrieval algorithms aim to overcome this challenge by leveraging mathematical and iterative approaches. These algorithms exploit the interplay between the measured intensity data and certain assumptions or constraints about the signal to iteratively estimate and refine its phase distribution.

Through a series of calculations and updates, the algorithms gradually converge towards a solution that recovers the original phase information. Even though the use of a reference aperture allows to encode the phase in FTH experiments, a trade-off needs to be considered as its size limit the resolution but it needs to be large enough to have a brightness comparable with the object hole. Thus, to overcome the resolution limit imposed by the aperture size, phase retrieval process can be applied. The reconstructed image will finally have a signal proportional to ⟨m z (x, y)⟩ as the X-rays have a normal incidence (along z).

Principle

Let's introduce the principle of phase retrieval algorithm, a presentation based on the comprehensive review of Marchesini [START_REF] Marchesini | Invited article: A unified evaluation of iterative projection algorithms for phase retrieval[END_REF]. The algorithm implementation and the related data preparation are covered more in depth in Appendix G. Let ρ(r) be the density function of the object under study with r the coordinates in the real space. The intensity I(k) of the diffraction pattern only depends on the modulus of the Fourier transform (FT) ρ(k), and not on the phase, yielding in the reciprocal space:

I(k) ∝ | ρ(k)| 2 = ρ(k) † ρ(k) (7.1)
Thus, its real space expression corresponds to the autocorrelation of the density function:

I(r) ∝ ρ(r) ⋆ ρ(-r) (7.
2)

The objective of the phase retrieval algorithm is thus to solve for ρ(r) or ρ(k) using the experimental constraints, thus recovering the phase signal. For the hologram to be properly sampled, the coherence width of the X-rays1 needs to be about twice the dimensions of the object in real space [START_REF] Spence | Coherence and sampling requirements for diffractive imaging[END_REF] (Shannon theorem), i.e. to be larger than the autocorrelation spread.

Iterative phase retrieval algorithms rely on two different constraints, one in each space (real and reciprocal), so that it goes back and forth between the two spaces while exacting projections in Hilbert spaces to respect the two constraints. The algorithms try to find an intersection between the sets solving each constraint:

1. Support constraint (real space): the object has a finite spatial extension that can be delimited, a volume called its support. It corresponds to the places in which the light goes through (object and reference holes). The first iterative process to apply those conditions was proposed by Fienup in the early 1980s [START_REF] James R Fienup | Reconstruction of an object from the modulus of its Fourier transform[END_REF], adapted from the Gerchberg-Saxton algorithm that has been used in electron microscopy [START_REF] Gerchberg | Practical algorithm for the determination of phase from image and diffraction plane pictures[END_REF].

To write concisely the algorithm iterations, we introduce several operators. In the real space, the projector P s sets to zero the component outside the support S while leaving the rest unchanged:

P s ρ(r) =        ρ(r) if r ∈ S 0 otherwise (7.3)

Our recipe

Many algorithms have been developed after the ER one that propose different approaches to the phase retrieval process. Depending on the situation, different algorithms can allow for a quicker and better convergence, avoiding local minima. Their combination is also a good approach to speed up the convergence. In our case, we have used the oversampling smoothness (OSS) algorithm [START_REF] Jose A Rodriguez | Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities[END_REF] and the solvent flipping (SF) algorithm [START_REF] Marchesini | Invited article: A unified evaluation of iterative projection algorithms for phase retrieval[END_REF].

The latter is similar to the ER algorithm: the projector P s is replaced by its reflector R s = 2P s -I with I being the identity matrix. The iterative expression is thus:

ρ n+1 (r) = R s P m ρ n (r) (7.6)
The OSS algorithm is based on the HIO (hybrid input-output) algorithm [START_REF] James R Fienup | Phase retrieval algorithms: a comparison[END_REF] that relies on non-linear feedback theory. The following quantity is first computed:

ρ ′ (r) =        P m ρ n (r) if r ∈ S (I -βP m ) ρ n (r) otherwise (7.7)
with β the convergence parameter between 0.5 and 1. The result is then refined with frequency filters applied in Fourier space with a 2D Gaussian whose standard deviation decreases over time.

ρ n+1 (r) =        ρ ′ (r) if r ∈ S F -1 F (ρ ′ (r)) e -0.5k/α otherwise (7.8)
With α the width of the Gaussian filter that is linearly decreased over the iteration process, thus reducing more and more the influence of the high frequency information outside the support.

Phase retrieval recipe

Left-and right-polarization hologram were first normalized with respect to each other, centered.

The beamstop was smoothed and any constant offset present was removed (see Appendix G for more details). After preprocessing the data, we begin by reconstructing the CL hologram with a combination of the OSS algorithm (250 iterations) followed by the SF one (200 iterations), which typically yielded a relative error with respect to the experimental hologram of 0.1%. The convergence parameter β started from 0.9 and decreased progressively with an arctan trend down to 0.5 over the OSS iterations. The filter parameter α is decreased linearly from the image size down to 0. Then, the solution for CL polarization was used as a starting guess for the CR polarization image which thus only went through the SF algorithm (200 iterations).

Pixels of the recorded hologram carrying unreliable values (e.g. the pixels shadowed by the beamstop used to protect the camera from the direct beam) were not subjected to the amplitude constraints and their value was left free to change. For each polarization, an average of the 20 reconstructions with the lowest errors has been extracted. From that, magnetic contrast is extracted by computing the difference of the two polarized images (which nullify the topographic contribution).

Loss of low frequencies

In FTH, the reconstructed signal provides information about the vertical component of the magnetization averaged over the thickness. However, due to the intensity of the direct beam, a beamstop needs to be used in the center of the hologram. Since a hologram corresponds to information in the Fourier space, the low frequencies information are lost from this missing signal.

This unrecorded information limit the possibility of quantitative imaging, as the impact of the missing data varies depending on the magnetic state being examined. Although the signal obscured by the beamstop is reconstructed during the phase retrieval process for FTH, it is not considered reliable enough for quantitative analysis, as it depends on the specific reconstruction parameters.

Imaging a single gradient multilayer

In this part, we image an SG multilayer with parameter X 1 = 1.8 nm, S = 0.1 nm and N = 13 repetitions with t N Pt = 3 nm. In the two following subsections, the FTH measurements and the reconstruction of the magnetization images have been performed within the frame of a collaboration with Riccardo Battistelli and Felix Büttner from Helmholtz-Zentrum Berlin für Materialien und Energie and University of Augsburg, Germany.

The out-of-plane field dependency measured with FTH is presented in Figure 7.3. The magnetization was saturated out-of-plane beforehand in positive field. Each image corresponds to the real part of the reconstructed signal obtained from phase retrieval processes. The object hole (of 800 nm diameter) containing the magnetic information has a contrast linked to the average m z over the thickness. Near remanence, ellipsoidal objects are visible in the aperture. As the OH is relatively small, statistics cannot be extracted as the number of textures is small. However, they seem to have a typical size close to 200 nm near remanence. Increasing the magnetic field in absolute value results in a contraction of the cocoons: at -181 mT, they are now all circular with a reduced size (closer to 120 nm in diameter).

Near saturation, at -237 mT, a large number of cocoons have disappeared and the remaining ones are still shrinking in size. The field dependency observed is in agreement with the previous MFM study led on a larger area (see Subsection 4.1.2). The MFM amplitude decreases as the field increases and this is compatible with textures that are more and more confined. Here, due to the missing frequencies hidden behind the beamstop, we cannot be fully quantitative between images. However, within a same frame the contrast of several textures can be compared. The loss of low frequencies also explains the circular artefacts visible at high magnetic field. Indeed, with few or no magnetic textures (near saturation), the information mostly resides at low frequency and is thus hidden by the beamstop. As it is reconstructed imperfectly, it yields this typical circular artefact. Despite this shortcoming, FTH allows to average the magnetization over the thickness with a lateral resolution of about 20 nm using a non-perturbative approach. For the SG multilayer, it is coherent with our previous observations made with MFM.

Imaging a double gradient multilayer

For the study of DG multilayer, FTH sets itself as a powerful technique as the X-ray transmission should yield distinct signatures for different spin textures. It also probes the bottom gradient which was not possible with surface sensitive technique such as MFM. The multilayer under consideration corresponds to S = 0.1 nm, X 1 = 2.0 nm, N = 13 layers, M = 15 repetitions, Y = 1 nm and t M Pt = 3 nm. Regarding the Pt in the gradient part (t N Pt ), it is set to 3 nm except in the last top seven layers of the top gradient in which it is 2 nm. For this structure, the gradient blocks correspond to 80% of the total magnetization. Thus, a single skyrmionic cocoon should have a contrast of about 40% of a 3D stripes. This difference should be resolved experimentally, however the distinction between two stacked cocoons and a 3D worms might be more subtle.

Increasing the magnetic field increases the vertical confinement and decreases the density, thus helping to resolve individual features. The difference between a single cocoon and other textures (3D worms or double cocoons) is at least 40% relative change in the signal which is easily visible in the images (red against white contrast). In Figure 7.4.a, FTH reconstructed real-space images of the object hole are displayed at different OOP fields starting from a demagnetized state, prepared with a demagnetization process using a field (decreasing from 1.2 T) tilted at 45 • with respect to the sample normal. At first, only two dark red 3D stripes are visible which extend from the top to the bottom of the multilayer. As the field increases, they get narrower and some white dots, with a weaker contrast, start to appear at 260 mT. As the signal scales with ⟨m z ⟩, the presence of two contrasts is directly linked to the existence of two types of objects with different vertical extension. The weakly contrasted dots thus correspond to spin textures with a smaller height, i.e. to skyrmionic cocoons. As the contrast is much weaker than the remaining stripe, they are likely to be single cocoon, either in the top or bottom gradient. At 320 mT, the stripe has transformed into a columnar skyrmion, still accompanied by single cocoons. Furthermore, the images at 342 and 348 mT put forward the transition between this columnar skyrmion and two vertically aligned cocoons (as highlighted with the pink circles) which have a stronger contrast than the single cocoon also visible in those frames. The evolution of the magnetic textures and the associated contrasts within a same frame observed with increasing magnetic field already demonstrates the existence of different magnetic textures with a transmission technique. To observe the nucleation of cocoons, the other branch of the hysteresis has been recorded, starting from saturation and decreasing the field. In Figure 7.4.b, close to saturation (µ 0 H = 366 mT), a few cocoons are visible and more nucleate when lowering the field at 327 mT.

The density continues to rise until the cocoons begin to interact. Looking closer at the transitions between 306 and 280 mT, the apparent merging of two pairs of skyrmionic cocoons can be observed, as evidenced by the colored ellipses. Indeed, the cocoons appear to be attracted to one another. As the contrast is doubled by the merging, this implies that one resides in the bottom gradient and the other one in the top gradient (at the same horizontal position). This transition, which places them one on top of the other, allows to reduce the dipolar energy. As the field decreases further (for instance at 208 mT), additional aligned cocoons appear and become more elongated in the horizontal plane. At remanence, it is likely that the smaller ellipsoidal objects are pairs of cocoons stacked upon each other (one in each gradient) while the worms extend over the full thickness of the multilayer (as supported by the previous MFM and micromagnetic simulations in Chapter 5). However, the difference in contrast is not sufficient to separate them as it only represents a 20% deviation in the signal. Moreover, the magnetic state is very compact and dense which yields an intricate hologram. Increasing the magnetic field increases the vertical confinement and decreases the density, thus helping to resolve individual features.

Attractive interaction between cocoons

Various transitions have been observed in the field sequence of Figure 7.4, namely a columnar skyrmion transforming into two cocoons or the horizontal alignment of two isolated cocoons. The latter transition is interesting as this unequivocally demonstrates that the two cocoons reside at different vertical level of the multilayer. Indeed, if they were in the same gradient, they would repulse each other to lower their dipolar interaction. We thus take a closer look at this transition using micromagnetic simulations. In Figure 7.5.a, line profiles are shown at different fields to illustrate the transition between the single cocoon state to the double one, evidencing the associated change in contrast. Interestingly, at first, only one of the cocoons is moving towards the other (see black arrow on the left graph) which suggest that the latter is pinned. However, once merged, the two aligned cocoons have slightly moved away from their previous position, a motion likely due to interactions with the other magnetic objects. This peculiar attractive interaction as well as the change in contrast confirms the 3D nature of the skyrmionic cocoons. In Figure 7.5.b, the total energy estimated from simulations for two cocoons, one in the bottom gradient and one in the top, is shown as a function of their distance (at 300 mT). In this case, the maximum force is close to 0.16 nN. Due to the stray field, the minimum of energy is obtained when the two are superimposed. However, experimentally, for instance near 282 mT, two close cocoons are measured that do not coalesce thus there must be some local interaction that overcomes the stray field. As a plausible hypothesis, we put forward the role of the anisotropy that can display spatial inhomogeneities due to the growth conditions. To test that hypothesis, in mumax 3 , we separate two cocoons by 150 nm and create a zone with a lower anisotropy than the rest of the simulation space (see red dotted square in Figure 7.5.c with an anisotropy 0.8 times as large as in the rest of the simulation).

Then we save the stray field emanating from the top cocoon and apply it to the bottom cocoon so that only the pinned one may move. After relaxation, we notice that the cocoon did not move at 325 mT whereas they aligned themselves at 300 mT, similarly to what is observed experimentally (at 280 and 282 mT). Ideally, one would create a random distribution of the anisotropy to further confirm the hypothesis but because of the complex structure of our multilayer which require multiple simulation regions, we reach mumax 3 limitations. However, this simulation works as a proof of concept and hints that the anisotropy can play an obstructing role in the merging of the cocoons. This result might also have been obtained for local variations of the DMI, M S or the exchange.

High resolution image

Additional FTH measurements have been performed at synchrotron PETRA-III (Germany) on the P04 beamline. The DG multilayer under consideration corresponds to S = 0.1 nm, X 1 = 1.6 nm, N = 13 layers, M = 15 repetitions, Y = 0.9 nm, t N Pt = 3 nm and t M Pt = 3 nm. As shown in previous This might arise from local pinning points where the anisotropy is lower as previously discussed. However the two clearly distinguishable contrasts evidence once again the 3D nature of the skyrmionic cocoons.

Skyrmionic cocoons in X-ray holography

The observation of different contrasts in transmission proves the existence of various objects with dissimilar vertical extension. Moreover, the attractive interaction at play between two skyrmionic cocoons confirm that they reside in different parts of the DG structure, one in each gradient. The holography experiments also verify their coexistence with more usual textures that are stable over the full thickness.

Temperature dependency and size evolution

Following the FTH study that demonstrated the existence of two populations of magnetic textures inside the DG multilayer, HERALDO measurements have also been performed on a similar structure.

Here, we consider the structure defined by S = 0.1 nm, X 1 = 2.0 nm, N = 13 layers, M = 15 repetitions, Y = 1 nm, t N Pt = 3 nm and t M Pt = 3 nm. To investigate a more general behavior, a larger object hole (2 µm diameter) has been fabricated, allowing more textures to be observed. In order to investigate the impact of the change in temperature on the different magnetic parameters, HERALDO imaging has been performed both at 80 K and room temperature.

HERALDO processing

The processing of the HERALDO data is slightly different from the FTH holograms as the reference aperture is a slit and not a hole. A detailed description of the process is presented in Appendix G.

Here, we present the main steps to obtain a real space image. After preprocessing the raw holograms (centering, normalization, smoothing the beamstop), a differential operator is applied on the far field diffraction intensity before retrieving, using a Fourier transform, the complex value image in the real space [START_REF] Guizar | Direct image reconstruction from a fourier intensity pattern using HERALDO[END_REF]. Contrarily to the coherent diffraction imaging (CDI) process and particularly the phase retrieval approach, the algorithm is deterministic and non-iterative [START_REF] Sg Podorov | A non-iterative reconstruction method for direct and unambiguous coherent diffractive imaging[END_REF]. The reference wave is provided by the scattered wavefront at the edge of the slit so that the resolution is limited by the sharpness of the edges.

The process is schematically described in Thus, a differential filter L is applied along one of the two slits (Figure 7.7.b), defined as [START_REF] Guizar | Direct image reconstruction from a fourier intensity pattern using HERALDO[END_REF]:

L = cos α ∂ ∂x + sin α ∂ ∂y (7.9)
with x, y the axis of the hologram and α the angle of the slit with respect to the x axis. Numerically, the corresponding matrix L can be computed as:

L i,j = cos α(X i -X 0 ) -sin α(Y j -Y 0 ) (7.10)
with X 0 , Y 0 the position of the central pixel of the hologram. A second filter is added to cancel the parasitic noise (stripes) induced by the second slit in the final reconstruction. It is either a binary filter that sets the values near the slits to zero or a gaussian filter, the latter being more efficient (see Appendix G). Finally, a Fourier transform of the resulting hologram is taken (Figure 7.7.c). After optimizing the phase factor, the magnetic information can now be observed in the object hole. Notably, at positive fields, the cocoons appear to be larger and fewer than at room temperature. A transition between a single cocoon and a paired one is visible between the 2 nd and 3 rd frame, as indicated by the significant change in contrast in the black squares. According to magneto-transport measurements, the nucleation in the strong PMA layer should happen near 30 mT. Thus, at remanence, a few 3D stripes are visible as well as some elongated pairs of cocoons which correspond to the smaller objects. Finally, decreasing the magnetic field below zero does not induce the nucleation of skyrmionic cocoons: the 3D stripes simply shrink and disappear which is a significant difference with the 300 K evolution.

To complement the holography measurements, MFM measurements have been performed without an external field at room temperature and at 77K (Figure 7.8.b). As the latter is more sensitive to the magnetization close to the surface, it can be correlated with the holography results that relate to the average magnetization of all the magnetic layers. At 300 K, the same characteristic pattern of stripes with smaller white dots in between is obtained. As the signal is originated from the divergence of the stray field, the weakly contrasted dots have to correspond to skyrmionic cocoons present in the top gradient of the structure and not to fully columnar skyrmions. Thus, it follows that the isolated objects detected near remanence in HERALDO are paired cocoons. At 80 K, the phase signal mainly detects the domain walls which could be due to the magnetization of the tip flipping or due to an increase of the sample's anisotropy [START_REF] Belliard | Investigation of the domain contrast in magnetic force microscopy[END_REF]. However, it still displays two types of structures and appear similar to its high temperature counterparts. This can be expected as no magnetic field was applied during or after the cool down so that the magnetic state was left undisturbed, i.e. frozen in, and might be metastable. In To verify the impact of anisotropy and saturation magnetization variations at low temperatures (using K u = 0.93 MJ/m 3 and M S = 1.26 MA/m), simulations were conducted. In Figure 7.10, we show simulated holography images obtained by averaging m z along the vertical direction, weighted by the thickness of each magnetic layer, for the magnetic parameters corresponding to 80 and 300 K. This was done for various magnetic fields, starting from positive saturation and going down to negative saturation for both temperatures, similar to the experimental sequence. At 300 K, the simulated behavior is similar to the experimental results. Initially, numerous cocoons nucleated as shown by the weakly contrasted textures at 300 mT. Some cocoons were aligned horizontally (one on top of the other), as evidenced by stronger contrast. Decreasing the field increased the number and size of cocoons until they formed pairs (one in each gradient of the multilayer). At remanence, paired cocoons are observed along with a 3D worm that extended over the full thickness, as indicated by the black contrast. Finally, decreasing the magnetic field on the opposite side of the hysteresis leads to the creation of isolated cocoons (paired or not) with the opposite polarity.

