Laurent Ranno

Anne Pierre

Laurent Cagnon

Nora

Vitoria Thibaut

Philippe David

Sébastien Dufresnes

Valérie Guisset

Laurent Del Rey

Gwenaëlle Julié

Didier Dufeu

Daniel Lepoittevin

Denis Maillard

Sébastien Pairis

Richard Haettel

Christophe Hoarau

David Barral

Bruno Fernandez

Thierry Fournier

Fairy Fingers

" Motte

Simon Le Denmat

Julien Jarreau Pour

Alessandro At Elettra

Mathilde Bérard

Aurélie Laurent

Angélique Simoès

mes encadrants, en passant par Sandrine, Ségo, Michal Olivier À Christophe

Bea Yann

Karim

Clément, Marine, Ioan Ketty

Émilie Lorenzo

Chloé, Magali Pauline Gilles

Olivier Maillet

Anne Sylvain

Elie Lamirand

Claire Hamelin

Alex Le Fruit

Mat Fabre

Alexandre " God

Artaud Et N'oublions Pas

Edouard Benj Canals

David Jésus-D'or

Anne Ju

Pierre Bernand-Mantel

Grand Giroux

Stefania Jan

Merci Timothée

Bibiane Vincent Audrey

Maxime Adrien Mathias

Julien, Étienne Samantha Simon

Alix " Préz

" Tiran

Elie Robin

Rémi ! Y'

J'adore écrire, et pourtant, rédiger ce qui suit m'intimide. Peut-être qu'après avoir écrit tout mon saoul dans le reste de ce manuscrit et créé un pavé bien épais, je redoute de m'éparpiller, ne serait-ce qu'au sujet de l'exercice d'écrire des remerciements. Après tout, ces remerciements seront certainement imparfaits, notamment parce qu'on ne s'attaque pas à ce genre de choses tous les jours... Cependant, avant d'aller trop loin dans le méta, revenons-en à ce qui importe.

Puisqu'il est hors de question pour moi de ne pas faire de ces remerciements quelque chose de vraiment personnel, une question se pose nécessairement : qu'est-ce que je veux exprimer ici ? Quelles sont mes attentes ?

Je crois que j'ai envie de rendre hommage aux personnes sans qui ce travail de doctorat aurait été moins simple, plus fastidieux, mis en péril, voire impossible. Mais aussi sans lesquelles ce travail aurait été moins drôle, moins stimulant, moins passionnant. De là, je tiens ensuite à célébrer ce qu'il y a eu autour de cette thèse : les bons moments au laboratoire qui n'ont rien eu à voir avec le boulot, et tout ce qui s'est construit en-dehors de l'Institut Néel. Des instants conviviaux à Grenoble, à Lyon, à tout un tas d'endroits ; des liens qui se sont tissés ou étendus, et qui ont contribué à la beauté de ces années de doctorat. Ces années, j'ai le sentiment qu'elles m'ont vu changer à pas mal d'égards, et cela m'amènera à regarder en arrière, vers la base, vers ce dont je suis parti. Et même si cela sonne comme une évidence, le soutien de mes proches à travers cet étrange passage qu'est la thèse a été déterminant, rassurant. Cela me tient à coeur de ne surtout pas l'oublier.

Si cette sacrée aventure a pu se terminer comme elle s'est terminée, c'est évidemment grâce à mon jury en premier lieu. Daniel Lacour, Ursula Ebels, Jörg Raabe et André Thiaville, merci pour votre lecture critique de mon travail, les questions posées pendant la soutenance et... votre patience pour avoir tenu les quelque deux cent cinquante pages de mon manuscrit ! De même, un grand merci, Olivier, pour ta patiente et bienveillante relecture de ma thèse avant l'envoi au jury. En fait, il n'a pas été question de patience qu'à ce moment-là ; je crois avoir eu la possibilité d'apprendre beaucoup par moi-même pendant la thèse grâce à la patience dont tu as fait preuve à mon égard. La rédaction du manuscrit en est sans doute l'exemple le plus flagrant... Merci également Olivier pour ton encadrement, ta confiance, ta rigueur et ton exigence scientifique, ainsi que ton intuition. Enfin, merci d'avoir donné "[ton] accord moral" pour qu'un grand dadais hirsute, en retard au premier rendez-vous et dégoulinant de neige fondue (pendant quelques minutes) vienne faire une thèse avec toi ! Je voudrais aussi remercier toutes les personnes qui m'ont aidé avant la soutenance, en particulier pour les répétitions. Stefania, Jan, Laurent, Benjamin et Christophe, ça a été une chance pour moi de bénéficier de votre écoute attentive et de vos conseils. Et merci Sylvain et Jean-Christophe pour vos encouragements ! Il y a au laboratoire de nombreuses personnes envers qui je suis reconnaissant pour ces années en thèse. Je l'ai toujours dit, l'Institut Néel est à mes yeux un très bon cadre scientifique mais aussi humain, et ça a été un réel plaisir pour moi que d'y évoluer et d'y travailler.

En ce qui concerne le travail, c'est bien sûr à Olivier que je pense d'abord puisque tout a commencé avec lui. Ensuite : Christophe, je rends hommage à ton soutien moral (et alimentaire, à grand renfort de graines notamment), tes conseils, les longues discussions dans l'un ou l'autre de nos bureaux, et les X manips qu'on a fait marcher ensemble. Mention spéciale aussi à la serveuse catalane terrorisée pendant le temps de faisceau à Alba ! Parmi les autres proches collaborateurs, j'en viens maintenant à toi, Jean-Christophe : je te suis très reconnaissant pour toutes les discussions que nous avons eues, ton aide absolument cruciale via les simulations numériques, ton soutien pour mes enseignements au CPP... sans oublier les fous rires dans le bureau et les thés pris en K-fêt ! C'est probablement une bonne chose que d'en venir à la K-fêt puisque K-fêt le lundi matin à 9h veut dire... MMNMM ou Monday Micro-and NanoMagnetism Meetings. Je garderai en mémoire ces fameuses réunions de début de semaine, mais surtout la convivialité et la bonne humeur (si, si, même avant le café !) dans l'équipe MNM. Merci Jan,

CVD Chemical Vapour Deposition. 56 DC Direct Current; applies to regimes in electronic systems where the control voltage(s) and/or current(s) are kept constant. By extension, this denomination may also apply to a frequency range many decades below the working frequency(/ies), e.g. when the latter is(/are) in the gigahertz range.. 31,63,94,113,123,[START_REF] Hoefer | Thermo-optic properties of gadolinium garnet laser crystals[END_REF]192,222,231,232 EDX Energy-Dispersive X-ray analysis; this spectroscopic technique involves a Scanning Electron Microscope (see SEM) and relies on the emission of photons from the sample after excitation by the primary electron beam, Auger electrons or other causes for inelastic scattering. Element-specific edges can be detected as peaks in an EDX spectrum, indicating the presence of a certain element; in some cases, it is even possible to carry out quantitative composition analyses. 125,126,187,190 EELS Electron Energy Loss Spectroscopy is an electron-microscope-based technique relying on inelastic scattering of electrons inside the investigated sample, and their subsequent energy-resolved detection. Electrons impinging on the detector with an energy smaller than the primary electron beam energy originate from such inelastic scatterings. The spectroscopy allows the specific detection of elements thanks to well-defined edges in the acquired spectra which correspond to the binding energy of certain atomic shells of these elements [3]. 147

fcc Face-Centered cubic; an ABAB type of stacking of layers in the 2D cubic lattice system, A and B being offset so that each node lies above or below the center of a square of the neighbouring layers. 64

FEBID Focused Electron-Beam-Induced Deposition is an additive, bottom-up fabrication process relying on the decomposition of precursor molecules. This reaction is triggered (mainly) by the primary electron beam of a scanning electron microscope, while the molecules are provided by a directional gas injection nozzle (see GIS). xii, 56,[START_REF] Weber | Magneto-optical properties of KTb 3 F 10 and LiTbF 4 crystals[END_REF][START_REF] Ranno | Magnetometry on Gadolinium Gallium Garnet[END_REF]148,[START_REF] Hertel | Multiscale simulation of Bloch point dynamics in thick nanowires[END_REF][START_REF] David | Microwave Engineering -Third Edition[END_REF][START_REF] Hasnain | Dispersion of Picosecond Pulses in Coplanar Transmission Lines[END_REF]153,[START_REF] Bohm | Optical dispersion of gadolinium gallium garnet (GGG)[END_REF]158,159,160,162,163,164,165,166,172,174,176,177,185,195 FIB Focused Ion Beam. 56,192 FIBID Focused Ion-Beam-Induced Deposition is a technique similar to FEBID, the main difference being that the cracking up of molecules containing the desired material is performed by an ion beam instead of an electron beam. 56,[START_REF] Bohm | Optical dispersion of gadolinium gallium garnet (GGG)[END_REF]157,160,172 GGG Gadolinium Gallium Garnet; a transparent (in the visible range) paramagnet of composition Gd 3 Ga 5 O 12 . This material and substituted variants thereof have been widely used as substrate for the growth of other garnets such as yttrium iron garnet (YIG). It possesses an intrinsic magneto-optical rotation that is relatively low compared to other rare-earth-based (substituted) garnets. 90,91,92,93,94,96,97,98,101,104 GIS A Gas Injection System is notably of use in a scanning electron microscope for deposition of material assisted by a focused electron/ion beam. It allows a directional precursor gas flow towards the beam spot. See FEBID. xii, 147

GSG Ground-signal-ground; this acronym refers to a transmission line category featuring one signal electrode surrounded by two grounded electrodes. An example is the coplanar waveguide (see CPW). 112,113,124,[START_REF] Rao | Polyether Ketones[END_REF] hcp Hexagonal Close-Packed; an ABAB type of stacking of layers in the 2D hexagonal lattice system, A and B being offset so that each node lies above or below the center of a triangle of the neighbouring layers. 64

IBE Ion Beam Etching. 56, 189

LEEM Low-Energy Electron Microscopy is an imaging technique relying on electrons escaping the sample surface with a low (i.e. a few dozen eV at most) energy [4]. Experimentally, the sample is impacted by a beam from an electron gun brought at a high potential with respect to the ground, but differing from that of the sample only by the start voltage (see STV). The latter is moderate, a few volts to the dozen of volts in order of magnitude, so that mostly the surface is probed in LEEM. Indeed, the electrons have too low an energy to reach deep into the sample. 151

MFM Magnetic Force Microscopy. 31,68,69,122,123 MOKE The Magneto-Optical Kerr Effect constitutes a probe of a sample's magnetization by way of a change in the polarization of light reflected off the sample surface. Typically, microscope setups using this effect involve a linearly polarized laser focused onto the sample and whose reflected beam is analysed. Depending on the incidence angle, the changes in linear polarization (angle and/or amplitude) reflect distinct components of the local magnetization vector.. 31,188 PCB Printed Circuit Board is a generic name for a variety of supporting elements in electronic circuit design and fabrication. It consists of an insulating layer onto which one, two (on both sides) or more (on the inside, in a multi-layered manner) layers of conductor have been laminated. The machining of such supports allows to create tracks, pads, ground planes and so on which serve as basis for the electronic circuit to be built upon. 110,123,232 PEEK PolyEtherEtherKetone is a plastic material that is compatible with ultra-high vacuum and absorbs little water (which is convenient for such applications). It is also interesting thanks to its high mechanical stiffness and resilience, as well as a strong resistance to a wide range of chemicals. 81, 104

PEEM PhotoEmission Electron Microscopy, in other words a microscopy technique relying on electrons extracted from the sample after excitation of the latter by irradiation with light. It may for instance be applied in the case of synchrotron X-Ray radiation, where the tunable photon energy allows for element-specific investigations thanks to well-defined absorption edges. xiii, xiv, 31,32,66,67,68,75,76,77,78,79,80,81,82,122,123,[START_REF] Weber | Magneto-optical properties of KTb 3 F 10 and LiTbF 4 crystals[END_REF]148,[START_REF] David | Microwave Engineering -Third Edition[END_REF][START_REF] Gupta | Microstrip Lines and Slotlines[END_REF][START_REF] Hasnain | Dispersion of Picosecond Pulses in Coplanar Transmission Lines[END_REF]153,[START_REF]ASM International Handbook Committee, Properties and Selection: Nonferrous Alloys and Special-Purpose Materials[END_REF][START_REF] Hoefer | Thermo-optic properties of gadolinium garnet laser crystals[END_REF][START_REF] Bohm | Optical dispersion of gadolinium gallium garnet (GGG)[END_REF]157,158,159,160,161,162,163,165,166,167,168,169,170,171,172,177,185,186,188,189,190,191,193,194,195,197,198,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,231,232,233,236 Permalloy A commonly used name for a family of iron-nickel alloys with atomic composition close to 80% nickel and 20% iron. It is interesting as a model soft magnetic material in the sense that in the vicinity of this composition, both the magnetostriction (at or above 80% nickel) and the magnetocrystalline anisotropy (below 80% nickel) of FeNi vanish [5]. 13,19,22,23,24,25,27,36,65,66,80,123,213,214,215,216,235 redox Short for reduction-oxidation; may apply to any pair of reducer and oxidizer in the most general case, or to a couple of reducer and oxidizer (the former oxidizing into the latter and the latter reducing into the former).

62, 63

RIE Reactive Ion Etching. 58, 59

SEM Scanning Electron Microscope; this type of instrument relies on a focused beam of high-energy electrons (several kilovolts usually) to scan a sample surface, with a resolution that can reach the range of 10 nm.

As the name implies, it is a scanning microscope, and due to the nature of the probe, the investigation of insulating samples is less straightforward than that of conducting ones. xii, 57,58,60,61,66,125,[START_REF] Ranno | Magnetometry on Gadolinium Gallium Garnet[END_REF]148,[START_REF] Gupta | Microstrip Lines and Slotlines[END_REF][START_REF] Hasnain | Dispersion of Picosecond Pulses in Coplanar Transmission Lines[END_REF]157,158,159,160,169,173,174,186,187,189,190,191,192,193,194,195,234 SMA SubMiniature A connector type, whose development dates back to the 1960s. This light and small (the female end is about 6.35 mm in diameter) connector is nowadays very common in microwave applications. 107, 112

STV The Start Voltage is defined in PEEM as an offset in electric potential of the sample surface with respect to the nominal high voltage that is applied to the experiment. It allows to tune the energy of the secondary electrons which contribute to the image. xii, 76,81,82,[START_REF] Gupta | Microstrip Lines and Slotlines[END_REF]210,214 STXM Scanning Transmission X-Ray Microscopy is, as the name implies, a scanning probe technique relying on an X-Ray beam. The latter is focused thanks to a Fresnel lens onto a small (a few tens of nanometres) spot and the intensity transmitted through the sample is measured. In practice, this is a synchrotron-based technique, with stringent requirements in terms of sample transparency, but not as demanding as PEEM with respect to high-vacuum and the presence of magnetic fields at sample level (for the sake of electron collection). xiv, 75,78,171,185 TE Transverse electric; this label refers to propagation modes (in this manuscript of a transmission line) only with zero longitudinal electric field. Such a mode may only propagate above a certain cut-off frequency which might happen to be zero, and is strongly dispersive by nature. The dispersion relation of such a mode is non-linear because of geometry-dependent boundary conditions on the propagating fields on one hand, and coupling between the longitudinal component of the magnetic field and the other field components on the other hand. 113,114 TEM Transverse electromagnetic; this label refers to propagation modes of an electromagnetic wave with zero longitudinal electric and magnetic fields. As opposed to transverse electric or transverse magnetic modes, TEM modes have a linear dispersion relation (neglecting dispersion in propagation medium/transmission line parameters). Most often in this manuscript, TEM modes will be those of a high-frequency transmission line. 113,114 TM Transverse magnetic; this label refers to propagation modes (in this manuscript of a transmission line) only with zero longitudinal magnetic field. Such a mode may only propagate above a certain cut-off frequency which might happen to be zero, and is strongly dispersive by nature. The dispersion relation of such a mode is non-linear because of geometry-dependent boundary conditions on the propagating fields on one hand, and coupling between the longitudinal component of the electric field and the other field components on the other hand. 113, 114

TVW The Transverse-Vortex Wall is the domain wall type encountered in magnetically soft cylindrical nanowires in the small diameter regime; the latter is delimited by its upper bound of ca. 7∆ d [1], beyond which the only other domain wall type (the Bloch Point Wall, see BPW) has a lower energy. ∆ d is the dipolar-exchange length, the only relevant intrinsic length scale in a soft, infinite magnetic cylinder. Towards very low diameters, the Tranverse-Vortex Wall is well described by the 1D domain wall model (see e.g. Thiaville and Nakatani [6]), and thus reminiscent of transverse walls found in soft flat nanostrips. This simple magnetic texture acquires complexity as the diameter increases: it develops curling both along the domain wall core and along the wire axis beyond the above-mentioned threshold [1]. xi, 23,24,26,27,30,79,185,213,214,

INTRODUCTION

General introduction

Research in solid-state magnetism at ever smaller length scales has benefited from a range of applicative possibilities for over forty years, notably towards information technology. A widespread example is the hard-disk drive. As progress in fabrication techniques such as lithography and sputtering (among many others), was unlocking further reduced sample sizes, novel physical phenomena appeared. Over the past thirty years, the fields called nanomagnetism and spintronics have known a strong upturn, with flagship research topics such as field-or current-driven domain wall motion or the discovery (and applications) of the Giant MagnetoResistance effect. The physical impact of reduced dimensions is that crossovers in terms of sample size can be defined between different regimes. These cross-overs involve length scales intrinsic to magnetism [1], and it is their typical order of magnitude of at most a few (tens of) nanometres that defines cross-overs when going to the nanoscale. The concerned phenomena include equilibrium magnetization domain patterns, spin-dependent transport, magnetization dynamics, and many other topics. Most studies so far have pertained to magnetic samples as can be produced by standard lithography techniques and thin film deposition, that is to say flat objects.

Within this research area, a topic that has been raising and still raises strong interest is that of domain wall motion. From soft strips patterned out of thin films under magnetic field or electric current [2,3] to ultra-thin and narrow conduits with perpendicular magnetic anisotropy [4], a wide variety of systems has been investigated in theory, with micromagnetic simulations, and in experiments. However, they can usually be described within two dimensions of space due to their reduced thickness preventing variations of magnetization in the third dimension. Therefore, an interest in three-dimensional samples has risen [5].

Examples of such samples are given by cylindrical nanowires, which can be fabricated with tunable diameters and composition thanks to nanoporous alumina templates [6]. This geometry has also been investigated with the help of theory and numerical simulations [2,7,8], yet experimental reports on studies of such individual nanostructures are scarce. In particular, cylindrical nanowires have been predicted to undergo magnetization reversal following one of two scenarios, each corresponding to a different type of domain wall, and stark differences in their dynamics are expected but have not yet been observed. These domain wall types are stable or metastable with respect to one another, with two well-defined regimes in terms of wire diameter [7]. Below a threshold of about seven times the dipolar-exchange length, the domain wall configuration with lowest energy is the so-called transverse(-vortex) wall, with a shape reminiscent of transverse wall from strips [7]. Its field-driven behaviour has been predicted to be qualitatively identical to that of transverse walls in strips far above the Walker breakdown [2,9], with a low mobility due to the mainly precessional motion.

Above the aforementioned diameter threshold, another domain wall type becomes the lowest-energy state, namely the Bloch point domain wall [7], sometimes referred to as vortex wall in literatures. While the Bloch point wall does support (on its cross-section) an orthoradial magnetization winding reminiscent of magnetic vortices e.g. in soft flat strips [2], it cannot be wholly assimilated to these objects as its center, the Bloch point, does not feature a core with a well-defined magnetization. To the contrary, magnetization vanishes at this location, which makes this domain wall truly singular from a micromagnetic point of view. Another view on the matter, based on topology applied to physics of continuous, ordered media, consists in viewing the Bloch point as topological defect.

When subjected to an applied field, the Bloch point wall is expected to move with mobilities orders of magnitude above its low-diameter counterpart, the transverse-vortex wall [2,8]. Moreover, its dynamical steady state has been predicted to feature two well-defined regimes in terms of applied field value. Micromagnetic simulations have determined the existence of a threshold field below which the above-mentioned magnetization winding can only assume one sense of rotation, the latter being only defined right-handed by the direction of motion (and not for instance by the domain wall polarity) [2]. Below the threshold, the two sense of rotation are allowed, albeit with starkly different mobilities. These peculiar responses to an applied magnetic field raise significant interest for this domain wall, in addition to its singular nature.

Several reports from theory or simulations about the field-driven dynamics of domain walls in soft nanowires exist but experimental confirmation of the above-mentioned features is yet to come. The noteworthy investigation by Cristina Bran and coworkers [10] using X-ray Magnetic Circular Dichroism (XMCD) coupled to PhotoEmission Electron Microscopy (PEEM) focused on cylindrical nanowires with sizeable magnetocrystalline anisotropy. It is also the case with the systems investigated by Yurii P. Ivanov and coworkers [11][12][13] even though in certain studies shape anisotropy dominates [12,13]. It is in this context that I have started my experiments targeting notably the dynamics of Bloch point walls, with the help of shadow XMCD-PEEM to monitor the domain wall configuration before and after motion. Within the group I have worked with, first steps in this directions have been made concerning the elaboration and characterization of suitable nanowires, including the first XMCD-PEEM observations of the two types of domain walls which had been predicted and studied with micromagnetic simulations. These investigations were the topic of the PhD theses by SANDRINE DA COL and SÉGOLÈNE JAMET [14,15]. The imaging of domain walls was purely static, no magnetic field was applied during these experiments to try and move them. Furthermore, to the best of my knowledge, no experimental work since then was reported concerning the field-driven dynamics of transverse-vortex walls or Bloch point walls. In this thesis, I will report on the experimental work I have conducted with the goal of probing the dynamics of domain walls in (soft) ferromagnetic cylindrical nanowires, under magnetic field.

Manuscript outline

This document is divided into four chapters. The first one aims notably at presenting elements of theoretical background concerning micromagnetism. This introductory first chapter then moves on to a review on domain walls in soft, cylindrical nanowires. In particular, the two afore-mentioned domain wall types, the transverse-vortex wall and the Bloch point wall, are described with the help of micromagnetic simulations as well as elements belonging to topology applied to continuous ordered media. This chapter then ends with a review on the experimental state of the art pertaining to domain walls in cylindrical nanowires.

In the second chapter, the focus moves on to instrumentation I have worked on in order to apply magnetic fields to such nanowires. As a transition between Chapter 1 and the present chapter, the starting point is a description of the electrochemical method used by my colleagues to fabricate the cylindrical nanowires that I have worked on. Since the experimental observation technique relevant for my work has been a synchrotron technique named Xray Magnetic Circular Dichroism coupled to PhotoEmission Electron Microscopy (XMCD-PEEM), the remainder of this second chapter deals with a presentation of this microscopy, followed by my developments related to instrumentation and the application of magnetic fields.

One of the XMCD-PEEM studies I have taken part in during my thesis is then reported in Chapter 3. It quite differs from the others due to the original sample that has been investigated. While the latter was effectively a soft ferromagnetic, cylindrical nanowire, its fabrication and geometry are of a totally different kind with respect to the samples described in Chapter 2. Indeed, this nanowire was produced by colleagues from the University of Zaragoza with Focused-Electron-Beam-Induced Deposition (FEBID), a technique akin to 3D printing which allowed a tailored geometry featuring bends. This geometry enabled the observation of one domain wall during the first of their kind XMCD-PEEM experiments on this sample. In terms of materials, this nanostructure was a core-shell nanowire with platinum coating (as protection against oxidatoin) around a cobalt core.

This thesis' last chapter reports my XMCD-PEEM investigations of the cylindrical nanowires produced by my colleagues. First, the sample preparation steps taken before the experiments are discussed. Thereafter, I present my experimental results concerning the statics of these nanostructures as well as the dynamics of the domain walls they host. The dynamics is not time-resolved, only observed before and after application of a magnetic field. The observations are in partial agreement with the theoretically predicted Bloch point domain wall's selection of winding, yet suggest a more complex picture of this phenomenon. They also provide evidence for a micromagnetic process that had been predicted by neither theoretical nor simulation-based works, namely transformation between domain wall types. Some insight into this dynamical process is provided with the help of micromagnetic simulations. In a last section, I describe an implementation of instrumental developments I have led in order to apply nanosecond-scale pulses of magnetic field to nanowires during XMCD-PEEM experiments.

MICROMAGNETISM AND DOMAIN WALLS

I Elements of micromagnetism in static equilibrium

I.1 Energies in magnetism

The description of a static equilibrium at finite temperature requires the knowledge of the relevant contributions to the system's free energy [1]. In this first section, a sample of volume V , with surface denoted ∂V , and possessing a finite spontaneous magnetization M s at a fixed temperature T (in practice room temperature) will be considered. A further assumption is made, namely that of a uniform magnetization magnitude, so that the unit vector -→ m( r, t) = -→ M ( r, t)/M s with the same direction as magnetization is enough to describe the sample's configuration. This theoretical frame is that of micromagnetism [2], which is a continuous medium approach. While the assumption || -→ m|| = 1 generally holds, a case where it breaks down will be described in section III.3. It must be kept in mind that unless otherwise mentioned, the following discussions will be restricted to ferromagnets described by the theory of itinerant magnetism.

I.1.a Exchange interactions

The first, most obvious energy term to consider is the reason behind ferromagnetism. In transition metals and their alloys, a convenient theoretical frame of description is the Stoner criterion [3]. The starting point is the band diagram of a metal without spontaneous magnetization, and the crucial question is: from the usual band filling, how does an asymmetry build up between the two bands associated to up and down spins? It turns out that a socalled band splitting I (a few tenths of electron-volt [3] typically) may appear if the total system's energy variation is negative. Here, two adverse contributions arise: on one hand, the splitting allows some electrons of definite spin to lower their energies thanks to hopping (in a quantum-mechanical description), and on the other hand, there is an increase in energy because some electrons close to the Fermi level E F occupy higher-energy states than previously. The portion m of splitting achieved, i.e. the proportion of stabilization through hopping, is then a measure of magnetization as imbalance between up and down spins.