As expected, the evolution at 80 K differs due to the change in magnetic parameters. Nucleation occurs at a lower field, close to 200 mT, and the cocoons appear to be larger and less numerous than in the 300 K case. The nucleation in the strong PMA layers occurs at 25 mT numerically which is close to the transition observed in transport measurements (Figure 7.9.a). Note that the nucleation is assisted by inserting a skyrmion in the middle layers at different fields. At remanence, a 3D worm is observed next to cocoons larger than at 300 K. The density of cocoons seems smaller than at 300 K but still somewhat higher than experimentally observed, which may arise from uncertainty in the magnetic parameters or the numerical nucleation process. The same types of contrast are visible, corresponding to either single cocoons, paired cocoons, or a 3D worm. At negative fields, the simulations are consistent with the experiments. A few cocoons of polarity are still visible at -100 mT, but no other nucleation occurs once they disappear at lower fields. This is a significant difference from the measurements and simulations at 300 K, which is numerically reproduced here.

Cocoon size distribution

Thanks to the large field of view (2 µm), the object hole accommodates many features which have been fitted, when possible, with a two-dimensional Gaussian model for every field. An example with the coexistence of isolated and paired cocoons is studied in Figure 7.11.a, at 300 K and 240 mT, as shown in the inset. The average full width at half maximum (FWHM) and the amplitude of the Gaussian for each object are displayed in the histograms and the right panel shows a typical fit that was performed on a cocoon. The average sizes are ranging from 50 nm up to 130 nm diameter (coherent with the 90 nm diameter estimated from magnetotransport measurements Subsubsection 6.2.3.4). Similarly, the amplitudes spans different values, the lowest corresponds to single cocoons whereas the highest are more likely to be associated with paired cocoons. Then for each magnetic field, the quartiles of the average FWHM has been extracted and shown in Figure 7.11.b at 300 and 80 K, where the 1 st and 3 rd quartiles serve as error bars to frame the median value while the color encodes the density. The characteristic diameter at low temperature displays a stronger dispersion, reaching sizes above 200 nm whereas at room temperature it mostly stays below 130 nm. In both cases, we can notice the almost linear increase of the size when decreasing the field from positive saturation.

Probing at different energy edges

To further ensure the 3D nature and the coexistence of different spin textures, DG multilayers with different ferromagnetic elements have been optimized. Thanks to the energy tunability of the synchrotron light, it allows to probe different vertical regions of the multilayers. In this case, the gradient parts of the DG structure contain only Co (S = 0.1 nm, X 1 = 1.8 nm, N = 13 layers, t N Pt = 2 nm) while the middle part also has some permalloy (NiFe). More precisely, it consists of (Co0.3|NiFe1.1|Al1.4|Pt3) 15 with the seed of Co necessary for the PMA. More details on the anisotropy and the field dependency of such multilayers are given in Appendix D.

Given the structure of the DG multilayer, using X-rays near the Co L 3 edge (778.6 eV) yields a magnetic contrast mostly representative of the gradient parts while near the Ni L 3 edge (854.7 eV) it relates only to the strong PMA layers. In Figure 7.12, the field dependency after an OOP saturation is given for the two different energies. At remanence, at the Co edge, worms are observed along with small white dots in between. However, at the same field for the Ni edge, only the worms are visible.

As it corresponds to the middle layers, it means that the worms are stabilized over the full thickness of the sample while the dots are not present in the strong PMA layers. They thus correspond to skyrmionic cocoons that are either in the bottom or in the top part of the structure. This confirms that they have a vertical extension at least limited by the strong PMA layers.

After increasing the field (200 mT), the 3D worms and the cocoons reduce in size, the latter being only visible at the Co edge. At 300 mT, skyrmionic cocoons are still stable along with columnar skyrmions, easily identifiable thanks to the image at the Ni edge. Finally, near saturation, the middle layers are saturated while cocoons are still stable in the remaining magnetic layers. Two contrasts can be seen at the Co edge, indicating the presence of either pair of cocoons or single ones.

Probing Co and Ni edges

The HERALDO measurements on DG multilayer presenting two distinct ferromagnetic elements (Co and NiFe) demonstrate unequivocally that the skyrmionic cocoons do not extend in the middle layers. At higher field, it also evidences their coexistence with columnar skyrmions that extend in all the layers.

Three-dimensional reconstruction: X-ray laminography

Even though the 3D nature of the skyrmionic cocoons has been proved and observed using many concomitants techniques, the precise magnetization distribution has not been obtained experimentally.

To overcome this shortcoming, X-ray laminography, a generalized approach to tomography, has been used to reconstruct the vectorial field of the magnetization.

Introducing 3D tomography

Laminography principle

X-ray laminography is a rapidly developing technique making use of soft X-rays to recover the 3D distribution of the magnetization. With it, remarkable reconstructions have already been achieved in bulk magnets [START_REF] Witte | From 2D STXM to 3D imaging: Soft x-ray laminography of thin specimens[END_REF][START_REF] Donnelly | Three-dimensional magnetization structures revealed with X-ray vector nanotomography[END_REF][START_REF] Donnelly | Experimental observation of vortex rings in a bulk magnet[END_REF] or in ferrimagnets [START_REF] Hermosa-Muñoz | 3D magnetic configuration of ferrimagnetic multilayers with competing interactions visualized by soft X-ray vector tomography[END_REF][START_REF] Hermosa | Bloch points and topological dipoles observed by X-ray vector magnetic tomography in a ferromagnetic microstructure[END_REF] or even for the time-resolved study of vortex [START_REF] Finizio | Three-dimensional vortex gyration dynamics unraveled by time-resolved soft x-ray laminography with freely selectable excitation frequencies[END_REF]. Recently, a magnetic skyrmion has also been reconstructed in a magnetic multilayer [START_REF] Raftrey | Quantifying the topology of magnetic skyrmions in three dimensions[END_REF]. It relies on the acquisition of many projections measured at different incidence angles which, coupled to a reconstruction algorithm, allow to recover the magnetization vectorial field. As shown in Figure 7.13.a, the sample plane is tilted at an incidence angle θ = 45 • relative to the beam. Due to the tilted incidence angle, the X-rays probe as much out-of-plane component than in-plane ones, along the k vector of the incident photon. Then, STXM projections are acquired at different rotation angles around the laminography axis. Measuring along a 360 rotation allows the full components of the magnetization to be probed. In Figure 7.13.b, those projections are shown in Fourier space for the tomography and the laminography. The blue shape corresponds to the so-called missing wedge or cone, i.e. the volume that cannot be measured experimentally due to the setup constraints. For the laminography, the missing cone mostly corresponds to information along the z axis which thus should have a smaller resolution than the in-plane ones. Its volume is however much smaller than for the tomography case.

Reconstruction principle

The reconstruction of the magnetization is a simple iterative process based on a gradient descent process [START_REF] Donnelly | Tomographic reconstruction of a three-dimensional magnetization vector field[END_REF][START_REF] Di | Threedimensional tomographic imaging of the magnetization vector field using Fourier transform holography[END_REF]. Let x, y be the coordinates of the detector (the image) plane. At each step, the following error metric is to be minimized:

ϵ n = ϕ x,y P ϕ n (x, y) -P ϕ exp (x, y) 2 (7.12)
with ϕ the rotation angle, P ϕ exp (x, y) the measured set of projections and P ϕ n (x, y) the current guess projection:

P ϕ n (x, y) = z R ϕ m n • z (7.13)
with R ϕ the rotation matrix. Then, the current magnetization field can be updated as:

m n+1 = m n -β ∂ϵ ∂m n (7.14)
with β the convergence parameter. This process is repeated until reaching the chosen convergence criterion, yielding a magnetization guess whose projection correspond to the experimental ones.

Sample preparation

As STXM is a transmission technique, the multilayers have been grown on a Si 3 N 4 membrane (200 nm thick, 0.5 × 0.5 mm 2 ). To proceed to the reconstruction, the algorithm requires projections that have been aligned thanks to the topography which would not be possible with plain films. Thus, the multilayers have been patterned into nano or micro dots using electron-beam lithography. The presence of sharp boundaries allows to easily align the different projections. Two DG multilayers have been prepared for those experiments. The first one is similar to the one studied with HERALDO (S = 0.1 nm, X 1 = 2.0 nm, N = 13 layers, M = 15 repetitions, Y = 1 nm, t N Pt = 3 nm and t M Pt = 3 nm), while the second one has a step of 0.2 nm (S = 0.2 nm, X 1 = 1.7 nm, N = 13 layers, M = 15 repetitions, Y = 1 nm, t N Pt = 2 nm and t M Pt = 3 nm) to check the impact of the confinement. We call the former one DG1 and the latter DG2.

The experiments have been realized on the PolLux beamline at PSI synchrotron (Switzerland) at the Co L 2 edge (796 eV). For the reconstruction, we benefited from the help of Claire Donnelly (Max Planck Institute for Chemical Physics of Solids, Dresden, Germany) and used her algorithm implemented on the beamline [START_REF] Donnelly | Tomographic reconstruction of a three-dimensional magnetization vector field[END_REF]. The reconstruction parameters are discussed in more details in Appendix I along with the experimental parameters. The multilayer under consideration has a considerable amount of Pt which limits the X-ray transmission and is also relatively thick, hence the weak contrast. However, because of the limited resolution along the vertical axis, its structure is a trade-off to ensure that we have 3D textures sufficiently elongated to be resolved by the optics of the laminography setup.

Reconstructed magnetization field in a DG multilayer

We first focus on the DG1 structure (S = 0.1 nm, X 1 = 2.0 nm, N = 13 layers, M = 15 repetitions, Y = 1 nm, t N Pt = 3 nm and t M Pt = 3 nm).

2D magnetic state

In Figure 7.15, the magnetic state after an OOP saturation is presented for different cases at remanence. First, we have the characteristic magnetic state made of 3D worms and skyrmionic cocoons in the plain film observed with MFM. A similar state is measured in a microdot (

X-ray laminography of skyrmionic cocoons

The reconstruction of the magnetization distribution obtained with the X-ray laminography experiments is the first 3D experimental observation of skyrmionic cocoons and 3D worms in magnetic multilayers. It unambiguously confirms the 3D nature of cocoons that was inferred from micromagnetic simulations and other concomitants experimental techniques (MFM, transport, holography, STXM...).

with α the vorticity component, ϵ αβγ the Levi-Civita tensor summed over the three components (x, y, z). The integral of the magnetic vorticity over a closed loop corresponds to the 3D topological number (Eq. 4.3). Near singularities such as Bloch points, the magnetic vorticity suddenly changes sign. Thus to identify such singularities, we consider the extrema of the divergence of Ω.

In The reconstruction assumes that magnetization is continuous in the material under consideration whereas the real medium is discontinuous as it is a multilayer. Thus, the reconstructed magnetization is averaged over the magnetic and non-magnetic layers and since reversal of the magnetization occurs, Bloch points are expected to appear. However, they might not be real as in reality they are likely to be located in non-magnetic layer according to the micromagnetic simulations (see Subsection 4.1.2). As the resolution estimated with FSC is well above the interlayer distance, their precise location cannot be determined experimentally.

Simulations based on the experimental reconstructions

To exploit the laminography measurements further, the reconstructed magnetization was used as input for micromagnetic simulations. To this end, the magnetization was simply interpolated to recover a finer grid. Due to the large size of the reconstructed magnetization, 5 × 5 nm 2 cells were used for simulating a cylinder of 2.42 µm diameter without periodic boundary conditions. As the real dot is in reality 3.5 µm in diameter, differences are expected near the edge as the boundary conditions are different. This effect should be negligible near the central area. Moreover, the chirality of the domain wall do not resemble the experimental one. For a skyrmionic cocoon, it switches from CW Néel to non-chiral to CCW Néel from top to bottom, similarly to previous simulations (see Section 5.2). The laminography technique might thus not be sufficient to resolve the domain walls with accuracy, whether it is a lack of resolution or an averaging phenomenon that cancels the Néel parts in transmission. However, the fact that the simulations display similar spin textures and do not change significantly from the reconstructed magnetization supports the validity of the latter.

In Appendix J, the field dependency inside the micro dots is discussed, evidencing the impact of the edge effects.

7.2.4

Second reconstruction: DG with step 0.2 nm

Chapter 7 Summary

Advanced X-ray techniques have been employed in double gradient multilayers to investigate the 3D spin textures. Holography measurements, such as FTH or HERALDO, reveal distinct contrasts in transmission, directly associated with 3D worms and skyrmionic cocoons. Magnetic transitions and interactions are detected, providing evidence for the 3D nature of the cocoons. The attractive interaction promotes their horizontal alignment, minimizing the dipolar energy and indicating their presence at different vertical levels. HERALDO size analysis shows that the typical median diameter of cocoons is approximately 100 nm across the entire field range. Additionally, using X-rays at two energy edges in optimized structures with different ferromagnetic elements confirms the limited vertical extension of cocoons, which terminates at strong PMA layers. STXM characterization of larger areas yields similar conclusions. Three contrasts are observed at different magnetic fields, corresponding to three magnetic configurations: 3D worms, paired cocoons, or single cocoons. Initial investigation of current pulses indicates that magnetic transitions can be achieved depending on the field and voltage, with partial worms dividing into multiple skyrmionic cocoons for instance. Further research is required to fully understand the impact of current in more motion-optimized DG multilayers for future applications. Furthermore, the first 3D reconstruction of skyrmionic cocoons in DG multilayers is accomplished through X-ray laminography. This breakthrough confirms the coexistence and 3D nature of worms and cocoons in microdots at remanence, providing direct experimental evidence for their presence.

8

C o n c l .

Stabilization and characterization of 3D spin textures in magnetic multilayers

O

ver the past few years, there has been a remarkable surge of interest in the field of 3D nanomagnetism. This field focuses on magnetic nanostructures and intricate spin textures that extend beyond two effective dimensions and vary along the third dimension, perpendicular to the plane of the magnetic film. Before the start of my PhD, pioneering demonstrations of 3D magnetic textures such as bobbers [START_REF] Redies | Distinct magnetotransport and orbital fingerprints of chiral bobbers[END_REF]1], torons [START_REF] Ao Leonov | Homogeneous and heterogeneous nucleation of skyrmions in thin layers of cubic helimagnets[END_REF]2], hopfions [START_REF] Liu | Binding a hopfion in a chiral magnet nanodisk[END_REF] or truncated skyrmions [START_REF] Mandru | Coexistence of distinct skyrmion phases observed in hybrid ferromagnetic/ferrimagnetic multilayers[END_REF][START_REF] Oguz Yildirim | Tuning the coexistence regime of incomplete and tubular skyrmions in ferro/ferri/ferromagnetic trilayers[END_REF] had been demonstrated or predicted. However, significant challenges still needed to be overcome, particularly for their characterization and their stabilization in functional materials. Most of these newly discovered textures have been detected in chiral magnetic crystals or nanostructures, which strongly limits their potential applications. Additionally, characterizing 3D magnetic textures was and remains problematic as it requires probing the magnetization over the thickness. This task often relies on high-end X-ray techniques that are rapidly developing but needs access to synchrotron light. Within this context, my PhD research has focused on stabilizing and characterizing novel 3D spin textures in magnetic multilayers. Multilayers have emerged as a prominent platform for engineering new magnetic objects due to their flexibility and the ability to control their magnetic properties.

By carefully selecting materials and adjusting the parameters of the layers (such as thicknesses and interfaces), it becomes possible to exert a considerable degree of control over the magnetic interactions, providing a great malleability to engineer 3D magnetic objects. Once the stabilization of new 3D spin textures, namely the skyrmionic cocoons, has been achieved, their characterization was thoroughly pursued by many concomitant approaches (see Figure 8.1). This research aligns with the current trend of developing techniques to characterize more easily the third dimension of the magnetization, which remains a challenging endeavor even today.

Summary

In Chapter 3, I have provided an introduction to magnetic multilayers, focusing on growth and characterization techniques. Emphasis has been placed on investigating interface quality and material properties through various structural methods. The aim was to ensure minimal intermixing and prevent contamination. Then we have delved into the magnetic properties of the Pt|Co|Al trilayer, serving as a foundation for most multilayers in the thesis. Additionally, a first experimental and numerical investigation of aperiodic multilayers with variable ferromagnetic thickness have been conducted to explore the impact of non-uniform anisotropy. A noteworthy structure, the single gradient (SG) In Chapter 4, I have offered an in-depth characterization of the SG structure using magnetic force microscopy (MFM) and micromagnetic simulations. The research has primarily revolved around understanding its field dependency and the influence of the structure parameters. A novel spin texture, termed the skyrmionic cocoon, has been put forward, featuring an ellipsoidal shape with limited vertical extension between the top and bottom magnetic layers. Those numerical predictions have been confirmed by the signal decrease in MFM with the field variation. A study of the energetics of the system has revealed that those objects are mainly stabilized thanks to the dipolar interaction, overcoming the effective in-plane anisotropy. The topology of the cocoons has been discussed, using a 2D approach to respect as much as possible the mathematical premises of topology, and has yielded similar properties as for the "standard" skyrmion stabilized in periodic multilayers. We have further investigated their chirality through X-ray resonant magnetic scattering (XRMS) experiments, although the quantitative, and even qualitative analysis of the signal remains a challenge due to the multiple inversions of the dichroism.

Expanding on the SG structure, I have introduced in Chapter 5 the double gradient (DG) multilayer, enabling the coexistence of skyrmionic cocoons and more conventional columnar skyrmions. It relies on the use of strong perpendicular magnetic (PMA) layers to modulate the textures over the films normal direction. This new structure has been characterized, similarly to the SG one, with extensive MFM studies combined with micromagnetic simulations. The presence of two contrasts in the former has been related to the two objects existing in the simulations, the 3D worms (or skyrmion) and the cocoons. The parameter space of the DG multilayer (thicknesses, step, spacer) has been partially explored with MFM to understand qualitatively the changes induces by those variations. For instance, for low thicknesses of Co, the coexistence of skyrmions and cocoons is harder to obtain than for higher thicknesses (above the spin reorientation thickness). The chirality has also been probed with XRMS but the signal (with 4 or 6 nodes) has proven to complex to be analyzed without numerical support.

Considering the complexity of characterizing 3D spin textures, magnetotransport measurements are utilized in Chapter 6 to probe the average magnetization. These measurements have been successfully correlated with micromagnetic simulations with a high accuracy (a few percents deviation), suggesting that the textures are precisely captured. Therefore, the presence of multiple spin textures with different vertical extensions is convincingly inferred. The magnetotransport measurements have also been used to dress a magnetic phase diagram based on the transitions visible in the derivative of the longitudinal and transversal resistances. Finally, it has been shown that the different spin textures can be detected based on their electrical signatures using Hall measurements with or without complementary imaging, a valuable property for potential applications. Finally, in Chapter 7, I have focused on synchrotron measurements using soft X-rays, enabling the retrieval of information about the full magnetization. Various transmission experiments, including holography and scanning transmission X-ray microscopy, are performed to study DG structures at different magnetic fields and X-ray energies. Notably, reconstructions of the magnetization for a DG multilayer with different ferromagnets (Co, NiFe) have unambiguously confirmed the limited vertical extension of the cocoons. Moreover, experimental reconstruction of the 3D magnetization distribution from laminography experiments solidifies the existence of skyrmionic cocoons and their coexistence with other spin textures. This unique property holds significant potential for various 3D skyrmionic applications.

In conclusion, this thesis has presented highly promising spin textures, meticulously studied through numerical simulations and experimental investigations. The methodologies applied here can readily be adapted to explore other 3D textures that may be stabilized in other innovative multilayer architectures.

Notably, the complex magnetic multilayers, such as the DG structure, hold great potential for advancing 3D skyrmionic devices. The tunability of these multilayers, coupled with the degree of freedom offered by the third dimension, opens up exciting possibilities for future developments in this field.

Perspectives

The exploration of 3D nanomagnetism holds great promise due to its added dimensionality and increased degrees of freedom. In our research on magnetic multilayers, we have successfully stabilized a particular 3D object called the skyrmionic cocoon. The cocoon's confinement can be controlled by adjusting the thickness of the strong perpendicular magnetic anisotropy (PMA) layers. Building upon this tunability, there is potential to discover and stabilize new 3D objects with innovative multilayer architectures, such as the split gradient or the encapsulated gradient (discussed in Chapter 5). So far, our focus has predominantly been on the Pt|Co|Al system. However, the knowledge gained from this investigation can be extrapolated to other multilayer stacks, like Pt|NiFe|Al, as demonstrated through holography. Exploring different metals or considering antiferromagnetic couplings may lead to the discovery of other intriguing properties. For instance, cocoons could be antiferromagnetically coupled, either within one cocoon or between two halves of a single cocoon. These possibilities open up new avenues for research and could unveil novel phenomena in the field of 3D nanomagnetism.

Extensive investigations of the static properties of spin textures in aperiodic multilayers (SG and DG) have been conducted using various complementary approaches (see Figure 8.1), confirming the 3D nature of the cocoons. Now, it would follow to examine their electrical properties, both in terms of detection and current-induced dynamics. Preliminary studies have shown promising results for their detection, but further research is crucial to explore their potential applications. Additionally, understanding and controlling their motion is essential for optimizing future devices. In DG structures, which typically involve a large number of repetitions (e.g., 41 repetitions), it may be necessary to reduce the overall thickness to minimize losses during current pulses. For this purpose, DG examples with fewer repetitions (e.g., 15 repetitions as shown in Chapter 5) may be more suitable and worth exploring.

Additionally, it would be interesting to have a dominant sense of rotation in order to facilitate motion 1which could be achieved by reducing the number of layers or using different materials (to modulate the DMI strength). To develop reliable devices, it is imperative to control the nucleation of cocoons and columnar skyrmions as well. Several approaches can be considered, such as using notches in tracks or a sharp triangular contact layer to locally increase the current density. Additionally, precise control over the vertical position of nucleation is essential.

If motion and nucleation can be controlled, a racetrack structure with multiple memory states becomes possible, leveraging the 3 rd dimension of the system, as illustrated in Figure 8.2. For instance, a single cocoon could encode two states depending on its vertical position, a pair of cocoons could represent another state, and a columnar skyrmion could encode a fourth state. Combining these with the state without any spin texture, a DG multilayer could offer five distinct states. Furthermore, by stacking multiple SG structures together, the tunability of the multilayer allows for additional vertical positions for cocoons, potentially leading to a scalable structure with even more memory states. This scalability is a crucial feature for potential applications. As we delve into 3D nanomagnetism, we find ourselves exploring uncharted territories due to the increased number of degree of freedom. Apart from the practical applications, there is much to be understood from a theoretical perspective. Techniques such as holography imaging and magnetotransport have revealed various magnetic transitions, such as those between skyrmions and cocoons or cocoons and 3D worms. To predict the behavior and instabilities of these intriguing spin textures, a more detailed understanding of these transitions is essential. One approach to achieve this understanding is through studying the paths of minimal energies, often employing the geodesic nudged elastic bands (GNEB) method [START_REF] Pavel F Bessarab | Method for finding mechanism and activation energy of magnetic transitions, applied to skyrmion and antivortex annihilation[END_REF]. By analyzing the energy landscape and transition pathways, we can gain insights into how the spin textures transform and interact with one another. For example, comprehending these transitions could enable the controlled merging of two cocoons into a skyrmion or vice versa. It is worth noting that, unlike 2D objects, multilayers with spatially separated layers, hence relieving the constraint of a continuous magnetization texture, cannot be studied using the topological considerations used in continuous 3D magnets. Consequently, their topological properties cannot be adequately addressed, a shortcoming that can be interesting to explore. Understanding and characterizing the topological aspects would be crucial in understanding better the properties of these spin configurations.

In summary, the magnetic multilayers explored in this thesis mark a crucial milestone in advancing the engineering of 3D spin textures and achieving control over their position along the normal of the films. Leveraging the tunability of these multilayers, a wide array of structures can be envisioned beyond the aperiodic ones studied thus far. The next natural progression is to delve into the current-induced properties of the skyrmionic cocoons, specifically their nucleation and motion, as this knowledge is pivotal in moving towards potential applications like a racetrack memory with multiple states. By further investigating these aspects, we can pave the way for exciting advancements and practical implementations in this field. wall cut is plotted for simulations conducted with a cell size of 1 nm (black) and 2 nm (red). As anticipated, using a smaller cell size ensures a smoother rotation of the magnetization, with a maximum rotation of 14 • . For the larger cell size, the largest rotation is around 28 • , which remains below the acceptable limit. This analysis confirms that the micromagnetic simulations adhere to the required smooth rotation criteria, even in the presence of narrow domain walls in the strong PMA layers of the DG structure.

A.2 Example of a typical script

To facilitate the replication process, we offer a detailed example that illustrates the construction of an aperiodic magnetic multilayer using the dilution technique. The provided script demonstrates the creation of a DG structure, starting from an out-of-plane initialization with added noise, and compute the relaxed magnetic states for different external field. As the simulation cells possess fixed dimensions throughout the simulation space, we utilize the dilution trick to define the magnetic parameters for ferromagnetic layers with varying thickness. This example can be easily adapted to study an SG structure as well.

Definition of parameters and geometry

The initial step involves setting the magnetic parameters, including exchange, saturation magnetization, anisotropy, and DMI, as well as defining the thicknesses and properties of the layers in the multilayer structure. Subsequently, the geometry of the simulation space is established. The layer thicknesses are determined based on the actual interlayer distance and a reference thickness that corresponds to a dilution factor of 1. It should thus correspond to the thickness for which the exchange and the saturation magnetization were measured.

D

A p p e n d i x

Optimization of NiFe aperiodic multilayers

In order to leverage the energy sensitivity of XMCD measurements, multilayers incorporating permalloy (NiFe) were optimized. By utilizing X-rays, different contrasts can be achieved when probing at different absorption edges. To demonstrate the 3D nature of skyrmionic cocoons unequivocally, this approach was employed in DG multilayers. Specifically, Co was utilized in the SG blocks, while NiFe was chosen as the material for the strong PMA layers (see Subsection 7.1.6). In this section, we present the optimization of such structures. Although various compositions of NiFe exist, only a single composition was used for the growth of these multilayers: 80% Ni and 20% Fe.

D.1 Obtaining PMA in thin NiFe layers

For the composition of interest to us, NiFe is a soft magnet characterized by a near zero-uniaxial anisotropy [START_REF] Bozorth | The permalloy problem[END_REF][START_REF] Waeckerlé | Evolution and recent developments of 80% Ni permalloys[END_REF]. In Figure D.1, some hysteresis loops are presented which corresponds to three repetitions of the Pt|NiFe|Al stackings. This system was selected to check the evolution of the anisotropy with the NiFe thickness (t NiFe ). For the three thicknesses measured here (0.8, 1.1 and 1.4 nm), the anisotropy does not display any easy axis, effectively corresponding to a near zero anisotropy. This poses a challenge for stabilizing a strong perpendicular anisotropy, which is necessary for our DG structures. To induce perpendicular anisotropy, proximity effects can be utilized by incorporating a thin Co seed layer, even with a thickness less than an atomic layer, inspired by the approach described in [START_REF] Juge | Skyrmions in synthetic antiferromagnets and their nucleation via electrical current and ultra-fast laser illumination[END_REF].

In

Processing X-ray holograms

In this appendix, the methods used to process the holograms are presented for Fourier transform holography (FTH) and X-ray holography with extended reference by autocorrelation linear differential operator (HERALDO).

G.1 Preparation of the holograms

Starting from the raw data, we develop here the main first steps common to both the FTH and the HERALDO holograms, applied for each helicity.

1. Centering the hologram. It is primordial to properly center the hologram as this defines the origin of the Fourier space, i.e. where the zero frequency is. If the hologram is not adequately positioned, artefacts will be visible in the final image.

2. Normalization by the background. This step is done to ensure that the two helicities have a similar signal (which might be dissimilar depending on the degree of polarization of the light). Align the mean XMCD Fluctuations are obtained for the XMCD signal between different projections which might be linked to instabilities of the beam or experimental problems. To reduce this phenomenon, the projections were corrected to obtain a similar average XMCD in the central part of the dot under consideration.

I.2 Redeposition from etching

In the individual projections of STXM with a 45 In the first zone (blue), the redeposited materials act like a screen to the X-rays as they still propagate in the multilayer afterward, yielding a weaker contrast. In zone 2 (red), the X-rays directly go through the multilayer, unobstructed by the edges. Finally, in the zone 3 (green), only the resist edges are probed with a non-zero magnetic signal.

J

A p p e n d i x

Edge effects in aperiodic multilayers

Using the laminography reconstruction, a magnetization tensor can be extrapolated to use in the micromagnetic simulations (going from 20 × 20 × 20 nm 3 voxels to 2 × 2 × h nm 3 depending on the interlayer spacer). The corresponding relaxed magnetic state was presented in The density of cocoons increases with the field as the 3D worms are shrinking. At 200 mT, an effect of edge repulsion starts to be visible as most spin textures are disconnected from it. Increasing the field further strengthen this effect and at 300 mT, the cocoons and columnar skyrmions that are stabilized are at least 500 nm away from the edge. The dimension of the dot are relatively large but it evidences the impact of the border on the textures in its vicinity.

Figure 1 -

 1 Figure 1 -Skyrmions et nanomagnétisme 3D. a) Distribution (simplifiée) des spins pour un skyrmion colonnaire. b) Structure typique d'une multicouche magnétique. FM : ferromagnétique. HM : métal lourd. c) Distribution schématique des spins en 2D et 3D. d) Exemples de structures magnétiques en 3D, de gauche à droite : bobber [1], toron [2] et hopfion [3].

Figure 2 -

 2 Figure 2 -Dépendance sous champ magnétique hors du plan pour un simple gradient. a) Mesures MFM à différents champs magnétiques. b) Simulations micromagnétiques correspondantes représentées avec des isosurfaces (en rouge, m z = -0.8, en blanc, m z = 0 et en bleu foncé, m z = 1).

Figure 3 -Figure 4 -

 34 Figure 3 -Dépendance sous champ magnétique hors du plan pour un double gradient. a) Mesures MFM à différents champs magnétiques. b) Simulations micromagnétiques correspondantes représentées avec des isosurfaces (en rouge, m z = -0.8, en blanc, m z = 0 et en bleu foncé, m z = 1).

Figure 1 . 1 -

 11 Figure 1.1 -List of spintronics devices categorized in two depending on the prime effects they are based on: TMR/GMR (Mott type) or spin-orbit interactions (Dirac type). From [4].

Figure 1 . 2 -

 12 Figure 1.2 -Basics of skyrmions. a) Illustration of a Néel skyrmion with a negative core. b) Schematic

Figure 1 . 3 -

 13 Figure 1.3 -Towards skyrmion-based devices. a) Schematic illustration of a racetrack memory [40]:

Figure 1 . 4 -

 14 Figure 1.4 -Emergence of 3D nanomagnetism. a) Examples of geometries illustrating the possibilities

Figure 1 . 5 -

 15 Figure 1.5 -Zoology of 3D spin textures. a) Comparative illustration of a bobber next to a skyrmion

Figure 1 . 6 -

 16 Figure 1.6 -Three-dimensional experimental reconstructions of the magnetization. a) Reconstruction

2. 3 . 1 X 25 2. 3 . 2 X 28 I

 31253228 -ray absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -ray Magnetic Circular Dichroism (XMCD) . . . . . . . . . . . . . . . . . . . . . . . . . . n this chapter, I introduce the theoretical notions necessary to understanding the results and discussions presented in this thesis. The dynamic equation of the magnetization is first presented along with the different energy terms taken in considerations in thin films and magnetic multilayers.

Figure 2 . 1 -

 21 Figure 2.1 -Illustration of the magnetization dynamics from the LLG equation. a) Precessional, b) dissipative and c) combined motion. From [89].

Figure 2 . 2 -

 22 Figure 2.2 -Demagnetizing (stray) field illustration. a) Vector fields of the magnetic flux B = µ 0 (H demag + M ) with schematical surface charges σ m . b) Different magnetic domains configurations illustrating the stray field importance, ordered from the highest to lowest dipolar energy (left to right).

Figure 2 .

 2 Figure 2.2.b, different ideal cases are represented with their stray field. The right-most distribution of the magnetization limits most efficiently the dipolar energy. However, it is not ideal regarding the other interactions, notably because of the symmetric exchange cost of the domain walls.

  ij the Liftshitz invariants and D the micromagnetic DMI constant that depends on the thickness. The chirality encouraged by the DMI lies within the sign of D: for negative D (resp. positive), the counterclockwise (resp. clockwise) rotation of the magnetization is favored.

Figure 2 . 3 -

 23 Figure 2.3 -Dzyaloshinskii-Moriya interaction (DMI). a) Illustration of the DMI arising at the interfaces between a ferromagnetic element (gray) and a heavy metal (blue). From [10]. b) Importance of the stacking order: the chirality reverses with the stack (CW: clockwise, CCW: counterclockwise). c) Engineering the DMI in magnetic multilayers: additive contributions of two different heavy metals (HM) layers encapsulating the ferromagnetic (FM) one.

Figure 2 . 4 -

 24 Figure 2.4 -Skyrmion type depending on their core polarity P and their helicity ϕ h . Adapted from [107].

  Referred to as a skyrmion tube or columnar skyrmion, the chirality of the skyrmion can change throughout the structure depending on the relative strengths of the DMI and the dipolar interaction. The DMI, originating from interfacial effects, promotes a Néel chirality, either counterclockwise (CCW) when the DMI is negative or clockwise (CW) otherwise. For the Pt|Co|Al stacking (written from bottom to top layer), extensively studied in this thesis, the DMI favors CCW Néel walls as D < 0. In order to minimize the dipolar energy, it is always energetically favorable to introduce CW Néel caps in the top layers and CCW ones in the bottom layers, as depicted in Figure 2.5. In the intermediate layers, Bloch walls are formed as a transitional structure between the two rotational senses. The specific distribution of CW and CCW Néel walls depends on the relative strengths of the DMI and the dipolar interaction [108]. In the absence of DMI, the dipolar interaction would lead to a symmetric distribution of CW and CCW Néel walls (Figure 2.5.a). However, as the magnitude of D increases in absolute value, the proportion of CCW Néel walls increases (Figure 2.5.b).

Figure 2 . 5 -

 25 Figure 2.5 -Vertical profile of a hybrid skyrmion evidencing the flux closure with the evolution of the domain wall chirality for a) D ∼ 0 and b) D < 0.

Figure 2 . 6 -

 26 Figure 2.6 -Illustration of the winding number w S 1 for mappings π 1 (S 1 ) (S 1 → S 1 ) counting the

Figure 2 . 7 -

 27 Figure 2.7 -Singular topological defects for different dimensionalities of the real (d) and spin space (p). Defects exist for d ′ = d -p ≥ 0. Adapted from [111].

Figure 2 . 8 -

 28 Figure 2.8 -Different Bloch points configurations topologically equivalent. Adapted from [113].

Figure 2 . 9 -

 29 Figure 2.9 -Smooth magnetic textures of the π 2 (S 2 ) homotopy group. a) Anti-skyrmion. b) Trivial

Figure 2 . 10 -

 210 Figure 2.10 -Total scattering cross section (and mean free path) in Co arising from different processes: coherent (Rayleigh), incoherent (Compton), photo-absorption (σ ph ) and pair or triplet production. Adapted from [114].

Figure 2 . 11 -

 211 Figure 2.11 -Illustration of the XMCD two-steps model. Adapted from [119].

Figure 3 . 1 -

 31 Figure 3.1 -Introduction to magnetron sputtering. a) Schematic process of deposition. b) Illustration of the magnetic confinement improving the plasma stabilization in a planar magnetron. c) Target operating at high temperature with a glowing plasma. (b-c) extracted from [124].

3. 1 .

 1 Precise and controlled growths via magnetron sputtering coupled to a ring of magnets on the outside. It allows to confine the secondary electrons on a given trajectory (see Figure3.1.b), parallel to the surface, which increases the ionization efficiency, allowing to reach higher deposition rates. Interestingly, this confinement effect can be clearly seen as the target is getting etched deeper and deeper (see Figure3.1.c).

Figure 3 . 2 -

 32 Figure 3.2 -Reflectivity curve on a thin film. 100 seconds deposition of Pt on SiO 2 (2.5 mbar, 180 mA, λ = 1.5409 Å). The inset shows the Fourier transform of the counts used to determine the thickness. The schematic represents the sample structure with exaggerated roughness.

Figure 3 . 3 -

 33 Figure 3.3 -Reflectivity curves and fits of magnetic multilayers. a) For Pt5|(Pt3|Co1.7|Ru1.4) 5 |Pt3.

  7|Al1.4|Pt3, to evaluate the possible intermixing at the interfaces and characterize a potential oxidization of Co. The corresponding HAADF image (Figure 3.4.a), with and without filtering, highlights the different transitions. It appears that most interfaces are not perfectly flat as there are some height differences (close to 0.3 nm) which is consistent with the roughness extracted from XRR.

Figure 3 . 4 -

 34 Figure 3.4 -STEM measurements performed on Ta5|Pt8|Co0.7|Al1.4|Pt3. a) HAADF image (left) without and (right) with a high-pass filtering. b) Chemical profile along a vertical cut of the different elements probed with EDX. c) EDX maps of Ta, Pt, Co, Al and O from left to right. From [137].

Figure 3 . 5 -

 35 Figure 3.5 -3D chemical topography reconstructed with atomic probe. a) Multilayer prepared with FIB into a tip. Reconstruction of b) Pt5|(Pt3|Co1.8|Al1.4) 5 |Pt3 and c) Pt5|(Pt3|Co1.8|Ru1.4) 5 |Pt3. Each dots corresponds to an atom of the corresponding color-coded material.

Figure 3 . 6 -

 36 Figure 3.6 -Distribution of M S × t Co from OOP AGFM data measured by Yanis Sassi [131] in black and my own in orange.

Figure 3 .

 3 7.a, examples of hysteresis loops recorded with AGFM on Ta5|Pt3|(Co t Co |Al1.4|Pt3) 3 are shown. The signal is normalized by the value at saturation, i.e. M S , thus corresponding directly to

Figure 3 . 7 -

 37 Figure 3.7 -Effective anisotropy in Ta5|Pt3|(Co t Co |Al1.4|Pt3) 3 from AGFM measurements. a) Outof-plane hysteresis for different Co thicknesses (normalized to correspond to the vertical component of the magnetization m z .). b) In-plane (IP) and out-of-plane (OOP) hysteresis for t Co = 1.5 nm. c) Evolution of K eff × t Co using two different determination method of K eff .

Figure 3 .

 3 8.a, BLS spectra for different incidence angles are shown with two clearly defined Stokes and anti-Stokes peak in each case. In Figure 3.8.b, the same data is shown for θ = 60 • as a function of the absolute value of the frequency which evidences the shift (2f DMI ) between the two peaks. By fitting the width and the center of each peak with an asymmetric Lorentzian function, f DMI can thus be extracted for each incidence angle. This would be sufficient to obtain an estimate of the D value but for more accuracy it is extracted for all the different wave vectors, finally yielding f DMI = -0.05224 k sw (see Figure 3.8.c). Using Eq. 3.5, it follows that: D = -0.89 ± 0.05 mJ/m 2 =⇒ D s = D × t Co = -0.80 ± 0.05 pJ/m (3.6) using M S = 0.995 MA/m measured in this sample and γ = 184 GHz/T. The value is consistent, although slightly lower, with comparable studies in the recent literature based on domain wall velocities for which D s = 0.91 and 1.21 pJ/m [150] or D s = 0.87 pJ/m [151]. The Pt|Co|Al stacking thus gives rise to a non-negligible DMI which helps to stabilize non-collinear spin textures.