Thus, the balance between these adverse contributions results in magnetization being defined on one hand as the difference in numbers of up and down spins (counted by integrating over the bands until the Fermi level), and as the adjustable proportion of stabilization brought by splitting, to be balanced with the above-mentioned energy cost. This self-consistent definition leads to the Stoner criterion: I • ρ(E F ) > 1, where ρ(E) is the density of states at electron energy E. In a hand-waving approach: starting from zero magnetization, if the system reaches a trial value m, then self-consistently, this results first in a separation between the bottoms of both bands of m • I, and also in an energy balance for the full spin-up and spin-down bands. If for this total balance, it costs too much to split the whole bands and thus equilibrium is reached for m < m, then magnetization is reduced with respect to the trial values. Repeating the process leads to null magnetization: this is the case where the Stoner criterion is unfulfilled. On the other hand, if for the total balance, the increase of energy due to the occupation of higher levels is outmatched by the stabilization m • I, then the outcome is an equilibrium at m > m since it is still favourable to split the bands further. This case of Stoner criterion fulfilment leads to a spontaneous non-zero magnetization, and covers the cases of iron, cobalt, and nickel [3]. Another view on the onset of ferromagnetism considers the "molecular field" (theorized by French physicist Pierre Weiss in 1907 [3]), that is to say an additional contribution to the local magnetic field -→ H proportional to magnetization. This contribution, gathering the above-mentioned electronic effects, leads to an increase in susceptibility: any small field creates magnetization in response, but this response in turn enhances the local field which promotes stronger magnetization, and so on. In such an approach, if the strength of this molecular field (bearing some analogy with the above-mentioned band splitting) is large enough, the magnetizationless state becomes unstable as its susceptibility diverges and ferromagnetism emerges.

As we have seen, the core ingredient was "electron hopping". Without going into the quantum-mechanical formalism describing this, let us keep in mind that a more detailed treatment of d-band electrons in a tight-binding approach leads to a description called the Heisenberg model [3], with a hamiltonian Ĥexch reading:

Ĥexch = -J <i,j> - → S i • -→ S j , (1.1)
where the sum runs once over all bonds between lattice sites i and j, each bearing a dimensionless magnetic moment -→ S i or -→ S j with fixed length 1 S. J is called the exchange integral, with dimensions of energy. Its sign determines the type of magnetic ordering: a positive sign leads to ferromagnetism, a negative one to antiferromagnetism.

It is now time to zoom out and go to length scales at which the details of the material's lattice are no longer important. In other words: let us move on to a mesoscopic scale a which a continuous medium approach is relevant. We will be only interested in slowly-varying orientations of our magnetic moment, so that Taylor expansion of the above Eq.(1.1) should lead to a description of the energy cost associated to non-uniform configurations. More details in this regard can be found in section A 1 .1; the important result is that for an isotropic ferromagnet, deviations from the uniformly magnetized state lead to an energy density cost E exch that reads:

E exch ( r) = -A • - → m( r) • - → - → m( r) , (1.2)
where r is the position vector, -→ is the vector Laplace operator, -→ m = -→ M /M s is the reduced magnetization vector with unit length (M s is still the material's spontaneous magnetization), and A is the so-called exchange parameter, exchange stiffness or exchange constant, with dimension of energy per unit length. Typical values are on the order of magnitude of a few or ten pJ/m [3][4][5][6]. A more intuitive and equivalent formulation can be obtained after integration by parts provided the resulting contributions on the sample's surface can be neglected:

E exch ( r) = A • i,j ∂m i ∂x j r 2 , (1.3)
where m i is the i-th components in cartesian coordinate of the unit vector -→ m, and x i the i-th cartesian coordinate.

This formulation clearly shows that any deviation from the uniform state result in an energy cost. As a side note, it must be noted that the "isotropic material" hypothesis is key. In systems lacking inversion symmetry [3,7] or in ultrathin films with non-equivalent interfaces [8,9], another exchange parameter arises, this time related to antisymmetric coupling between neighbouring magnetic moments. This results in a tendency not to align these with one another but to favour angles of 90 • between them. Termed Dzyaloshinskii-Moriya interaction strength, this property has recently led to novel and interesting phenomena in thin-film magnetism [8][9][10][11][12][13][14]. These go however too far beyond the scope of the present work.

I.1.b Magnetic anisotropies

So far, we have tackled why a ferromagnetic order might emerge, but not which direction it may choose. In other words: our magnet is yet fully isotropic. Despite the above reasoning regarding the isotropy of exchange interactions, even in cubic crystalline systems a certain anisotropy originating from the lattice may still be present [15]. Actually, there is a wide variety of functional forms for the so-called magneto-crystalline anisotropy depending on the considered lattice. The latter's symmetries constrain the possible special directions of this magneto-crystalline anisotropy: there are hard axes/planes corresponding to an energy cost if magnetization points along them, and easy axes/planes corresponding to an energy gain.

In practice, the functions representing the associated energy densities are assimilated to their second-order or fourth-order expansions in the components m i of -→ m since higher-orders are in most cases not measurable experimentally [15]. As an example, the general (but already truncated) magneto-crystalline anisotropy energy density E m.c. in a cubic crystal reads:

E m.c. ( r) = K c,1 (m 2 x m 2 y + m 2 x m 2 z + m 2 y m 2 z ) + K c,2 m 2 x m 2 y m 2 z
(1.4) 1 We have implicitly made the assumption of negligible quantum fluctuations holding in the large moment limit.

where K c,1 and K c,2 are the material's magneto-crystalline anisotropy constants. They can be distinct not only in absolute value but also in sign. In terms of origin, whatever the crystal system, magneto-crystalline anisotropy stems from interactions on the atomic scale, be it spin-orbit coupling, crystal-field effects, or dipole-dipole interactions on the atomic scale [3]. In a manner of simple illustration, Fig. 1.1 represents a fictive anisotropic ferromagnet, with easy magnetization directions corresponding to the long bonds' direction in the tetragonal unit cell. Apart from the above-mentioned volume anisotropy, there exist further anisotropic contributions to the ferromagnet's free energy originating from the interfaces. [3,16]. Due to its origin, its influence on the system's static equilibrium is only sizeable in samples such that the proportion of surface magnetic atoms is not negligible with respect to atoms in the volume. The archetypal examples are (ultra-)thin films [16], in particular out of cobalt between other elements such as the heavy metals platinum or palladium. tetragonal lattice system hosting ferromagnetism (blue arrows) under the influence of a magnetocrystalline anisotropy. On each site (red ball), the system's magnetic moment points preferentially along the vertical bonds, as suggested by the elongated halos.

These volume and surface effects exist already in perfect, defectfree and unstressed samples. In practice, many fabrication techniques 2are associated with internal stress that is most visible in (ultra-)thin films [6,16,17]. The associated mechanical strains may themselves influence the material's magnetic anisotropies through magnetoelasticity [6,18]. The general magnetoelastic contribution to the free energy is of the form E mag.-el. :

E mag.-el. ( r) = i (B 1 • u ii ( r) m 2 i + j =i B 2 • u ij ( r) m i m j , (1.5
) where u ij is a mechanical strain tensor's component [19] and the constants B 1 , B 2 are called the magnetoelastic coupling coefficients [18]. As can be seen in Eq. (1.5), they are seen to play the role of magneto-crystalline anisotropy constants tuned by the material's strain. Since they embody a direct coupling of magnetism to the physical lattice, they are also dependent on the sample fabrication process in addition to its composition [17].

Since the simulations and experiments reported in this work will deal with ferromagnetic systems in which the magneto-crystalline and surface anisotropies are respectively sought small and negligible due to the sample dimensions, I will not go into further details in this respect. The systems of interest are labelled "soft" ferromagnetic systems, as opposed to "hard" ferromagnetic systems which feature a sizeable anisotropy of bulk or surface origin. Also, magnetoelasticity will be neglected unless otherwise specified.

I.1.c Dipolar interactions

Dipolar couplings are best understood at first glance by considering point magnetic dipoles. The field -→ H µ ( r) created at an arbitrary point r by such a dipole µ with norm µ placed at position r 0 reads [20]:

- → H µ ( r) = 3 µ • e e -µ 4π|| r -r 0 || 3 , (1.6)
where e = ( rr 0 )/|| rr 0 || is the unit vector pointing from the point dipole to the point with position r. Now, if at that position another point dipole with magnetic moment µ 2 with norm µ 2 is present, the interaction energy E 12,dip. associated to the pair of moments reads:

E 12,dip. = -µ 0 - → H µ ( r) • µ 2 = - 3 µ • e e • µ 2 -µ • µ 2 4π|| r -r 0 || 3 (1.7)
To better understand this interaction, let us assume that in spherical coordinates, µ and µ 2 are along the zdirection, and denote θ the angle from µ to e. Eq.(1.7) now reads: If the angle θ is such that µ 2 is in the blue-shaded area, the parallel alignment is energetically favourable; if cos 2 (θ) > cos 2 (θ c ) as in the red-shaded area, the alignment becomes unfavourable.

The important element is here that the interaction energy may change sign i.e. lead to favourable or unfavourable parallel alignment depending on the sign of 3 cos 2 (θ) -1. This leads to the definition of the special angle θ c = acos(1/ √ 3) 54.73 • such that for cos 2 (θ) > cos 2 (θ c ) = 1/3, E 12,dip. (θ) < 0 and parallel alignment of µ and µ 2 is favoured. Fig. 1.2 illustrates this angular behaviour with our point dipoles µ (purple) and µ 2 (black) at different positions (dashed yellow lines). Regardless of the strength |E 12,dip. (θ)|, its sign is represented by the blue or red colour for negative and positive signs respectively. As along as the magnetic moment µ 2 is in the blue-shaded area, its parallel alignment with µ leads to an energy gain 3 , whereas this alignment comes at an energy cost if µ 2 is in the red-shaded area.

The dependence of this interaction over the distance between magnetic dipoles leads to a fast decay with increasing spatial separation, in the case of point dipoles. Based on this pair interaction, we may now come back to a finite-volume sample: dipolar interactions arise between all pairs of infinitesimal volumes dV making up the whole ferromagnetic volume V . Any point r with reduced magnetization -→ m( r) feels a dipolar field -→ H d ( r) deriving from Eq.(1.6):

-→ H d ( r) = M s ˚ 3 - → m( u) • ( r -u) • ( r -u) 4π|| r -u|| 5 - - → m( u) 4π|| r -u|| 3 d 3 u (1.9)
Integration of the resulting interaction energy over the whole sample leads to its total dipolar energy:

E dip. = - µ 0 2 ˚-→ H d ( r) • -→ M ( r)d 3 r , (1.10)
where the factor 1/2 avoids double-counting of a pair of infinitesimal volumes. From this deceptively compact expression results one of micromagnetism's most difficult issue. Indeed, the computation of the dipolar field at one position requires an integral over the whole sample volume. Then, computing the system's total dipolar energy requires yet another integral over the volume. As a consequence, except in rare cases or under stringent approximations, no simple analytic results can be obtained for this energy. Before moving on to a few such examples, let us try and get some physical insight into this (so far) very mathematical treatment 4 of a magnetic interaction. As a first step, let us consider a truly infinite sample (in all directions) with uniform magnetization, -→ M ( r) = -→ M 0 , and choose an arbitrary position r 0 . For any distance r, the shell of magnetic volume comprised between a distance r and r + dr from r 0 , the contribution d --→ H d,r from this volume to the dipolar field at point r 0 is exactly zero. This follows from the uniform magnetization assumption and Eq. (1.6), or it can be seen from the zero integral over θ in Eq. (1.8). Now comes the crucial assumption of an infinite sample. The above allows us to associate zero dipolar energy to our chosen point r 0 , but only the infinite sample dimensions enable to do the same for all sample points. Based on this, we have learned that a truly infinite and uniformly magnetized ferromagnet comes at null cost in dipolar energy, but on the other hand, this result is fragile as it breaks down as soon as the "infinite sample" assumption is removed. This leads us to think of sample boundaries as particularly important for dipolar interactions. Since sample surfaces are important, let us illustrate their role with evidently finite ferromagnetic samples, namely hard magnetic beads. Fig. 1.3 shows a sequence of three subsequent photographs of a self-assembled ring of hard magnetic beads. On the first image, the circular equilibrium form is shown. On the second one, pressure is applied on both sides of the ring, causing it to distort without breaking. As can be seen on the right hand side image, releasing the pressure is followed by a return to the ring's original circular shape.

These beads only interact through dipolar interaction. A schematic representation of a pair of adjacent beads is given in Fig. 1.4, with their uniform magnetization in black and the uncompensated "poles" that one can imagine in a dumbbell picture of magnetization as dipolar vector. Rotating one bead with respect to the other leads to poorer "pole" compensation, and an increase in energy as demonstrated in Fig. 1.3. As a result, the means by which dipolar interactions come into play seems to be these "poles". Now that we have gained a bit of practical insight in a (literally) hand-waving manner, let us confirm it more rigorously. Since we have seen that variations in the orientations of magnetization as well as boundaries appear as the sought causes, we shall try and make them appear. The starting point is the maxwellian definition of magnetic field -→ H , magnetic induction -→ B and magnetization -→ M : .11) In the absence of applied magnetic field and in a static regime, the only magnetic field is the dipolar field -→ H d . Since we are interested in variations, let us use the absence of divergence of -→ B :

- → B = µ 0 ( - → H + -→ M ). ( 1
→ • - → B = 0 = µ 0 → • -→ H d + µ 0 → • -→ M .
(1.12)

We now see that in an way analogous to electric charge densities for the electric field, there exist volume sources for the dipolar field -→ H d , namely -→ • -→ M , hereafter called magnetostatic volume charge density ρ m . One can show that a magnetostatic surface charge density σ m exists on the sample boundaries, it reads -→ M • n, where n( r) is the outward-pointing unit vector normal to the system surface at point r [15]. Thanks to this complete description of the field's sources 5 , one can derive the following expression for -→ H d [3]:

-→ H d ( r) = ˚V ρ m ( u) • ( r -u) 4π|| r -u|| 3 d 3 u + ‹ ∂V σ m ( u) • ( r -u) 4π|| r -u|| 3 d 2 S. (1.13)
With this, we easily retrieve the above examples: the infinite, uniformly magnetized sample has no boundary thus no magnetostatic surface charge density and all derivatives of magnetization are zero, thus no magnetostatic volume charge density either. Furthermore, the uniformly magnetized beads are seen to interact only trough their magnetostatic surface charges. In this latter example, as a matter of fact, the dipolar field's calculation at any point can be carried out. This case, that of a uniformly magnetized sphere, is the simplest case among the few examples in which the effect of dipolar interaction has an analytical form without any other assumption than that of uniform magnetization: the cases of ellipsoids [21], with the important limits constituted by the infinite cylinder and the infinite plane slab. For such simple systems, the dipolar field inside the sample, also called demagnetizing field, is uniform and can be expressed as [3,15,21]:

-→ H d,ellips. ( r) = -N • -→ M ( r) (1.14)
where N is the demagnetizing tensor of the considered body. Its coefficients are dimensionless, moreover it is a symmetric 3 × 3 matrix, and it has the interesting property Tr N = 1 [21]. While (complicated) formulae exist for general ellipsoids [21] and (significantly simpler) the simpler ellipsoids of revolution [3], I will only present the three most common and easily determined cases: the sphere, the infinite plane slab, and the infinite cylinder.

• Due to the sphere's symmetry, the three diagonal coefficients 6 of N must be equal. Since their sum is 1, each of them is equal to 1/3.

• In the infinite slab, as long as (uniform) magnetization lies in the slab's plane, there are neither surface nor volume magnetostatic charges, therefore, these situation are associated to null demagnetizing field. It follows that the two in-plane coefficients of N are zero, implying that the remaining one, corresponding to out-of-plane magnetization, is unity.

• The same argument can be used along the infinite cylinder's axis. Due to rotational symmetry around that axis, the two other demagnetizing coefficients must be equal, and therefore each is equal to 1/2.

In all the cases where this formalism applies, one can easily see that demagnetizing fields induce an energy density that reads:

E dip ( r) = -µ 0 -→ H d ( r) • -→ M ( r)/2 = µ 0 M 2 s 2 - → m( r) • N • - → m( r) ≡ K d - → m( r) • N • - → m( r) . (1.15)
If we summarize the above, it now appears that diverging volume magnetization configurations as well as components of magnetization normal to the same surface lead to costs in dipolar energy. As a result, a trend to avoid magnetostatic charges appears; when magnetization is uniform, dipolar interactions lead to preferred orientations depending on the sample's boundaries. The latter phenomenon is referred to as shape anisotropy: when possible, magnetization follows the sample's edges 7 . Furthermore, we have introduced a characteristic energy density scale: µ 0 M 2 s /2 = K d , termed the dipolar constant. In several common ferromagnetic materials at room temperature, M s is on the order of a several hundreds to about one thousand kA/m, leading to diplar constants from about 1.6 × 10 5 kJ/m 3 to 1.4 × 10 6 kJ/m 3 . impose → × -→ H d = -→ 0 . Thus, the dipolar field is uniquely determined by its source. 6 One can always make all others equal to zero, since N is symmetric and real, it can be diagonalized in an orthonormal basis of eigenvectors, which we set as frame. 7 Again, the discussion assumes no other sources of magnetic anisotropy, otherwise their corresponding costs must be taken into account.

I.1.d Applied field and Zeeman term

Finally, we come to the well-known effect of an externally applied magnetic field -→ H a . For a classic magnetic point dipole (i.e. a small loop of current) with moment µ, the presence of the applied field results in a torque -→ Γ = µ × µ 0 -→ H a [20]. This leads to a precession of the dipole around the applied field. Furthermore, the force acting on the dipole reads [20] -→ F = → µ • µ 0 -→ H a . In a uniform applied field, combining both leads to the picture of precession relaxing (to be realistic) towards equilibrium in the minimum of the associated potential energy

-µ • µ 0 -→ H a .
This is the well-known Zeeman term describing the change in energy as a function of relative magnetic moment and field orientations. In a ferromagnetic sample, the associated energy density straightforwardly reads:

E Zeeman ( r) = -µ 0 M s -→ H a ( r) • - → m( r).
(1.16)

I.2 Domains and domain walls

With the different energy terms from previous section in mind, we may now have a look at somewhat realistic systems and determine their equilibrium micromagnetic configurations. As a starting point, let us consider a ferromagnetically soft disk of radius R and thickness t, with R quite larger than t. Based on the above, a question immediately arises. Magnetization should mostly lie in the disk's plane due to the difference in the demagnetizing coefficients associated to the in-plane orientations on one hand and to the out-of-plane orientation on the other hand. Indeed, the magnetostatic surface charges are larger if magnetization points out-of-plane with respect to in-plane. We yet have to determine whether

• the exchange interaction dominates and the disk is uniformly magnetized,

• or dipolar interactions dominate, in which case magnetization should wind around the disk's core, following the sample edges and thus creating no magnetostatic surface charges. In addition, the orthoradial magnetization pattern creates no volume magnetostatic charges.

These two situations are shown schematically in Figs. 1.5.a) and b), respectively. Qualitatively, from its functional form, the exchange interaction is rather expected to dominate at "small scales", whereas dipolar interactions have been seen to scale up with the system's volume and/or surface, so they should be the leading term at "sufficiently large" sample dimensions. In other words, there should be a physical length scale ∆ relevant for the transition between these two regimes. Dimensional analysis on Eq.(1.2) and Eq. (1.15) suggests that this length scale should be such that:

A ∆ 2 ∼ K d ⇐⇒ A∆ ∼ K d × ∆ 3 .
(1.17 Based on this, we expect the transition from the uniform magnetization pattern in Fig. 1.5.a) to the non-uniform pattern in Fig. 1.5.b) to occur when the radius R is on the order of the length scale ∆ d now defined as:

∆ d = A K d = 2A µ 0 M 2 s .
(1. 18) This quantity is called the dipolar-exchange length. In terms of orders of magnitude, it usually ranges from a few to a few tens of nanometers, an example pertaining to a soft magnetic material being Permalloy with ∆ d 5 nm. Based on this physical length scale, we find a first reason why non-uniform magnetization patterns can occur. We may now wonder how magnetization varies spatially when rotating from one orientation by 180 • , as is the case in Fig. 1 To answer this, let us change sample and choose an infinite, homogeneous system characterized by a uniaxial magneto-crystalline anisotropy, that is to say a volume energy density of the form E m.c. = K u (1m 2

x ), with K u a positive constant. This is easily seen to favour m x = ±1, hence the name. Let us now assume that for all points in space with coordinate z → -∞, m x = -1 and for z → +∞, m x = +1. Now that the effectively one-dimensional problem's boundary conditions have been set, we may try and derive the magnetization profile along the y-axis. Our situation is sketched in Fig. 1.6.

Since the sample is infinite, magnetostatic surface charges are irrelevant. Furthermore, since the problem is one-dimensional, the divergence of magnetization reads: → • -→ M ( r) = M s dm z /dz. From this, it is clear that rotating magnetization around the y-axis leads to magnetostatic volume charges, whereas rotation around the zaxis does not. This wall configuration is the well-known Bloch domain wall [22]. Thus, we will assume that m z is zero everywhere, which can be more rigorously justified in a more detailed approach [15]. With this, we can safely forget about dipolar interactions for now.

We must find the equilibrium configuration -→ m(z) satisfying our boundary conditions. Since || -→ m|| = 1 and m z = 0, we can use spherical coordinates to write m x (z) = cos [ϕ(z)] and m y (z) = sin [ϕ(z)]. Based on Eq.(1.3) and our choice of uniaxial anisotropy, we may write the system's total energy per unit surface E as:

E = ˆ+∞ -∞ A dϕ(z) dz 2 + K u sin 2 [ϕ(z)] dz.
(1. 19) In order to find the function ϕ such that E is minimum, we need to use functional derivation which in the case amounts to Euler-Lagrange formalism. I will not delve into the function derivation formalism but instead suggest some bibliographical references. The reader is invited to have a look at the concise and intuitive presentation made by Feynman and Hibbs [23] for a first contact, then possibly at the Wikipedia page on functional derivatives [24] (which goes into quite some detail and appears correct at the time of writing, while remaining understandable), and finally at the thorough Appendix A in the monograph by Parr and Yang [25].

The key idea remains simple: finding an extremum of the quantity E requires setting its derivative δE δϕ(z) with respect to the variable function ϕ to zero for all z. This leads us to:

δE δϕ(z) = -2A d 2 ϕ dz 2 (y) + 2K u sin [ϕ(z)] cos [ϕ(z)] = 0 ∀z. (1.20)
Re-arranging this equation, we can multiply by dϕ dz and integrate, taking into account the fact that at ±∞ ϕ(z) has vanishing derivatives and sin [ϕ(z)] → 0. In other words, we respect our boundary conditions which imply that at infinity, there is no energy cost associated with exchange nor with anisotropy. This allows us to write:

A dϕ dz (z) 2 = K u sin 2 [ϕ(z)] ∀z, (1.21)
which expresses an equipartition of exchange energy and magneto-crystalline anisotropy energy at all points.

Paying attention to signs, we may write:

dϕ dz (z) = - sin [ϕ(z)] ∆ ∀z, (1.22)
where the Bloch parameter ∆ = A/K u , with dimension of a length, bears significant resemblance to the dipolarexchange length defined earlier. Changing variables from ϕ to tan(ϕ/2), one arrives at the magnetization profile in terms of angle:

ϕ(z) = 2atan exp - z ∆ (1.23)
This profile, termed Bloch wall profile, is represented in Fig. 1.7, with m x as colour code. The transition from m x -1 to m x +1 occurs over a length on the scale of the Bloch parameter ∆. The regions z → ±∞ feature uniform magnetization and are thus called magnetic domains. Correspondingly, the non-uniform region in-between is called a domain wall. It is noteworthy that the magnetization rotation with m y = 0 i.e. in the (xOz) plane results in a domain wall called Néel wall.

x y z A further valuable feature of this model is that one can analytically calculate the domain wall extent and its energy E. Straightforward integration of Eq. (1.19) leads to an energy 4 √ AK. As for the width, several conventions exist. One may define it e.g. as the integral of m x , leading to π∆, or the integral of m 2

m x → -1 m x → +1
x following Jakubovics [26], leading to 2∆. In appendix section A 1 .2, I present a small extension of this one-dimensional model with an arbitrary angle between the two uniform domains. This simple approach could for example model a thin nanostructure with high aspect ratio consisting of two misaligned segments, the domain wall sitting in the bend.

Magnetic domains and domain walls, as well as magnetization textures confined close to sample boundaries [27,28] come in a wide variety and an exhaustive description goes far beyond this work. Much more thorough reviews on the subject can be found for instance in the books by Hubert and Schäfer [15] or by Coey [3]. Before the focus narrows in on a few examples directly related to this work, I will move on to the action of an externally applied field on a theoretically simple object: a hard ferromagnetic particle with uniform magnetization. At this stage, the described equilibria were all devoid of externally applied magnetic fields. Let us add this ingredient and consider now the effect of an externally applied field on a simple ferromagnetic system: a hard magnetic particle with uniaxial (magneto-crystalline) anisotropy and assumed uniform magnetization. Fig. 1.8 presents the situation, with the applied field -→ H a at a certain angle θ H with respect to the system's easy magnetization axis (dashed line). In the general case, the angle from this axis to magnetization -→ M is non-zero, it is denoted θ.