Figure 3 . 8 -

 38 Figure 3.8 -Extracting the DMI constant from BLS measurements in Ta5|Pt8|Co0.9|Al1|Pt3. a) Magnon BLS spectra at different incidence angles. b) Illustration of the shift f DMI for θ = 60 • . c) Linear fit of the frequency difference f DMI as a function of the magnon wave vector k sw .

Figure 3 . 9 -

 39 Figure 3.9 -Skyrmions in the multilayer Ta5|Pt8|(Co1.9|Al1.4|Pt3) 5 |Pt3. a) AGFM measurement of the OOP hysteresis loop, normalized to correspond to the vertical component of the magnetization m z . b) MFM phase maps measured at different OOP magnetic fields (see orange dotted lines).

  towards manipulating the anisotropy to engineer 3D objects were done on a multilayer with a monotonous variation of the Co thickness. Indeed, the first case (I) corresponds to Ta5|Pt3|(Co[1.8:0.1:2.4]|Al1.4| Pt3) which at remanence only shows a distribution of worms moderately aligned. The second state (II) corresponds to two stacked repetitions of the previous structure and it shows a distribution of elliptic isolated objects at remanence. Building on this successful observation of isolated textures at remanence, the multilayer Ta5|Pt3|(Co[2.4:-0.1:1.8]|Al1.4|Pt3)|(Co[1.9:0.1:2.4]|Al1.4|Pt3) with a similar trend of ferromagnetic thickness was also grown and measured (III). However, after the OOP saturation, the stable state only shows a worm distribution. The multilayer with a symmetric distribution of Co thickness was also studied. This corresponds to Ta5|Pt3|(Co[1.8:0.1:2.4]|Al1.4|Pt3)|(Co[2.3:-0.1:1.8]|Al1.4|Pt3): the bottom layer is the thinnest and we increase the thickness up to the middle layers with 0.1 nm step before going back down symmetrically.

Figure 3 . 10 -

 310 Figure 3.10 -MFM investigation of aperiodic multilayers with variable thickness of Co, grown on a seed of Ta5|Pt3 with repetitions of Pt3|Co X|Al1.4. The Co thickness evolution is plotted below each phase map. The measurements were performed after an OOP saturation without any external field. Each image is a 2 × 2 µm 2 with a scale bar of 500 nm.

Figure 3 .

 3 [START_REF] Moreau-Luchaire | Additive interfacial chiral interaction in multilayers for stabilization of small individual skyrmions at room temperature[END_REF].a), we can observe barrel-shaped skyrmions that are columnar: they extend from the bottom interface to the top layer. The situation changes drastically as we introduce a spatial dependency for the anisotropy along the vertical direction: when it has a maximum (Figure3.11.b), some textures appear to be cut in the middle as the PMA is too strong to sustain the existence of domains. Inversely, if the anisotropy has a minimum in the middle layer (Figure3.11.c), i.e. the SG structure, the objects are confined in the bottom layers and disappear from the top ones, introducing a third dimensionality 1 . The last case, a hybrid one with a more elaborated trend of the anisotropy but built from the SG structure (similar to state VI in Figure3.10), evidences the ability to confine 3D objects making use of layers with strong PMA.

Figure 3 . 11 -

 311 Figure 3.11 -Micromagnetic simulations evidencing the importance of the anisotropy for the stabilization of 3D textures. For each case, the relaxed state is displayed with isosurfaces (left), with m z = 0 in white and m z = -0.8 in red, with a vertical cut of m z along representative textures and the evolution of the uniaxial anisotropy (right). For clarity, the vertical dimension is scaled by a factor 5. a) Uniform anisotropy (skyrmionic multilayer). b) Maximum and c) minimum anisotropy in the middle layer. d) Effect of strong PMA layers. An external OOP field of 250 mT is applied and the parameters of Table 3.4 have been used.

4 C h a p t e r Skyrmionic cocoons in single gra- dient multilayers CONTENTS 4 . 1 First approach of skyrmionic cocoons 52 4. 1 . 1 53 4. 1 . 2 55 4. 1 . 3 61 4. 1 . 4 66 4. 2 Topology in a discontinuous medium 67 4. 2 . 1 67 4. 2 . 2 68 4. 3 Investigating the effective chirality 69 4. 3 . 1 70 4. 3 . 2 72 4. 3 . 3 75 4. 4

 441521153125513611466267216722683693170327233754 Optimization of the single gradient multilayers . . . . . . . . . . . . . . . . . . . . . . . . Revealing skyrmionic cocoons with an out-of-plane magnetic field . . . . . Impact of the thickness gradient on the vertical confinement . . . . . . . . . . . Resilience to an in-plane magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-dimensional lattice-based topological number . . . . . . . . . . . . . . . . . . . . . Three-dimensional approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XRMS principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Understanding the dichroic signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effective chirality from angular maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary 79

Figure 4 . 1 -

 41 Figure 4.1 -Properties of Single Gradient (SG) multilayers with parameters S = 0.1 nm, X 1 = 1.7 nm, N = 13 layers and t N Pt = 3 nm. The Pt (resp. Al) thickness remains fixed at 3 nm (resp. 1.4 nm). a) Evolution of the Co thickness for a typical SG multilayer, defined by the number of layers N , the thickness step S, and the starting thickness X 1 . b) Hysteresis loops recorded with AGFM for an IP (top) and OOP field (bottom). The field was increased up to ± 1.4 T to ensure the saturation.

Figure 4 . 2 -

 42 Figure 4.2 -Study of different single gradient (SG) structures at zero field after an out-of-plane saturation. MFM phase maps for a) Variations of X 1 with the other parameters fixed (S = 0.1 nm and N = 13 layer) and for b) Variations of S and N with X 1 = 1.8 nm and X center = 2.4 nm fixed. c) Average long axis, density and circularity extracted for different X 1 . The insets in the bottom left corners correspond to simulated MFM phases.

Figure 4 . 3 -

 43 Figure 4.3 -Out-of plane field dependency of a typical SG multilayer (S = 0.1 nm, X 1 = 1.7 nm, N = 13 layers, t N Pt = 3 nm). a) Phase maps obtained with MFM accompanied by simulated ones (1×1 µm 2 ) at the bottom left corner. The scales shown in the right image are common to all MFM maps. The same scanning parameters and tip were used for all measurements. b) Corresponding micromagnetic simulations after an out-of-plane saturation (with random noise) displayed with isosurfaces (in red, m z = -0.8, in white, m z = 0 and in dark blue, m z = 1).

FieldTable 4 . 1 -

 41 (mT) Diameter (nm) Density (obj/µm 2 Average density and shape information extracted from the field dependency of the SG structure (Figure4.3).

  Figure 4.5.a,b,c the layer-resolved magnetization profiles simulated at 0, 225 and 275 mT respectively. These profiles clearly evidence the evolution from a fully columnar skyrmion to a vertically confined object. The resulting local texture changes significantly for µ 0 H ⊥ = 275 mT: the skyrmion tube evolves into a smaller objectas its core shrinks and then disappear in the outer layers. Based on this vertical confinement and their peculiar profile, we name skyrmionic cocoon this texture. In the top part (layers 11 to 13), the magnetization is pointing up with a slight tilt (⟨m z ⟩ > 0.99 for the cocoon presented in Figure4.5.c) toward the cocoon's core that is induced by the dipolar field. Directly above the cocoon, in the layers 9 and 10, this inclination is stronger (⟨m z ⟩ > 0.85) which yields a vortex-like texture with no singularity as the spins rotate continuously keeping m z > 0 everywhere. Note the absence of Bloch points (BP) at the extremities of the cocoons, allowed by the discontinuous distribution of the magnetic materials in the multilayer along the z direction. Simply put, a Bloch point corresponds to a volume where the magnetization is void, i.e. a zone where the spins cancel each other and that impacts the topology of the system under consideration.

Figure 4 . 4 -

 44 Figure 4.4 -Skyrmionic cocoons: evidencing their vertical confinement in a SG structure (275 mT).

Figure 4 . 5 -

 45 Figure 4.5 -Layer resolved simulated magnetization of the SG multilayer (S = 0.1 nm, X 1 = 1.7 nm, N = 13 layers, t N Pt = 3 nm). Field evolution of a given texture (encircled in Figure 4.3) with layer cuts of the magnetization at a) 0 mT, b) 225 mT and c) 275 mT. The vertically confined object at 275 mT corresponds to a skyrmionic cocoon. d) Horizontal cuts of different layers at remanence.

Figure 4 .

 4 Figure 4.6.a, the different energies (dipolar, DMI, exchange, anisotropy and Zeeman) are plotted in

Figure 4 . 6 -

 46 Figure 4.6 -Micromagnetic energies in a SG structure. a) Energies in the different layers at 0 mT (left) and 275 mT (right). Distribution of the energies on a given object at remanence in the 3 rd and 8 th layer (starting from the bottom) with b) relative and c) the same scale.

  Figure 4.7.a) while for the other two, it disappears from the outer layers and acquire a characteristic ellipsoid shape (Figure 4.7.b). The asymmetry of the confinement is due to the DMI (a positive DMI would favor the confinement in the top layers). Thus, overall, the use of gradients facilitate the stabilization of isolated objects which can be strongly confined under the appropriate conditions.

Figure 4 . 7 -

 47 Figure 4.7 -Average height of the magnetic textures obtained in micromagnetic simulations for the a) uniform, b) SG and c) RSG structures with various S parameters at different magnetic fields and thicknesses with N = 13 layers. Depending on the circularity of the textures, the marker is either an open square for C < 0.9 or a plain circle for C ≥ 0.9. A zero height corresponds to a uniform magnetization. Below each graph is an associated vertical cut of the vertical component of the magnetization (m z ) of an axisymmetric object at an external field of 250 mT and an average thickness close to 2 nm.

  4|Pt3) 13 According to the previous numerical study, negligible vertical variations are expected along the vertical direction of the tubular skyrmions. This should correlate with almost no evolution of the MFM signal with the field. Looking at the left column of Figure 4.8.a, the phase maps shown the nucleation of circular objects starting close to 250 mT and reaching a tightly packed distribution near 230 mT.

  Figure 4.9.a, line cuts through several skyrmions are plotted at 243 mT with almost no variations in the signal. Thus they are likely to have a similar vertical extension. • S = 0.1 nm with Ta5|Pt3|(Co[1.8:0.1:2.4]|Al1.4|Pt3)|(Co[2.3.0:0.1:1.8]|Al1.4|Pt3) (⟨t Co ⟩ = 2.08 nm) In this case, the nucleation starts at a lower field, near 220 mT, as shown in the middle column. Note that the phase scale is half the previous one. The second or third image, at 221 or 216 mT, display various objects that appear to have slightly different contrasts (white or red dots). Therefore, we observe textures that have a lesser contrast and thus probably a stronger vertical confinement since the stray field picked up by the magnetic tip is reduced. Contrarily to the first sample, it thus corresponds to skyrmionic cocoons. Examples of line cuts through such objects are shown in Figure 4.9.b at

Figure 4 . 8 -

 48 Figure 4.8 -Probing the cocoons extension with MFM for SG multilayers (⟨X⟩ = 2.1 nm, N = 13 layers, t N Pt = 3 nm) with different S. The measurements have been performed under identical conditions with the same magnetic tip. a) Phase maps at different OOP magnetic fields for S = 0, 0.1, 0.2 nm. Note that the phase scale for S = 0 nm is larger than for the other cases. b) Line cuts through the various textures indicated in (a) with the dotted black lines. For S = 0.2 nm, the signal has been smoothed to highlight the signal of the textures. The black ellipses in (a) indicate the position of the line cuts presented in Figure 4.9.

  2 nm with Ta5|Pt3|(Co[1.5:0.2:2.7]|Al1.4|Pt3)|(Co[2.5.0:0.2:1.5]|Al1.4|Pt3) (⟨t Co ⟩ = 2.03 nm) For this large value of the thickness step, the nucleation is further shifted to lower field as the first dots seen by MFM appear close to 180 mT. The phase scale is the same as for S = 0.1 nm to allow for an easier comparison. Thanks to the line cuts at 185 mT, we can detect objects with signals close to 0.05 degrees, smaller than for the previous case. It thus suggests a stronger vertical confinement than in S = 0 and 0.1 nm, as expected from the simulations. At the smallest field (160 mT), the cocoons appear more clearly but their contrast is still relatively faint. The contrast thus increases with decreasing field, indicating the extension of the objects. As we are limited by the field range, we cannot go lower to study further the trend for this architecture but the results on this field range are already coherent with the simulations (Figure 4.7). Similarly to S = 0.1 nm, profiles of the signal reveals variations in the contrasts of the different objects with similar diameters (Figure 4.9.c), implying that they have different vertical extension.

Figure 4 . 9 -

 49 Figure 4.9 -Line cuts of the MFM signal at a given magnetic field through different textures for a) S = 0 nm, b) S = 0.1 nm and c) S = 0.2 nm. The numbers indicate the apparent diameter equal to the FWHM of a Gaussian fit. The cuts positions are identified on Figure 4.8 with the black ellipses.

  Figure 4.3 (Ta5|Pt3| (Co[1.7:0.1:2.3]|Al1.4|Pt3)|(Co[2.2:-0.1:1.7]|Al1.4|Pt3)) after an out-of-plane saturation to observe its behavior as shown in Figure 4.10.

Figure 4 . 10 -

 410 Figure 4.10 -MFM study of SG under an external in-plane field (S = 0.1 nm, X 1 = 1.7 nm, N = 13 layers, t N Pt = 3 nm after an out-of-plane saturation). The measurements are performed starting from remanence up to 400 mT before going back to zero field. Phenomena of tip reversal is visible in the first frames so that only the domain walls are visible.

Figure 4 .

 4 [START_REF] Moreau-Luchaire | Additive interfacial chiral interaction in multilayers for stabilization of small individual skyrmions at room temperature[END_REF].a. From positive field down to -100 mT, N sk remains constant at a negative value in the bottom layers up to the layer 6.Above that it becomes close to zero before decreasing back down in the last top three layers. It reaches values greater than ±1 because we have multiple objects in our simulation space, 16 in this case (as shown in Figure4.11.b), and each of them contributes by ±1 so the extremal values equal ±16. In the layers 7 to 11, N sk drops to zero. This corresponds to the layers in which the domain walls are uniformly pointing in the same direction, having no well-defined chirality. Below -100 mT, the same trend is noticeable apart from an inversion of the sign due to the magnetization reversal in the core of the objects.

Figure 4 . 11 -

 411 Figure 4.11 -Field study of the 2D topological number in the SG structure (S = 0.1 nm, X 1 = 1.7 nm, N = 13 layers, t N Pt = 3 nm). a) Map of the topological number in every layers at different magnetic fields. b) Corresponding simulated states represented with isosurfaces (m z = 0 in white, m z = -0.8 in red and m z = 0.8 in blue). The field is swept from positive saturation to negative. At 200 mT, 16 cocoons are stabilized.

Figure 4 . 12 -

 412 Figure 4.12 -Calculation of the 3D topological charge, neglecting the non-continuity of the magnetization along the growth direction. a) Selection of a cubic bounding box around a cocoon fully buried in the magnetic multilayer. A slice of the magnetization inside the box, indicated by the dashed line, is shown next to the bounding box. Two topological charges of opposite sign compensate, leading to q = 0. b) Case when the cocoon emerges to the surface of cubic box, yielding a skyrmion texture on its side. c) Impact of the discretization on the 3D topology.

Figure 4 .

 4 Figure 4.13 -X-ray resonant magnetic scattering (XRMS) principle. Experimental configuration and definition of the angles of interest. The beam propagates in the (yz) plane and hits the sample with an incidence angle θ.

  CL -I CR . To allow for comparison between the various recorded images and to correct systematic variations, this difference will be normalized by the sum, yielding what we will indifferently call the XMCD or asymmetry ratio or normalized difference (I CL -I CR ) / (I CL + I CR ). The following maps therefore correspond to the normalized difference. Moreover, to account for the projection related to the photon incidence θ, the images have been geometrically corrected by a factor sin θ along the vertical direction of the maps.

Figure 4 .

 4 Figure 4.14a) Energy scan near the Co absorption edges measured on the SEXTANTS beamline at SOLEIL synchrotron. b) Positions of the extrema depending on the chirality of the magnetic textures.Extracted from[START_REF] Chauleau | Chirality in magnetic multilayers probed by the symmetry and the amplitude of dichroism in x-ray resonant magnetic scattering[END_REF].

Figure 4 . 15 ,

 415 Figure 4.15, Néel patterns with different chiralities are displayed. A pattern with a minimum (resp. maximum) at 90 • and the opposite at 270 • corresponds to a CCW (resp. CW) Néel chirality as evidenced by the left (resp. right) signal.

Figure 4 . 15 -

 415 Figure 4.15 -Illustrative XRMS examples at the SEXTANTS beamline (SOLEIL, France).

Figure 4 .

 4 Figure 4.16.b and its 2D Fourier transform yields a blurry ring in reciprocal space. It allows to retrieve a periodicity p MFM = 239 ± 40 nm -1 . The excellent agreement between these two methods confirms that the magnetic state probed by XRMS corresponds to the lattice of isolated objects.

Figure 4 . 16 -

 416 Figure 4.16 -Extraction of the magnetic periodicity at 160 mT for Ta5|Pt3|(Co[1.8:0.1:2.4]|Al1.4| Pt3)(Co[2.3:0.1:1.8]|Al1.4|Pt3). a) XRMS map measured at θ = 19 • , ℏω = 778.5 eV of the SG mounted on a permanent magnet. b) MFM phase map at 160 mT with an inset showing the magnitude of the 2D Fourier transform.

Figure 4 . 17 -

 417 Figure 4.17 -Reflectivity curves measured off resonance (ℏω = 770 eV) and near resonance (ℏω = 778 eV) for both helicities on a SG multilayer (S = 0.1 nm, X 1 = 1.8 nm, N = 13 layers, t N Pt = 3 nm) mounted on a permanent magnet providing 160 mT out-of-plane at the sample position. Missing data are visible for high incidence angles when the detection diode measures negative values. The curves have been shifted for clarity.

Figure 4 .

 4 [START_REF] Yu | Variation of skyrmion forms and their stability in MnSi thin plates[END_REF].a, the normalized difference is shown for five different angles that are evidenced on Figure4.17 with dotted orange lines for the four smallest ones. The size of the resulting image differs from the geometrical correction needed to account for the projection along the y axis. For θ = 10 and 19 • , the negative (resp. positive)

Figure 4 . 18 -

 418 Figure 4.18 -Representative XMCD ratio maps at different incidence angles (shown on Figure 4.17

  with dotted orange lines). a) Scattering patterns. b) Example of the data processing for θ = 24.1 • . c) Signal obtained after the ring integration.

plotted in Figure 4 .

 4 [START_REF] Yu | Variation of skyrmion forms and their stability in MnSi thin plates[END_REF].c. Notice the drop in all signals near ψ = 270 • that corresponds to the beamstop and should thus be ignored. For all the incidence angles, the signal is relatively high with an XMCD ratio of at least 10%. The inversion of the dichroism is clearly visible with the sign inversion of the purple curve corresponding to θ = 43 • .

Figure 4 . 19 -

 419 Figure 4.19 -Summary of XRMS measurements in a SG multilayer. a) Map of the asymmetric ratio in function of the incidence angle θ and the azimuthal angle ψ. Micromagnetic simulations of the SG at 150 mT displayed with b) layer resolved cuts and a c) vertical cut of m z (along the dotted line in b) to evidence the domain wall winding.

5 C 1 81 5. 1 . 2 82 5. 1 . 3 87 5. 1 . 4 95 5. 2 Probing 3D textures with XRMS 96 5. 2 . 1 96 5. 2 . 2 99 5. 2 . 3 103 5. 3

 518112821387149529621962299231033 First approach of the DG multilayer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unveiling the coexistence of magnetic textures . . . . . . . . . . . . . . . . . . . . . . . . Thickness optimization of the DG structure . . . . . . . . . . . . . . . . . . . . . . . . . . . Exploring other multilayer architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chirality from micromagnetic simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Investigating a stripe state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cocoons and 3D worms: a multinodal signal . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary 105

  .1.a for Ta5| Pt3| (Co[2.0:0.1:2.5]|Al1.4|Pt2)(Co[2.6:0.1:2.0]|Al1.4|Pt2)|(Co1.0|Al1.4|Pt3) 15 |(Co[2.0:0.1:2.5]|Al1.4| Pt2)(Co[2.6:0.1:2.0]|Al1.4|Pt2). To shorten it, it can be described with S = 0.1 nm, X 1 = 2.0 nm, N = 13 layers, M = 15 repetitions, Y = 1 nm, t N Pt = 2 nm and t M Pt = 3 nm.

Figure 5 .

 5 Figure 5.1.b, measured with AGFM.The IP loop displays a smooth variation of the magnetization along with the field sweep. However, the saturation field is relatively high (near 500 mT) due to the difficulty to pull the magnetization of strong PMA layers in the film plane. Their impact is even more noticeable in the OOP loop. Indeed, at 60 mT, an abrupt step is visible (black arrows) that is related to the strong PMA layers. Those represent 20% of the total magnetic thickness. If they were to flip with the magnetic field, a transition of 0.4 in M/M S would be expected (see schematic jumps on the OOP cycle). Here, it is closer to 0.1 indicating that the magnetization only flips partially: the jump thus corresponds to the partial nucleation of domains in those intermediate layers. This is

Figure 5 . 1 -

 51 Figure 5.1 -Properties of a double gradient (DG) multilayer with parameters S = 0.1 nm, X 1 = 2.0 nm, N = 13 layers, M = 15 repetitions and Y = 1 nm. The Al thickness remains fixed at 1.4 nm. The Pt in the SG parts is t N Pt = 2 nm and 3 nm in the middle repetitions (t M Pt = 3 nm). a) Evolution of the Co thickness for a typical DG multilayer. b) Hysteresis loops recorded with AGFM for an IP (top) and OOP field (bottom). The field was increased up to ± 1.5 T to ensure the saturation.

Figure 5 . 2 -

 52 Figure 5.2 -DG multilayer OOP field dependency toward saturation. a) Experimental MFM phase maps measured after a 30 • demagnetization (away from the normal). b) Relaxed states of micromagnetic simulations, displayed with isosurfaces (in red, m z = -0.8, in white, m z = 0 and in dark blue, m z = 1), at various magnetic field starting from a magnetization saturated at 60 • from the normal. The magnetic field is applied perpendicularly to the sample.

Figure 5 . 3 -

 53 Figure 5.3 -DG multilayer field dependency from saturation. MFM phase maps at different magnetic field following the field steps indicated on the half hysteresis cycle below (following the black arrows). Due to experimental constraints, the sample is saturated manually with an external magnet to transition between state I and II. Tip reversal artefacts are visible in the right most image.

Figure 5 . 4 ,

 54 we present the MFM images recorded at remanence after different initialization processes and the corresponding simulated MFM images. First, saturating the sample magnetization in-plane (IP) results in stripes with various branching giving a single MFM contrast. This indicates that the textures are columnar and thus go through all the layers. No skyrmionic cocoons are stabilized by this process. In case of an initialization with an OOP field, two different contrasts are clearly visible (white and gray) in the MFM images. The related simulations told us that such a case is representative of the presence of two different objects as previously evidenced. Finally, the case of a tilted field during the initialization leads to a configuration containing distorted stripes with dots of lower contrasts in their midst.

Figure 5 . 4 -

 54 Figure 5.4 -Magnetic configurations obtained after different magnetic initializations. For the measurements (resp. the simulations), the magnetization was either demagnetized (resp. saturated) out-of-plane (OOP), in-plane (IP) or with a tilted magnetic field, 60 • away from the normal (resp. 30 • ). All images share the same length scale, displayed in the bottom right image.

Figure 5 . 5 -

 55 Figure 5.5 -In-plane field evolution of the DG multilayer going to saturation (top row) and going to remanence (bottom row). The sample was measured after an OOP saturation.

Figure 5 . 6 -

 56 Figure 5.6 -Change of the magnetic configuration with different magnetic thickness X 1 . MFM images measured near remanence (a small bias field around 15 mT is applied to avoid tip flipping phenomena) after an OOP saturation. The phase amplitude for each map is indicated near the bottom right corner. The tip might differ from one measurement to another.

Figure 5 . 7 -

 57 Figure 5.7 -Field dependency for X 1 = 1.6, 1.8, 2.0 and 2.2 nm after an OOP saturation. The phase amplitude for each map is indicated near the bottom right corner. The magnetization was saturated out-of-plane before the measurements.

Figure 5 . 8 -

 58 Figure 5.8 -Minor loop example to obtain the coexistence of cocoons and skyrmions for X 1 = 1.6 nm.

Figure 5 . 9 -

 59 Figure 5.9 -Impact of t N P t on the magnetic configurations. MFM phase maps representative of the out-of-plane field dependency. The phase amplitude for each map is indicated near the bottom right corner.

Figure 5 . 10 -

 510 Figure 5.10 -Impact of M on the magnetic configurations. MFM phase maps representative of the out-of-plane field dependency. The phase amplitude for each map is indicated near the bottom right corner.

  To this end, several cases are presented in Figure5.11. For S = 0.2 nm, which should yield a stronger vertical confinement (based on the study on SG presented in Subsubsection 4.1.3.1), interesting mixed states are obtained at remanence. After a demagnetization with a tilted field (first image), elongated cocoons (partial worms) are stabilized inside the stripe domains. After an out-of-plane saturation (second image), a similar observation is made with worm domains. Increasing the magnetic field at 100 mT provokes the contraction of the cocoons into more circular textures but both up and

Figure 5 . 11 -

 511 Figure 5.11 -Impact of S on the magnetic configurations. MFM phase maps representative of the out-of-plane field dependency. The phase amplitude for each map is indicated near the bottom right corner. The magnet icon indicates that an external saturation with a magnet was performed previously to the measurement.

Figure 5 . 12 -

 512 Figure 5.12 -Reducing the number of layers. MFM phase maps representative of the out-of-plane field dependency. The phase amplitude for each map is indicated near the bottom right corner. The magnet icon indicates that an external saturation with a magnet was performed previously to the measurement.

First, we introduce

  what is later on called encapsulated gradient, which consists of a single gradient sandwiched between M repetitions with strong PMA magnetic layers as illustrated in Figure5.13.a. In the (b) panel, MFM measurements are shown for different M value. Similarly to the previous study, double contrasts are visible, with red and white worms. This evidences two types of objects: the ones that extend the full thickness and the ones that are only in the gradient part of the structure. The difference of the two different contrasts evolves linearly with M , which confirms this hypothesis. It also corresponds to the micromagnetic simulations presented in Figure 3.11.d. For instance, the state at 180 mT for M = 5 shows a lattice of cocoons which is confined between uniformly magnetized layers pointing up. The OOP hysteresis (Figure 5.13.c) appears squarer than for the DG.

Figure 5 . 13 -

 513 Figure 5.13 -Encapsulated gradient structure. a) Schematic structure and example of the thickness evolution. b) MFM study for different M values. The rest is defined with S = 0.1 nm, X 1 = 1.8 nm, N = 13 layers, Y = 1 nm, t N Pt = 3 nm and t M Pt = 3 nm. c) AGFM cycles for M = 15.

5. 2 .

 2 Probing 3D textures with XRMS (Figure 5.14.c) is similar to the DG one, with a step near remanence corresponding to the formation of domain in the strong PMA layers.

Figure 5 . 14 -

 514 Figure 5.14 -Split gradient structure. a) Schematic structure and example of the thickness evolution. b) MFM study for different M values. The rest is defined with S = 0.1 nm, X 1 = 1.8 nm, N = 13 layers, Y = 1 nm, t N Pt = 3 nm and t M Pt = 3 nm. c) AGFM cycles for M = 15.

Figure 5 .

 5 15.a). A vertical cut across one characteristic domain wall is presented in Figure 5.15.b 2 . The red arrows represent the magnetization component in the (xz)

Figure 5 . 15 -

 515 Figure 5.15 -Domain walls profile in DG after an IP saturation. a) 3D relaxed states represented with isosurfaces (m z = 0.8 in red and m z = 0 in white). b) Vertical cut of the magnetization along the green line in (a). The indicated scale is linked to the distance along x. The spacer between two ferromagnetic layers is 5.4 nm.

  16.a). The domain walls display the same characteristic as the one observed in the IP case. For the cocoons, it is identical to their stabilization in SG as shown in (Figure5.16.b). Indeed, a similar evolution is obtained for both cocoons: CCW Néel in the bottom layers and CW Néel in the top ones. In the middle layers, the magnetization is uniformly pointing up.

Figure 5 . 16 -

 516 Figure 5.16 -Domain walls profile in DG after an OOP saturation. a) 3D relaxed states represented with isosurfaces (m z = 0.8 in red and m z = 0 in white). b) Vertical cut of the magnetization along the green line in (a). The indicated scale is linked to the distance along y. The spacer between two ferromagnetic layers is 5.4 nm.

Figure 5 . 17 -

 517 Figure 5.17 -2D field study of the topological number in DG structure. a) Map of the topological number in every layers at different magnetic fields. b) Corresponding simulated states represented with isosurfaces (m z = 0 in white, m z = -0.8 in red and m z = 0.8 in blue). The field is swept from positive saturation to negative.

Figure 5 . 18 -

 518 Figure 5.18 -Reflectivity curves measured off resonance (ℏω = 650 eV) and near resonance (ℏω = 777.5 eV) for both helicities on a DG multilayer at remanence. The curves have been shifted for clarity.

Figure 5 . 19 -

 519 Figure 5.19 -Illustrative XRMS examples at the BOREAS beamline (ALBA, Spain).

Figure 5 .

 5 Figure 5.20.b. Those representative patterns display multiple inversions as θ is increased, probing deeper into the multilayer. Those inversions of the dichroism are correlated with changes in the chirality which are expected based on the micromagnetic simulations (Figure 5.15). At low incidence angles, i.e. in the top magnetic layers, a clockwise Néel chirality is observed with a maximum (in red) at ψ = 90 • . Then the effective chirality goes back and forth between CW and CCW Néel, although it demonstrates more inversion than expected from the previous study. This already demonstrates the complex evolution for DG multilayer while evidencing the evolution of the chirality through the thickness.

Figure 5 . 20 -

 520 Figure 5.20 -XRMS study of a DG multilayer after an IP demagnetization. a) MFM image at remanence with its Fourier transform in the inset. b) Scattering pattern of the normalized difference recorded at 777 eV at various incidence angles (indicated on the reflectivity curves with the orange dotted lines in Figure 5.18). c) Corresponding radial integrals.

Figure 4 .

 4 [START_REF] Yu | Variation of skyrmion forms and their stability in MnSi thin plates[END_REF] in the previous images. The results are plotted in Figure5.20.c. While there is a clear decrease of the envelope of the signal with the incidence angle, it evidences the multiple inversion of the signal with a satisfying signal to noise ratio.The full angular dependency of the asymmetric ratio is shown inFigure 5.21.a with θ ∈ [5.5 • , 43.5 • ].At high incidence angle (θ > 30 • ), there appears to be no dichroism with a signal close to zero. A low incidence angles, the effective chirality is predominantly CW Néel up to θ = 14 • . In between, the aforementioned inversions of the dichroism are visible. They are also evidenced in the line cuts along ψ = 90 and 270 • in Figure5.21.b. The line cut at 180 • shows that the dichroism remains close to zero on the range of angles that were probed.

Figure 5 . 21 -

 521 Figure 5.21 -XRMS signal map of a DG demagnetized in-plane. a) Normalized difference measured at different incidence angle θ radially integrated over for all ψ. b) Line cuts for ψ = 90, 180 and 270 • .

  .

  14.b) due to the domain wall chirality. For CW Néel, the signal cancels for ψ = 0 and 180 • .

Figure 5 . 22 -

 522 Figure 5.22 -XRMS signal for θ = 7.75 • . a) Resulting asymmetric ratio for different in-plane rotation of the sample. b) Ideal example illustrating the impact of the orientation of the stripes over the scattering pattern. c) Extrapolation of the scattering pattern for a disorganized magnetic state (worms).

Figure 5 . 23 -

 523 Figure 5.23 -XRMS signal map of a DG demagnetized out-of-plane. a) Normalized difference measured at different incidence angle θ. The inset shows the magnetic state measured with MFM with a scale bar of 1 µm. (The scattering patterns have been cropped to the center region in both directions). b) Signal maps over different incidence angles (ℏω = 777 eV).

6 C 1 107 6. 1 . 2 110 6. 1 . 3 114 6. 1 . 4 119 6.2 Detecting skyrmionic cocoons 120 6. 2 . 1 120 6. 2 . 2 121 6. 2 . 3 128 6.3 Summary 133 T

 61107121101311414119120211202212123128133 Probing the average magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In a single gradient multilayer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In a double gradient multilayer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Establishing a magnetic phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resistance contribution of a single object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simultaneous imaging and electrical measurements . . . . . . . . . . . . . . . . . . . . Statistical data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . he existence of skyrmionic cocoons in single gradient (SG) and double gradient (DG) magnetic multilayers has been established in Chapter 4 and Chapter 5 with thorough experimental study to support their existence and to characterize their properties. For now, most of the knowledge was brought by MFM imaging, a technique mostly sensitive to the top layers as the contrast arises from

Figure 6 . 1 -

 61 Figure 6.1 -Device used for Hall measurements. a) Experimental referential. b) Optical microscope image of devices patterned with UV lithographies. The voltage probes used in the experiments are indicated on the bottom track (orange and green contacts).

Figure 6 . 2 -

 62 Figure 6.2 -Hall measurements of R xx (θ) with a rotating field in the (xz) plane (ϕ = 0 • ) in red and the (yz) plane (ϕ = 90 • ) in black for a) the SG structure and b) the DG one. A magnetic field of 650 mT (resp. 3T) for the SG (resp. the DG).

  .3.a along with the resistance computed from the micromagnetic simulations. To facilitate the comparison, the experimental values have been centered around zero by subtracting the offset R 0 xy and normalized by the high field value of R xy for θ = 0 • , noted R θ=0 • AHE . For the track under consideration, R θ=0 • AHE = 107 mΩ. By definition, it also corresponds to half the amplitude of the anomalous Hall effect. This normalization of R xy allows to compare it directly to the average m z .

Figure 6 . 3 -

 63 Figure 6.3 -R xy (H) measured in the (yz) plane (ϕ = 90 • ) in the SG multilayer. a) Normalized experimental curves (full markers) and corresponding values extracted from simulations (open markers) for various θ. The magnetic field is swept from positive to negative values. b) Representative relaxed states of the simulations for θ = 0 • , a 1.024 × 1.024 µm 2 area displayed with isosurfaces (m z = 0 in white and m z = -0.8 in red).

Figure 6 . 5 -

 65 Figure 6.5 -Average squared magnetization component per layer extracted from micromagnetic simulations for positive fields for ϕ = 0 • . For ϕ = 90 • , m x and m y are simply exchanged.

  .6.a, R xy (H) are shown for various θ angles after normalization to compare with the simulations. For the DG, the half amplitude of the AHE is R θ=0 • AHE = 40.5 mΩ. The open symbols corresponding to the average ⟨m z ⟩ extracted from the simulations are coherent with the experimental values and trends, almost superimposing perfectly. Except for θ = 90 • , a jump is measured near 50 mT that is reproduced numerically at a slightly higher field (at 75 mT). To verify its origin, the relaxed state of the simulations are displayed in Figure 6.6.b at significant fields for θ = 0 • .

Figure 6 . 6 -

 66 Figure 6.6 -R xy (H) measured in the (yz) plane (ϕ = 90 • ) in the DG multilayer. a) Normalized experimental curves (full markers) and corresponding values extracted from simulations (open markers) for various θ. The magnetic field is swept from positive to negative values. b) Representative relaxed states of the simulations for θ = 0 • , a 1.024 × 1.024 µm 2 area displayed with isosurfaces (m z = 0 in white and m z = -0.8 in red).

Figure 6 . 7 -

 67 Figure 6.7 -R xx (H) in the DG multilayer. Experimental curves (full markers) and corresponding values extracted from simulations (open markers) for various θ. The magnetic field is swept from positive to negative values. a) ϕ = 90 • . The inset zooms around 50 mT for θ = 0 • (as evidenced by the gray box), near the nucleation in the PMA layers, with a relative scale in mΩ. b) ϕ = 0 • . The two black curves have the same amplitude.

Figure 6 . 8 -

 68 Figure 6.8 -Study of the anomalous and planar Hall effects in the DG structure. a) Raw data of R xy (H, θ = 0 • ) (left) and R xy (ϕ) under a saturating field (right) plotted with the same scale. The red line marks the half amplitude of each curve (the expected zero). b) Planar cut of a skyrmionic cocoon with a Néel domain wall in presence of an OOP magnetic field. The circled green annotations correspond to the sign of the PHE contributions.

Figure 6 . 9 -

 69 Figure 6.9 -Sensitivity of the magnetotransport measurements for R xx (H, θ = 0 • ) measured on the SG sample. a) Experimental curve and simulated points at 200 mT. b) Vertical cut of the magnetization showing the modified numerical states. A state: unmodified. B (resp. C) state: top (resp. two) layer uniformly pointing up. D state: top two layers and bottom one uniformly pointing up. The change between A and B is subtle: in A there is in-plane components in the top layer due to the dipolar field of the cocoons while in B the magnetization is uniformly out-of-plane in the top layer.

Figure 6 . 10 -

 610 Figure 6.10 -Sensitivity of the magnetotransport measurements for R xx (H, θ = 0 • ) measured on the DG sample. a) Experimental curve and simulated points at 325 mT. b) Vertical cut of the magnetization showing the modified numerical states. E state: unmodified. F state: only the skyrmionic cocoons present in the bottom gradient.

Figure 6 . 11 -

 611 Figure 6.11 -Phase diagram from transport measurements. a) R xy (H) and b) R xx (H) and its derivative with θ = 0 • , ϕ = 90 • (OOP field). The background colors in the upper part correspond to the magnetic phases as predicted by the simulations. The colors in the lower part corresponds to the phases associated with transitions in the derivative. In red (U): uniform magnetization, in blue (C): cocoons only, in green (C+W): cocoons plus 3D worms). The width of the magnetic phases and the transition fields (background colors) are the same in both graphs.

Figure 6 . 12 -

 612 Figure 6.12 -Topography of tracks with Ta contacts for Hall measurements measured with AFM. Track a) I and b) II measured on different samples with the same structure.

  .13.a, two different measurements with different field steps and current are shown in track I. The black curve has a resistance measured almost every 0.1 mT which allows to observe several sharp jumps in the resistance. As R xy scales with m z , those correspond to the annihilation of up domains or non-collinear textures. To understand the evolution of the signal, MFM images are provided in Figure 6.13.b that have been measured concomitantly during the field cycle, at fields indicated by the orange dotted lines. Some jumps are correlated with the imaging process, showing that the magnetic tip might disturb the magnetization and thus induces transitions in the resistance.