This context is referred to as the Stoner-Wohlfarth model [29]. Considering a volume anisotropy constant K and a particle volume V the system's total energy reads: E = -µ 0 M s V H a cos (θ Hθ) + KV sin 2 (θ).

(1.24)

Dipolar interactions are left out or effectively absorbed into this anisotropy constant. It is important to note that under the "uniform magnetization hypothesis", this frame is adapted to describing soft ellipsoids, provided K is replaced by K d corrected by the proper demagnetizing coefficients [30]. In particular, one can easily show that this anisotropy field reads M s /2 for the infinite cylinder. This will be important to keep in mind in Chapter 2, section I.5.a and Chapter 3, section III. 1.

With respect to the calculation of the Bloch wall's profile, the goal is here to determine the static equilibrium angle θ. Introducing adimensionate variables e = E/(KV ), h = µ 0 M s H a /(2K), the model reads: e = -2h cos (θ Hθ) + sin 2 (θ).

(1.25)

Let us focus on the case θ H = π, starting from θ = 0. The minima of energy are straightforwardly defined by de/dθ = 0 and d 2 e/dθ 2 ≥ 0. We have:

de dθ = -2 sin (θ) • h -cos (θ) (1.26)
d 2 e dθ 2 = -2h cos (θ) + 2 cos (2θ) (1.27) This leads to three extrema, θ 1 = 0, θ 2 = acos(h) and θ 3 = π. Based on this, we see that the stable equilibrium position θ 1 remains until h = 1 ⇐⇒ H a = 2K/(µ 0 M s ) ≡ H K , where H K is defined as the anisotropy field. At this stage, this position becomes unconditionally unstable. While θ 3 is always a stable equilibrium as expected from both Zeeman and anisotropy terms, we see that the intermediate extremum θ 3 is always an energy maximum until it merges with θ 1 when h → 1. To further emphasis this point, one can calculate the energy barrier from θ 1 to θ 2 i.e. e(θ 2 )-e(θ 1 ), it reads (1-h) 2 : it therefore decreases to zero when h → 1. Of course, the behaviour at negative h is equivalent to swapping θ 3 and θ 1 .

d 2 e dθ 2 θ 1 = 2(1 -h) (1.28)
In summary, as expected, a sufficiently high field leads to destabilization of the equilibrium position associated with Zeeman energy cost in favour of the other equilibrium position. Mathematically, the Stoner-Wohlfarth particle's magnetization is shown to abruptly change its orientation by 180 • right when the applied field reaches the anisotropy field. This process, which is not described in this approach purely based on energetics, is called magnetization switching. Since all of magnetization rotates in unison, such switching is often referred to as coherent rotation [3,15,27].

For the sake of completeness, it is important to note that the cases where θ H is arbitrary still bear analytical resolution [3,15]. In the general case, the behaviour of the equilibrium angle θ eq (H a , θ H ) is less abrupt. Still, a magnetization switching i.e. a discontinuity in angle is observed at all field angles θ H except when the applied field is exactly along the particle's hard axis. A convenient graphical representation is that of the hysteresis loop [3], consisting in the plot of the system's magnetic moment or magnetization versus the externally applied field. A few examples are reproduced in Fig. 1.9, adapted from [3] (see Fig. 7.11 p.248).

The hysteretic character is clear from the extended range of bistability. Apart from the anisotropy field, two important quantities are apparent for each hysteresis curve, namely the amount of magnetization M r still in the direction of applied field when the latter is ramped down to zero. This is called remanence, and it is seen to go to zero (as expected) when the field is along the hard axis. This reduced quantity is highlighted in blue. Also, the field such that the system's magnetization in the applied field direction is zero is called coercive field H c . In some cases (see e.g. the 10 • curve), this field corresponds to the switching field H sw such that there is a discontinuity in magnetization angle.

Hysteresis loops can be acquired on a wide variety of magnetic systems through a diversity of methods [31][32][33][34][35][36] whose description goes too far for the purposes of this introduction to micromagnetism. For more details in this regard, the reader is referred to monographs [3,15] in which both measurement methods as well as information to be gained from hysteresis loops are described.

II Magnetization dynamics

After tackling the different contributions to a ferromagnet's free energy in order to understand the static equilibrium configurations in the absence of external stimulus, we have just seen an example of perturbation of such equilibria, namely the application of a magnetic field. The Stoner-Wohlfarth particle has shown in section I.3 that even a simple model leads to strong non-linearity (hysteresis is one example) and discontinuities as revealed by magnetization switching.

It is now time to go beyond this static-only picture of micromagnetism and address the dynamics. As a first step, let us describe the switching process in more detail. Then, we will come to the general frame within which magnetization dynamics is theorized. [37] in a ferromagnetic cylinder (transparent gray). Magnetization progressively cants away from the wire axis under the influence of an applied field.

II.1 Non-uniform magnetization reversal modes

In the previous example, the particle subjected to an external field was assumed to always retain uniform magnetization; there was only one degree of freedom to describe the whole system. However, we know from section I.2 that non-uniform magnetization patterns may spontaneously exist in sufficiently large samples. Also, even in homogeneously magnetized systems, one can imagine that transient localized inhomogeneous textures such as the Bloch domain wall presented in section I.2 may provide a path for switching associated with a smaller energy cost than the Stoner-Wohlfarth particle's coherent rotation. Finally, a description of such phenomena must evaluate the role of temperature, since energy barriers between well-defined states have already been encountered in section I.3. Paths to magnetization reversal, also called reversal modes, have been investigated among others in ellipsoids such as cylinders so as to ease the re-introduction of dipolar interactions into the model [37][38][39]. The principle is to find magnetization patterns superimposed as small perturbations on the starting equilibrium configuration such that at a certain field called nucleation field H N , the initial state becomes unstable against growth of these perturbations [37]. In other words, a larger amplitude of such a mode leads to an energy gain until the linearised equations of state no longer hold and the global micromagnetic state changes.

Analytically, a non-uniform reversal mode now referred to as curling [37] has been determined [40] to cost less energy than coherent rotation for sufficiently large sample dimensions. For a given distance with respect to the (central) axis along which the field is applied, this mode has magnetization rotate by an increasingly large angle. An illustration is provided in Fig. 1.10. Therefore, on such a cylindrical shell, magnetization now tends to wind clockwise/anticlockwise around the system's axis. Such a pattern has the advantage of flux closure, that is to say the creation of less magnetostatic charges 8 . This alleviation of the reversal mode's cost in dipolar energy allows it to occur at fields lower than the Stoner-Wohlfarth switching field if the sample is wide enough. As was guessed in section I.2, the relevant length scale at which this transition occurs is on the order of the dipolar-exchange length ∆ d [37].

Physically speaking, the cost in exchange energy caused by this non-uniform pattern decreases with increasing lateral dimensions, until the energy gain from flux-closure allows this new mode to take over. Interestingly, although we are dealing with a reversal mode, the situation on the wire axis is unclear. Magnetization must switch eventually, but in the cylinder's case represented in Fig. 1.10, symmetry must be broken to allow rotation. Otherwise, if magnetization remains along the axis, the only possible switching is achieved by a discontinuous full reversal, as noted by Arrott and cowokers [38] in 1979. This singular reversal mode will be of central interest for this thesis, and discussed in more detail in section III.3.

II.2 Thermally assisted quasistatic switching

Even though we are leaving aside this case for the moment, it clearly appears that localized non-uniform magnetization patterns can play a significant role in magnetization reversal. Yet, we must keep in mind that so far thermal agitation has been left out. Whatever the energy barrier(s) that a ferromagnetic system must overcome so that magnetization reversal occurs, if these become comparable to k B T , where k B is Boltzmann's constant, then our (actually 0 K-)picture should break down. Statistical mechanics at fixed temperature [1] implies that transitions become possible even for non-zero barrier(s). Already here, we see the need to further depart from a purely static/time-averaged view.

Many phenomena in micromagnetism, from microscopic to macroscopic scales, are dynamical by essence9 [3,41,45], sometimes with time scales spanning several orders of magnitude [41,44]. When transitions associated to energy barriers are considered, a frequent approach consists in assuming a so-called Arrhenius-Néel law to be valid [27,41], stating that the typical time τ over which the energy barrier with height E will be overcome reads:

τ = τ 0 exp E k B T (1.31)
where the barrier height may depend e.g. on an applied field. Such an assumption has found experimental support (see Ref. [27] for a review on this subject), and has even led to confirmation of the Stoner-Wohlfarth model's energy barrier in magnetic nanoparticles at low temperatures [46]. In the case where reversal is not uniform, experiments suggest [27] that energy barriers associated to a nucleation volume V N smaller than the sample volume lead to good agreement between theory and experiments. This volume typically has dimensions on the order of the relevant domain wall width [27,41,45] or the size of crystallites in the sample [41]. In practice, taking these phenomena into account also allows a more accurate prediction of coercive fields than the zero-temperature approach, which yields too high results [37].

II.3 Precession and effective field

While we have delved deeper into magnetization dynamics, so far only energetics has guided us in tackling switching, reversal and nucleation processes. Let us now go one step further and consider magnetization dynamics on the relevant time and space scales. From section I.1.d , we remember that an applied field exerts a torque on a magnetic moment, and leads to an energy term tending to align the moment along the field. What we lack, in essence, is on one hand the link between torque on magnetization and magnetization's time derivative, and the dissipation necessary for the (observed) relaxation towards the above-mentioned equilibrium.

To put it briefly: the former is magnetization's other nature as angular momentum. This double-sided nature is most easily seen by analogy with a current loop: the system of charged particles creating this current possesses an angular momentum proportional to the current intensity, therefore proportional to the loop's magnetic moment.

Extending this classical result to the motion of electrons around atomic nuclei associated with their quantummechanical orbital then spin angular momenta l and ŝ [3] allows one to write: μl = γ l and μs = g s γŝ, where μl and μs are the quantum-mechanical magnetic moment operators associated to l and ŝ. γ = -e/(2m e ) is called the gyromagnetic ratio, m e being the electron's mass and e > 0 being the elementary charge. g s 2.00 is the Landé factor associated to the electron's spin moment [3]. The quantum-mechanical composition of these angular momenta leads to the full electronic magnetic moment. Going over to a piece of transition metalbased ferromagnet, in which the contribution of orbital momentum is weak ("quenched") [3] leads to an effective gyromagnetic ratio γ very close to -e/(m e ) g s γ. Based on this, we arrive (classically) at the following time evolution of angular momentum l: .32) This precession equation lacks a reason why the magnetic moment should relax towards a minimum of energy. Thomas Gilbert has proposed in 1955 [47] a functional form that is today the wide-spread approach to the physically expected damping, as a modification of the form initially proposed by Landau and Lifshitz [48] motivated by inconsistencies in the strongly damped regime [49]. The resulting equation of motion, therefore named Landau-Lifshitz-Gilbert (LLG) equation, reads:

d(γ • l) dt = µ 0 |γ | -→ H a × µ = d µ dt . ( 1
d µ dt = γ 0 -→ H a × µ + α || µ|| µ × d µ dt , (1.33)
where γ 0 = µ 0 |γ |. Here, the dimensionless quantity α, often named Gilbert damping, is usually much smaller than one. In order to preserve the length of µ, the term responsible for damping must be orthogonal to -→ µ . Its dependence on d µ/dt is reminiscent of viscous friction. Since α 1, magnetization precesses a large number of times before reaching equilibrium. This dynamics is illustrated on Fig. 1.11, with the magnetic moment's trajectory in black. Starting from µ ⊥ -→ H a at rest, i.e. the field is turned on stepwise, it is straightforward to see from Eq.(1.33) that the initial angular velocity is γ 0 H a .

-→ If we now come back to a ferromagnetic body, since magnetization is the volume density of magnetic moments, it is straightforward to write down the LLG equation for the reduced magnetization -→ m. To do so, we must consider of course any externally applied field, but also take into account all the contributions to the free energy from section I.1. This implies defining an effective field associated to these terms in a consistent manner: the definition must yield the externally applied field from the Zeeman term or the dipolar field itself (without factor 1/2) from the dipolar energy density, for instance. The sum of all these effective fields will induce precession following Eq. (1.33). Summing all contributions, one arrives at the LLG equation in its used form:

H a µ - → e z - → e x - → e y
d - → m dt ( r, t) = γ 0 --→ H eff ( r, t) × - → m( r, t) + α - → m( r, t) × d - → m dt ( r, t) ,
(1.34) where the effective field --→ H eff is the sum of all above-mentioned fields resulting from the different energy terms. To be more precise, it reads:

--→ H eff ( r, t) = - 3 i=1 1 µ 0 M s • δE tot. δm i ( r, t) - → e i ≡ - 1 µ 0 M s • δE tot. δ - → m( r, t) ,
(1.35) where the -→ e i 's are the cartesian frame's unit vectors, and E tot. is the ferromagnetic sample's total free energy.

We are once more confronted to functional derivation, yet the interpretation of Eq. (1.35) as "field leading to a generalized Zeeman term" is straightforward.

In Eq.(1.34), the damping is still a phenomenological parameter, the theoretical prediction of which is difficult. All mechanisms leading to energy losses such as eddy currents in metals [49], coupling to the lattice [50,51], electron-electron interactions [52] and spin currents [53,54] to name a few, manifest as a contribution to this damping. In a recent careful and thorough investigations, Weindler and coworkers [55] have even shown that the local micromagnetic configuration may (significantly) enhance damping.

Nevertheless, this formalism's successes are numerous. Spin waves [42], ferromagnetic resonance i.e. the peak in precession amplitude around a static field occurring when a perpendicular AC field has exactly the precession frequency [42], domain wall motion (with adapted damping if need be) [56], magnetization switching in nanostructures [57], interplays with electric currents and spin-orbit physics [58] are only a few examples of dynamical phenomena understood in this theoretical frame.

It must be mentioned that the LLG equation is, unless corrected, a null temperature picture. A frequent reintroduction of thermal agitation is based on the fluctuation-dissipation theorem [59,60] for self-consistency. Another approach uses the so-called Landau-Lifshitz-Bloch equation taking not only this theorem into account but also relaxes the constraint || -→ M || = M s .

II.4 Dynamics of micromagnetic textures

We have so far considered the dynamics of either a single point magnetic dipole or of magnetization at one point in a ferromagnet, but what of localized objects such as the Bloch wall from section I.2? It turns out that under hypotheses of motion of a rigid object at constant velocity, an integrated form of the LLG equation can be derived to describe the motion of a localized micromagnetic texture as a particle: the Thiele equation [61]. This equation, which can be adapted for instance to take into account inertial effects 10 [63], has been particularly used in the context of vortex motion [64] and recently that of skyrmion motion [65]. Some insight regarding this approximate frame of description and the derivation of the Thiele equation(s) is presented in the section A 1 .3. The practical advantage of applying this formalism is that only a few degrees of freedom are sufficient to characterize the motion of, say, a domain wall. One challenge is determining the force resulting from e.g. the externally applied field. Fortunately, the case of 180 • domain walls is simple in this respect. For a uniform field -→ H a applied along one magnetic domain, the Zeeman term is expected to lead to growth of this domain, therefore domain wall motion. Now, considering any sample section S swept from one domain to the other, one can integrate the force exerted on magnetization that has been defined in section I. It must be pointed out however that the hypothesis of rigid object moving at constant velocities is not necessarily verified. In addition to creating a force leading to forward motion, an applied stimulus such as a magnetic field may for instance distort the considered micromagnetic object dynamically. A very well-known example is that of soft, flat strips and the so-called Walker breakdown [63,66]. The domain walls in these systems, namely the transverse and vortex walls [67], can undergo periodic structural transformations if the applied field is large enough. This transformations are in turn associated with oscillations in the wall velocity, as known from simulations [63] and experiments [56]. An example is provided in Fig. 1.12, adapted from [63]. A field-driven vortex wall in a soft strip is seen to change structure in time as vortices and antivortices are periodically injected from the strip's side.

III Domain walls in nanowires

In this section, the focus will move to domain walls in soft ferromagnetic nanowires of cylindrical cross-section, unless otherwise specified. Due to shape anisotropy (see section 1.9), magnetically soft cylinders are expected to host magnetization aligned with the wire axis. If we now enforce boundary conditions with opposite uniform domains, the outcome can be either a so-called head-to-head or tail-to-tail domain wall, "head" and "tail" being understood as the point and butt of the arrow representation of the magnetization vector. Furthermore, contrary to the Bloch wall from section I.2, there must be magnetostatic volume charges since the magnetization component along the wire varies as a function of position on the wire axis. Therefore, these domains walls are termed "charged". Both situations are schematically represented in Fig. 1.13, along with the magnetostatic volume charges.

-→ m -→ m The question of their internal configuration remains. In the following, we will start by the vanishing diameter limit to describe domain walls in effectively one-dimensional wires. Then, the system diameter will progressively increase and we will describe the emerging features as they arise.

III.1 One-dimensional model

Even the limit of very small diameter with respect to the dipolar-exchange length ∆ d , it seems a priori dangerous to approximate the dipolar interactions as a purely local effect. Indeed, the afore-mentioned domain wall's magnetostatic charge behaves like a monopole11 Q = 2πR 2 M s [67] where R is the wire diameter. Still, full micromagnetic simulations of soft nanowires with square cross-section by André Thiaville and Yoshinobu Nakatani [63] reveal that the equilibrium configurations of domains obtained with cross-section side a equal to, or about seven times the material's dipolar exchange length, show magnetization profiles with small or negligible transverse variations. Furthermore, they are in good qualitative agreement with a Bloch wall profile [63] and the agreement is even quantitative for a = ∆ d . The relevant anisotropy constant is K = N cyl. K d = µ 0 M 2 s /4, where N cyl. = 1/2 is the infinite cylinder's transverse demagnetizing coefficient. In terms of field, this approximation is equivalent to stating that everywhere in the nanowire, the dipolar field reads

-→ H d ( r) = -M s /2 × -→ m t
, where -→ m t is the reduced magnetization perpendicular to the wire axis.

The corresponding domain wall configuration is schematically represented in Fig. 1.14. The domain wall is seen to be a spread magnetic moment perpendicular to the wire axis, hence it is referred to as transverse wall. It must be noted that all of the system's magnetization lies within one (here vertical) plane.

From the point of view of theory, there is no energy gain in twisting part of the domain wall out of this plane; this was confirmed by simulations [63]. As a result, the one-dimensional domain wall model seems to be fully described by the azimuthal angle ϕ describing in spherical coordinates the domain wall core's orientation, and the Bloch wall profile, which can be parametrised as an angle θ(z), still in spherical coordinates, with respect to the z-axis.

This constant azimuthal angle remains a valid assumption even if a transverse volume anisotropy of the form E ani.,t = K 1 sin 2 (ϕ) [63]. The domain wall width 12 and energy per unit surface become A/(K + K 1 sin 2 ϕ) and 4 A(K + K 1 sin 2 ϕ) respectively. This additional leads furthermore to a strong difference in field-driven dynamics, as shown by Thiaville and Nakatani [63] in their review based among others on the one-dimensional x y z

m z → +1 m z → -1
Fig. 1.14: Schematic representation of a head-to-head domain wall in a cylindrical nanowire. This is based on the onedimensional model [63] in which magnetization is invariant on a cross-section, and has a Bloch wall profile.

(1D) model investigated by Schryer and Walker [66]. For calculations-related details, the reader is referred to the afore-mentioned publication [63]; let us compare the cases of zero and non-zero transverse anisotropy, starting with the former. In both situation, the starting point is a relaxed domain wall configuration under zero field; the latter is turned on stepwise at t = 0. Both a variational approach [63] and a description "à la Thiele" lead to the following features of 1D domain wall motion under field 13 of arbitrary strength:

• the domain wall's azimuthal angle ϕ is a linear function of time, and the angular velocity is proportional to the applied field,

• and the domain wall velocity is immediately proportional to this angular velocity and the Gilbert damping α.

Due to the fact that α 1, forward motion and angular precession occur at very different paces. The first observation is simply interpreted as the precession of the domain wall's transverse moment in the applied field, which is always perpendicular to it (this is reminiscent of section II.3, p.18). On the other hand, in this picture of precessing moment, we must recall that relaxation of -→ m along the applied field, which here corresponds to growth of the domain along the field (and therefore to the domain wall motion) only proceeds through damping. This is why the domain wall velocity is proportional to α and the applied field. Quite often, one defines the domain wall mobility as the ratio of velocity over applied field. Simulation results in an atomistic approach supporting this evolution have been obtained by Wieser et al. [68]. By contrast, any finite transverse anisotropy leads to the apparition of three very distinct regimes as a function of applied field. The relevant scale is given by αK 1 /(µ 0 M s ) ≡ H W , referred to as the Walker field [63]. Defining h = H a /H W , the 1D model with transverse anisotropy leads to:

• an inertial behaviour in which the angular and translational accelerations can be discontinuous but the velocities cannot; the domain wall has acquired a mass as theorized by Döring [62], inversely proportional to K 1 ,

• a mobility inversely proportional to α while h < 1, with an azimuthal angle that is static in the steady-state,

• a sharp drop in average domain wall velocity when h 1, with marked velocity oscillations,

• and a terminal regime h 1 in which domain wall velocity and angular velocity go to limits corresponding to the isotropic case.

The transition between the low-field regime (h < 1) and high-field regime (h 1) is termed the Walker breakdown. The physical reason why a field ∝ K 1 is the relevant scale is simple: the Walker field corresponds to the limiting case in which the anisotropy field associated to K 1 still stops the domain wall core's precession. Below this field, the domain wall's configuration reaches a fully rigid state, which can be shown within Thiele's formalism [63] to lead to propagation with a mobility inversely proportional to α. Beyond the Walker field, the domain wall core precesses periodically but this precession is far from harmonic, and the coupling between angular and translational velocity results in the latter having a low mean value and strong oscillations. In the regime far above the Walker field, the torques due to the transverse anisotropy have a negligible impact on the core's precession and can be neglected. As a result, the isotropic case is recovered when h 1. As a conclusion, it is clear that any finite transverse anisotropy leads to the existence of a low-field regime in which the mobility is higher than above the Walker breakdown by a factor 1/α 2 1. Furthermore, one should be cautious when considering the isotropic case. With the above-mentioned proportionality between Döring mass and inverse transverse anisotropy in mind, it is clear that the limit K 1 → 0 (corresponding to the isotropic case) leads to a diverging domain wall inertia and not to "massless" domain walls [69]. Physically, this is all the clearer with the interpretation of the Döring mass as a measure of the domain wall's distortion when its velocity changes 14 : in the isotropic case, the domain wall endlessly distorts as it precesses, whereas its distortions are finite below the Walker field when K 1 is finite [63].

III.2 Transverse(-vortex) wall

If the nanowire diameter or cross-section side (for square sections) is increased with respect to ∆ d , inhomogeneities develop in the directions perpendicular to the wire axis [63,67]. In particular, retaining a Bloch-wall-based description, the apparent domain wall width is seen to vary along the wall's core [63,67]. The reason is that increased lateral dimensions allow the exchange cost of non-uniform configurations to decrease, and these in turn allow a decrease in dipolar energy. To understand why, let us consider the shape assumed by domain walls when the lateral dimensions are (significantly) larger than the dipolar-exchange length. Two examples are reproduced from [67] and [63] showing a distinctly triangular core shape for this head-to-head domain wall. c) Similar cut for a wire with 35 nm side and same material parameters [63], featuring a tail-to-tail domain wall. As expected ( -→ m → --→ m), the core shape is also triangular. Cell size 5 nm × 5 nm × 5 nm. Image reproduced from [63].

The non-uniform apparent width is readily explained by the triangular wall core. In fact, the whole domain wall can be seen as a composition of two 90 • domain walls and the core as a small transverse domain. This more complex structure allows to alleviate magnetostatic energy costs [67]. Although much caution is needed in the comparison, this is a trend which can be understood for domain walls without magnetostatic volume charges; qualitatively, the explanation is that the domain wall energy is a convex function of the wall angle i.e. has a positive second-order derivative with respect to this variable. This is hinted at by the generalization of the static 1D domain wall model I propose in appendix section A 1 .2. In this approach, two 90 • walls cost about 60% of the energy associated with only one 180 • domain wall. Again, this hand-waving argument is only qualitative. Moreover, the crucial difference is that this treatment either neglects dipolar interactions or treats them in a purely local way. Also, it is theoretically not suited for two domain walls at a finite distance from one another.

Therefore, the first departure from the 1D domain wall description as the lateral system size increases is an asymmetry in the wall core shape. Further increase in nanowire diameter leads to an interesting configuration transition described in detail by JAMET et al. [67]: the apparition of curling features in the vicinity of the domain wall core. We remember curling from the discussion of magnetization reversal modes in section II.1, see e.g. Fig. 1.10. JAMET et al. have shown that a continuous transition occurs at a wire diameter of ca. 7∆ d . Immediately above this diameter, there is a rapid rise in magnetization curling both along the wire axis and along the domain wall core, as shown e.g. in Fig. 8.b) from Ref. [67]. This is illustrated in Fig. 1.16, showing views of a so-called transverse-vortex wall (TVW) in a nanowire with 70 nm diameter (ca. 14∆ d ) and Permalloy material parameters. This micromagnetic configuration was obtained by JEAN-CHRISTOPHE TOUSSAINT with the finite-elementsbased program FeeLLGood [70].