Figure 6 . 13 -

 613 Figure 6.13 -Hall measurements coupled with MFM in a DG structure (track I) with an increasing magnetic field. a) Examples of R xy measurements (absolute value) with I = 1.5 mA. b) Correlated MFM images measured at the fields indicated by the orange dotted lines. The green ellipses serve as guide to show the evolution between consecutive frames. The last image outlines the active area (between the Ta contacts) for clarity.

Figure 6 . 14 ,

 614 Figure 6.14, R xy is presented in a reduced field range near saturation for two cycles in conjunction with the associated MFM images. Smaller transitions, of the order of 100 µΩ are noticeable as evidenced by the blue values. We first look at the top graph and the corresponding phase maps. Starting from the last magnetic state (1) with the two isolated columnar skyrmions, the increase of the magnetic field leads to the disappearance of one of them (state 2), correlated with the 103 µΩ jump. Then, in the frame 3, the tip erases the remaining skyrmion, which appear larger than the first one and more centered with respect to the contacts, for a contribution of 125 µΩ. Interestingly, a last jump is visible with a smaller amplitude (35 µΩ) that could correspond to the annihilation of a texture farther away or a cocoon in the bottom gradient. Already, a clear correlation between the two skyrmions and R xy is visible.

Figure 6 . 14 -

 614 Figure 6.14 -Electrical signal of the annihilation of columnar skyrmions in the DG structure (track I). a) Evolution of R xy near saturation (zoom of the curves of Figure 6.13). The green (resp. blue) numbers corresponds to various jump heights (in mΩ) identified (resp. unidentified) on the MFM images. b) Corresponding MFM images. The green ellipses evidence the transitions between consecutive frames.

  15.b. To compute the latter, R xy values have been smoothed first by wavelet denoising (using the implemented wdenoise Matlab function). Looking at dR xy , transitions can easily be identified as the minimum (indicated by the red filled dots) and we can notice already the recurrence of many steps close to 20 µΩ.

Figure 6 . 15 -

 615 Figure 6.15 -Hall measurements in track II coming from saturation (decreasing the field in absolute value). a) Consecutive measurements near the nucleation of cocoons. The curves have been shifted for clarity. The arrows indicate the position of MFM imaging. b) Zoom for three of the previous curves along with the difference between consecutive data points (dR xy ) to highlight the transitions.

Figure 6 . 16 -

 616 Figure 6.16 -Correlation between R xy and nucleation events. Systematic identification of the steps in R xy (left) due to nucleation events observed with MFM (right). The orange lines are positioned at the field of the MFM images. The arabic numbers indicate the identified transitions in the R xy curves. The transparent gray boxes pinpoint the location of the Ta contacts.

  extract useful information from the numerous cycles measured (see some examples in Figure 6.17.a), we rely on the numerical detection of the different steps. To this end, R xy is first smoothed by wavelet denoising and corrected by subtracting the slope near saturation, rendering the steps sharper.

Figure 6 . 17 -

 617 Figure 6.17 -Numerical detection of R xy transitions. a) Typical curves measured coming from saturation. b) Zoom on the first curve with its derivative to illustrate the numerical detection of the steps. The blue line corresponds to a numerical threshold defined by ⟨dR xy ⟩ -0.5σ(dR xy ). The peak heights is refined by taking into accounts the additional points under the refined threshold ⟨dR xy ⟩ -0.375σ(dR xy ).

Figure 6 . 18 -

 618 Figure 6.18 -Statistical measurements of R xy in patterned tracks in a DG structure for the nucleation of skyrmionic cocoons (decreasing the field coming from saturation). a) Full centered R xy loop corresponding to each track. The orange bands indicate the field range corresponding to the statistical measurements. b) Histograms of the transitions detected for three different tracks obtained by cycling in a reduced range of magnetic field. The quantity represented is dR xy times the area A, allowing comparison between the different tracks. The Gaussian outlines serve as guide to the eyes (with the central value in green).

Figure 6 .

 6 [START_REF] Finocchio | Magnetic skyrmions: from fundamental to applications[END_REF].b, the field dependency of the transitions are plotted to check if they occur at regular field. At -162 mT, it seems that a single cocoon nucleates in many cases while at -160 mT a pair is regularly stabilized. Moreover, near -158.5 mT, transitions often happens but with different heights this time. For lower fields, the distribution is more dispersed, although it appears denser near ∆R c , 2∆R c and 4∆R c . The values observed in this case are coherent with the ones from the full field range and the other tracks, yielding ∆R c = 166 ± 20 µΩ.µm 2 . It is further impressive as it was measured in different devices of various widths which causes many fluctuations. Note that there is a factor 2 between the MFM and the statistical study that arises from experimental considerations. Additionally, the field are shifted towards lower value as the calibration is different. Those considerations do not change the conclusions drawn with each technique.

Figure 6 . 19 -

 619 Figure 6.19 -Statistical measurements of R xy in track V focusing on a reduced field range (-163 to -150 mT). a) Histogram of the numerically detected transitions. The Gaussian outlines serve as guide to the eyes (with the central value in green). b) Field dependency of the transitions. The ellipses indicates the transitions repeating at the same field over different cycles.

  considering the dominating effects, namely the spin Hall magnetoresistance (SMR) and the anisotropic magnetoresistance (AMR), in correlation with the micromagnetic simulations the variations of the transversal and longitudinal resistance (R xy , R xx ) have been understood, both for the SG and the DG. No fitting parameter was used in the comparison as all constants were experimentally measured, ensuring the reliability of the numerical values. Moreover, it allows to create a phase diagram in DG multilayers using the derivatives of the experimental signals which is coherent with the numerical predictions, providing an alternative without the need for imaging. This put forwards three main phases, as observed already with MFM: the nucleation of cocoons, then their coexistence with 3D

Figure 7 .

 7 1.a for the FTH configuration. It shows the X-ray propagating along z through the object hole (OH) and one reference hole (RH), thus creating an interference pattern recorded by a digital sensor (CCD camera). Thanks to the XMCD effect, the difference between the hologram recorded for each X-ray polarization h CL -h CR yields information regarding ⟨m z (x, y)⟩, i.e. m z averaged over z, that is through all the magnetic layers.

Figure 7 . 1 -

 71 Figure 7.1 -Holography principle. a) Schematic of the Fourier transform holography (FTH). b) SEM images of a multilayer prepared for FTH (top) and for HERALDO (bottom). OH: object hole, RH: reference holes (evidenced by the red circles), S: slit.

2 .

 2 Modulus constraint (reciprocal space): the intensity, i.e. the squared modulus, is fixed by the experimentally recorded hologram (except in the center where a beamstop reduces the direct beam, hiding the lowest frequencies).

Figure 7 . 2 -

 72 Figure 7.2 -Iterative phase retrieval process. Left: Initialization with the experimental amplitude zoomed near the center and an initial guess for the phase (obtained from the inverse FT of the support). Right: Main loop for the error reduction algorithm showing the successive steps with states near convergence. The gray images are in the real space and correspond to the real part of the hologram. In the Fourier space, the modolus is shown.

Figure 7 . 3 -

 73 Figure 7.3 -FTH measurements performed on a SG multilayer (X 1 = 1.8 nm, S = 0.1 nm, N = 13 repetitions, t N Pt = 3 nm) at different out-of-plane fields. The object hole has a 800 nm diameter, thus a pixel is approximately 20 × 20 nm 2 . The reconstructions were processed by R. Battistelli.

Figure 7 . 4 -

 74 Figure 7.4 -FTH imaging performed with a 800 nm object hole in a DG multilayer. Dependency with a) increasing and b) decreasing magnetic field. The colored ellipses follow the same textures over different fields to highlight the associated transitions. A bilinear interpolation has been performed on the reconstructions, increasing the number of pixels by a factor 1.25.

Figure 7 . 5 -

 75 Figure 7.5 -Study of the transition between two isolated cocoons. a) Linecuts of the signal of skyrmionic cocoons in the green ellipses of Figure 7.4. The black arrow indicates the relative motion of the cocoons between the two successive fields. The position of the linecut has been changed between the two graphs but the inset shows the same sample area (a circle of 200 nm radius). b) Total and demagnetization energy as a function of the cocoon distance obtained from micromagnetic simulations. c) Relaxed state starting from two cocoons separated by 150 nm at two different magnetic fields. The stray field of the top cocoons is applied to the isolated bottom cocoon. The red dotted square corresponds to a 20 nm artificial pinning region where the anisotropy is decreased by 20% from the nominal value.

Figure 7 . 6 -

 76 Figure 7.6 -High resolution image at 200 mT of a DG structure coming from saturation. The red circle shows some examples of misalignment of paired cocoons.

Figure 7 . 7 ,

 77 starting from the recorded hologram in the Fourier space. Due to the presence of two orthogonal slits, a vertical and a horizontal line are cutting the holograms. As shown with the FT in Figure 7.7.a, no magnetic information is recoverable as such.

Figure 7 . 7 -

 77 Figure 7.7 -Processing the HERALDO holograms. The gray images are in the real space and corresponds to the real part of the reconstruction. The modulus of the holograms (CL-CR) and of the filters are shown in Fourier space.

Field and temperature

  dependencyHERALDO measurements have been performed at 300 K and 80 K while sweeping an OOP magnetic field from positive saturation to negative. Representative maps of the vertical component of the magnetization in the object hole are presented in Figure7.8.a (the full field dependency is available in Appendix H). At 300 K (top row), starting at saturation at 330 mT, skyrmionic cocoons have nucleated when lowering the field as shown with the image at 240 mT. Two contrasts are detected, for single or paired cocoons. Based on the hysteresis measurements, when decreasing the field from positive saturation the middle layers with large PMA should only accommodate domains for fields lower than 125 mT, which thus prohibits the existence of full columnar skyrmions above that field.Those observations thus confirm the presence of different 3D textures. Then, at 100 mT, large stripes begin to propagate into the field of view, corresponding to 3D stripes extending over the full thanks to the previous MFM and micromagnetic studies (Chapter 5). At remanence, a mixed state of 3D stripes and paired cocoons is observed. Upon the decrease of the magnetic field, the stripes of up magnetization get narrower and paired cocoons emerge in their stead. Finally, at -270 mT, we can observe a few sparse skyrmionic cocoons along with a few columnar skyrmions, near the bottom left for instance. Note that the two saturation states (with only low frequency signal) exhibit similar backgrounds due to the beamstop. At 80 K (bottom row in Figure7.8.a), a field of 330 mT is not sufficient to saturate the magnetization.

Figure 7 .

 7 [START_REF] Waldrop | More than Moore[END_REF].c, the field dependency of the density of isolated objects is plotted for the two temperatures. At 300 K, it is in agreement with the phase diagram predicted with magneto-transport measurements (see Subsubsection 6.1.4), as the cocoons start to nucleate near 280 mT, hence the increase in the density, and the nucleation of domains in the PMA layers near 125 mT lowers the number of cocoons. It thus drops from nearly 15 objects per µm 2 to about 3 objects per µm 2 below 50 mT. The density at high negative field depends on the way the 3D stripes disappears so variations are expected depending on the magnetic history. At 80 K, the density does not exceed 4 objects per µm 2 over the whole field range. The observed differences can be attributed to the evolution of the magnetic parameters with the temperature that can be probed thanks to magneto-transport measurements. Complementary images are presented in Appendix H at 300 and 80 K in the absence of a magnetic field.

Figure 7 . 8 -

 78 Figure 7.8 -HERALDO imaging performed with a 2 µm object hole. a) Field dependency measured at 300 (top row) and 80 K (bottom row). The colored shapes put forward examples of paired or single cocoons cocoons as well as transitions between the two states. b) MFM images (3 × 3 µm 2 ) at remanence at room temperature (left) and 77 K (right). (At low T, the tip magnetization switches at domain walls, but not at the edges of cocoons). The length scale is common to (a). c) Evolution of the density of isolated objects based on HERALDO measurements.

Figure 7 . 10 -

 710 Figure 7.10 -Simulations of holography images obtained by sweeping the magnetic field for the two cases studied in the main text. To simulate at room temperature (resp. at 80K), K u = 0.82 (0.93) MJ/m 3 and M S = 1.22 (1.26) MA/m were used. In both cases, D S = 2.34 pJ/m and A = 18 pJ/m.

Figure 7 . 11 -

 711 Figure 7.11 -Size study of the isolated objects observed with HERALDO based on two-dimensional Gaussian fits. a) Size and amplitude distribution at 240 mT. The images on the right display an example of experimental data along with the corresponding fit. The two dotted lines respectively correspond to each of the FWHMs which are then averaged for each object to yield the size histogram. The white scale in both images is 60 nm long and pixels are 15 × 15 nm 2 . The inset image shows the magnetic state in the 2 µm object hole. b) Field evolution of the median FWHM (Q2) along with the color coded density at 300 and 80 K. The error bars correspond to the first (Q1) and third quartile (Q3) of the average FWHM as illustrated in (a).

Figure 7 . 12 -

 712 Figure 7.12 -HERALDO measurements on a DG multilayer with different ferromagnetic elements: Co in the gradients parts and Co/NiFe in the strong PMA layers. OOP field dependency after an OOP saturation at the Co edge (top) and at the Ni edge (bottom). The object hole is 5 µm wide. The last pair of image (at 350 mT) was measured after another field sequence.

Figure 7 . 13 -

 713 Figure 7.13 -Introduction to X-ray laminography. a) Setup (from[START_REF] Witte | From 2D STXM to 3D imaging: Soft x-ray laminography of thin specimens[END_REF]). b) Filling of the Fourier space based on the projections plane measured in the real space (from[START_REF] Holler | Three-dimensional imaging of integrated circuits with macro-to nanoscale zoom[END_REF]). The missing volumes are represented in blue.

Figure 7 . 14 -

 714 Figure 7.14 -STXM images recorded on the DG1 multilayer (S = 0.1 nm). a) Different dots probed at normal incidence (θ = 0 • ). b) Examples of projections measured with the tilted incidence angle (θ = 45 • ). The dot corresponds to the leftmost one in (a). The red circles evidence the rotation of the "hook" feature visible in the central part of the dot. The arrow indicates the direction of the X-rays wave vector (k).

Figure 7 .

 7 15.b) which shows skyrmionic cocoons even near the edges of the dot. The state is reminiscent of the STXM image which displays similar contrasts (Figure 7.15.c). With the laminography experiment, we thus expect to reconstruct the 3D worms and the skyrmionic cocoons visible in those images.

Figure 7 . 15 -

 715 Figure 7.15 -Typical magnetic state at remanence after an OOP saturation. MFM measurement in a) plain film and b) a microdot. c) STXM measurement on a different dot. The dotted circle represents the cylinder reconstructed via laminography.

Figure 7 . 17 -

 717 Figure 7.17 -Three-dimensional view of the laminography reconstruction displayed with the isosurface m z = 0. The horizontal cut corresponds to the magnetization of the bottom layer. Examples of columnar skyrmions or paired cocoons are circled.

Figure 7 . 18 -

 718 Figure 7.18 -Close-up on 3D textures near the center with the isosurface m z = 0. The quiver correspond to a) m z inside the isosurface and b) m x in the domain walls. The inset in (a) shows the position of the zoom on the complete reconstruction.

Figure 7 .

 7 Figure 7.20.b indeed corresponds to a singularity in the reconstructed magnetization.

Figure 7 . 20 -

 720 Figure 7.20 -Bloch points (max of ∇ • Ω) and anti-Bloch (min of ∇ • Ω) points found near the central region of the dot. a) Magnetization and extrema of the divergence of the magnetic vorticity. b) Interpolated representation of the Bloch point indicated by the black square. The inset shows a schematic of Bloch point (BP) and anti-Bloch point (ABP).

Figure 7 . 21 -

 721 Figure 7.21 -Relaxed magnetic state initialized from the laminography reconstruction. a) Horizontal cuts of the magnetization in representative layers. b) 3D representation with isosurface m z = 0. The dark circles evidence some differences with the experimental reconstruction.

Figure 7 . 22 -

 722 Figure 7.22 -Laminography reconstruction with 40 projections measured in the DG2 multilayer (S = 0.2 nm, X 1 = 1.7 nm, N = 13 layers, M = 15 repetitions, Y = 1 nm, t N Pt = 2 nm and t M Pt = 3 nm).

8. 1 .

 1 Summarystructure, has emerged from this study, displaying promising capabilities for stabilizing zero-field skyrmions.

Figure 8 . 1 -

 81 Figure 8.1 -Main investigation methods used during this PhD for the study of 3D spin textures.

Figure 8 . 2 -

 82 Figure 8.2 -Racetrack concept with multiple encoded magnetic states thanks to 3D textures with limited vertical extension (FST: fractionnal skyrmion tube). From [207].

Figure A. 2 -

 2 Figure A.2 -Rotation θ = sin (m z ) in a domain wall of the strong PMA layers for different cell sizes.

Figure D. 1 -

 1 Figure D.1 -Impact of the NiFe thickness. AGFM hysteresis loops for Ta5|Pt3|(NiFe t N iF e |Al1.4| Pt3) 3 multilayers.

  Figure D.2, the AGFM cycles are displayed for different multilayers consisting of ferromagnetic layers Co t seed |NiFe0.8, where t seed ranges from 0 to 0.3 nm. A significant impact on the anisotropy is G A p p e n d i x

3 . 2 (G. 1 )IA p p e n d i x Optimization of the laminography reconstruction and additional data I. 1

 3211 Smoothing the beamstop. The beamstop is used to protect the detector from high intensities near zero frequencies. However, due to the sharpness of its edge (see Figure G.1.a,b), it can introduce artefacts in the real-space image. To smooth this edge, we use the function introduced in[START_REF] Streit-Nierobisch | Magnetic soft x-ray holography study of focused ion beam-patterned Co/Pt multilayers[END_REF] defined as:u(k) = 1 + a exp -(kk 0 ) 2 σwith a and σ the parameter to adapt to fit the beamstop. The latter is linked to the size of the filter while the former adapts the width of the smoothing area. A typical profile is shown in Figure G.1.c for a = 10 5 and σ √ ln a = 25 pixels. Following this pre-processing, different treatments are applied depending on the exact nature of the measurements, either FTH or HERALDO. Experimental and reconstructions details For the main reconstruction presented in Chapter 7 for a DG structure of step 0.1 nm, a single helicity STXM frame represents a 4.25 × 4.25 µm 2 with 20 × 20 nm 2 pixels for a dwell time of 20 ms. A 30 nm Fourier zone plate was used and both exit slits were set at 20. The energy was set at the L 2 Co edge (796 eV). Depending on the reconstruction, angle steps of 7.2 • for 50 projections or 14.4 • for 25 projections were used. The reconstruction of the magnetization was performed using the algorithm implemented by Claire Donnelly on the PolLux beamline at PSI, based on a gradient descent process [204, 205] (see Subsubsection 7.2.1.1). The programs are used to properly align the different projections to use them as an input for the reconstruction algorithm. Two additional processes were implemented to improve the quality of the reconstructions as explained below. Beam deconvolution The individual STXM projections are deconvoluted with a Gaussian filter of standard deviation 37.5 nm (width of the point spread function of the beam).

  • incidence angle, a loss of signal was observed in different part of the dot (see Figure I.1.a). This arises due to the redeposition of the multilayer materials during the etching process. In Figure I.1.b, a schematic of the X-rays propagation to understand the different areas.

Figure 7 . 21 .

 721 Here, we present the field evolution of the magnetization in a microdot (2.4 µm diameter) to observe the effect of the edge on the spin textures. The projection of the average m z along the normal is shown at different OOP magnetic field in Figure J.1. Three contrasts are observed in the field sequence that corresponds to the different magnetic objects: 3D worms or columnar skyrmion (black), paired cocoons (dark gray) and single cocoons (light gray).