The top row's views focus on cuts containing the wire axis and the wall core. While the latter is itself very narrow, the domain wall extends over a width significantly larger than the wire diameter 15 , still retaining a somewhat triangular shape reminiscent of Fig. 1.15. The novel feature arising from the increased diameter > ∆ d [67] are the longitudinal and transverse curling highlighted in Fig. 1.16.a). The reason behind this curling is again flux closure [67], and geometrical constraints lead to the appearance of a vortex and antivortex on the wire surface. They correspond to the entry and exit point of magnetic flux through the wire; in other words, the intersection of the wall core with the wire surface. This is highlighted by the component of magnetization normal to the wire surface -→ m • n in Fig. 1.16.d-f). These two well-defined objects will be of crucial importance in Chapter 4, section

V.3.
In terms of field-driven motion, the TVW's behaviour is very similar to that of the transverse wall from section III.1's 1D model, at least at low applied magnetic fields. Further insight will be provided in Chapter 4 section V.3. 15 More details on the behaviour of domain wall widths in cylindrical nanowires have been reviewed in Ref. [67].

a) b) c) e) f ) d) 0 -1 1 m z 0 -1 1 m x x y z 0 -1 1 -→ m• n 0 -1 1 -→ m• n x y z x y z x y z 0 -1 1 -→ m• n x y z 0 -1 1 m y x y z

III.3 Bloch point domain wall

Micromagnetic simulations have shown that magnetization reversal in soft cylindrical nanowires may proceed with a starkly different domain wall type, provided the wire diameter is large enough [71]. Instead of a well-defined core perpendicular to the nanowire, this other domain wall supports small normal magnetization -→ m • n, as well as rotational symmetry around the wire axis. Furthermore, the plane perpendicular to the wire axis that contains the domain wall's center appears to be an antisymmetry plane for the longitudinal component of magnetization (in static equilibrium) [67].

Combining the constraints of rotational symmetry and the plane antisymmetry for magnetization around the wire axis immediately leads to the conclusion that the intersection between this plane and the wire axis, i.e. the domain wall's very center, cannot possess a finite magnetization. As such, this peculiar point cannot be described accurately by micromagnetism, which posits the norm of the magnetization vector as constant [2]. In other words, the reduced magnetization -→ m cannot be defined at this point, which corresponds to a micromagnetic singularity.

It turns out that such singular points have been predicted as early as 1965 in an outstandingly important publication by Ernst Feldtkeller [72], now available in English [73]. In the context of a soft cylinder's magnetization reversal, Feldtkeller showed that transient states featuring a micromagnetic singular points must exist. The simple and powerful argument is based not on symmetries (contrary to the previous paragraph) but on the mathematical image of the sample surface on the unit sphere by magnetization, here considered as a mathematical map. To put it differently, the presence or absence of a micromagnetic singularity within a given volume is (graphically) assessed only based on the portion of the unit sphere covered by magnetization when sweeping over all this volume's boundary surface. If that covered area on the unit sphere wraps the latter once or more, then a point with zero magnetization must exist somewhere in the volume. Actually, this is a reasoning using arguments and results from algebraic topology, even though the publication by Feldtkeller pre-dates the broader use of topology in condensed matter; see e.g. the Nobel-awarded works by Kosterlitz and Thouless [74,75] and/or the introduction to the thorough review by Mermin [76].

We will come back to topology in section III.4. Let us however keep in mind that on this basis, this new domain wall type already is starkly different from the transverse(-vortex) walls or (anti-)vortex walls from strips. Indeed, none of these domain walls contain such a singular point for magnetization as a ferromagnet's order parameter. Such a point is today referred to as Bloch point 16 , thus leading to the name Bloch point wall (BPW) [63]. Since such a feature lies outside the realm of micromagnetism, this theoretical frame cannot fully capture it; still, let us have a look at the domain wall configuration in Fig. 1.17.

b)

0 -1 1 0 -1 1 m x 2 -→ m• n a)
x z Fig. 1.17: Views of a tail-to-tail BPW's equilibrium micromagnetic configuration simulated with FeeLLGood [70] in a cylindrical nanowire with Permalloy material parameters. The wire diameter is 70 nm, the scale bars 50 nm. The mesh size was 5 nm. a) Cut through a plane containing the wire axis. Magnetization perpendicular to this plane (m x ) is highlighted as colour code. The Bloch point itself sits between simulated mesh nodes, which is why the wall's center appears to have a finite (and oddly oriented) magnetization. b) Representation of magnetization on the wire surface (represented as green cylinder). The orthoradial winding of magnetization, here right-handed with respect to the z-axis, is even more apparent, as well as the rotational symmetry around the wire axis. The weakness of magnetization's component normal to the wire surface -→ m• n is clear: even multiplied by a factor two, it is barely visible as colour code. Visualizations showing this can be found in Ref. [67]. Simulation credit: JEAN-CHRISTOPHE TOUSSAINT.

The node at the very center seems to be in contradiction with the above symmetry-based reasoning but this is only an artefact due to the Bloch point's singular nature. Several works report that this object sits as far as possible from the simulated nodes with simulated magnetization [63,[77][78][79][80], and this is also the case here: the Bloch point is inside a tetrahedral mesh cell. The rotational symmetry around the wire axis is quite apparent from the m x component, which is antisymmetric on Fig. 1.17.a)'s cut view with respect to the wire axis. Cuts views at any angle around the wire axis would lead to similar visualizations. At the middle position on the wire axis, magnetization is almost perfectly orthoradial at all radii in the wire and on the surface. This is shown by the surface view on Fig. 1.17.b), which reveals the small component of surface magnetization perpendicular to the wire axis, -→ m • n, as colour code. Even after multiplication by a factor 2, this quantity is barely visible. After its observation in nanowire reversal simulations, the BPW has been simulated at rest in nanowires with square [63] and disk cross-sections [67]. It was notably shown that this domain wall's energy was higher that that of the transverse(-vortex) wall until a threshold lateral dimensions. In the case of cylindrical nanowires, both domain walls have the same energy for a diameter of ca. 7∆ d [67], which we remember to be the diameter beyond which curling features develop in the TVW. A similar value (ca. 6.2∆ d ) was found in square nanowires [63]. Beyond the threshold, the BPW always has a lower total energy than the TVW [67]; once more, this is a result of energy gain allowed by flux-closure, in the present case thanks to the BPW's orthoradial winding.

Initial state

Steady-state motion at the initial stage of application of a magnetic field -→ H a along the wire axis, towards the reader, and above the selection field H sel . The brown arrows represent the corresponding torques on the winding. In addition to forward motion, the applied field promotes inequivalent distortions for the windings. Based on Refs. [63,81], the right column shows the steady-states. The magnetization from the winding right-handed with the direction of motion (top) has canted but the winding remains the same. On the other hand, the opposite winding has switched (dashed purple arrows) to become right-handed with respect to the BPW velocity -→ v DW .

µ 0 -→ H a µ 0 -→ H a - → v DW µ 0 -→ H a µ 0 -→ H a - → v DW
In Fig. 1.17, a certain winding is represented, but it is crucial to note that the opposite winding, that is to say left-handed with respect to the z-axis, has the same energy. This is easily demonstrated by 180 • -rotation around any axis perpendicular to the wire axis, leading to the opposite winding while preserving the domain wall polarity. Therefore, this domain wall's degree of freedom with two discrete values appears to be rather anecdotal at first sight. It turns out that this is only true in static equilibrium: during field-driven motion, this winding plays a crucial role, as was first shown in nanowires with square cross-sections [63] and later on in cylindrical nanowires [81].

Let us focus on a head-to-head BPW, since the tail-to-tail BPW's case can easily be deduced by symmetry. Fig. 1.18 shows a schematic cross-section of the domain wall in the plane perpendicular to the wire axis that contains the Bloch point. The latter is represented as a "+" sign reflecting the volume magnetostatic charge density at this location. The left column in Fig. 1.18 shows the two winding possibilities, subjected to an external field µ 0 -→ H a applied along the wire axis, "towards the reader". This leads to a domain wall velocity in the same direction. The torques from the applied field on the windings' magnetization is represented in brown. In the case of an initial winding right-handed with respect to this velocity -→ v DW (top row), the field-induced distortion consists in an inward canting of magnetization, as shown in the right column. If the initial winding is left-handed with the direction of motion (bottom row), the field tends to cant magnetization outward and not inward. A strong asymmetry emerges because in the first case, canting leads to increased magnetostatic volume charges 17 , whereas here the canting leads to reduced volume charges. While this distortion also induces larger surface charges, which oppose the canting, it still proceeds if the applied field is large enough [63,81] until the winding switches and becomes right-handed with -→ v DW . There is indeed a threshold in applied field H sel for this phenomenon to occur.

The same reasoning with the opposite domain wall polarity and/or a field in the opposite direction leads to the following "selection rule" at sufficiently high field: the BPW adopts a winding that is right-handed with the direction of motion. Furthermore, it has been shown [63,81] that even the steady-state velocities below the threshold field H sel depend on the BPW winding. The winding left-handed with the direction of motion has a lower mobility and reaches a plateau in velocity before the winding switches at H sel ; at least, this is the case in square nanowires [63]. A similar behaviour was observed in vortex-like domain walls in soft nanotubes [82], which are in essence BPWs without micromagnetic singularity. Since the physics at play pertains to dipolar interactions rather than the Bloch point itself, qualitative comparison should remain meaningful.

Right-handed

Left-handed The field-driven features of the BPW are summarized in Fig. 1.19: the above-mentioned difference in mobility, the selection of winding at H sel , and the velocity plateau beforehand for the initially left-handed winding are clearly visible. Above H sel , the velocity seems to saturate: this behaviour was also observed in cylindrical wires by Hertel [81] and attributed to the excitation of spin-waves, often termed spin Cherenkov effect or magnonic regime [77,81,82]. In a hand-waving view, this results from the high velocities reached by the BPW, attaining the group velocities of spin waves in the system. The domain wall at rest can be viewed as a superposition of all spin wave modes with certain amplitudes. When the BPW moves sufficiently fast, some translating spin wave modes reach their propagative group velocity. As a result, they are less damped and tend to extend along the wire: thus, some of their amplitude "flows away" from the domain wall. This excitation mechanism gives insight into why the BPW velocity shows a much less steep increase with field. At even higher fields (applied inductions of 200 mT), the domain wall structure has been reported to further distort [83] and spontaneously emit/leave behind pairs of Bloch point around "magnetic droplets" which eventually annihilate.

H sel
Irrespective of its high-field behaviour, the BPW clearly shows a dynamical behaviour markedly distinct from that of the TVW. The mobilities have been predicted to scale with 1/α due to the rigid steady-state motion [63,68,81] as expected from an approach "à la Thiele". Furthermore, the predicted selection of winding is quite a remarkable property. Leaving aside vortex-like domain walls in soft nanotubes (which are close relatives of BPWs) [82], to my knowledge, no other domain wall configuration subjected to static magnetic fields display such a well-defined, once-only switch of a bivalued degree of freedom.

In addition, since the Bloch point is the only stable singularity allowed in ferromagnetism [63,76] (an elegant reasoning on dimensions presented for instance by Braun [84] demonstrates this), the BPW in soft cylindrical nanowires is perhaps the simplest domain wall type hosting a micromagnetic singular point. Now that we have reviewed this domain wall's dynamics, it is time to examine its singular nature from a different perspective: that of algebraic topology.

III.4 Elements of topology applied to micromagnetism

Algebraic topology applied notably to condensed matter systems has risen in the 1970's and known a noticeable upturn in recent years, especially in micromagnetism. It has provided invaluable insight into phase transitions [74,75], band structure theory [85,86], and liquid crystals [87] to name a few topics. As for micromagnetism, despite an early acknowledgement of topology as a useful tool [88], there has been a recent topology-related uproar particularly due to the interest in magnetic skyrmions/bubbles 18 [12,14,65,90,91]. In many instances, references to a so-called "topological protection" of such micromagnetic objects are made [90,92,93], even though other authors [94] suggest more caution. This section does not aim that high, but it will base on two thorough publications [76,84] to tentatively convey a consistent picture of algebraic topology applied to micromagnetism without straying too far from mathematical rigour.

As a first word, it must be said that the primary goal of algebraic topology is classification. More specifically, the tool of use is the so-called homotopy theory, which will be discussed in a moment. To quote Nathaniel D. Mermin: "homotopy theory provides the natural language for the description and classification of defects in a large class of ordered systems" [76]. Let us illustrate the classification methodology on a specific example and then generalize it. Considering the stick and torus in Fig. 1.20, one easily sorts them in the following categories: "elongated object without any hole" and "round object with a hole in the middle". This first classification is unsatisfactory because the "elongated" and "round" features are irrelevant. Squashing the stick into a sphere, a plate, etc. still leaves it in the same first category, similarly to stretching or twisting the torus. In other words, "smooth" or rather continuous distortions do not alter their fundamental nature 19 , which is to (not) have a hole, a feature not as easily described as it seems. Think of twisting the torus on and on like a rubber band, for instance.

A more efficient sorting is achieved with homotopy theory's favourite tool, namely closed loops [76,84]. In the present case, they can be thought of as other tori in space. Quite clearly, if we focus on the stick, any closed loop that can be imagined may be displaced and distorted into any other closed loop. Sliding over or around the stick is no problem, as suggested by Fig. 1.21. The situation, however, is very different in the torus' case. If we consider a loop closed around the torus like a chain link, it can be smoothly moved around and distorted into any other loop that is also closed around the torus, but that is it. It is impossible to transform it into any of the loops that are not closed around the torus, as suggested by the right hand side of Fig. 1.21. Just like in the stick's case, the latter type of loops can all be transformed into one another without any trouble, so that one is just as good as another, or rather equivalent to any other. . . Let us take a step back. By choosing a simple tool for classification, the difference between the stick or any similar object (plate, ball. . . ) and the torus (or a steel nut, or a cup. . . ) directly appears as the number of sets of equivalent closed loops that can be defined with respect to these objects. Moreover, these sets are quite insensitive to the details of the considered objects. As a result, the use of closed loops allows to characterize an object as a whole, irrespective of features which can (dis)appear through continuous transformations.

The non-equivalent ensembles of closed loops are referred to as homotopy classes [76,84]. If we come closer to micromagnetism, such loops must be seen as mathematical maps: from the sample (real) space into the order parameter space, i.e. the set of values that can be assumed by the order parameter, here -→ m. Such a map builds correspondence between one closed loop in real space and a closed loop in the order parameter space. Let us consider a very simple but important example for magnetism to illustrate this: a 2D sample with magnetization constrained to lie in the plane. Fig. 1. 22 shows three such cases. The left column illustrates a closed loop on which the local magnetization is drawn. To the right, the (mathematical) image of this real-space loop in the order parameter space, which is here the unit circle. In a first case, magnetization encountered on the real-space loop is uniform; in other words, all its points have the same order parameter. Thus, this loop's image is one point on the unit circle; we have a first example of a trivial configuration [76]. The second row is only slightly more complicated: magnetization tilts one way then the other way by the same amount on both half of the real-space closed loop. Following magnetization's point leads to the order parameter space image: a closed loop. For the sake of clarity, its two halves are vertically offset although they correspond to the same points. This case's interest is that if one imagines applying a strong magnetic field along the mean direction, then eventually the first row's situation should be recovered. Correspondingly, this amounts to squashing the order parameter space loop into a single point. It must be kept in mind that the proper order parameter space loop (not the one represented here for visualization's sake) really is a circle arc, a bent segment. Similarly to the above stick and plate, these two magnetic configuration are therefore (topologically) equivalent. This is only to say that one can be distorted into the other using only continuous transformations, but the strength of this description is that it encompasses all that lies in the portion of plane enclosed by the real-space loops, without even knowing what it is [76].

Let us consider the third row of Fig. 1.22 to better realize this. Here, the configuration is fully radial and reminiscent of a vortex. Straightforwardly, the image on the unit circle wraps around it exactly once. This means that no matter what we do, this image cannot be shrunk to a point with only continuous transformations. In other words, this radial configuration cannot be smoothly transformed into the first row's configuration. Since we have assumed that magnetization is continuous, shrinking the real-space closed loop should not break this once-winding around the unit circle. Still, this must break down eventually 20 : the closed loop eventually reaches a point where magnetization's derivatives diverges [76].

Sample space (with 1D loop)

Order parameter space Taking a step back, we have arrived at a key result: a simple reasoning on a closed loop encompassing a discontinuous point from an arbitrary distance allows to detect this singular point 21 . It must be noted that apart from all the loops encompassing this singular point, there is the set of all other closed loops which do not include the point. Assuming there are no other discontinuities, they are equivalent to the closed loops from the first and second row. Thus, we retrieve the classification applied to the stick and torus from Fig. 1.21. Instead of holes, the essential features are now singular points for magnetization; in the following, they will be referred to as topological defects. As was just shown, such objects have a tremendous influence since they can be "detected" by closed loops arbitrarily far away.

In a more practical view, examining a given configurations with closed loops enables to detect and locate any singular points based on the question: does magnetization wind once (or more) around the order parameter space? This question is that of the so-called degree of the closed loop as mathematical map [84]. While the above example was in two dimensions both for sample space and order parameter space 22 , the reasoning holds for three-dimensional samples and micromagnetism i.e. a reduced magnetization -→ m living on the three-dimensional unit sphere. The order parameter now being bidimensional, so are the closed loops, which correspond to closed surfaces. As a brief mathematical excursion, the reason why only closed surfaces may be relevant in the search for topological defects in micromagnetism is the existence of an isomorphism between the 2-dimensional fundamental homotopy group π 2 of the three-dimensional unit sphere S 2 and the set of integers [84]. The reader, if not familiar with this terminology or result, is invited to omit the previous sentence for a first reading and to read the review by Braun [84] so as to become acquainted with the formalism of homotopy groups, which I will not introduce here for the sake of brevity.

Adapting the previous two-dimensional examples shows that if magnetization as sampled on a closed surface winds once around the unit sphere, then a point singularity exists somewhere in the volume encompassed by this closed surface. This is shown in Fig. 1.23, showing a Bloch point's vicinity and the latter's representation on the unit sphere.

Sample space (with 1D loop) Order parameter space It is important to note that finding a non-zero integer number of windings implies that at least one singularity exists. However, null winding does not mean that not singularities exist in this volume: two opposite defects can also lead to the same result 23 . The reader is referred to the publication by Mermin [76] for a more thorough description.

At this stage, it is useful to introduce the computable quantity N that reflects the above-mentioned "number of windings" and thus allows to detect topological defects. It is sometimes referred to as (Chern-)Pontryagin index [95], topological charge [81,[96][97][98] or number [12,99], skyrmion number [84,100], or winding number [76,96,101], which will be the case in the following. Its expression for the case of a unit vector on the unit sphere defined in a threedimensional sample (hereafter referred to as S 2 winding number), is given in all generality for instance by Braun [84]. One can show that in cartesian coordinates, it reads:

N = 1 4π ¨∂V - → m • ∂ - → m ∂x × ∂ - → m ∂y dx ∧ dy + ∂ - → m ∂x × ∂ - → m ∂z dx ∧ dz + ∂ - → m ∂y × ∂ - → m ∂z dy ∧ dz (1.36)
For a rigorous definition of the differential forms (d's) and the outer product (∧), the reader is referred to e.g. Ref. [102]. I will only state without proof that e.g. dx ∧ dy corresponds to the scalar product between the outward-oriented, unit vector normal to the surface ∂V and the vector (dxdy) -→ e z ( -→ e z being the cartesian z-axis' unit vector), with (dxdy) the infinitesimal surface element spanned by the usual differential elements dx and dy. this quantity corresponds to the surface covered on the unit sphere by the unit vector -→ m, divided by 4π.

This winding number is a powerful tool in the investigation of simulated micromagnetic domain wall configurations. Since it assumes only integer values (within numerical error), it is conceptually easy to apply it to the localization of Bloch points for instance as a function of time during a simulation. An example of such use will be provided in Chapter 4 section V.3. For the sake of completeness, I must add that computing the winding number of the full volume in which a TVW is simulated leads to zero. In other words: the TVW contains no topological defect, as expected.

As a conclusion to this section, it must be kept in mind that except for the pictures of applications of fields to distort magnetic configurations, physical dynamics and energetics have been completely absent from the discussion. Moreover, the fact that two topologically inequivalent configurations cannot be transformed into one another by continuous transformations seems to imply some sort of robustness because notably of exchange interactions. However, that is already a physicist's reasoning on top of topological considerations: homotopy theory is mathematics, it does not aim at discussing stabilities or "protections". A footnote by Nathaniel D. Mermin seems here most fitting to me: "(. . . ) it is important to bear in mind that in many cases the topological analysis provides only a framework, into which a subsequent study of energetics must be fitted to arrive at a full understanding". Caution is therefore needed in the real, experimental world.

III.5 Short review on experimental investigations of domain walls in cylindrical nanowires

Individual soft wires hosting domain walls have so far been mostly investigated with Magnetic Force Microscopy (MFM). For a detailed introduction to this scanning probe technique, the reader is referred to the book by Hubert and Schäfer [15]. After dispersion of nanostructures on a substrate, these can be inspected one by one, by contrast with methods such as magnetometry in arrays. Thanks to its spatial resolution on the scale of a few tens of nanometres and its sensitivity to stray fields, MFM is a commonly used microscopy in the context of nanowires [103][104][105][106][107][108][109]. . MFM is sensitive to stray fields acting on the probe through interaction with the thin ferromagnetic coating on its tip. This probe, which is usually a silicon cantilever, is mechanically excited at its resonance frequency; in this condition, the sensitivity to external forces is optimal. These forces are measured by the resulting offsets in resonance frequency or the oscillation's phase. The interpretation of MFM images is not immediate as not vector magnetization but only one component of dipolar fields from the sample is measured [15].

Furthermore, the technique is invasive to some extent due to the stray fields from the MFM tip and measurements have been shown to affect the magnetic configuration of e.g. nanostrips or cylindrical nanowires [110,111] in certain conditions, demonstrating the need for suitable probes [105].

Beyond static or remanent imaging [103,106,112], MFM has been implemented in combination with the application of DC magnetic fields to investigated the reversal [108,109], domain wall nucleation and pinning [105,107]. However, confrontation of simulated MFM contrast from domain walls with experimental measurements [105,113] seems to show that deducing the domain type with this technique is challenging at best. Moreover, since the two senses of winding of a BPW generate the same vertical stray fields (as can be seen by wire rotation around a vertical axis by 180 • ), this fine but crucial micromagnetic feature cannot be revealed with MFM. Therefore, an experimental technique providing a higher level of details on a sample's magnetic configuration is needed to conduct field-driven domain wall motion experiments, if the goal is to infirm or confirm the predictions from simulations (which were described in section III.2 and section III.3).

Other techniques, such as Lorentz microscopy imaging or electron holography within a Transmission Electron Microscope [114,115], or magneto-optical imaging using the Magneto-Optical Kerr Effect (MOKE) [15,116] , have also been employed in several experimental works [112,[117][118][119][120][121]. Transmission-Electron-Microscope-based imaging is non-invasive, allows the quantitative retrieval of magnetic inductions [120,121] but its interpretation becomes complex and often requires simulation for samples starkly different from thin films [122]. These methods rely on the phase acquired by the electron beam as it passes through the ferromagnetic sample. The physical picture is that electrons get deflected in a direction perpendicular to the local induction due to the Lorentz force. As a result, measurements of the variations in beam deflection or of the acquired phase allow the reconstruction of magnetic inductions.

(Focused) MOKE on the other hand is more often used for the acquisition of hysteresis loops [106,107,112,119] when applied to cylindrical nanowires, since the resolution is limited by the light's wavelength to several hundred nanometres. Magneto-optical methods rely on the interaction between typically a linearly polarized light beam and a ferromagnet. When reflected off a ferromagnetic sample's surface for instance (Kerr geometry), the light's polarization may undergo rotation as well as become elliptical [15], and these changes can be interpreted in terms of magnetization orientation. In such experiments, the technique is non-invasive provided that not too much power from the beam is absorbed by the sample, which may result in heating.

Further information can be acquired with a recently pioneered synchrotron technique, namely shadow XMCD-PEEM. X-ray Magnetic Circular Dichroism (XMCD), as the name implies, is a differentiated absorption of X-ray light as a function of the beam's circular polarization state resulting from magnetization [123]. PhotoEmission Electron Microscopy (PEEM) on the other hand is not a physical phenomenon but an experimental "photon in, electron out" technique most relevant for surface studies [124]. XMCD has been experimentally measured in the late 1980's [125], and PEEM using synchrotron radiation has been implemented in the early 1990's [126,127]. The majority of experimental works combining XMCD and PEEM have focussed on thin films and patterned structures thereof [124,128]. In this context, the seminal work by Judith Kimling et al. [129] demonstrated in 2011 that using the transmission through semi-opaque samples of circularly polarized X-rays could overcome the usual restriction of PEEM to thin samples. Since this limitation arises from the escape depth of the photoelectrons used for imaging, the approach consisted in having the X-ray beam first traverse the magnetic sample, then give rise to photoelectrons on the sample's substrate. The name "shadow" XMCD-PEEM refers to the strong, resonant absorption of X-rays through matter: it is modulations in this geometrical-optics shadow's intensity that reflect magnetism, this time from the sample volume and not just its surface.

APPENDICES FOR CHAPTER 1

A 1 .1 From the Heisenberg model to the micromagnetic formulation of exchange interaction

We have seen in the section I.1 that the coupling between neighbouring atomic sites from which ferromagnetism results can be written in terms of the ferromagnetic Heisenberg model with hamiltonian H exch :

Ĥexch = <i,j> -J - → S i • -→ S j = -JS 2 <i,j> - → s i • - → s j , (A 1 .1)
where the sum < i, j > runs once over all nearest-neighbour bonds, -→ s i and -→ s j are unit vectors denoting the orientations of -→ S i and -→ S j , whose (classical) length is S. The usual formulation has J > 0 (for ferromagnets) occurring as an energy per bond, and S a dimensionless number. Let us go from this discrete description to a continuous one by taking the above-mentioned orientations to be defined everywhere. If one Taylor-expands the -→ s j vector in the continuum limit, where we consider a function -→ s ( r) of the continuous position variable r, it reads:

- → s j = - → s ( r + - → a j ) - → s ( r) + - → a j • → - → s ( r) + 1 2 l - → a j • H(s l ) - → a j - → e l , ( A 1 .2)
where -→ a j is the vector pointing from site i to site j, -→ e l is the l-th unit vector in the cartesian base (as in the above section), and H(f ) is the hessian matrix of the function f . Owing to Schwarz' theorem, it is symmetric. Being real, it possesses a eigenbasis of orthogonal unit vectors. In this basis, it is diagonal and its trace is still s l , where is the scalar Laplace operator. Now, we must use the assumption that we will be considering only an isotropic material for the (micromagnetic) exchange interaction. This is taken to mean two things:

• considering any arbitrary micromagnetic configuration -→ m 0 ( r) and any point -→ r i , the system's total exchange energy (at the continuous medium level) is invariant under all rotations with fixed point -→ r i ,

• at any point -→ r i , the micromagnetic exchange energy density is isotropic.

As a result, for a given configuration -→ s ( r) and thus a given matrix H(s l ), the outcome of our result must be the same as the average over all possible orientations of our cartesian basis (corresponding to the -→ e j 's). This is helpful because we no longer need to worry about the respective orientation of the eigenbasis of H(s l ) and our cartesian basis. When we sum over the bonds (at one site i corresponding to r), we also average over their orientation, so each of them contributes to the three eigenvalues of H(s l ) with a factor 1/3 times the squared length of the vector. As a result, each lattice vector contributes (on -→ e l ) as its squared length divided by three, times s l . Let us denote a such that:

a 2 = 1 3 j,distinct dir. - → a j 2 , (A 1 .3)
in other words the length whose square is a third of the sum of the squared lengths of the bond vectors of distinct directions. Let us denote z is the number of nearest neighbours. Thanks to the space inversion symmetry, all first-order derivatives vanish in the sum over bonds so that the exchange energy E exch δV corresponding to the evaluation of our hamiltonian Ĥ at site r in a cell 24 of volume δV reads:

E exch δV = -z × JS 2 - 1 2 × z × JS 2 × 2a 2 z - → s • - → - → s . (A 1 .4) E exch δV = -JzS 2 -JS 2 a 2-→ s • - → - → s . (A 1 .5)
In the second term, we have accounted all bonds (hence the first z), which contribute on average a 2 /(z/2) considering the definition of a: there are z/2 distinct bond directions. Therefore, 2a 2 /z is the squared effective length per bond. Our result can be checked in the ferromagnetic state, whose total energy per site must be the number of bonds times the energy per bond, which is -z × JS 2 = -JzS 2 , consistently with the above equation. Since we are interested in departures from the uniform ferromagnetic state, we can drop the constant term and write the energy density E exch as:

E exch = - JS 2 a 2 δV - → s • - → - → s . (A 1 .6)
The energy density we have found is exactly of the form:

E exch,micromag = -A - → m • - → - → m (A 1 .7)
which is used in micromagnetism, with A the usual exchange stiffness and -→ m the normalized magnetization vector.

A 1 .2 Domain wall in a bent 1D system

Let us consider a thin cylindrical nanostructure with an angle: If θ(s) is the angle between the fixed z direction and the local magnetization unit vector -→ m(s), θ 0 > 0 is the half angle defining the bend, and s the curvilinear abscissa describing the structure, one can write the local energy density as:

s θ 0 - → m(s) z θ(s)
E(s) =        A dθ ds 2 + K sin 2 [θ(s) -θ 0 ] if s ≤ 0 A dθ ds 2 + K sin 2 [θ(s) + θ 0 ] if s ≥ 0 (A 1 .8)
A and K stand for the material's exchange stiffness and uniaxial anisotropy constants respectively. It is worth noting that there is a head-to-head solution with -→ m along -z at the bend, which we can safely reject as lowestenergy state since it will obviously cause more pronounced variations of θ. One can show (see end of section) that its energy is higher than that of the profile sketched above. Thus, we will only consider the aforementioned headto-head domain wall configuration of lower energy, the tail-to-tail domain wall of lower energy being obtained by adding π to the θ(s) function.

The problem is complete with the given boundary conditions: θ(s → ±∞) = ∓θ 0 and (by symmetry) θ(s = 0) = 0. One obtains the domain wall profile by minimizing its total energy per unit surface E DW θ, dθ ds :

E DW θ, dθ ds = ˆ+∞ -∞ E(s)ds (A 1 .9)
This minimization is imposed by setting the functional derivative δE δθ(s) to 0 for all s. This amounts to:

δE δθ(s) =          -2A d 2 (θ(s) -θ 0 ) ds 2 + 2K sin [θ(s) -θ 0 ] cos [θ(s) -θ 0 ] = 0 for s ≤ 0 -2A d 2 (θ(s) + θ 0 ) ds 2 + 2K sin [θ(s) + θ 0 ] cos [θ(s) + θ 0 ] = 0 for s ≥ 0 (A 1 .10)
Let us focus on the left side (s ≤ 0) and write u(s) = θθ 0 . Denoting ∆ = A/K, one can write:

2∆ 2 d 2 u ds 2 = sin (2u) (A 1 .11)
Multiplying by the derivative of u yields:

∆ 2 d ds du ds 2 = d ds sin 2 (u) (A 1 .12)
In other terms, the equipartition of anisotropy energy and exchange energy still holds in the bent wire case, as expected. Keeping in mind that sin [u(s)] ≤ 0 and du/ds ≤ 0 for this wire section, this leads to:

∆ du ds = + sin u (A 1 .13)
Introducing the new variable t(s) = tan u(s) 2 , knowing that sin u(s) = 2t(s)/[1 + t(s) 2 ] this differential equation can be solved after a bit of trigonometry and yields:

∆ dt ds = t (A 1 .14) hence: t(s) = A exp s ∆ (A 1 .15)
The boundary condition for s = 0 is then:

A = t(0) = tan 0-θ 0 2 = -tan θ 0 2
. With this, one obtains an equation for θ(s):

tan θ(s) -θ 0 2 = -tan θ 0 2 exp s ∆ (A 1 .16)
Taking into account the domains of definition of the involved trigonometric functions, this can be rewritten as:

θ(s ≤ 0) = θ 0 -2 arctan tan θ 0 2 exp s ∆ (A 1 .17)
In the same manner, one can obtain:

θ(s ≥ 0) = -θ 0 + 2 arctan tan θ 0 2 exp - s ∆ (A 1 .18)
This can be summed up in one formula, using sgn(s) = s/|s|:

θ(s) = sgn(s) -θ 0 + 2 arctan tan θ 0 2 e -sgn(s) s ∆ (A 1 .19)
In the special case where θ 0 = π/2, we obtain:

θ(s) = - π 2 + 2 arctan e -s ∆ ∀s (A 1 .20)
and retrieve the usual Bloch wall profile. Using equation A 1 .13, one can calculate the domain wall width ∆ DW . It reads:

∆ DW = ˆ+∞ -∞ sin 2 [u(s)]ds = -2 ˆ+∞ 0 sin[u(s)]∆ du ds ds = +2∆[cos(θ -θ 0 )] +∞ 0 = 4 sin 2 θ 0 2 ∆ (A 1 .21)
The definition used is close to that of Jakubovics [26] but without prefactors. The idea is to be able to use the scaling laws for the domain wall width derived by SÉGOLÈNE JAMET et al. [67]. As expected, this width yields the usual 2∆ for θ 0 = π/2, which corresponds to the straight wire case. In the general case, the energy reads:

E DW (θ 0 ) = 8 sin 2 θ 0 2 √ AK = 2 sin 2 θ 0 2 E(π/2) (A 1 .22)
The reduction in domain wall energy (for all θ 0 < π/2) with respect to the straight wire case only occurs because the exchange energy cost continuously goes down and the anisotropy axis always follows the system's local axis. Using the same methods, one can compute the energy E DW and width ∆ DW of the two (degenerate) domain wall configurations of higher energy (i.e. pointing "against the bend"); they read:

E DW = 8 cos 2 θ 0 2 √ AK E DW E DW = cot 2 θ 0 2 > 1 ∆ DW = 4 cos 2 θ 0 2 ∆ ∆ DW ∆ DW = cot 2 θ 0 2 > 1
Just to have an order of magnitude, the factor cot θ 0 2 2 is 5.8 for 2θ 0 = π/2. As stated previously, even though the physics at such diameters is more complex than a one-dimensional problem, the experimental results should hint at the trend displayed by the width and energy of the domain walls. This trend has been illustrated notably by the experiments on Permalloy strips and micromagnetic simulations by Lewis and coworkers [130].

A 1 .

Thiele formalism

The following calculations are adapted from Wysin's [131,132]. In these references, the Thiele equation is derived with either an inertial term and no damping [131] or vice-versa [132]. In this document, our goal is to combine both approaches for the sake of generality. The starting point is to rewrite the Landau-Lifshitz equation in an equivalent form. This equation reads:

d - → m dt = γ 0 --→ H eff × - → m + α - → m × d - → m dt (A 1 .23)
where γ 0 = µ 0 |γ|, γ is the gyromagnetic ratio, -→ m = -→ M /M s is the reduced magnetization, and α is the Gilbert damping constant. Let us now introduce the net field --→ H net following Wysin [132], whose expression is:

γ 0 --→ H net = γ 0 --→ H eff + d - → m dt × - → m -α d - → m dt (A 1 .24)
The interpretation of this field becomes a bit clearer when imposing the condition that -→ m be always aligned with it, in other words:

γ 0 - → m × --→ H net = - → 0 (A 1 .25)
In fact, this is a similar to a transformation of frame such that locally, magnetization stands still. Given a certain magnetization texture and its time evolution, there is a (spatially varying) frame in which there is no dynamics but still, even if looking at magnetization (anywhere) in this frame does not tell what is going on, stating that -→ m precesses around the real fields plus the term with d -→ m/dt × -→ m contains all the time evolution. In a way, this resembles classical mechanics of solids and transformations into non-galilean frames where the situation is clearer because the object of interest no longer moves: this often generates so-called "pseudo-forces" similarly to the way the field-like term d -→ m/dt × -→ m arises in the frame where magnetization stands still. The latter equation reads:

d - → m dt • - → m • - → m -- → m • - → m • d - → m dt + γ 0 - → m × --→ H eff -α - → m × d - → m dt = - → 0 (A 1 .26)
Exploiting the constant (with respect to time) norm of -→ m, one can rewrite this as:

d - → m dt = γ 0 --→ H eff × - → m + α - → m × d - → m dt (A 1 .27)
which is exactly equation (A 1 .23). Thus, the two following equations:

γ 0 --→ H net = γ 0 --→ H eff + d - → m dt × - → m -α d - → m dt (A 1 .28) - → m × γ 0 --→ H net = - → 0 (A 1 .29)
are equivalent to the Landau-Lifshitz equation. Thus, the net field --→ H net appears to be some sort of sum of all elements dictating the magnetization dynamics.

A 1 .3.1 Use the alignment of magnetization and the net field Our goal is now to use the equivalent form of the Landau-Lifshitz equation to derive an equation of motion for an arbitrary magnetization profile. For this, we will use two properties of the net field: its alignment or proportionality with magnetization, and its expression in terms of effective field, magnetization and magnetization's time derivative.

Considering the above, one may now write [132] for any position x = (x, y, z) = (x 1 , x 2 , x 3 ) and time t:

--→ H net = β( x, t) - → m (A 1 .30)
Now let us apply -→ m on the matrix Ā such that:

Ā =   
   = ∂ i m j (i,j)∈(1,2,3) 2 (A 1 .31)
This matrix is in fact the transposed jacobian matrix of -→ m. One can write:

Ā • - → m = i j (∂ i m j )m j - → e i = 1 2   ∂ 1 ( - → m 2 ) ∂ 2 ( - → m 2 ) ∂ 3 ( - → m 2 )   = - → 0 (A 1 .32)
In the case, we are exploiting the constance of the norm of -→ m with respect to space, in contrast to the previous section where time was considered. The above result still applies upon replacement of -→ m with --→ H net since both vectors are collinear. The advantage is that we may now connect the time evolution of magnetization (through the Landau-Lifshitz equation contained in --→ H net ) with variations of magnetization in space. In other words, we are about to mix all the ingredients necessary for the study of propagating magnetization profiles.

Equations (A 1 .30) and (A 1 .32) allow us to write:

Ā • γ 0 --→ H net = - → 0 (A 1 .33)
So far, we have stated on one hand that -→ m 2 is constant with respect to time and space, and rewritten the time evolution of -→ m. Then, in equation (A 1 .3.1), we have combined these constraints on -→ m 2 with this rewritten time evolution. The equation (A 1 .32) implies that Ā is an intrinsic projector onto the plane normal to -→ m. Thus, using the net field which contains all the dynamics leads to a kind of conservation equation. Using the definition of the net field, this reads:

Ā γ 0 --→ H eff + d - → m dt × - → m -α d - → m dt = - → 0 (A 1 .34)
The reader is invited to prove that this equation (A 1 .34) can be obtained by projection of the LL equation onto the matrix Ā. The trick is to use the fact that since it is a projector onto the plane normal to -→ m, the vector product can be performed before or after applying the matrix to the field-like terms since their components along magnetization do not contribute in the end.

One way to view what we have done so far with a bit more physics is the following: starting from the (local) frame in which magnetization stands still at a given point, we are trying to connect to the points in space right next to our starting point through the spatial derivatives that are contained in Ā. Thus, we should be able to look at the dynamics of an extended object. At this stage, nothing has been said regarding either the distribution of magnetization or the contents of the effective field --→ H eff . We will now make the following two hypotheses:

• There is a magnetization profile in translation

• One or several contributions to the effective field exert a force on the magnetization profile, resulting in the translational motion

The first hypothesis implies that -→ m can be specified as follows:

- → m( x, t) = -→ m 0 x - - → X (t); - → V (t) (A 1 .35) where - → X (t) = -→ X 0 + ´t -∞ - → V (
u)du is the position of one particular point of the magnetization profile in the laboratory frame, and -→ m 0 is the function describing the profile we are interested in. This function may depend on -→ V as a parameter; for instance in the 1D model of domain walls [63], provided there is a transverse anisotropy, the domain wall width depends on the velocity through the tilt angle of the domain wall with respect to the easy axis. The position -→ X defines the moving frame following the translational motion of the profile. This motion happens according to a velocity vector -→ V (t) which may vary in time: this will lead to inertial effects. It is very important to note that this choice implies a further hypothesis, namely that magnetization does not depend on the acceleration d -→ V /dt. Another way to view this is to state that the transitory regime between two different velocities -→ V 1 and -→ V 2 may well feature details not included in this model, which will yield an inertial mass independent of the profile's acceleration 25 . Along this line, the model should be valid for small variations -→ V 1 --→ V 2 occurring over large time scales so that the (vanishing) acceleration can be safely neglected.

Keeping this in mind, let us turn to the effective field and our second hypothesis. Provided that we start from a relaxed micromagnetic configuration, only external drives may induce the motion of a magnetization profile. Obviously, exchange interactions, magnetocrystalline or surface anisotropies, as well as dipolar interactions play a role in the sense that they (along with the external drive) shape the dynamic equilibrium state. Yet, on their own, they cannot lead to a steady translational motion which requires a constant power intake due to the Gilbert dissipation. On this basis, one can therefore assume that the force exerted on the magnetization profile only depends on the external drive, such as an externally applied magnetic field. This reasoning is somewhat similar to classical mechanics, where the sum of internal forces of a rigid object must cancel out.

The well-known case of the gyrotropic motion of vortices in dots hints at the limit of this statement and of our model. Indeed, in this case, dipolar interactions are absolutely essential since they are the main source of the repulsion coming from the edges of the sample. However, one should first remember that this motion proceeds under constant non-zero acceleration. Then, in such cases, since calculating their effect is impossible analytically, the way to proceed is the following: dipolar interactions are known to contribute to the force exerted on the profile, but left out in a first step. Then, one can derive the Thiele equation and only afterwards re-include them as an empiric term of proper symmetry. Since one can safely expect dipolar interactions to create an energy landscape with a minimum at the dot's center and increasing slopes towards the edges, the physics should be captured by an ad-hoc potential well that can be added after all the calculation effort is done. Nevertheless, even in this case, one should keep in mind that if the stimulus is turned off in a realistic case with damping, dipolar interactions on their own are not able to sustain steady-state motion.

III.2.b Deriving the force from the system's total energy

From equation (A 1 .34), we may guess that the force will emerge from the term Āγ 0 --→ H eff , and in fact we will demonstrate just that. One first way to find out how exactly both are related is based on the usual definition of a force as

- → F = - → X E.
In our frame of a moving frame, however, one needs to step carefully as far as the coordinates on which → X acts are concerned. Indeed, these must be the coordinates of the "center of mass" of our object, in other words -→ X . In this regard, equation (A 1 .35) imposes for any indices i and j:

∂m j ∂x i = - ∂m j ∂X i (A 1 .36) Let us denote → f = i (∂f /∂x i ) - → e i and → • - → B = i ∂B i /∂x i .
Using the chain rule for functionals [25], we may write any component F i of the force -→ F :

F i = - ∂E ∂X i = - ˚δE δ - → m • ∂ - → m ∂X i d 3 u = -µ 0 M s ˚--→ H eff • ∂ - → m ∂x i d 3 u. (A 1 .37)
We have used the definition of the effective field as:

-µ 0 M s --→ H eff = δE/δ - → m.
At this stage, one needs to remember that the model can only be valid if brutal evolutions i.e. large accelerations are excluded.Even if we write later on time derivatives of the velocity, the relevance of these terms in the end result only holds for vanishing accelerations. With this, the force reads:

- → F = -µ 0 M s ˚ i,j ∂m j ∂x i H eff,j - → e i d 3 u = -µ 0 M s ˚Ā • --→ H eff d 3 u (A 1 .38) A 1 .3.

Derivation of the Thiele equation

We start from equation (A 1 .34). Using the indices, one may write it as:

i,j,k,l ∂m j ∂x i γ 0 H eff,j + jkl dm k dt m l -α dm j dt - → e i = - → 0 (A 1 .39)
where abc is the Levi-Civita symbol. Now, using equation (A 1 .35), we may explicit the time derivatives:

dm k dt = p - ∂m k ∂x p V p + ∂m k ∂V p dV p dt (A 1 .40)
It is at this stage that the limits of our model show most. Indeed, only the assumption of a magnetization profile independent of its acceleration allows the above equation to be exact. In principle, other terms involving higher-order time derivatives of the velocity should be present. Each of them would result in an additional term in the Thiele equation, but with a less and less straightforward physical meaning. In the following, no such term will be kept as we shall use our "vanishing accelerations" hypothesis. Thus, for all indices i, equation (A 1 .39) now reads:

j,k,l,p ∂m j ∂x i γ 0 H eff,j + jkl - ∂m k ∂x p V p + ∂m k ∂V p dV p dt m l -α - ∂m j ∂x p V p + ∂m j ∂V p dV p dt = 0 (A 1 .41) j,k,l,p ∂m j ∂x i γ 0 H eff,j -jkl ∂m k ∂x p m l V p + α ∂m j ∂x p V p + jkl ∂m k ∂V p m l -α ∂m j ∂V p dV p dt = 0 (A 1 .42) j,k,l,p γ 0 ∂m j ∂x i H eff,j -jkl m l ∂m j ∂x i ∂m k ∂x p V p + α ∂m j ∂x i ∂m j ∂x p V p + jkl m l ∂m j ∂x i ∂m k ∂V p -α ∂m j ∂x i ∂m j ∂V p dV p dt = 0 (A 1 .43) j γ 0 ∂m j ∂x i H eff,j - j,k,l,p jkl m l ∂m j ∂x i ∂m k ∂x p V p +α j,p ∂m j ∂x i ∂m j ∂x p V p + j,k,l,p jkl m l ∂m j ∂x i ∂m k ∂V p -α ∂m j ∂x i ∂m j ∂V p dV p dt = 0 (A 1 .44) j γ 0 ∂m j ∂x i H eff,j -   l m l   j,k,p ljk ∂m j ∂x i ∂m k ∂x p V p     + α j,p ∂m j ∂x i ∂m j ∂x p V p + p   l m l   j,k ljk ∂m j ∂x i ∂m k ∂V p   -α j ∂m j ∂x i ∂m j ∂V p   dV p dt = 0 (A 1 .45)
If we now multiply by µ 0 M s /γ 0 , we begin to recognize the first term as the local density of force exerted on the magnetization profile.

- j -µ 0 M s ∂m j ∂x i H eff,j - µ 0 M s γ 0 p - → m • ∂ - → m ∂x i × ∂ - → m ∂x p V p + α µ 0 M s γ 0 p ∂ - → m ∂x i • ∂ - → m ∂x p V p + µ 0 M s γ 0 p - → m • ∂ - → m ∂x i × ∂ - → m ∂V p -α ∂ - → m ∂x i • ∂ - → m ∂V p dV p dt = 0 (A 1 .46)
We can now return to the vector equation and analyze the different terms as we recognize matrix-vector products:

--µ 0 M s Ā • --→ H eff - µ 0 M s γ 0 - → m • ∂ - → m ∂x i × ∂ - → m ∂x p • - → V + α µ 0 M s γ 0 ∂ - → m ∂x i • ∂ - → m ∂x p • - → V + µ 0 M s γ 0 - → m • ∂ - → m ∂x i × ∂ - → m ∂V p -α ∂ - → m ∂x i • ∂ - → m ∂V p • d - → V dt = - → 0 (A 1 .47)
In the above equation, we see three new operators acting on the profile's velocity or its acceleration. The first one is related to the gyrovector, the second one is the (symmetric) dissipation matrix, and the third one is the inertial mass of the profile.

We may now integrate over the whole sample volume. Considering our hypothesis of uniform translational motion, our model implies that in at least one spatial direction, the sample volume is infinite. One can write the Thiele equation as:

M • d - → V dt = - → F + G • - → V + αD • - → V (A 1 .48)
where the components of the tensors read:

G ij = µ 0 M s γ 0 ˚ - → m • ∂ - → m ∂x i × ∂ - → m ∂x j d 3 x (A 1 .49) M ij = µ 0 M s γ 0 ˚ - → m • ∂ - → m ∂x i × ∂ - → m ∂V j -α ∂ - → m ∂x i • ∂ - → m ∂V j d 3 x (A 1 .50) D ij = - µ 0 M s γ 0 ˚∂-→ m ∂x i • ∂ - → m ∂x j d 3 x (A 1 .51)
and as in equation (A 1 .38), the force vector reads:

- → F = -µ 0 M s ˚ Ā • --→ H eff d 3 x (A 1 .52)
The effect of the gyrotensor G can be rewritten using a gyrovector -→ G (see the corresponding section A 1 .3.5):

- → G = - µ 0 M s γ 0 ˚     - → m • ∂ -→ m ∂x 2 × ∂ -→ m ∂x 3 - → m • ∂ -→ m ∂x 3 × ∂ -→ m ∂x 1 - → m • ∂ -→ m ∂x 1 × ∂ -→ m ∂x 2      d 3 x (A 1 .53)
With this tool, the Thiele equation can be written in one of its most usual forms:

M • d - → V dt = - → F + - → G × - → V + αD • - → V (A 1 .54)
As can be seen, the motion in our frame of approximation is Newtonian with a well-defined inertia that reflects the distortions of the profile -→ m( r) when the object acquires velocity. There is a certain force acting on the object as well as a damping term ensuring that a (possibly static) steady-state is reached in the end 26 . It is easy to see that D has the proper sign, that is to say negative, in a 2D-sample case 27 , using the fact that as a symmetric real matrix, it can be diagonalized in an orthonormal basis of eigenvectors. The 3D-sample case eludes the author but physically, it must work. Last but not least, the term associated to the gyrovector -→ G does influence the motion direction but does no work. To say that it acts on the micromagnetic object just like a magnetic field acts on an electrically charged particle is true. . . and somewhat ironic. More seriously, tracing back its origin, one can see that the gyrotensor is a consequence of the precessional motion of magnetization: the second term in equation (A 1 .39), which is itself the vector product between -→ m and d -→ m/dt in the net field --→ H net , in other words, the ingredient leading to the precession of magnetization around the effective field in the LLG equation. Therefore, the gyrovector represents an internal effect of the micromagnetic object on itself (hence, it cannot do work in our rigid approximation) because of the dynamics it obeys. As will be seen in the following section, in a hand-waving manner, it is non-zero if the micromagnetic object's configuration covers a finite surface on the unit sphere. This is why for instance a "simple" Bloch/Néel wall cannot experience such an effect 28 .