Figure J. 1 -

 1 Figure J.1 -Weighted average of m z projected along the normal in a microdot in presence of an out-of-plane magnetic field. The initial state corresponds to the interpolation of a laminography reconstruction.

. 18 2.2 A practical introduction to topology 20

	C h a p t e r
	2	Basic concepts of nanomagnetism in magnetic multilayers
		CONTENTS
	2.1 Magnetization relaxation in magnetic multilayers	10

2.1.1 Landau-Lifshitz-Gilbert equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.2 Energy contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.3 Skyrmion chirality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Topology basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.2 Bloch points and other singular topological defects . . . . . . . . . . . . . . . . . . . . 22 2.2.3 Topology of the magnetic skyrmions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 Interaction with X-rays 25

Table 2

 2

	Interaction	Effect	Energy density
	Heisenberg exchange	Collinearity	A (∇m) 2
	DMI	Chiral tilting	D L (x) zx + L (y) zy
	Dipolar	Flux closure domains	-µ 0 M S m • H demag /2
	Anisotropy	Collinearity along easy axis	K eff (m • z) 2
	Zeeman	Alignment with external field	-µ 0 M

.1) which can be determined thanks to the Landau-Lifshitz-Gilbert equation in the micromagnetic approach. S m • H ext

Table 2 . 1 -

 21 Magnetic interactions in multilayered systems.

Table 2 . 2 -

 22 Definitions of the dimensions and associated contours.

	Symbol	Interpretation	Values	Contour dimensionality
	d	Real space dimension	1,2,3	n = d -1
	p	Number of m components 1 (Ising), 2 (XY), 3 (Heisenberg)	r = p -1

Towards aperiodic magnetic multi- layers CONTENTS 3.1 Precise and controlled growths via magnetron sputtering 30 3.1.1 Principle of magnetron sputtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1.2 Calibration with thin films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 Structural characterization of periodic multilayers 33 3.2.1 Reproducibility of the growths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2.2 Imaging the atomic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3 Breaking the periodicity to tune the magnetic interactions 38 3.3.1 The Pt|Co|Al trilayer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3.2 Standard skyrmionic magnetic multilayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.3 First designs of intricate magnetic multilayers . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.3.4 Controlling the vertical extension with an external magnetic field . . . . . . 49 3.4 Summary 51

	C h a p t e r
	3

.1. Those correspond to a set measured at a given time since the rates evolve as each target is etched deeper, hence the need for regular calibrations.

	Material Rate (Å/s) Current (mA) Power (W)
	Pt	4.41	180	80
	Co	0.86	125	46
	Al	1.81	150	55
	Ta	1.12	150	44

Table 3 .2 -Thickness

 3

			Sample 1		Sample 2
	Materials	Ta Pt Co Pt	Ta Pt Co Au
	Nominal thickness (nm)	10	8 1.7 3	10	8	1.7	5
	Fitted thickness (nm)	9.2 8.1 1.6 3	9.2 8.1 1.6 5.2
	Roughness (nm)	0.3 0.3 0.5 0.5	0.4 0.5 0.5 0.4

and roughness determined with XRR.

  see Appendix A). The relaxed states after an out-of-plane initialization are represented with isosurfaces for four different OOP magnetic fields. At zero field, considering a planar cut, an irregular lattice of circular objects and more elongated ones is found. Both kinds are columnar, i.e. they are extending from the bottom to the top interface. At µ 0 H ⊥ = 175 mT, the objects are smaller and more circular, resulting in a configuration similar to what can be found in standard skyrmion multilayers

Table 6 .

 6

1 -Experimental constants extracted from R xx (θ)

Table 6 . 2 -

 62 Statistical Hall measurements: characteristic of the tracks and associated measurements.

	Track W Track (µm) W Ta (µm) Area (µm 2 )	Field (mT)	I (mA) # Cycles # Transitions
	III	2.4 ± 0.3	2.3 ± 0.3	5.1 ± 1.3	200 → 180	2	37	961
	IV	2.4 ± 0.3	2.3 ± 0.3	5.1 ± 1.3	±200 → ±175	2.5	72	830
	V	1.4 ± 0.3	2.3 ± 0.3	3.2 ± 1.1	-160 → -115	1.5	64	839

Table 7 . 1 -

 71 For the SG, K u = 0.83 ± 0.03 MJ/m 3 at 80 K which is coherent as it does not have strong PMA layers that increase the anisotropy. Measurements in another DG at additional temperatures are presented in Appendix E. Magnetic constants in a DG structure at different temperatures obtained from magnetotransport and SQUID measurements.

	Temperature (K) M S (MA/m) K u (MJ/m 3 )
	80	1.26 ± 0.02	0.93 ± 0.03
	300	1.22 ± 0.02	0.82 ± 0.03

The skyrmion size and chirality of its edge can slightly vary depending on the layers so that it is not strictly invariant over the thickness.

2.1. Magnetization relaxation in magnetic multilayers

We know from previous studies[START_REF] Legrand | Spatial extent of the Dzyaloshinskii-Moriya interaction at metallic interfaces[END_REF] that the growth itself might disymmetrize the interfaces so that a small DMI contribution might appear, however reduced by one order of magnitude.

This approximation is even less valid in magnetic multilayers, strongly discretized along the vertical axis.

A sphere of dimension p -1 is naturally embedded in a space of dimension p. In other words, for a p-vector order parameter, the manifold of internal states or contour is of dimension p -1 since the amplitude is constant.

2.2. A practical introduction to topology

For hopfions, the base manifold has actually a higher dimension than the target one: it belongs to the homotopy group π3(S 2 ).

Skyrmions can also have N sk = -1, like anti-skyrmions. They can be differentiated by their vorticity, i.e. the configuration of the spin rotation. If the vertical component of the skyrmion magnetization in Figure2.9.c was reversed (as well as the background), it would correspond to N sk = -1.

2.3. Interaction with X-rays

According to the dipole selection rules, transitions to s shells are also theoretically allowed but it was shown that the d transitions dominate[START_REF] Ebert | L-edge x-ray absorption in fcc and bcc cu metal: Comparison of experimental and first-principles theoretical results[END_REF].

3.3. Breaking the periodicity to tune the magnetic interactionsexchange. This shift allows to recover an estimation of the DMI strength[START_REF] Legrand | Spatial extent of the Dzyaloshinskii-Moriya interaction at metallic interfaces[END_REF]:

A drop of mz in the middle layers (whiter background) is also noticeable in the vertical cut. It arises from the stronger in-plane effective anisotropy that pulls the magnetization away from the normal. This does not impede the stabilization of out-of-plane spin textures

4.1. First approach of skyrmionic cocoons

Regarding the contrast, mumax 3 automatically sets the color scale to match the amplitude of the output so that it does not appear to change, contrarily to expectations.

The tip remains magnetized along the magnetic field (along z) during the measurements so it cannot be responsible for the variations of the signal.

The disappearance of the spin textures from the top layers primarily stems from the system's asymmetry induced by the DMI. When the sign of the DMI is reversed in simulations, they vanish from the bottom layers first.

Those simulations were run again with smaller cell size (1 nm 2 square), yielding similar observations.

Two field modules are available on our MFM setup: the first one allows to scan from -150 mT to +150 mT while the second one adds a permanent magnet, thus accessing the range 150 mT to 300 mT.

We consider the magnetization discontinuous along z due to the distance between two ferromagnetic layers and the absence of exchange coupling (due to the non-magnetic spacer in between).

4.3. Investigating the effective chirality

5.1. Optimization of the double gradient structure

The vertical scale is not representative of the physical system as the spacers are not included and the ferromagnetic layers have slightly different thicknesses.

K eff should be used instead of Ku but for an in-plane anisotropy this becomes ill-defined. Thus, we approximate the width using this formula.

In the simulations, rotating ϕ by 90 • is equivalent to exchange mx and my.

6.2. Detecting skyrmionic cocoons

Signal from spin textures nucleating farther away from the active area also contribute but it scales with the inverse of the distance to the contacts and are thus neglected here.

If the light is only partially coherent, the coherence function of the illuminating wavefield can be reconstructed to limit reconstructions failures and artefacts[START_REF] Clark | High-resolution three-dimensional partially coherent diffraction imaging[END_REF].

7.1. Measuring in transmission: X-ray holography

Domains wall with opposite sense of rotation want to move in opposite directions.

The seed of Co (0.3 nm) used to obtain the proper anisotropy with the permalloy is not considered here as its signal in XMCD measurements would likely be very weak.

Remerciements Remerciements

with the corresponding numerical values. The latter are obtained from the same previous simulations, this time considering the in-plane components and using the experimental coefficients (R AMR , R SMR ) determined previously. The resulting predicted magnetoresistance curves are in good agreement with the experimental resistance as shown on Figure 6.4, similarly to what we find for the Hall resistance (see Figure 6.3).

The discrepancies visible in the (yz) and the (xz) plane are of similar amplitude but they appear larger for the latter case due to its reduced amplitude (and scale). To highlight the scale difference, note that the black curve have the same amplitude on both graphs. Typically, the maximum error between the simulations and the experiments is less than 10 mΩ. Note that a different track was measured for the two planes resulting in a different R 0 xx . Hereafter, we discuss the diverse trends observed in these measurements (Figure 6.4), while bearing in mind the negative contributions of the SMR and AMR to the signal. The average square components of the magnetization per layer are shown in Figure 6.5 to help with the analysis.

• ϕ = 0, 90 • and θ = 0 • Starting from large positive field, the magnetization is saturated along z and, as no textures nucleate at first when decreasing the field, the resistance R xx remains constant. Near 250 mT, skyrmionic cocoons start to appear and their boundaries or domain walls (DW) include in-plane components in the magnetization, both m x and m y contributions (see Figure 6.3.b for the relaxed numerical states).

Those contributions are relatively small with ⟨m 2

x ⟩ ≈ ⟨my 2 ⟩ < 0.25 (see Figure 6.5), but as the AMR and the SMR have negative contributions, the resistance drops accordingly. As the cocoons and their DW expand, this contribution increases until they reach their maximum size at remanence, hence the minimal resistance. The inverse reasoning holds for negative magnetic field which contracts the magnetic textures and their DW, thus reducing their negative contributions.

Preparation of the samples

To allow the X-rays to go through the sample, it is imperative to deposit the multilayers on a substrate that is transparent as much as possible to the X-rays in the relevant range of energy. Thus, they have been grown on the top side of a Si 3 N 4 membrane (200 nm thick, 0.5 × 0.5 mm 2 ) with a back side covered with a 1.1 µm-thick aluminum-gold mask multilayer Ta5|(Au25|Al2.5) 40 that was milled away with focused ion beam (FIB). The multilayer structure allows to reduce the strain and hence the buckling of the membrane. The mask is opaque to the X-rays and thus only allow them to pass through the holes prepared with FIB. Some examples are presented in Figure 7.1.b for FTH (top) and HERALDO (bottom) with SEM (scanning electron microscopy) images, evidencing the reference holes for the former and the slits for the latter. In our multilayers of interest, for FTH, the reference holes are typically 30 to 50 nm wide with an OH of 800 nm in diameter while in HERALDO the slit is 30 nm wide and 5 µm long for an OH of 2 µm. For HERALDO, two slits are prepared as it could be used to acquire projections while tilting the sample, thus allowing to reconstruct the 3D magnetization.

The dimensions of the apertures are a trade-off between the resolution and the contrast, as the large the reference hole the more light there is but the resolution is more limited. Additionally, the aperture have an upper limit that corresponds to the coherence length of the X-rays.

Unless mentioned otherwise, the FTH and HERALDO measurements have been performed on the SEXTANTS beamline at SOLEIL synchrotron using a 2048 × 2048 pixels Peltier cooled CCD camera and with X-rays near the L 3 Co edge (at 778.5 eV). Thanks to the undulators, a polarization efficiency for circular or linear polarization superior to 99% can be reached.

Measuring in transmission: X-ray holography

The second projector P m is defined in the reciprocal space to apply the modulus constraint. It replaces the modulus of the current iteration by the experimental one: (k) with P m = F -1 P m F (7.4)

with ϕ(k) the reconstructed phase, I(k) the experimental intensity, F -1 and F the inverse and forward Fourier transforms respectively. The operator P m is a projector on the nonconvex set of the magnitude constraint. Finally, as it is an iterative process, it requires a starting point ρ 0 that is usually taken as the experimental modulus multiplied by a random phase factor. Otherwise, it can be set as the inverse FT of the support mask after adjusting its amplitude to match with the experimental one (see Appendix G).

Using those operators, the Gerchberg-Saxton algorithm, often called the error reduction (ER) algorithm, can simply be written at the iteration ρ n+1 as:

ρ n+1 (r) = P s P m ρ n (r) (7.5)

Illustration of the iterative process

An example of the iterative loop is shown in Figure 7.2 with experimental data to illustrate the previous algorithm. First, the initialization is performed by using the experimental diffraction pattern multiplied by a phase matrix obtained from the inverse FT of the support. The center of the hologram is hidden by a beamstop (the horizontal line with a circle in the center) to protect the CCD camera from the most intense signal. This initial state ρ 0 (k) is then fed into the main loop. From Eq. 7.5, the FT is first computed to go back to the real space. This yield an intense central circular signal that is surrounded by different reflections of the object hole. The central area arises from the autocorrelation of the object holes and the reference holes separately. Then, the correlation of each reference hole with the object hole gives a pair of twin images on opposite sides from the center. Those transmission images directly give information on the magnetic structures [START_REF] Eisebitt | Lensless imaging of magnetic nanostructures by X-ray spectro-holography[END_REF][START_REF] Streit-Nierobisch | Magnetic soft x-ray holography study of focused ion beam-patterned Co/Pt multilayers[END_REF][START_REF] Latychevskaia | Phase retrieval methods applied to coherent imaging[END_REF]. This state is then projected onto the support with P s , here on the top left OH image along with the four different RH. Finally, this is transformed back into the reciprocal space before switching out its modulus by the experimental one (P m ), apart for the beamstop pixels which are left free.

Note that the pixels hidden behind the beamstop, i.e. the low frequency information, are recovered during the iterative process. Finally, the error reconstruction is simply computed as the relative difference between the amplitude of the reconstructed hologram and the experimental one. The solutions are unfortunately not unique so that it is important to repeat the phase retrieval process several times starting from a different initial state to ensure the convergence.

Temperature dependence of the anisotropy and consequence on the stability of the cocoons

To investigate the evolution of the magnetic properties at low temperature, magneto-transport measurements have been performed on the DG structure (S = 0.1 nm, X 1 = 2.0 nm, N = 13 layers, M = 15 repetitions, Y = 1 nm, t N Pt = 3 nm and t M Pt = 3 nm) measured with HERALDO as well as on the corresponding SG sample (S = 0.1 nm, X 1 = 2.0 nm, N = 13 layers, t N Pt = 3 nm). The multilayers have been patterned using UV lithography into Hall bars of dimensions 20 × 100 µm 2 before being measured in a field setup with an electromagnet able to reach 0.65 T. In Figure 7.9.a, R xy (H, θ = 0 • ) is displayed at 300 K (top) and 80 K (bottom) showing an increase of the saturating field, a more open loop and a transition happening at lower field when lowering the temperature. Indeed, the jump near 30 mT at 80 K corresponds the formation of domains in the strong PMA layers as observed with the previous holography sequence. To draw more quantitative information, R xy (θ) has also been studied (Figure 7.9.b) at 650 mT, with θ the angle of the field with respect to the sample normal. For the DG, the presence of a sharp transition due to the reversal of the middle layers is visible.

To estimate the change of anisotropy, we assume a simple model considering only an interfacial uniaxial anisotropy axis:

Fitting R xy with this model yields an average effective anisotropy as all layers do not have the same magnetic properties because of the thickness gradient. The results for the DG multilayer are presented in Table 7.1 along with the M S measured with SQUID in the same multilayer.

STXM projections

The DG1 multilayer (S = 0.1 nm) has been investigated first in STXM with a normal incidence to probe the different magnetic states in several dots. Such projections are shown in Figure 7.14.a evidencing two types of contrast that arise from the 3D worms and the skyrmionic cocoons.

In Figure 7.14.b, the incidence angle is 45 • and two projections are shown for different rotation angles measured on the first dot. The characteristic features are still identifiable as evidenced by the red circles, however the signal in the top left part appears more blurry. This weaker contrast is due to the redeposition of the resist edges during the etching of the multilayer. It can reach more than 500 nm height and thus act like a screen for the X-rays arriving near the dot edge (see Appendix I for a schematic). The central area of the dot thus contains the most reliable information. The bottom right signal arises from the X-rays that go through the redeposited material only (and not the multilayer).

Reconstructing the magnetization in 3D

To obtain a complete set of projections, we focus on the dot of 3.5 µm diameter introduced in As a more explicit representation of the magnetization distribution, the isosurface m z = 0 is shown in Figure 7.17 which delimits the transitions from one domain to another, allowing an easier identification of the different spin textures. It evidences the 3D worms that extend from the bottom to the top layers while inside the red domains, pairs of skyrmionic cocoons are present that do not extend in the middle layers (black ellipses). Some columnar skyrmions (yellow ellipses) are also visible.

Estimation of the laminography resolution

In various imaging fields, evaluating the resolution is a crucial aspect of the experiments. To this end, Fourier shell correlation (FSC) is widely used for microscopy imaging and three-dimensional reconstructions [START_REF] Van | Fourier shell correlation threshold criteria[END_REF]. For this, the data set is divided into two independent sets, in our case made of 25 projections each, and the reconstructions are processed separately. Then, in Fourier space, the cross-correlation between the two independent data sets is computing over spherical shells of increasing size as a function of the spatial frequency k:

with F j (k) the Fourier transform of each subset and k i the i th shell. Many criteria exist to establish the resolution based on the previous computation [START_REF] Van | Fourier shell correlation threshold criteria[END_REF] but in our case, we use the 1/2 bit criterion that corresponds to a signal to noise ratio of 0.41. The FSC computed for each components are displayed in Figure 7.19 along with the different resolution (r x , r y , r z ). It is relatively constant along each direction, with an average of 75.7 nm. As the single gradients are nominally 86.8 nm thick, the resolution should thus be sufficient to resolve the cocoons at remanence as they extend over the full gradient.

However, it might not be enough to resolve the domain wall chirality as well as the singularities in the magnetization. Note that it is only an estimation of the resolution and that it depends on the used criterion. Interestingly, even with only 25 projections, skyrmionic cocoons are still visible in the 3D reconstructions, supporting the validity of the complete reconstruction.

Magnetic vorticity

The laminography reconstruction, if the resolution was high enough, would allow the magnetic vorticity to be calculated. It is defined as [START_REF] Donnelly | Experimental observation of vortex rings in a bulk magnet[END_REF]:

A

A p p e n d i x

Micromagnetic simulations of aperiodic magnetic multilayers

This appendix contains all the essential information required to replicate the micromagnetic simulations using mumax 3 [START_REF] Vansteenkiste | The design and verification of MuMax3[END_REF] as described in this manuscript. It also includes technical specifications and validations.

A.

A.1.1 Dilution of the magnetic parameters

In mumax 3 simulations, the voxels within the simulation space have fixed dimensions, making it challenging to define layers with varying thickness, which is a concern for our aperiodic multilayers.

To address this issue, the magnetic parameters are diluted by a factor that is proportionate to the thickness of the respective layer. This dilution factor, denoted as f = X i /L z , is determined using the vertical dimension of the cells L z . Then the micromagnetic parameters are adjusted accordingly [START_REF] Lemesh | Accurate model of the stripe domain phase of perpendicularly magnetized multilayers[END_REF]:

This approach establishes effective parameters for the magnetic layers, considering their nominal thickness, which enables the study of aperiodic multilayers, including single or double gradient structures.

By accounting for the varying thicknesses, we can accurately analyze and investigate such complex magnetic configurations.