Perhaps one of the best examples to gain an intuition as to the gyrovector's action is looking at a vortex domain wall in a soft strip, under a magnetic field aligned with that strip. If one considers magnetization close to the vortex core, along the perpendicular to the strip direction, then one sees that this magnetization points either along the field or anti-parallel to the field, depending on the side. An infinitesimal amount of time later, magnetization on both sides will point slightly to one side. Indeed, magnetization at these two points is starting to assume the same direction (to that same side) in a first step, then to rotate by 90 • to form the uniform domain after passage of the vortex. Hence, the time derivative on those two sides points to one side and thus the time derivative at the core itself must also point to that side. Therefore, the vortex core must shift in that direction. If we want to interpret this formula, it is easier to consider the particular case of a two-dimensional sample, say, an infinite plane P perpendicular to the z direction. The integrand then needs to be multiplied by a t • δ(zz 0 ) Dirac delta distribution, where z 0 is the coordinate of the plane of interest, and t its thickness. Then, since there are no more any variations of -→ m along the z direction, the integral becomes:

- → G (2D) = - µ 0 M s t γ 0 ¨P      - → m • ∂ -→ m ∂y × ∂ -→ m ∂z - → m • ∂ -→ m ∂z × ∂ -→ m ∂x - → m • ∂ -→ m ∂x × ∂ -→ m ∂y      dx dy = - µ 0 M s t γ 0 ¨P    0 0 - → m • ∂ -→ m ∂x × ∂ -→ m ∂y    dx dy = - µ 0 M s t γ 0 ¨P - → m • ∂ - → m ∂x × ∂ - → m ∂y - → e z dx dy (A 1 .55)
With this, we see that without any assumption regarding the distribution of magnetization, in the 2D case, the gyrovector becomes orthogonal to the plane of the sample. Furthermore, if we leave out the µ 0 M s t/γ 0 prefactor, we see that the integral defining the gyrovector takes the same form as the winding number N for a 3D unit vector field -→ s defined within an arbitrary surface δV encompassing a given volume V ∈ R 3 . Indeed, one can write:

N = 1 4π ˆδV - → s • ∂ - → s ∂x × ∂ - → s ∂y (dx ∧ dy) + - → s • ∂ - → s ∂x × ∂ - → s ∂z (dx ∧ dz) + - → s • ∂ - → s ∂y × ∂ - → s ∂z (dy ∧ dz) (A 1 .56)
where the differential forms dx ∧ dy and so on yield the projection of the unit vector locally orthogonal to δV onto (mnemonically) the base vector of R 3 generated by the vector product of the two base vectors corresponding to the form, in this example -→ e z = -→ e x × -→ e y . The integration is carried out over the whole surface, even though no infinitesimal surface element of the type dx dy is specified, for the sake of generality.

Coming back to the gyrovector, in the case of our 2D sample, we would write the full winding number as:

N (2D) = 1 4π ˆδV - → m • ∂ - → m ∂x × ∂ - → m ∂y - → e z dx dy (A 1 .57)
Since the winding number represents a (signed) surface covered by the vector field on the unit sphere, divided by the surface of the latter, we can attribute the same interpretation to the integrals defining the gyrovector's components; thus, taking the 4π prefactor into account:

- → G (2D) = - µ 0 M s t γ 0 S z - → e z (A 1 .58)
Here, S z is the surface covered by our magnetization distribution from the whole plane P on the unit sphere. In the case of a vortex, for instance, S z = ±2π as the magnetization distribution covers half the unit sphere 29 . If we imagined adding to our sample a plane P with a normal vector along -→ e x , then our gyrovector would read:

- → G = - µ 0 M s t γ 0 (S x - → e x + S z - → e z ) (A 1 .59)

III.4.b More general case

We may also imagine rotating the normal vector to P while keeping the basis ( -→ e x , -→ e y , -→ e z ): it is clear that the covered surface on the unit sphere should be shared on the x, y and z components of the gyrovector. Let us take the example of a rotation of our basis vectors around the x direction by an angle θ, as illustrated by the following scheme: The basis after rotation is denoted ( -→ u x , -→ u y , -→ u z ). Either by full calculation or simple projection, one can write:

θ - → e z - → e y - → e x = -→ u x - → u z -→
- → G = - µ 0 M s t γ 0 S z - → e z = - µ 0 M s γ 0 S z (sin θ -→ u y + cos θ -→ u z ) (A 1 .60)
Indeed, after changing the new coordinate system (and recognizing y as y cos θz sin θ), the integral expression reads:

- → G = - µ 0 M s t γ 0 ¨P - → m • ∂ - → m ∂x × ∂ - → m ∂y sin θ -→ u y + - → m • ∂ - → m ∂x × ∂ - → m ∂y cos θ -→ u z dx dy (A 1 .61)
Now, if we come back to the discussion of equation (A 1 .56), we must remember that in the sum of terms constituting the winding number, there is a projection onto the local normal -→ n to the surface. In the rotated basis, a winding number would read:

N = 1 4π ˆP - → m • ∂ - → m ∂x × ∂ - → m ∂y (dx ∧ dy ) + - → m • ∂ - → m ∂x × ∂ - → m ∂z (dx ∧ dz ) + - → m • ∂ - → m ∂y × ∂ - → m ∂z (dy ∧ dz ) (A 1 .62)
where, in this case, dx

∧ dy = - → n • -→ u z = - → e z • -→ u z = cos θ, and dx ∧ dz = -- → n • -→ u y = -- → e z • -→ u y = -sin θ.
Thus, in a first step, we may make the intuitive remark that the projection of the gyrovector onto the normal to our (plane) sample surface yields the covered surface S z :

- → G • - → n = - µ 0 M s t γ 0 • S z (A 1 .63)
In a second step, we should note that there was a minus sign appearing in the differential form dx ∧ dz = sin θ, but this minus sign is in fact absorbed by the reversed order of ∂ -→ m/∂x and ∂ -→ m/∂z respectively in the winding number equation and in the definition of the gyrovector's second component.

III.4.c Conclusion

As a result, we can see that the geometrical interpretation of the gyrovector is the following: for a sample that is a 2D plane (or a collection of such planes), the projection of the gyrovector onto the normal vector to this plane (or sum of projections of each plane's gyrovector onto its respective normal vector) divided by µ 0 M s t/γ 0 corresponds to minus the surface covered on the unit sphere by the magnetization distribution of that sample. This interpretation still holds for samples of finite thickness as long as the magnetization distribution is uniform across that thickness because the latter is then only an additional prefactor. However, in the general case of a 3D sample with an non-uniform magnetization distribution, the geometrical interpretation no longer holds. Furthermore, it is now clear that for a 2D sample described by a non-flat surface, the geometrical interpretation does not hold. Indeed, since by construction the gyrovector is built up from both the local orientation of the sample surface and the locally covered surface on the unit sphere, the meaning of this vector (after the integration is performed) in this case can no longer be straightforward. In other terms, on the local scale of an infinitesimal surface element d 2-→ σ , there is an unambiguous correspondence between the orientation and norm of the gyrovector on one hand, and the orientation of d 2-→ σ and the covered surface on the unit sphere d 2 S on the other hand, but integrating over the whole sample mixes both the latter so that the total covered surface cannot be retrieved by a projection any more. Not even an averaged surface orientation may help recover the geometrical interpretation: the reader is invited to consider the case of equation (A 1 .59) as an example.

A 1 .3.5 From a gyrotensor to a gyrovector

The second term of equation (A 1 .47) reads:

- µ 0 M s γ 0 - → m • ∂ - → m ∂x i × ∂ - → m ∂x p • - → V (A 1 .64)
and thus can be written:

-G • - → V (A 1 .65)
where G is the gyrotensor, of (i, j)-th

component µ 0 Ms γ 0 - → m • ∂ -→ m ∂x i × ∂ -→ m ∂x j
. Let us check that this term can be rewritten --→ G × -→ V , where the vector -→ G 's i-th component reads:

G i = - 1 2 j,k ijk G jk (A 1 .66)
Denoting δ uv the usual Kronecker symbol, we may write:

- - → G × - → V = 1 2 µ 0 M s γ 0 a    i,b,j,k aib ijk - → m • ∂ - → m ∂x j × ∂ - → m ∂x k V b    - → e a = 1 2
µ 0 M s γ 0    a,i,b,j,k iba ijk - → m • ∂ - → m ∂x j × ∂ - → m ∂x k V b    - → e a = 1 2
µ 0 M s γ 0 a,b,j,k (δ jb δ ka -δ ja δ kb ) - → m • ∂ - → m ∂x j × ∂ - → m ∂x k V b - → e a = 1 2
µ 0 M s γ 0 j,k - → m • ∂ - → m ∂x j × ∂ - → m ∂x k V j - → e k - 1 2
µ 0 M s γ 0 j,k - → m • ∂ - → m ∂x j × ∂ - → m ∂x k V k - → e j = 1 2
µ 0 M s γ 0 j,k - → m • ∂ - → m ∂x j × ∂ - → m ∂x k V j - → e k + 1 2
µ 0 M s γ 0 j,k - → m • ∂ - → m ∂x k × ∂ - → m ∂x j V k - → e j = 1 2
µ 0 M s γ 0 j,k - → m • ∂ - → m ∂x j × ∂ - → m ∂x k V j - → e k + 1 2
µ 0 M s γ 0 j,k - → m • ∂ - → m ∂x j × ∂ - → m ∂x k V j - → e k = µ 0 M s γ 0 j,k - → m • ∂ - → m ∂x j × ∂ - → m ∂x k V j - → e k = µ 0 M s γ 0 i,p - → m • ∂ - → m ∂x p × ∂ - → m ∂x i V p - → e i = - µ 0 M s γ 0 i,p - → m • ∂ - → m ∂x i × ∂ - → m ∂x p V p - → e i = - µ 0 M s γ 0 - → m • ∂ - → m ∂x i × ∂ - → m ∂x p - → V = -G • - → V (A 1 .67)
Thus, the constructed vector -→ G , which will be called gyrovector from now on, acts on the velocity vector via a vector product in the same way as the gyrotensor. Let us now focus on the expression of G i the i-th component of the gyrovector:

G i = - 1 2 j,k ijk G jk (A 1 .68)
Due to the properties of the Levi-Civita tensor, this expression in fact has only one remaining term, as can be seen on the following example:

G 1 = - 1 2 j,k 1jk G jk = µ 0 M s γ 0 ˚ - 1 2 - → m • ∂ - → m ∂x 2 × ∂ - → m ∂x 3 + 1 2 - → m • ∂ - → m ∂x 3 × ∂ - → m ∂x 2 d 3 x = - µ 0 M s γ 0 ˚-→ m • ∂ - → m ∂x 2 × ∂ - → m ∂x 3 d 3 x (A 1 .69)
The other components can be deduced from cyclic permutations, so that:

- → G = - µ 0 M s γ 0 ˚     - → m • ∂ -→ m ∂x 2 × ∂ -→ m ∂x 3 - → m • ∂ -→ m ∂x 3 × ∂ -→ m ∂x 1 - → m • ∂ -→ m ∂x 1 × ∂ -→ m ∂x 2      d 3 x (A 1 .70)

INSTRUMENTAL DEVELOPMENTS FOR DOMAIN WALL MOTION IN CYLINDRICAL NANOWIRES

I Samples I.1 Brief overview of nanoparticle fabrication techniques

The methods used to fabricate nanostructures fall into two broad categories, a far from exhaustive1 overview of which will be presented in the following: the top-down and bottom-up approaches. These correspond to either starting from the bulk material and removing matter until the desired sample is obtained, or starting from scratch and progressively adding material with processes which spontaneously deliver the target shape. Typically, topdown techniques involve films of deposited material with lateral dimensions much higher than those of the target sample; these films are then patterned [1][2][3][4][5][6][7] into the desired shape. Strip structures [8], dots [9] or other patterns can readily be obtained, depending on the etching method: physical such as ion milling, chemical as in wet etching involving acids or bases. The use of optical or electron lithography allows the definition of two-dimensional patterns whose features are limited by the resolution of the lithography itself (the optical or electronic wavelength) and that of the etching process, such as Ion Beam Etching (IBE) or Focused Ion Beam (FIB). It should be noted that such methods can yield samples with a three-dimensional structure [10,11]. However, going over to samples with very high aspect ratios such as wire-like or tubular structures with low diameters (say on the order of a few hundred nanometers or below) is easier to achieve with an intermediate mould or dedicated fabrication methods.

An example of such a method with high versatility in terms of geometry is Focused Electron-Beam-Induced Deposition (FEBID) [12][13][14] or Focused Ion-Beam-Induced Deposition (FIBID). These two techniques are rather similar to three-dimensional printing and will be discussed in Chapter 3 (see section II). Along a different line, chemical methods are naturally suited: by selecting the proper chemical reaction and synthesis environment, nanoparticles with a wide variety of composition and geometry can be fabricated; a famous example being carbon nanotubes elaborated with Chemical Vapour Deposition (CVD) thanks to metal catalysts [15]. Beside these considerations, in the frame of bottom-up methods, advantage is often taken of self-assembly or self-organization processes so that large numbers of closely similar structures are produced. Generally speaking, such processes rely on the fact that at equilibrium (in statics or dynamics), the considered system reaches a configuration featuring a well-defined regular arrangement or pattern. Thus, it is possible for instance to fabricate vertical semiconducting GaAs nanowires [16] and control their structure [17] thanks to a Vapour-Liquid-Solid growth mode (involving a liquid gallium droplet) that is self-limited in the substrate plane. Another example is that of codeposition of two oxides under reductive conditions [18,19] leading to metallic nanowires inside an oxide matrix.

I.2 Nanoporous alumina membranes as moulds for elongated nanostructures I.2.a Introduction

In this work, most of the investigated samples have been prepared by colleagues of mine using moulds, namely nanoporous alumina membranes. The corresponding fabrication process will be described in the following.

The examples from the end of previous section share a common feature: the resulting elongated nanoparticles assume their shape not because their physical environment constrains them to do so, but because the physics and chemistry of their growth does (with the help of the experimental conditions). In other words, there is no mould into which material is added in order to take up all the available space and creating the mould's negative out of the desired material. This approach, however, has the advantage of allowing to check the sample's dimensions before it is actually fabricated. Along this line, porous membranes as templates for the fabrication of nanowires or nanotubes have been extensively investigated. In particular, nanoporous anodic aluminium oxide (AAO) among other oxides [20][21][22] has become a popular and versatile material [23][24][25] allowing the deposition of metallic nanowires [26,27] or nanotubes [28] via electrodeposition. Here, the self-organized feature is the hexagonal array of nanopores which are created by a process involving several steps in general. Starting from a polished aluminum foil, the sample first undergoes a mild anodization2 (in potentiostatic mode with a voltage on the order of several tens of volts) in an acid aqueous solution [26,30]. The aluminum is oxidized, and the resulting alumina is dissolved in the acid solution.

At first, there does not seem to be a reason why this process should lead to porous alumina, and even less so to ordered pores. In fact, the starting point for self-organization is an instability in the oxide layer arising from topography, provided the initial field is strong enough (irrespective of the applied voltage) [24]. In this regard, it is worth indicating that experimentally, there is a limitation in terms of applied voltage before the membrane fails because of breakdown (which then causes cracks and burns in the material); Lee and coworkers report for instance breakdown on pre-anodized aluminium membranes when the voltage was above 155 V [31]. Regions where the oxide layer (atop the metallic aluminium) is thinner lead to more focused electric field lines since they correspond to a smaller resistance between the two electrodes constituted by the electrolytic solution on one hand and the aluminium wafer on the other hand. The locally enhanced electric field favours the chemical [24] reaction of alumina dissolution at the interface with the electrolyte; as a result, the thinner the oxide layer is, the faster it is etched away. This process is self-enhancing because it creates holes or incipient pores where the electric field lines are more and more concentrated, further increasing the drive for pore formation.

It must be noted that the incipient pores feature a certain thickness of oxide at their bottom: it results from the fact that locally, the electric field becomes strong enough not only to sustain the differential etching but also to drive the migration of oxygen ions through the Al 2 O 3 layer. Thus, at the metal/oxide interface, the oxidation of aluminium can take place so that the dissolved alumina at the oxide/electrolyte interface is replaced. The incipient pores can therefore extend deeper and deeper since the dissolution and formation of alumina on both sides of the oxide layer at their bottom balance one another.

However, since our starting point was an aluminium surface with a certain roughness but no specific pattern, one ingredient is missing to explain the self-ordering of the pores. As a matter of fact, the electric-field-induced instability described above leads to an average interpore distance D int,E (this distance being measured between the centers of two neighbouring pores) of a few tens of nanometers, well below the periodicity of typically obtained arrays where D int easily reaches one or several hundreds of nanometers. Moreover, the corresponding pore ordering is poor [24], so that another mechanism must be at play. In fact, at the stage of field-assisted alumina dissolution, Jihun Oh and Carl V. Thompson showed [24] that another instability may take place and lead to the large-scale ordered array of pores, although the exact mechanism is not exactly clear. This second instability is of mechanical nature; one argument in favour of the interplay of mechanics with the pore formation is the difference in the densities of pure aluminium and of alumina: the ratio between both can range from 0.85 [32] to 1.9 [33]. Such differences lead to plastic deformations in the material, which occur with a particular wavelength D int,S . Following this new length scale, a plastic flow of the oxide layer appears at the bottom of the pore under the influence of the electric field, and balances the formation of new oxide in the steady-state.

This overall behaviour, whatever its origin, leads in summary to a first part of the anodization process associated with an incipient and thus rather poor pore ordering, and a second distinct part of the anodization process with larger typical length scale and better pore ordering. It is important to note that what is usually referred to as "second anodization" or re-anodization [34][35][36] consists in resuming anodization after removing the poorly ordered oxide layer, in order to benefit from the better pore ordering. Thus, the second anodization is not to be mistaken for the above-mentioned second regime during the anodization, which is characterized by the better ordering. Fig. 2.1 shows SEM views of AAO templates displaying this hexagonal long-range order.

The latter part originates (in the mechanical instability model) from the interaction between neighbouring pores, which is absent in a purely field-driven oxidation/dissolution mechanism. In a hand-waving picture, the created oxide must be pushed away (laterally at first) from the bottom of the pore to allow the latter to grow deeper. There are however other pores nearby, whose oxide flow will repel that of the pore we are considering. Therefore, the system finds an equilibrium when the pore distance is such that the lateral flows of oxide (under the drive of the electric field) do not block one another: if the pores are too close, then the growth is hindered because the aluminium oxide cannot be removed from the pore bottom by plastic flow. On the other hand, any "free space" will be filled because the pores on the boundary will feel an imbalance of force on their respective centers. Furthermore, this picture naturally leads to predicting a hexagonal lattice system for the nanopores array: since it is the most dense [37,38] arrangement for disks in a plane, the pores are expected to adopt this lattice system in order to optimize the pore formation process.

I.2.b From growing pores to nanoporous templates

In this regard, the possibility of self-organized nanoporous alumina as widely observed in the literature [31,34,36,39] seems easier to understand. Now, this property is usually exploited in a two-step process. The first anodization is carried out in order to yield an orderered array of pores starting from one side of the aluminium wafer: this array features however a number of poorly ordered, possibly branched pores [23,24] at the top interface with the electrolyte, from which the incipient pores were etched. Unless the branched pores are desired for further use [40], the anodization is stopped and the top alumina layer is etched away, for instance in phosphoric acid [29]. The output is thus an aluminium wafer with a regular pattern of somewhat spherical holes: the latter act then as nucleation points in the following anodization step, the "second anodization" or "re-anodization" step [34][35][36]. Provided that the experimental conditions match those of the hole pattern, the second anodization leads to pore formation following exactly the existing hexagonal array. In short, the first anodization and removal of the oxide layer result in a similar aluminium wafer as can be obtained with physical indentation. This method can be used directly on aluminium membranes to produce nanoporous AAO so as to relax both the constraints in terms of diameter but also in terms of array geometry: Masuda and coworkers [41] have achieved arrays of square or triangular (as opposed to the usual hexagonal cross-section) nanopores for instance.

Once the pore length is sufficient, the remaining bottom aluminium can be removed using a saturated HgCl 2 solution [29] so that a membrane consisting only of nanoporous AAO is obtained. However, the pores are still closed at the bottom, at this stage. The pores can be opened chemically in a diluted phosphoric acid solution; a compromise must be found in order to fully dissolve the shell at the pore bottom while preventing the rest of the membrane from being damaged. Another approach consists in using Reactive Ion Etching (RIE) [42]. After this pore opening, which is schematically represented in Fig. 2.2, a metallic layer is deposited on one side of the template to form an electrode for further electroplating. Its thickness is at least one or two pore diameters . The goal is to take advantage of the fact that the deposition method, usually electron gun evaporation, is not strictly directional and that the mean deposition direction should have some misalignment with the pores. As a result, starting from the rims of the pores, the metallic deposition should cover fully them. Typically, a seed layer of titanium is deposited first for the sake of adhesion, followed then by a thick gold layer so as to ensure a good and durable electrical contact. It must be stressed at this point that the target pore diameter D pore and average template porosity p or equivalently the interpore distance D int are not independent parameters. The porosity is defined as the proportion of pore cross-section in the total sample surface; it reads [43]:

p = π 2 √ 3 D pore D int 2 . (2.1)
As highlighted for instance in Sandrine Da Col's PhD thesis [29], the experimental conditions of electrolyte pH, composition, temperature, as well as the applied anodic voltage all influence several different processes including ionic migration, the nature and amount of ionic species in the pores (which may contaminate partly the alumina layer in the pores [23]), the plastic flow of alumina [24] as well as the formation and dissolution of alumina at the two interfaces. In turn, these dynamic processes interact with one another, and it is their complex balance that finally leads to a steady state with a given pore diameter and porosity. While there is an abundant literature [35,44,45] documenting what ranges of interpore distance and pore diameters can be obtained under which experimental conditions, it is important to keep in mind that the pore formation in AAO is not a straightforward process.

I.2.c Tailoring alumina templates along the pore length

Going beyond the process described above in order to obtain more complicated AAO templates can be achieved in several distinct ways, depending on the target array geometry. The example of branched pores, which are a problem in conventional templates, has already been mentioned [40]. Another possibility of tuning the pore geometry consists in modulated the diameter during the anodization. This was first achieved by Lee and coworkers in 2006 [31] with a change in the anodization voltage, the sample staying in the same electrolytic bath all the while. The modus operandi consists in forcing the porosity p to change while the interpore distance remains constant, thus resulting in a diameter change. It is known that the porosity of AAO decreases with the applied voltage [36,39]; thus the pore diameter D pore can be expected to decrease when the anodic voltage increases, taking Eq.(2.1) into account.

However, as was stated above, the dynamic equilibrium reached during the pore growth relies on several interconnected phenomena: therefore, changing only one experimental parameter should not allow to tune only one geometrical feature of the template [29]. Yet, in practice, the production of diameter modulation does not rely on the system reaching another equilibrium states dictated by both the initial interpore distance and the new applied voltage. The process actually involves a very steep if not step-like variation in the applied voltage. As a result, the equilibrium alumina barrier thickness at the bottom of the pore changes abruptly and thus the actual barrier thickness starts to change. When the electric field within the alumina reaches an equilibrium value, so does the barrier thickness t Al 2 O 3 . Considering how this thickness is directly linked to the interpore distance (in fact, D int = 2t Al 2 O 3 -D pore ) and the pore diameter, the latter will increase/decrease if the thickness decreases/increases so that the interpore distance remains constant. The argument for the latter's not being affected is the following: a change in the average interpore distance means that either new pores should appear or existing pores should close. Such a mechanism is hardly compatible with the creation of alumina at the pore bottom and its subsequent flow along the pore walls. On the other hand, a change in pore diameter does accommodate the change in barrier thickness while allowing the interpore distance D int to remain constant. In other words, the change of diameter follows the abrupt change in electric field as a "lesser evil". The Scanning Electron Microscope (SEM) image in Fig. 2.3, which features a multimodulated nanoporous alumina template produced by SmartMembranes GmbH R , shows an example of achievable diameter modulation pattern. There are many investigations on such diametermodulated templates reporting on quite a variety of tailored pore geometries [23,[46][47][48][49][50].

It should be noted that the production of diameter-modulated pores is also naturally possible if the electrolyte used for the anodization is changed. As was mentioned previously, the geometrical parameters of AAO depend on the composition of the solution [44], among other factors. With the anodic voltage as other degree of freedom, the (at least theoretical) possibility of finding two sets of experimental conditions with the same interpore distance but with a different equilibrium pore diameter arises. This is actually the approach chosen by Lee and coworkers [47]; the trouble is that for each diameter modulation, the electrolyte needs to be changed. With respect to the above-described technique which only relies on changes in the applied voltage, the method of Lee and coworkers is more tedious in the prospect of pores featuring many diameter modulations along their length.

In the prospect of diameter tuning, another method of widening the spectrum of template geometry consists in performing Atomic Layer Deposition (ALD). Several anodization steps can be combined with intermediate etching and ALD steps to change the diameters of the chosen pore segments. ALD allows the conformal deposition of materials (notably but not restricted to oxides) with a very fine control over the film thickness. Indeed, the film is formed atomic layer by atomic layer thanks to a process involving (for each layer) the subsequent saturation of the sample surface with reagents, which are chemisorbed on it. The excess reagents are pumped away. If the temperature conditions are suitable, then exactly one layer of material is produced in one cycle, and the saturation of the sample surface leads to a very conformal deposition. Coming back to AAO, since alumina itself is a wellknown material for ALD, this method and material are suitable for deposition on and more importantly into AAO. The principle for diameter tuning is simply to perform ALD cycles on an available alumina template of a given diameter that is too high for the later application. This method was shown [43] to allow a precise control over the pore diameter while preserving a low roughness [29].

As a matter of fact, this method is also used by the group of Julien Bachmann in Erlangen as a first step to 400 nm 400 nm Fig. 2.4: Schematized cross-sectional view of a nanoporous alumina template as produced e.g. in the group of Julien Bachmann (left) and SEM views illustrating the modulated pore geometry. On the left image, the pores are visible in lighter gray as if they were illuminated by a light source to the left of the camera, hence the darker areas corresponding to shadows. The difference in geometry between the two diameter modulations is highlighted by these shadows.

The membrane is oriented so that the first produced diameter modulation is the top one. On the SEM images, it must be noted that not all variations in apparent pore diameter reflect the geometry. Indeed, the AAO template is cut in order to obtain a side view, and the cut is not necessary flat. As a result, some pores are only partly imaged: the preparation method spreads the pore diameters and the observed values are lower bounds. SEM image credit: RAJA AFID.

produce trisegmented nanowires [50]. The two outer segments have the same or very close diameters, while the middle one is thinner. Starting from an ordered but not yet through-running pore array with a given diameter D 1 , the sample is subjected to an acid bath containing phosphoric acid, which enlarges the pore as it progressively dissolves the alumina. The first pore segment now has a diameter D 1 > D 1 In a following step, the pore growth is resumed under the conditions in which the array was created; therefore, a pore segment with diameter D 1 is bored further into the template. Once this thinner pore segment (which will ultimately become the middle pore section) has reached the target length, the template is subjected to ALD. A sheath of 5 nm SiO 2 is deposited on the pore walls, so that the diameters of both the first and second segments are reduced by the same amount. Then, the anodization is resumed. Provided that the SiO 2 shell is thin enough [50,51], a break through the very bottom of the ALD layer is possible, so that the pore growth resumes. Eventually, this latter part of the pore is enlarged (after pore opening on the corresponding membrane side) with phosphoric acid. As a result, a third segment thicker than the second one grows: tuning the electrolyte composition as well as the anodization voltage allow to tune D 2 so as to match D 1 . The overall process thus leads to an array of trisegmented pores whose two outer sections are equivalent. It must be noted however that owing to the two distinct processes involved in the diameter modulations, the latter are not symmetric: in practice, the modulation created after ALD of SiO 2 and subsequent acid-assisted breakthrough is steeper than the one obtained after resuming anodization on the acid-enlarged pores. This can be seen in Fig. 2.4, where a cross-section of such a tailored AAO template is sketched. The direction of pore growth is from top to bottom: the smoother diameter modulation is the upper one, and the more abrupt one is located at the bottom of the image.

I.3 Electrodeposition of nanowires I.3.a Reminders about electrodeposition in general

The nanowires whose investigations will be presented later on (see Chapter 4, section I) have all been fabricated by my colleagues using nanoporous alumina templates featuring diameter modulations [50]. The working electrode (anode) is typically a gold layer at the pore bottom. The reference electrode is saturated calomel electrode, while a platinum grid works as a counter-electrode. The electrochemical reaction taking place is the reduction of metallic ions in solution so as to form their corresponding metal according to:

M n+ (aq) + ne -→ M (s) . (2.2)
This is one case of reduction-oxidation reaction (or redox reaction) between an oxidizer Ox and a reducer Red, which reads in the general case:

mOx + pe - qRed. (2.3)
To this so-called half-reaction is associated a standard electrode potential U 0 Red/Ox , which is not a voltage. Its definition involves another reduction-oxidation reaction, namely that of aqueous hydrogen ions:

2H + + 2e - H 2(g) . (2.4)
This reaction, when taking place on a platinum electrode, defines the so-called standard hydrogen electrode. Its standard electrode potential U 0 H + /H 2 is defined as zero. Similarly to how a grounded electrode in electronics is not necessarily at null electric potential but must serve as a reliable reference, the standard hydrogen electrode with U 0 H + /H 2 = 0 V allows all other standard potential to be properly defined and measured 3 . Indeed, only electric potential differences are accessible in experiments, and provided that the potential of one electrode is constant (regardless of its value), the standard electrode potential of other redox couples can be measured with respect to the standard hydrogen electrode. At equilibrium, the voltage difference U between the electrode associated with the redox couple Red/Ox and the standard hydrogen electrode can be expressed as:

U = U Red/Ox -U H + /H 2 , (2.5)
where U Red/Ox and U H + /H 2 are the electrode potentials during the experiment. In the general case, U Red/Ox reads:

U Red/Ox = U 0 Red/Ox - RT pF ln a m Red a q Ox (2.6)
where a i is the chemical activity of species i, R is the is gas constant, F the Faraday constant and T the temperature.

It can be deduced from this that applying a voltage below the equilibrium potential corresponds to a decrease in the activity of the ratio a Ox a Red . In the case of the reduction of a metal M n+ according to equation (2.2), considering that the activity of a solid in solution is equal to one, we may write:

U M n+ /M = U 0 M n+ /M + RT nF ln [M n+ ] C 0 (2.7)
where C 0 = 1 mol/L is a reference concentration and [M n+ ] is the concentration of M n+ ions in solution. As a result, an applied voltage lower than the equilibrium potential leads to the deposition of the metal M.

Our discussion so far has not taken into account the pH of the solution. Indeed, considering that the activity of a gas is one, we may write for the hydrogen in the general case:

U H + /H 2 = RT 2F 2 ln [H + ] C 0 = - ln (10)RT F pH. (2.8)
In other terms, during a reaction, the standard hydrogen electrode has in fact a zero electric potential only at zero pH. However, the above discussion need only be corrected according to the right-hand side of equation (2.8). Indeed, we may write at equilibrium in the general case:

U = U 0 Red/Ox - RT pF ln a m Red a q Ox + ln (10)RT F pH. (2.9)
Effectively, the relevant electrode potential during an experiment is the standard one corrected by the activities (at equilibrium) of the redox couple Red/Ox as well as the solution pH. If the applied voltage (with respect to the standard hydrogen electrode) is lower than this, then the reduction of Ox takes place.

According to the above, several other processes apart from the reduction of M n+ may occur in an experiment if different redox pairs are present. In the (frequent but not systematic) case of an acidic bath [26,29,42,52], the reduction of hydrogen must be taken into account both as a decrease in efficiency of the electrochemical current, and as a reason for decrease in pH on a more local perspective (inside the pores). This reduction follows immediately from equation (2.8):

2H + + 2e -→ H 2 (g) .
(2.10)

It must be stressed that for the electrodeposition process, temperature [as can readily be seen from equation (2.6)] and bath composition play an important role. For instance, temperature may affect even the mechanical properties (such as tensile strength) of some depositions [53], and additives in the solution may help producing smoother surfaces: one mechanism is the transient formation of complex ions in the solution [54]. Furthermore, the drive for the reaction itself has a large impact, whether it is set and controlled as the current I between anode and cathode or as the voltage difference U . Generally, if the deposition is performed at a constant voltage U (in potentiostatic conditions), then there is a window of values in which the electrochemical process is most efficient for the target reduction reaction. In other words, the proportion of electrons that do reduce the desired metal and not protons is close to one in this range. If the absolute value of the voltage |U | is too large, then the efficiency drops as a larger and larger fraction of the current promotes not the metal reduction but that of hydrogen. This effect results from the non-linearity of the voltage-current relation during electrodeposition. In a qualitative way, the current corresponding to a given reaction varies exponentially with the applied voltage, and the threshold corresponds to the standard electrode potential. Therefore, if the applied voltage U is lowered starting from 0 V, the reduction of the oxidizers in solution is subsequently promoted in the order of increasing standard electrode potential. Indeed, the lower the standard electrode potential, the lower the threshold for the reaction, so that reductions with higher and higher standard electrode potential are activated in a sequential manner.

Let us consider the case where the deposition of a given metal is conducted at fixed current and therefore current density (in galvanostatic conditions) on the substrate J. Owing to the definition of the standard hydrogen electrode and the often negative values of the standard electrode potential of metals [55] (e.g. U 0

Co 2+ /Co = -0.28 V, U 0 Fe 2+ /Fe = -0.447 V, U 0 Ni 2+ /Ni = -0.257 V), even though the latter's concentrations in solution as well as its pH must be taken into account, the deposition voltage and current are often negative. For the sake of simplicity, we will assume this to be the case in the following discussion.

Irrespective of the above-mentioned question of efficiency of the reaction, other effects related to the quality of the deposited metal must be considered. One consequence of an increase in current density is to promote metal growth of course, but also the nucleation of new grains. Conversely, if the current density is low (in absolute value), then the present relief will correspond to local variations in electric field lines which in turn promote growth on top of the "hills" (where the lines are concentrated) and impede it at the bottom of the "valleys", and the rate of new grain growth is low. As a result, in the latter case, the deposition clearly amplifies any existing roughness, while in the second case this unstable behaviour is somewhat balanced by the nucleation of new grains.

From this point of view, it would seem that the higher |J|, the lower the deposition roughness should be. However, one should keep in mind the above-mentioned restrictions in terms of deposition voltage range. If |J| is increased, then so is |U |, which can lead to a drop in deposition efficiency and possibly the reversed effect on the grain size [56]. Moreover, it has been shown that copper deposition from sulfate baths for instance become porous and rough at increased current densities [54]. Such effects are then also function of the bath composition and especially the additives, if present; the behaviour of the deposition's quality with respect to the current density can change dramatically depending on them [54].

I.3.b Time-dependent deposition regime

With respect to the above discussion, there is yet an additional parameter on which to play in order to tailor the deposition. So far, only galvanostatic or potentiostatic deposition has been tackled, however, there is no a priori restriction to varying the drive for electrodeposition as a function of time 4 . Rather than sine waves, in most cases trains of rectangular pulses are used. In this case, for comparison with DC depositions (of the same thickness), the time-averaged current density J must be considered, not the amplitude of the pulses. In this respect, it can be shown that for a fixed average current density J = -5 × 105 A/m 2 in the case of electrodeposited palladium [57], pulsed plating with a pulse duration T c = 1 ms and a signal period τ = 5 s reduces the average grain size and therefore the film roughness. This experiment highlights what was suggested above regarding high current densities: in this case, the large (negative) current density sufficiently promotes nucleation so that film growth does not occur only at the tips of the existing relief.

The effect of short pulses is schematically to induce electrodeposition only for a short period, during which the solution in the vicinity of the film is depleted. As a result, during the large time interval τ -T c T c during which the applied current is zero, diffusion of species towards the depleted layer can take place and restore the equilibrium concentration. A contrario, if the pulse length T c is close to the signal period τ , depending on the peak current density, it may happen that the concentration of metal ions in the solution close to the interface to the deposited film goes to zero. In this case, the metal ions that are reduced later on do not come from the immediate vicinity of the film surface any more, and because of this, they feel the inhomogeneities in the distribution of electric field lines. As the latter are more concentrated around tips of grains and generally any sharp topographic feature, they promote film growth especially on its rough features. Thus, the roughness increases as the deposition goes on because of this self-sustaining process. A contrario, if enough time is left to diffusion so that the concentrations close to the film always reach a value close to the equilibrium one in the solution 5 , then the effect of the inhomogeneous field lines distribution becomes less and less important.

Furthermore, instead of simply alternating phases with null current density and with a given negative peak current density associated with deposition, one may choose to add pulses with reversed sign as another degree of freedom. While such pulses of positive current are of course detrimental to the average film growth (and thus to the deposition rate), they have two beneficial effects. In the same way that sharp, high topographic features locally enhance the electrodeposition process, they are also more prone to oxidation than their neighbouring low trenches. Therefore, similarly to how film flatness is unstable during electrodeposition with respect to roughness, the latter is reduced by a phase of current inversion. The second effect is related to the concentration of species involved in the deposition reaction near the film interface. Since the phase with positive current partially dissolves the electrodeposited material, it also tends to restoring the solution composition in the vicinity of the interface with the electrolyte.

I.3.c Alloys and crystal structure

The process so far was described in the perspective of deposition of one metal only. If the objective is an alloy of given composition, say M x M 1-x , then of course both must be present in the solution. During deposition, the respective concentrations of metals M and M will matter: if all other parameters are kept constant, the larger the concentration of M n+ ions, the more reduction events per second will occur according to equation (2.2) and so the deposition's composition changes because x increases. However, irrespective of the electrolyte condition, the condition for deposition of species i is that the applied voltage be low enough so that the reduction of its corresponding oxidizer Ox i is activated, as we have seen earlier. Therefore, if the electrolyte composition is fixed (as well as the temperature), the composition of the electrodeposited film is directly related to the chosen voltage [26,58,59]. In the case of AC deposition controlled by voltage, there is even evidence [60] for an influence of the working frequency on the material composition.

For a given application, it is important in the general case to control not only the composition of the deposited material but also the size and orientation of grains, as well as the material's crystal structure. Several factors influencing the size of grains were discussed earlier; as for their orientation and the crystal structure, it is worth mentioning that the electrolyte's pH may play a very important role. Indeed, it was shown for instance to influence the crystal structure of electrodeposited cobalt films on copper: the work of Gaigher and van der Berg [61] demonstrates the appearance of the hcp cobalt phase for an electrolyte pH ≥ 3, while only the fcc cobalt phase is present if pH ≤ 3. In the case of cobalt nanowires grown into track-etched polycarbonate membranes, the pH was shown [62] to allow control over the crystalline orientation of hcp cobalt grains: the c-axis could be set either parallel (for pH ≥ 5.8) or perpendicular (for 3.5 ≤ pH ≤ 5.3) to the wire axis.

Keeping in mind the small differences in composition for these electrolytes (all species in solution identical except for NaOH in the work of Darques and coworkers and NaCl in that of Gaigher and van der Berg 6 ), it must be noted that the same control parameter, in this case pH, did not lead to the same modification of the material. Arguably, there is no direct correspondence a priori between an electrodeposited material on a flat substrate and nominally the same material if electrodeposited into a nanoporous AAO membrane. The reason lies in the electrodeposition process, which relies on the diffusion of species. Keeping all experimental conditions constant, one can readily expect the electrodeposition of a given metal to be different whether the anode (where the deposition takes place) is flat and conductive everywhere, or covered with an insulating porous film as is the case with AAO or track-etched membranes. Indeed, in the latter case, there is an increased current density in the pores with respect to the flat electrode case. Beyond the already discussed influence of the current density, the pore geometry prevents the solution it contains to reach an equilibrium with the electrolyte outside all the more as its aspect ratio is high due to diffusion into the narrow channel. One consequence of this is that restoring the various ionic concentrations (and therefore pH) becomes more difficult because of the pore confinement. Furthermore, stirring the electrolyte as is often done for homogeneity has a smaller effect for the same reason.

As an intermediate conclusion, while there is a variety of parameters (electrolyte composition, temperature, and so on) that enable to tune the electrodeposited material in several regards, one must keep in mind that film growth on a flat conducting electrode and nanowire growth inside nanoporous templates are similar but distinct issues, even with the same target material. In the case of magnetic materials, this is all the more critical as properties such as magnetization [26,63] magnetocrystalline anisotropy [26,64,65], damping [63,66], and magnetostriction depend strongly on the composition [67] and microstructure [67,68].

I.3.d NiCo and NiFe nanowires

The nanowires investigated in this thesis have all been produced by electrodeposition into modulated nanoporous AAO templates. As was mentioned previously, there are two different processes involved in the creation of pores with a thin segment in-between two segments of larger diameter (see description of trisegmented nanowires produced in Erlangen, section I.2.c These modulations of the pore diameter result in two outer thick sections of the same diameter D thick and a middle section of diameter D thin < D thick . The purpose of the diameter modulations is to act as energy barriers for domain walls; as was suggested by the constant domain wall energy per cross-section surface from the 1D domain wall model discussed in Chapter 1, section III.1, the domain wall energy increases with the wire diameter. This trend has been demonstrated by micromagnetic simulations performed by JAMET and coworkers [69].

Moreover, if a domain wall is located in the thin section, it is easy to show that the diameter modulations feature magnetostatic charges whose sign is the same as that of the domain wall (the magnetostatic charge associated to a domain wall was introduced in Chapter 1, section III). As a result, there exists a repulsive interaction between these magnetostatic charges, thus reinforcing the role of diameter modulations. This is illustrated on Figure 2.5, where a trisegmented nanowire featuring a domain wall is schematically represented. If one first assumes that magnetization is uniform along the wire axis except in the domain wall, then there are magnetostatic surface charges (as described in Chapter 1, section I.1.c ) at the diameter modulations, where the projection of magnetization onto the local normal to the wire surface is non-zero. These charges are highlighted here in red and the sign associated to them, as is the domain wall sitting in the thin wire section. Quite clearly, one can see that reversing magnetization so that -→ M → --→ M , all charges change in sign and the conclusion remains the same: magnetostatics tends to repel domain walls from the diameter modulations. In a second step, one may lift the assumption of uniform longitudinal magnetization and allows magnetization to relax towards equilibrium. In this context, close to the wire surface in the modulations, magnetization is expected to tilt so as to better follow the wire surface under the influence of the strong local stray field.

In terms of compositions, two materials have been used: The segment length was chosen to be about 10 µm for all three wire sections. As for the thicker sections, the purpose was to decouple the diameter modulations from the wire tips, where domain wall nucleation takes chloride itself. place and where magnetization is no longer uniform. The length of the thin section also plays a decoupling role, this time between both diameter modulation, but most importantly it defines the "containment box" thanks to magnetostatics [70] within which domain walls can be pushed back and forth. In the perspective of experiments with pulses of field with, say, 10 ns duration or less, the thin section length must be sufficient to allow the domain wall motion to not be impeded by the diameter modulations. In other words, once the domain wall is set into motion, the maximum length it should travel must be less than the thin segment length so that the motion stops only because the field pulse has ended and not because the diameter modulation stop the domain wall. Considering how numerical simulations suggest that the Bloch point wall may reach velocities on the order of a kilometer per second under only a few millitesla [71,72], the distance δz travelled during the pulse can be evaluated to be δz ∼ 1000 × 10 • 10 -9 = 10 µm. It is this order of magnitude that prompted the choice of the thin segment length.

I.4 Recovery of nanowires from their template

Once the nanowires have been deposited inside the AAO, the latter has served its purpose as mould and must be removed so that individual structures can be addressed. There arises the need for a chemically selective method that can destroy the alumina matrix while preserving the nanowires. Considering the volume of template to be eliminated, wet etching presents itself as a viable route since it also allows chemical selectivity to some extent. To be more precise: for a given etchant and constant experimental conditions, dissolution rates of different materials can vary over several orders of magnitude. As a result, by choosing a solution which etches alumina away at a much higher rate than the nanowire material, the damage dealt to the samples can be reduced. Furthermore, it is possible to use ALD before the electrodeposition of wires to create a protective sheath around them, thus further enhancing the capability of the etching process to remove the matrix while preserving the nanowires' integrity. For example, Sebastian Bochmann and Julien Bachmann coat the membrane's pores with a thin SiO 2 layer before electrodeposition. When the template with nanowires is immersed in a sodium hydroxide solution, the alumina is readily etched away but the much higher resilience of silicon oxide to the etchant creates a time window in which the AAO itself is completely removed, while the SiO 2 ALD layer still covers the nanowires 7 . Experiments at Institut Néel [29] involved ALD of hafnium oxide HfO 2 , but the protective sheath appeared, upon XPS investigation at synchrotron Elettra [29], to neither remain over the whole nanostructures nor prevent the surface oxidation of iron in Permalloy nanowires.

The nanowires investigated in this work were all freed from their alumina membrane using etching in sodium hydroxyde, following by several rinsing steps in distilled water. Their role is to recover neutral pH as well as removing as much of the template's remains as possible. In a later step, the wires in solution are rinsed with isopropanol. The latter does not promote oxidation and has the advantage of being both available with a high purity and highly volatile. The latter quality is important for the next step, which consists in evaporating a droplet of nanowires-containing isopropanol solution on a silicon wafer. A support is needed, and as will be further explained in Chapter 2, section II.2, it is necessary for it to be non-insulating for the sake of conducting PhotoEmission Electron Microscopy (PEEM) experiments. Thus, regular clean room wafers are cut into square pieces with slightly less than a centimeter side for use as substrates for nanowire dispersion.

In order to easily locate and identify specific nanostructures for later investigation, as well as for the sake of navigation across the sample (e.g. with SEM), the Si wafers have received a pattern of alignment marks before the cutting. These marks were prepared using laser lithography with a positive resist (i.e. whatever insulated part of the resist film is removed upon development) and a pattern I developed. The starting point was an existing mask featuring a square unit cell repeated all over the mask. The only variation concerned a label in the way of a letter (e.g. "A") followed by a numeral index (e.g. "9"), the former indicating the column of the cell and the latter its row. The upper border of the unit cell was a dash-dotted line, and its content consisted in two sets of four bars each, both sets in a spiralling pattern. The sets came in two sizes: the larger one meant for easy and quick spotting, the smaller one meant for e.g. finer microscope alignment and finer positioning, as shown in Fig. 2.6. One important feature of the mask was that in practice only the horizontal bars indicated a direction to the naked eye. On the right hand side of Fig. 2.6, a realization of the gold mark pattern with a dispersion of nanowires is shown. The coloured areas are developped i.e. free of resist after the lithography, hence they correspond to the regions eventually covered with metal. Right: portion of an optical map of a silicon wafer with this mark pattern and a dispersion of nanowires. The latter appear as black rods on the image owing to light scattering. As can be seen from the sharp transition in the middle of the image, the map is constituted by stitching together frames with some overlap between one another I have implemented several changes to the mask in the course of several synchrotron beam times. Let us review the requirements for the "ideal" alignment marks pattern in our context of nanowire investigation:

Density: There must be high enough a density of marks so that precise and unambiguous positioning of the nanostructures can readily be achieved. However, the density mustn't be too high in order to not be in the way of the investigation, that is to say having many marks under nanowires. Even when using small field of views (e.g. 15 µm to 30 µm in PEEM), the spacing between marks must be reasonably small so that no time is wasted on wandering in-between at random.

Orientation:

The pattern must allow for unambiguous determination of the field of view's orientation with respect to the sample. Furthermore, there must be a way to easily orient the sample with respect to, say, a sample holder with a fixed direction in which a magnetic field can be applied. Indeed, in this example, the latter must be aligned with the investigated structures. This means in turn that if their orientation with respect to the mark pattern is known, then there must be significantly large marks to spot with the naked eye and to assist in putting the sample in the proper direction.

Navigation support: The alignment marks must provide clear and easily identifiable directions towards different types of neighbouring marks.

It can be easily seen in Fig. 2.6 that the spacing between neighbouring marks is typically over 100 µm in the original pattern. That is ample space for reducing the risk of a large proportion of nanowires lying on top of marks, but some areas are too empty for the sake of navigation. Indeed, a representative field of view of 30 µm as used in PEEM for navigation (see the red circle in Fig. 2.6) is quite small with respect to some empty spaces between marks e.g. above D5 in Fig. 2.6. As a result, navigation in the absence of features between marks can become quite difficult owing to the lack of landmarks. Empirically, it has been observed during use of the original design that the presence of more marks, with further indications of direction would be useful for navigation. Some empty spaces could still be filled with one more mark, so there was a need for new, distinguishable symbols. These are shown in Fig. 2.7. They indicate several new directions, have original shapes and furthermore lack inversion symmetry, so that spotting one of them allows for orientation of the field of view with respect to e.g. an optical map of the sample. The reason behind the need for broken symmetry is that PEEM images are horizontally mirrored and rotated (around the normal) with respect to the optical mapping images. Therefore, symbols without inversion symmetry allow an easy and reliable re-orientation of optical maps during experiments. Such maps, as suggested in Fig. 2.6, are composed of many overlapping photographs of the sample that are then stitched together so as to form an image of the whole substrate.

Furthermore, these new marks fill areas which were devoid of marks before, so as to ease navigation. On a larger scale, an important addition consisted of brackets at each corner, with their two sides of a different length. They allow one to distinguish two orthogonal in-plane directions x and y up to a 180 • rotation. That is why I added a blank space in the middle of two bracket arms on the same side: such a feature breaks the symmetry between that side and its opposite, thus allowing an unambiguous sample orientation.

Coming back to the marks that can be seen in Fig. 2.6, it can readily be seen that some of them may be difficult to tell apart, such as the 8's and the 6's in this case, or the 5's and the 9's. This is due to overlapping segments in their pattern. In addition, the rescaling of the alignment pattern as a whole may cause some features to go below the resolution of the laser lithography, that is to say ∼1 µm to 2 µm. Typically, small segments or holes in non-connected shapes may not come out and impede the identification of a mark such as an 8. Because of this, I have made anew many marks in order to avoid too small details. Aside from myself, the latest versions of this mask have been used by Michal Staňo and Beatrix Trapp, both in PEEM and AFM-MFM.

The actual marks are constituted of a bilayer of gold on titanium formed with electron gun evaporation on the patterned resist. Titanium merely serves as seed layer, while gold provides a durable top layer (with respect to oxidation) featuring a nice optical contrast with respect to silicon.

I.5 Sample demagnetization to increase the likelihood of domain wall presence I.5.a Introduction and context

In the perspective of observing and manipulating domain walls, it is necessary to prepare as many nanowires as possible with domain walls. It has been shown [43] that soft nanowires with bends can be reproducibly brought (at remanence) into a multi-domain state after the application of an in-plane field. The latter's strength was above the saturation field so that in a first step, the wire is in-plane magnetized. The direction of saturation is that of the normal to the wire in the plane, in the middle of the bends i.e. where the wire's curvature is maximal. Thanks to this direction, magnetization falls back to the tangent to the wire with opposite directions on the sides of the bend. As a result, a domain wall must be nucleated in the bend. This process has the advantage of reproducibility and tolerance with respect to the field orientation, provided that the applied field H a = || -→ H a || is larger than all the switching fields (in a Stoner-Wohlfarth approach [73]) of the different wire sections. Indeed, in this case, even if saturation is not reached, magnetization everywhere has a positive projection onto the applied field. In fact, that is the requirement for domain wall nucleation. When the field goes to zero, magnetization aligns with the tangent to the wire whose direction is closest: since that tangent changes direction across the bend, so does magnetization. The switching fields may be estimated by considering the distinct wire sections as Stoner-Wohlfarth particles with an effective anisotropy arising from dipolar interactions, and approximating them as infinite cylinders. The estimation then takes advantage of the N ⊥ = 1/2 demagnetizing factor of the cylinder.

This method however relies on bent nanostructures, which constrains its application to either a full dispersion of nanowires in a random direction (in the hope that several wires will have had bends in the proper direction) or to a limited number of well-identified objects. In the latter case, the field is aligned along the in-plane normal to the wires in the bends for optimal efficiency. Even though this strategy has proved itself [43], it has the disadvantage that domain walls are expected (see e.g. Chapter 1 appendix section A 1 .2) to have a lower energy when inside a bend compared to a straight wire section. Furthermore, that consideration takes into account neither the possibility of defects being created in the nanowire because of its bending, nor local changes in energy due to magnetostriction (if that is relevant for the considered material). Therefore, while domain wall nucleation is made easier thanks to the saturation process, there are several reasons to expect domain walls to be located in the bends with a significant pinning strength, which is a hindrance to the investigation of their dynamics.