A.1.2 Technical details and verification

Inducing nucleation in double gradient multilayer Achieving numerical nucleation in strong perpendicular magnetic anisotropy (PMA) layers can be challenging due to the high energy cost associated with domain formation. However, in order to accurately reproduce the magnetotransport measurements (see Chapter 6) as well as the MFM imaging (see Chapter 5), it is necessary for the strong PMA layers of the DG multilayer to accommodate domains. To facilitate nucleation at the desired field, a seed is employed. More specifically, a Néel skyrmion (of 15 nm radius) is introduced solely in the middle layers through direct manipulation of the magnetization tensor. This technique aids in achieving the required nucleation and allows to fit the magnetotransport measurement.

Impact of the cell size

To further validate the results, simulations were conducted for both the SG and DG structures using a smaller cell size. The purpose was to ensure that the relaxed magnetic state remained largely unchanged. In Setting the magnetic parameters In the second part, the magnetic properties of all the magnetic layers within the simulation are assigned using the dilution method. This ensures that the magnetic parameters are appropriately defined, taking into account the varying thicknesses of the individual layers.

Numerical impact of the DMI strength

In the Pt|Co|Al multilayer, the DMI was experimentally measured using Brillouin light scattering (see Subsubsection 3.3.1.3), yielding a value of D s = -0.80 ± 0.05 pJ/m. However, in the micromagnetic simulations, a higher DMI strength of D s = 2.34 pJ/m was employed. From the energy analysis presented in Figure 4.6, it is evident that the dipolar and anisotropy energies dominate over the remaining terms. Therefore, this discrepancy in the magnetic parameters is not expected to significantly impact the stability of spin textures such as cocoons and 3D worms. However, it could affect the chirality of the domain walls. Consequently, this appendix aims to verify whether the relaxed magnetization states and the chirality remain relatively unchanged. To align with the range of values reported in the literature for the Pt|Co|Al stacking [START_REF] Ajejas | Tuning domain wall velocity with Dzyaloshinskii-Moriya interaction[END_REF][START_REF] Kim | Chirality-induced antisymmetry in magnetic domain wall speed[END_REF], a DMI strength of D s = 1 pJ/m is used for this verification.

Evolution of the relaxed magnetization

In In summary, using a larger DMI strength than the experimental value only leads to a minor difference in the vertical position of the skyrmionic cocoons at high fields. However, it does not significantly alter the relaxed magnetic states: the spin textures and their chirality remain largely unchanged.

Other single gradient structures

Reversed single gradient

To corroborate the numerical study presented in Figure 4.7 about the vertical confinement, different reversed single gradient (RSG) structure has also been investigated with MFM (N = 13 layers, S = -0.1 nm, t N Pt = 3 nm). As presented in Figure 3.10, at remanence after an OOP saturation, a worm state is stabilized for X 1 ranging between 2.0 and 2.5 nm (see

Symmetric stack Pt|Co

To verify the importance of the dipolar interaction in the stabilization of isolated textures, an SG structure was grown without the aluminium layers (Ta5|Pt3|(Co [1.8:

The symmetric interfaces around the Co are not expected to result in a significant DMI contribution as the inversion symmetry is conserved. From previous studies [START_REF] Legrand | Spatial extent of the Dzyaloshinskii-Moriya interaction at metallic interfaces[END_REF], we know that the growth might disymmetrize the interfaces so that a small DMI contribution might appear, however reduced by one order of magnitude. In Figure C.1.c, the corresponding MFM image measured at remanence after an OOP saturation is displayed. Similarly to a typical SG structure (S = 0.1 nm and X 1 = 1.8 nm), it is possible to observe isolated objects, although they are more elongated. This supports the significant role of the dipolar interaction in their stabilization in SG structure. already observed for a Co seed thickness of 0.1 nm (red curve) compared to the case of pure NiFe (gray curve). As demonstrated by the out-of-plane (OOP) loops, the anisotropy is now oriented perpendicular to the film plane. However, to obtain a more square-shaped loop, a seed thickness of at least 0.2 nm is required (blue curve), as shown in the zoomed-in view of the OOP cycles. Thus, the incorporation of a Co seed layer enables the stabilization of perpendicular magnetic anisotropy in NiFe layers.

D.2 Aperiodic multilayers with NiFe

Using a seed of Co to induce a non-zero uniaxial anisotropy in permalloy layers allows to develop more aperiodic multilayers like the single gradient (SG) and double gradient (DG) structures. They are described with the same parameters used in the main text (X 1 , S, N , M , Y , t N Pt , t M Pt ) with different subscript for the Co or the Ni. In this section, the seed thickness is consistently set to 0.3 nm and utilized whenever NiFe is present in the structure, even if not explicitly mentioned.

D.2.1 SG structure

D.2.2 DG structures

After ensuring that NiFe-based SG could be optimized, it follows to expend this knowledge to DG multilayer. First, we optimized DG with only permalloy before developing hybrid multilayers with both Co and NiFe layers.

NiFe DG structure

Similarly to the SG structure, we first start by changing the bottom layer thickness X NiFe = 1.7 nm, a worm state is stabilized along with a few skyrmionic cocoons in between but their density is relatively low. Increasing the thickness, 1.9 or 2.1 nm, allows to obtain more cocoons at remanence. Similarly to Co-based DG, the Pt thickness is also a sensitive parameter as it impacts the dipolar interaction which is mainly responsible for the stabilization of 3D textures in those multilayers. For t N Pt = 2 nm, the number of cocoons is clearly superior than for t N Pt = 3 nm as they form chains between the worms. With MFM, both cocoons and 3D worms are clearly observed. For the inverse configuration (NiFe in the SG blocks and Co in between), the same conclusion holds even though the cocoons distribution appear more symmetric as they are present in both types of stripes. Asymmetric multilayers can also be probed, with NiFe only in the top SG block which yields similar observations. In This section presents the micromagnetic simulations that have been used to compute the magnetoresistance behaviors with the field in Chapter 6.

E.1.1 In the SG multilayer

In Figure E.1, the relaxed magnetic states are shown for different angles of the fields when sweeping from positive to negative fields. The SG multilayer under study corresponds to X 1 = 1.7 nm, N = 13 layers, S = 0.1 nm and t N Pt = 3 nm (with a spacer of 4.4 nm). The evolution of the magnetization has already been discussed in Subsection 6.1.2 and is summarized here. For θ = 0 • (a), cocoons that are vertically confined nucleate at 225 mT and, as the field is lowered, they expand in every direction. Decreasing the field from remanence contracts the isolated objects and confine them until their annihilation. For θ = 30 • (b), elongated cocoons nucleate first and they extend along the direction of the field (y), forming stripes at remanence. The stripes contract upon the decrease of the field, yielding very narrow cocoons at -175 mT, before saturation. Finally, for θ = 60 or 90 • , mainly stripes states whose extension evolves with the field are stabilized.

E.1.2 In the DG multilayer

The magnetic states discussed in Subsection 6.1.3 are presented in Figure E.2 for the different inclinations of the magnetic field. The DG structure is defined by S = 0.1 nm, X 1 = 2.0 nm, N = 13 layers, M = 15 repetitions, Y = 1 nm, t N Pt = 2 nm and t M Pt = 3 nm. For θ = 0 • (a), cocoons nucleate first in the SG blocks while the 3D worms appear near 75 mT. Decreasing the field from remanence shrinks the 3D worms until they become columnar skyrmions while the cocoons are still stable, even at -425 mT. For θ = 30 and 60 • , the magnetic states obtained with a non-zero field are quite complex however at remanence, 3D stripes are stabilized with skyrmionic cocoons inside (partial worms at 60 • ). For an in-plane field, stripes are stabilized and they extend vertically as the field decreases in absolute value.

E.2 Additional transport measurements

This section introduces complementary measurements to the DG presented in Chapter 6 as well as measurements on a DG with fewer layers.

E.2.1 High field measurements

In Figure E.3, magnetotransport measurements are presented for the DG structure studied in Subsection 6.1.3 (S = 0.1 nm, X 1 = 2.0 nm, N = 13 layers, M = 15 repetitions, Y = 1 nm, t N Pt = 2 nm and t M Pt = 3 nm), obtained with a different setup with higher field capabilities. In this case, the magnetic field was swept between ± 3T, ensuring the saturation of the magnetization. The raw data are displayed to verify the amplitude of the normal Hall effect (in R xy ) and the Lorentz magnetoresistance (in R xx ).

The former is expected to give rise to a term proportional to the field while the latter has a positive parabolic contribution (∝ H 2 ). No such effects is seen at high field: the resistances reach a plateau when the magnetization is saturated. Indeed, the variations remains relatively small, typically less than 5 mΩ. We thus neglected those contributions in our analysis.

E.2.2 Analysis of the nucleation peak in R xx

In Figure 6.7, the evolution of the longitudinal resistance with the field of a DG structure was presented. Interestingly, near 50 mT, a negative peak is observed (highlighted in the inset) that corresponds to the nucleation of domains in the strong PMA layers. R xx drops due to the negative contribution of the AMR and the SMR but surprisingly it goes up again after this decrease. To understand this phenomenon in more details, ⟨m 2 x ⟩ and ⟨m 2 y ⟩ are presented in Figure E.4.a,b,c for each block of the DG structure, i.e. the bottom and top gradients as well as the strong PMA layers. We can notice at first that the in-plane components contributions in the middle layers are smaller than for the gradient blocks, as expected from the higher anisotropy (and thus narrower domain walls). the transition for all layers, yielding an intricate evolution. Let us focus on the m y components which is mainly responsible for the change in the resistance due to the large SMR contribution. We can notice that the main changes in ⟨m 2 y ⟩ before and after the nucleation in the strong PMA layers occur in the middle layers of each gradients, i.e. where the spin textures have no well defined chirality. Thus, when the 3D worms nucleate, they disturb mainly the domain walls of those layers and as it extends upon the decrease of the field, the in-plane distribution is modified consequently.

The total average over all layers presented in

In this case, ⟨m 2 y ⟩ decreases but this process likely depends on the initial shape and evolution of the 3D worms. This might be influenced by the shape anisotropy, nucleation points or a misalignment of the magnetic field. For instance, it the track is relatively narrow, the domains might propagate along the x direction which would yield m y components (in agreement with the experiments). Additionally, if there is a slight offset of the field in the x or y direction, it could favor the propagation of domains in privileged directions. Thus, the exact shape of the transition might display a statistical behavior and is not a universal property, contrarily to its existence. For this particular case, it is however well reproduced numerically.

E.2.3 Reduced double gradient multilayer

In Figure E.5, transport measurements are presented for a DG multilayer with a reduced number of layers (X 1 = 2.0 nm, N = 5, S = 0 nm, Y = 1.0 nm, M = 5, t N Pt = 2 nm and t M Pt = 3 nm). Those are measured in the (yz) plane, thus favoring the SMR effect. For R xx , the trends are similar to the ones observed for a DG with more layers (see Figure 6.7) except that the loops appear more open. For an out-of-plane magnetic field, the signal drops when spin textures nucleate due to the emergence of in-plane components. Thus, the SMR and the AMR still have negative contributions to the longitudinal resistance. For fields with an in-plane component along y, the nucleation of out-of-plane textures decreases the overall m y component and is thus responsible for an increase of the resistance. To further leverage the magnetotransport measurements, phase diagrams can be extrapolated using the out-of-plane characterization (similarly to Subsubsection 6.1.4). In Figure E.6.a,b, R xy and R xx are presented along with their derivatives. Background colors are defined using local maxima extrema present in the derivatives, corresponding to transitions in the magnetic states. In this analysis, an additional phase is observed (in gray) compared to the study of a thicker DG multilayer conducted using a similar approach (Figure 6 This analysis reveals a novel behavior for DG structures with a reduced number of layers. As the system approaches saturation, a coexistence of skyrmions and cocoons is initially observed, followed by the annihilation of the cocoons. Eventually, the skyrmions transition into cocoons. This finding can be correlated with transport measurements to identify distinct magnetic phases and provide further insights into the system's magnetic properties.

E.2.4 Phase diagram of a single gradient

A similar procedure was applied to a single gradient multilayer (X 1 = 1.7 nm, S = 0.1 nm, N = 13 The phase diagram of an SG structure is thus more intricate to analyze but it still provides with information regarding the range of stability of the cocoons and the columnar objects. This difficulty arises from the smoother transition between cocoons and skyrmions in the SG multilayer.

E.2.5 Temperature evolution in a double gradient multilayer

This section aims to introduce additional measurements for the same DG (presented in Subsection 6.1.3), especially regarding its temperature dependency. The investigated structure corresponds to In both R xx and R xy , sharp transitions are visible around remanence which correspond to the nucleation of domains in the strong PMA layers. Before this transition, only cocoons are thus stabilized.

If we consider the opaque curves (when sweeping the field from positive to negative saturation), we can observe that this step starts near 50 mT at 300 K and shifts progressively to -50 mT at 12 K. This is coherent with an increase of the PMA in the middle layers.

For an easier interpretation of the magnetoresistance, we rely again on their derivative and the associated transitions as displayed in Figure E.9. The shift of the main peak, which is the jump linked to the nucleation of 3D worms, is clearly visible. We can also notice that the region of stability of cocoons, when going towards saturation from remanence, seems to shrink as the temperature decreases. This region corresponds to the slow decrease in R xy as indicated by the arrows before saturation (near -400 mT) or the dip in R xx at the same field. This trend was observed in Subsubsection 7.1.5.2: cocoons were observed at room temperature but not at low temperatures. A similar observation is made here, using only the magnetoresistances. Finally, the nucleation of cocoons coming from saturation occurs at lower field for lower temperatures. The range of stability for the cocoons when increasing the field from remanence towards saturation clearly shrinks as the temperature drops, similarly to the observation made on Figure E.9. This is additionally coherent with the measurements performed on DG structure with a higher anisotropy (lower X 1 ), which is equivalent to lower the temperature, that do not stabilize cocoons on the same branch of the hysteresis (see Subsection 5.1.3). The stability of cocoons with respect to the temperature is interesting to study to understand how they evolve with different magnetic parameters. The evolution of R xx and R xy as they disappear could also help to analyze their contribution to each magnetoresistance and thus to investigate more precisely the evolution of their spin configuration.

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Complementary XRMS measurements

The following measurements have been performed at the BOREAS beamline located at the ALBA synchrotron. There, the 2048 × 2048 pixels cooled CCD camera is located at 40.2 cm from the sample, covering an angular range of 2.1 degrees, with a pixel size of 13.5 µm. It records the signal coming from the MARES diffractometer [START_REF] Barla | Design and performance of BOREAS, the beamline for resonant x-ray absorption and scattering experiments at the ALBA synchrotron light source[END_REF] in ultra-high vacuum conditions with a beam of 50 µm diameter.

Thanks to the undulators, a polarization efficiency for circular or linear polarization superior to 99% can be reached. The measurements are performed near the Co L 3 edge (at 777 eV).

F.1 Study of a second DG multilayer with 0.1 nm A second DG multilayer has been probed with XRMS. It corresponds to the one that was imaged with FTH (Subsection 7.1.4) with S = 0.1 nm, X 1 = 2.0 nm, N = 13 layers, M = 15 repetitions, Y = 1 nm, t M Pt = 3 nm and t N Pt = 3 nm except in the last top seven layers of the top gradient in which it is 2 nm. The measurements were performed after an in-plane demagnetization, with the stripes aligned with the X-rays wave vector.

Angular dependency

In Increasing the incidence angle yields multiple inversion of the dichroism within a few degrees. This complexity likely arises from the evolution of the chirality predicted by the simulations (Figure 5.15).

The differences between the measurements for a constant t N

Pt over the thickness and the one presented in Figure F.1 are significant. Changing the spacer thickness, apart from modifying the magnetic properties, also impacts the reflections of the X-rays. For both studies, a numerical support is needed to correlate the scattering patterns with the magnetic configuration.

Energy dependency

On the same sample, the energy dependency of the scattering signal was also investigated at fixed θ. Changing the energy of the X-rays impact their scattering behavior (the reflection conditions) so that the scattering pattern can change significantly. In all cases, the XMCD ratio decreases as the energy is set farther away from the Co edge.

F.2 Investigation of a DG with step 0.2 nm

The thickness step is an important parameter to control the vertical confinement of the cocoons: the larger the step, the stronger the confinement (Subsubsection 4.1.3.1). Here, a DG structure with a step S = 0.2 nm (X 1 = 1.7 nm, N = 13 layers, M = 15 repetitions, Y = 1 nm, t M Pt = 3 nm and t N Pt = 2 nm) was investigated with XRMS after an OOP saturation. A mixed state of 3D worms and cocoons is stabilized after this magnetic preparation (Figure 5 Those cases do not correspond to a well-defined chirality: they likely arise from the intermixing of the signal of cocoons with 3D worms. Increasing the incidence angle yields more diffuse signal, with sometimes what appear to be the overlapping of two contributions, at 21.9 • for instance. For θ = 29 • , a CW Néel chirality is observed which is reversed at higher angles. As the signal is so intricate, no radial integral has been performed on those data. For now, no such complex signal has been reported in the litterature and no qualitative analysis can be made without numerical support.

G.2 FTH details

Definition of the support

For the phase retrieval algorithm, it is necessary to define the support which corresponds to the region in which the light goes through (the object and the reference holes). In

Missing pixels and initial guess

The final preparation for the phase retrieval process consists in determining the missing pixels to be reconstructed and to determine an initial guess for the phase. Before this step, a given percentile (2 nd in our case) of the amplitude is subtracted to the hologram. The missing pixels then correspond

with M the missing pixels matrices (equal to 1 for missing values), ϕ S (k) the phase extracted from the support S and I(k) the experimental amplitude. This input can then be used with a phase retrieval algorithm.

Example of phase retrieval implementation

Here, we present the implementation of several algorithms simply defined using the operators introduced in Subsection 7.1.2: the projector P s on the support and P m for the modulus constraint.

The following code simply defines two functions that perform the operators' action, with ρ the updated guess. We also present the OSS algorithm that was used for the reconstructions.

Reconstruction error

The error is computed as the relative difference between the current guess amplitude and the experimental one:

with i, j the pixel coordinates.

G.3 HERALDO details

After pre-processing the holograms, a differential filter is applied on the hologram to account for

G.4 Phase optimization

To maximize the contrast of the final reconstruction, either in FTH or in HERALDO, the phase can be adjusted to recover all the signal that might be dispersed between the real and imaginary part (see Figure G.5.a). To this end, the final complex valued image is multiplied by a phase factor:

H

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Additional FTH and HERALDO reconstructions

H.1 More FTH reconstructions in DG structures

The following FTH measurements have been performed at the DESY synchrotron (PETRA III).

No phase retrieval method was used to correct them as the initial contrast is already satisfying.

DG structure with X 1 = 1.6 nm

Here, we study the DG multilayer defined by X 1 = 1.6 nm, N = 13 layers, S = 0.1 nm, M = 15 repetitions, Y = 1 nm, t M Pt = 3 nm and t N Pt = 3 nm. Such structures have been characterized with MFM (Subsection 5.1.3) which showed that, following a monotonous variation of the field, the cocoons and skyrmions are stabilized after different magnetic history. This study is repeated here using a transmission technique to validate the previous observations. This field evolution is coherent with the previous MFM study.

DG structure with a reduced number of layers

Similarly, a DG with a reduced thickness (X 1 = 1.9 nm, N = 7 layers, S = 0.1 nm, M = 9 repetitions,

Three-dimensional view

To evidence the 3D nature, the isosurface of m z = 0 is presented in

List of figures List of figures

Appendix A: Micromagnetic simulations of aperiodic magnetic multilayers

List of tables List of tables

Chapter 2: Basic concepts of nanomagnetism in magnetic multilayers