Domain wall pinning [4,6,71,74,75] and the interaction between domain wall configurations and pinning sites [5,7] has been the focus of many numerical and experimental investigations [76][77][78], but it is complex phenomenon with many possible causes, among them structure roughness [71], material composition [79] and defects [6]. In experiments, there is a landscape of potential energy for the domain wall. It can be seen as a function U p (x) in a one-dimensional case where the domain wall position 8 along the conduit is x. This would correspond for instance to very thin nanowires. The case of wide strips is more complex because of the change of dimensionality: instead of a point-like object in very thin wires or a three-dimensional object [69] with a reasonably well-defined center [72,[80][81][82], domain walls are in this case better described by a line separating two two-dimensional regions of the sample [3,[83][84][85]. In the following, we will restrict ourselves to cases where a one-dimensional frame is relevant and follow a description inspired notably by the book by Ralph Skomski [86], based on the Becker-Kondorski model [87,88].

As can be deduced from the above, the potential energy U p (x) can be expected to feature many local extrema along the sample. Fig. 2.8 illustrates what such an energy landscape may look like. Here, only two minima at respective positions x 1 and x 2 are represented, each with its depth and width along the domain wall conduit. The latter is assumed to have a constant cross-section S as well as a uniform spontaneous magnetization M 0 . When a field H a is applied in the direction that corresponds to pushing the domain wall to the right, a force 2µ 0 M 0 SH a -→ e x [86] is exerted on the latter. Therefore, a linear function -2µ 0 M 0 SH a x is added to the energy landscape, whose minima are affected depending on their characteristics. Of course, this can be generalized to a section S(x) depending on the abscissa. From a mathematical point of view, if the maximum of ∂U p /∂x inside an energy well 9 is larger than 2µ 0 M 0 SH a , then the minimum associated to the energy well remains, albeit at a shifted position.

Conversely, if the above-mentioned maximum is lower than 2µ 0 M 0 SH a , then the minimum vanishes. Coming back to physics, the first case corresponds to a field below the depinning field H depin that can be associated to the considered energy well, while the second case the applied field has overcome the energy barrier and the domain wall is depinned.

I.5.b Working principle

8 The notion of domain wall position is not necessarily straightforward because of the domain wall width. In other words, the spatial extent of a domain wall along the conduit direction forbids to treat it as a point-like object with a unambiguous position. One possibility to lift this ambiguity consists in making use of the components of magnetization averaged over the conduit cross-section: their extrema (or those of their spatial derivatives) may indicate a reasonable domain wall position. Another possibility consists in viewing the domain wall as a spatial distribution of magnetostatic charges, whose barycenter is then the domain wall position itself. That approach is relevant for MFM, which is sensitive to these charges but not (directly) to the components of magnetization. 9 The "inside" of such a well is the segment between the two local maxima of |∂Up/∂x| that are the furthest apart.

Energy

x x 1 x 2 x 2 U p (x) U p (x) -(2µ 0 M 0 SH a )x

DW trajectory

Fig. 2.8: Schematic view of an energy landscape U p (x) for a domain wall along a one-dimensional conduit of cross-section S and spontaneous magnetization M 0 , based on the Becker-Kondorski model [87,88]. The positions x 1 and x 2 (blue curve) correspond to local minima of U p . Under the application of a field H a , the first miminum vanishes due to the slope -2µ 0 M 0 SH a corresponding to the force exerted on the domain wall [86]. The second one on the other hand still exists at the shifter position x 2 (orange curve). As a result, a domain wall starting at position x = 0 will eventually stop at x 2 , following the purple dashed trajectory.

Waveform generator

Output BNC cable to Kepco Stage Fig. 2.9: Waveform generator used to control the Kepco power supply.

The stage consists in two large electromagnets capable of producing a vertical induction reaching at least 1.4 T. The maximum induction µ 0 H M = B M depends on the choice of pole pieces as well as the drive current I d . The latter is provided by a Kepco power supply that can be controlled using the waveform generator; both instruments being located in the rack next to the stage. Depending on the needs, the Kepco can be either voltageor current-driven.

In order to perform an out-of-plane demagnetization, the current produced by the Kepco should be oscillating (with null mean value) at a given frequency f c while the amplitude or envelope of the oscillation slowly decreases. Starting from B M , it should decrease over a time scale T τ c = 1/f c so that the difference δB between one local field maximum and the next is as small as possible. Indeed, the demagnetization process will be all the more efficient as the number of steps between saturation at B M and zero field is large. Fig. 2.10 illustrates what a field profile as a function of time should look like.

I.5.c Operation

Effectively, the Kepco is used as an amplifier of the signal provided by the waveform generator. Thus, this output waveform must be tailored according to the desired time scheme of the demagnetization: the amplitude B M will then be set by several distinct elements:

• the amplitude chosen on the waveform generator itself,

• the position of the sample with respect to the pole pieces,

• the choice of control of the Kepco by current or by voltage. For the demagnetizations prior to my experiments, the typical maximum inductions obtained right atop a pole piece (as measured with a magnetometer) reached B max 1.4 T. The samples inside a plastic box were usually placed on the bottom pole piece for a few minutes while the field was being cycled with an amplitude B M so that the electronics warms up and reaches a stationary temperature state, or at least gets closer to the latter. The chosen frequency was on the order of f c 250 mHz. This choice is motivated by the inductive character of the stage: the system with two electromagnets has a cut-off frequency above which the maximum output current amplitude will be significantly lower than the Kepco's capability. Thus, this low frequency allows by default the full range of accessible inductions.

Electromagnets Pole pieces

Then, the setpoint in terms of current amplitude was chosen so as to be just above the threshold I lim at which the stage's maximum induction B max is reached. Above that point, the pole pieces are magnetically saturated, so that the applied field only increases with the Oersted field produced by the coils, but no longer with the stray field from the pieces. Therefore, the gain in applied field per unit current drops drastically beyond I lim , and can be considered negligible. Because of this, feeding in a current slightly above I lim before starting the actual demagnetization ensures to benefit from all of the stage's capability.

The demagnetization itself must be programmed according to a certain envelope, as was mentioned before. To put it differently, the carrier sinusoidal current fed to the electromagnets is modulated by a signal s(t) whose amplitude monotonously decreases from a maximum value s max to zero. Furthermore, the signal must remain strictly thereafter so that the sample can be removed without experiencing any other magnetic field. This constraint excludes exponential decays for instance; as a result, a sawtooth signal was used, the shape of which is illustrated in Fig. 2.12. s(t) decays to zero over the set time scale T , which in my case was usually 3 days. That time scale was recommended on the basis of experience gathered on the demagnetization of sensitive nanomagnetic systems: in-plane artificial artificial spin ices. In order to efficiently demagnetize such nanomagnets systems [89], slow or rather short-step demagnetization is required; it appeared that for a fixed carrier frequency f c , a time interval T of a couple of days (rather than one day or less) s(t) T t s max ∆T Fig. 2.12: Schematical representation of the sawtooth signal used to perform the demagnetization. yielded better results. Considering a zero-signal period ∆T of a couple of hours, a cycle of modulation is carried out with a frequency on the order of 1 µHz.

Considering that T f c = T /τ c 1, one can show that up to first order in 1/(T f c ) 10 -3 , the extrema and zeros of the applied field can be computed while the linearly decaying envelope is neglected. Based on the above, we may thus estimate δB as B(t) -B(t + 1/f c ) = B M τ c /T . This allows us to claim that the demagnetized systems feel field increments on the order of a millitesla.

Producing this signal with the waveform generator was done using a program developed by YANN PERRIN from Institut Néel and the apparatus' amplitude modulation.

II XMCD-PEEM and shadow imaging II.1 X-ray Magnetic Circular Dichroism

Dichroism, that is to say a difference in light absorption by a medium depending on the light's polarization state, results from symmetry breaking. In the case of samples with finite magnetization, the broken symmetry is rotation around the axis along which magnetization points: if light propagates along this axis with, say, circular right polarization, then propagation with a wave vector parallel to magnetization is not equivalent to propagation with a wave vector anti-parallel to magnetization [90]. The wave equation's eigenvectors as admissible polarization states are circularly polarized waves [90] with distinct refraction indices [91]. The magneto-optical Faraday and Kerr effects originate from this circular birefringence [92].

This interaction between solid-state magnetism and a propagating light wave is a consequence of the material's electrons' response [90,92]. While it is of broad use in the visible spectrum [93], such a behaviour is still found at higher and higher photon energies [92]. Due to the Kramers-Kronig relations [94], the presence of non-zero birefringence implies the existence of dichroism. When going over to wavelengths in the soft X-ray range, i.e. a few nanometres, it is no longer relevant to describe these phenomena in terms of in terms of permeability or permittivity tensors [92], but rather in terms of photons scattered by atoms [94,95]. The wavelengths are no longer very large with respect to the lattice spacings and only a quantum description of matter in the presence of light can explain all the features of scattering in this energy range. However, this need is even more severe when one considers the usual, resonant configuration. As stated above, circular dichroism exists at all energies but is usually very weak [94] except around resonances.

Generally speaking, the treatment of resonant scattering predicts a non-vanishing contribution to the linear absorption coefficient of X-rays µ that is odd in magnetization along the propagation direction, provided that the beam is circularly polarized [95]. This absorption coefficient is such that along the propagation direction z, the beam's intensity I obeys:

dI dz = -µ • I (2.11)
To put it differently, considering a unit propagation wave vector -→ e and the reduced magnetization -→ m, the linear absorption coefficient reads at resonance [95,96]:

µ = µ 0 + δµP( - → e • - → m) , (2.12)
where P is the degree of circular polarization, e.g. +1 for circular right polarization, -1 for circular left polarization, and 0 for linear polarization 10 . In the following, only fully circularly polarized beams will be considered, unless otherwise specified. µ 0 is the polarization-independent contribution to µ, and δµ is the resonant magnetic contribution to X-ray absorption. The key element in Eq.(2.12) is that for fixed incidence, µ( -

→ m, P)-µ(-- → m, P) = µ( - → m, P)-µ( - → m, -P) =∝ - → e • - → m
, in other words, the difference between the linear absorption coefficient for fixed magnetization and opposite circular polarizations or for fixed circular polarization and opposite magnetization is directly proportional to the magnetization component along the beam.

In order to evaluate dimensionless numbers, it is customary practice to compute asymmetry ratios or contrasts [97][98][99]. For the sake of simplicity, I will refer in the following to XMCD contrast C as the difference between intensities I L and I R collected with circular left and right polarizations, normalized by the sum of these intensities:

C = I L -I R I L + I R . (2.13)
As an example, if a beam traverses a thickness t of material and magnetization is parallel with the wave vector, the contrast reads:

C = exp [-(µ 0 -δµ)t] -exp [-(µ 0 + δµ)t] exp [-(µ 0 -δµ)t] + exp [-(µ 0 + δµ)t] = tanh(δµ • t). (2.14)
At this stage, we have only touched how resonant X-ray scattering can be taken advantage of to obtain a projection of magnetization. It is now time to tackle more specifically the case of the materials of interest in this work, namely ferromagnetic transition metals and their alloys. The idea is to provide a less general argument than the above for the existence of XMCD but describe the phenomenon in more detail, for a specific case. A further focus consists in considering only specific resonances, called L 2 and L 3 absorption edges [97,100]. These correspond respectively to electronic transitions from the 2p 1/2 or 2p 3/2 band to levels in the 3d band, at the Fermi level [94].

In the following, the most important features of these transitions will be reviewed in order to give a simple picture in which the sensitivity to magnetization associated to the L 3 and L 2 edges can be grasped. The principle is to narrow down the possibilities of transition events from the core 2p 1/2 or 2p 3/2 levels to the 3d states at the Fermi level with the corresponding quantum mechanical constraints, then examine the role of circular polarization.

As was mentioned earlier, these transitions cannot be accurately described outside a quantum mechanical frame since the involved electrons change quantum numbers. The tool to examine these transitions is Fermi's golden rule, and the light-matter hamiltonian [97]. During the transitions, the first counter-intuitive element is that the magnetic induction associated to the X-ray light does not play a role [97]. The reason for this is that the interaction term involving the induction depends on spin but not space/spatial variations, and the initial and final states are orthogonal. Then, the only remaining part of the interaction hamiltonian is treated in the dipole approximation, i.e. constant electric field over the atomic shell [96], which allows the corresponding matrix element involved in Fermi's golden rule to be rewritten with the electric dipole operator [97].

Since the light-matter interaction driving the transition does not involve spin, a first selection rule arises, namely ∆m s = 0, where m s is the secondary spin quantum number. In other words: the L 2 and L 3 transitions conserve the electronic spin. The next step consists in developing the dipole operator: since we know that only circular polarizations are relevant for XMCD, the electric field for light propagating along a cartesian z-axis is of the form:

-

→ E = E 0 ( - → e x ± i - → e y ), leading to an electric dipole operator pel. ∝ x ± iŷ ∝ ∓r • Y ±1 1 [97]
, where Y M L is the spherical harmonics from quantum mechanics.

The transition can only occur if the matrix element involving this operator as well as the initial and final state is non-zero. Since here in Y M L , L = +1 and M = ±1, we arrive at the second and third selection rules: ∆l = +1 and ∆m l = ±1, where l is the azimuthal quantum number and m l the magnetic quantum number. ∆l = +1 for a 2p→3d transition, while the change in magnetic quantum number reflects the transfer of photon orbital momentum to the electron. The curious reader may consult the publication by Humblet [101] or lecture notes by Wysin [102] about the photon's angular momentum. The key aspect is here that circularly left (right) polarized photons carry an angular momentum + (-) along the propagation direction [96,102], where I take left circular polarization to mean an electric field

- → E = E 0 ( - → e x + i - → e y ) propagating along + - → e z .
This convention, used by François Baudelet in Ref. [97], is opposite to that of e.g. Stöhr [96]. In the following, circular left and circular right polarized light will be denoted CL respectively CR. Now, what remains to be done is first to decompose the final 3d and initial 2p 3/2 or 2p 1/2 states, originally eigenstates of the total angular momentum operators Ĵ2 and Ĵz , in an eigenstate basis for the orbital and spin angular momentum operators. This allows the matrix elements for all cases (in terms of initial state and light polarization state) to be computed. For a full treatment, the reader is referred to the book chapter by François Baudelet [97]. The outcome are the L 3 and L 2 edges' probabilities

P L 3 ,L ↑ , P L 3 ,L ↓ and P L 2 ,L ↑ , P L 2 ,L
↓ of excitation of a electron with up spin or down spin to the Fermi level with CL polarization. They read:

P L 3 ,L ↑ = 0.625 P L 2 ,L ↑ = 0.25 P L 3 ,L ↓ = 0.375 P L 2 ,L ↓ = 0.75
The corresponding probabilities for CR polarizations are simply obtained by swapping the numerical values for spin up and spin down [97]. As a result and to give a example, let us consider a 3d ferromagnet with magnetization along the beam. This situation means down spin due to the quenching of orbital momentum and negative gyromagnetic ratio [103]. At this stage, the presence of magnetization is of course crucial; otherwise, an asymmetry in excitation probability would be washed away by the equality in densities of empty states (for up and down spins) at the Fermi level. If soft X-rays with an energy corresponding to the metal's L 3 edge are shone first with CL then CR polarizations, then the measured XMCD contrast defined in Eq.(2.13) will be

C ∝ P L 3 ,L ↓ -P L 3 ,L ↑ = -0.25.
To put it differently, in this picture, magnetization along the beam means that CL polarized light is less absorbed than CR polarized light: if the photon intensities after a certain sample thickness can be recovered, then an excess of CL polarized photons will be measured with respect to CR polarized photons. To the contrary, if the measurement is proportional to the photon absorption, for instance with photoemission, then the measurement associated to CL polarization will be weaker than the one associated to CR polarization, as shown by the minus sign in C ∝ -0.25.

Thus ends this first description of XMCD at the L 2 and L 3 edges of ferromagnetic 3d transition metals. Now that the important theoretical elements have been introduced, let us try and review them up in a less formal and more hand-waving way. As a reminder: the goal is an asymmetry ratio of the form of Eq.(2.13), calculated from the total transition rates measured with CL then with CR X-rays. It must be kept in mind that both polarizations excited both up and down spins.

When the energy of circularly polarized soft X-rays is increased, electrons more and more strongly bound to a nucleus may be excited to the Fermi level. When the photon energy matches the interval from the 2p 3/2 then the 2p 1/2 bands, a band resonance is observed [96] i.e. the transition probabilities are high. These are constrained by angular momentum conservation: since a photon with angular momentum ± along the propagation direction is absorbed while spin is conserved, selection rules for the three quantum numbers l, m l and m s arise. For a given circular polarization state, say CL, at the L 3 edge, it is more probable to excite a spin-up electron [97] than a spin-down electron because the former, when excited by light, "resemble more" the available final 3d states, qualitatively speaking. However, in order to calculate an asymmetry of absorption for instance, we must weigh these probabilities for CL light by the corresponding spin-dependent densities of states at the Fermi level [97] and sum them to obtain the total transition rate. This is required by Fermi's golden rule. Then, we must switch from CL to CR polarization, thus swap the computed probabilities before weighing, and calculate the normalized difference between transition rates as in Eq.(2.13). If the densities of state were equal, then each spin would have the same transition rates with CL or CR X-rays, so that the asymmetry would be zero from cancellation by pairs. Since in a 3d ferromagnet, there are more down spins than up spins for upwards magnetization [103], the weighing amounts to multiplication by a positive constant.

This picture of XMCD would be incomplete without an additional consideration about the initial states in the 2p bands. Due to the spin-orbit-coupling-induced splitting into a triplet (2p 3/2 ) and a singulet (2p 1/2 ) band, the considered transitions at the L 2 and L 3 edges involve electrons initially with orbital momentum aligned respectively antiparallel to the (conserved) spin. This is actually a crucial aspect. Indeed, in the absence of splitting, the L 2 and L 3 edges would merge; at first sight, this is not very worrisome, however, one can show [97] that they have opposite XMCD. In fact, one can go one step further [97] and show that the magnitudes of XMCD weighed by the electron populations of the triplet and singulet state are equal in absolute value but opposite in sign. This means that spin-orbit coupling is a further vital ingredient for non-zero XMCD.

Let us have a look at a concrete example to illustrate this. Fig. 2.13 shows a schematic representation of a CL X-ray beam (in orange) with a photon energy ω = E L 3 corresponding to the L 3 edge of a 3d ferromagnet. Both 2p bands are represented with their spin and orbital momenta s and l, but only electrons from the triplet 2p 3/2 band are excited to the Fermi level at electron energy E F . The final states are in the 3d bands (also shown with their spin); here, the bands close to the Fermi level correspond to those of iron, as calculated with Density Function Theory (see [104] for an introduction) by Zahn and Mertig [105]. The s, p, and d bands are shaded in black, light gray and dark gray respectively. The corresponding densities of states (calculated with Density Functional Theory [104]) in dark gray were taken from [105]; the light gray and dark shaded areas correspond to the densities of state associated to p and s electrons.

E F E hω = E L 3 E L 3 2p 1/2 2p 3/2 s l 1 eV ρ ← ρ →
Along the electron energy axis E, the Fermi level at E F is indicated with a vertical line. The asymmetry in density of empty states at that energy is illustrated with the down and up spin densities ρ ← and ρ → in green.

According to the above, up spins are predominantly excited with respect to down spins, as schematically illustrated by the different thicknesses of the turquoise arrows representing the transitions. Due to the selection rules, there are only transitions to d bands, whose available densities of states ρ ← and ρ → are highlighted in green. In this case, with magnetization along the X-ray beam, the total transition rate with CL polarization is moderate because the predominantly excited electrons have few available states. If we now switch to CR polarization, the predominantly excited electrons are promoted to levels with a high amount ρ ← of free states, so that the transition rate is higher. As a result, the asymmetry ratio between the transition rates calculated following Eq.(2.13) leads to a negative number, as was argued on p.74. Now that the physics underlying XMCD at the L 2 and L 3 edges of 3d ferromagnetic metals is (hopefully) clearer, the focus should move on to the issue of measuring this effect. So far, we have only mentioned the measurement of "intensities" or of transition rates without further precision. However, as was pointed out on p.74, there is a crucial difference between measuring a photon intensity after traversing a certain sample thickness on one hand, and measuring a signal reflecting transition rates or absorption on the other hand. Indeed, for the same contrast convention and the same sample and beam configuration, these two methods lead to opposite contrasts. It turns out that this is exactly the difference between shadow or transmission XMCD-PEEM 11 as pioneered by Judith Kimling [107] and conventional XMCD-PEEM. This will be described in more detail in the next section, after an introduction to PEEM as an experimental technique. Image taken from Ref. [96].

II.2 PhotoEmission Electron Microscopy

As the name implies, PEEM is an electron microscopy involving photoemission, that is to say the excitation of electrons by light. This being said, the electrons that are collected to form an image are secondary electrons rather than the photoelectrons directly created by photon absorption [98]. The PEEM's working principle is illustrated in Fig. 2.14, adapted from Ref. [96], here in the case of an X-ray beam as probe. The electrons leaving the (flat) sample are accelerated by the high voltage difference (values from 12 kV to 20 kV are used across synchrotrons Soleil, Alba, and Elettra) between the sample and the objective lens. Further in the electron microscope's column, an aperture allows a selection in electron kinetic energy, rejecting higher-energy electrons which are not properly focused [96]. It is possible to also implement a selection in electron emission angle via another aperture ("contrast aperture") so as to reduce image aberrations. In the end, the magnified image is projected onto a screen and recorded. The corresponding fields of views are on the order of a few micrometres to a few tens of micrometres.

The light source can be in the ultra-violet or X-ray range; PEEM using synchrotron radiation has been implemented in the early 1990's [108,109]. A strong interest in PEEM has always been its surface sensitivity [108,110], arising from the low escape depth of electrons. The electronic mean free path in ferromagnetic transition metals is on the order of 1 nm to 3 nm [98], while the photon mean free path inside such materials in the soft X-ray range is above ca. 10 nm [111,112]. As a result, while photons can penetrate rather far into matter, a very large number of collisions would be necessary for photoelectrons created so deep to emerge.

The detailed processes following photon absorption and leading ultimately to secondary electrons leaving the studied sample will not be thoroughly reviewed here; the reader is invited to consult the book chapter by Schneider [98] for more insight on the matter. Schematically, in a first step, the core hole resulting from the excitation of an electron to the Fermi level is filled. This decay process leads to a large gain in energy, which is transferred to Auger electrons. The latter propagate in the solid, undergoing inelastic scattering and thus creating a so-called secondary electron cascade with a broad kinetic energy spectrum [98]. The minimum kinetic energy required to exit the sample is given by its work function, tuned by the start voltage (STV). This is a bias of typically a few volts applied to the sample in addition to the microscope's high voltage [113], and has a strong impact on imaging [99]. Obviously, the sample substrates in PEEM must be conducting for the sake of imaging and proper application of the STV. Another limitation concernes the lateral size of samples: in practice, less than a square centimetre surface is available to imaging, and the sample substrates can have sides of at most ca. one centimetre. Further details will be given in section III.1.

It must be stated that at this stage, the electron collection at a given point on the sample surface strongly depends on topography [98]. This results from the concentration of electrostatic field lines around sharp geometrical features or to put it differently, the influence of the sample's topography on the electric potential. It has been predicted and observed experimentally [98] that non-flat samples may display sizeable contrast that solely results from the surface topography. more detailed review, to the article by Bonetti [106].

As for the image acquisition, since the X-ray beam is focused on a spot that is larger than the field of view, a full-field image is obtained without any scanning. In fact, live imaging can be readily achieved [96], though in practice image integration times ranging from several seconds to several minutes are used to improve the signalto-noise ratio. For a long integration time, the latter is usually divided into segments of a few seconds each, the reason being mechanical drifts. Since navigation over the sample is performed by lateral sample holder shifting, slow drifts resulting from mechanical relaxations cause the field of view to move "slightly" even after all manual shifts are over. Depending on the conditions, notably the delay between mechanical stop and image acquisition start, such drifts can reach several micrometres. Therefore, if a single image is acquired over too long a time scale, the integrated drift is sizeable and can no longer be corrected. That is why the subdivision of a (sometimes required) long integration time into intervals short with respect to the drift time scales allows the acquisition of a single image to be almost unperturbed by drift. Ultimately, the acquired snapshots must of course be drift-corrected before averaging is performed.

Owing to the time structure of X-ray beams in many synchrotrons , X-ray PEEM can be performed at much smaller image acquisition times, down to 100 ps to 50 ps [114]. This is of particular importance for studies of dynamics in micromagnetism, since the relevant time scales may reach 10 ns to 0.1 ns [115][116][117][118] and below [119][120][121], depending on the system and experiment. The archetypal experiments taking advantage of the time-resolved XMCD-PEEM capabilities of synchrotron light sources are vortex core motions in patterned thin film elements under pulsed or alternating stimuli [114]. These are not restricted to magnetic fields or electric currents [74], as was recently illustrated by Michael Foerster in his investigation of magnetization dynamics in Ni dots induced by acoustic waves [122].

Overall, the majority of experimental works combining XMCD and PEEM have focussed on thin films and patterned structures thereof [98,114], be it square/rounded dots or stripes. This is in line with the surface sensitivity of PEEM and the two-dimensional nature of such systems; no missing information is hidden in the sample's volume.

This is qualitatively very different in sufficiently thick cylindrical systems, as has been illustrated e.g. in Chapter 1, section III.2. The existence of three-dimensional magnetization patterns calls for a technique with the capability to probe magnetization not only on the sample surface but also in the volume.

II.3 Shadow XMCD-PEEM imaging

It is useful to remember at this point that even at X-ray energies corresponding to the L 2 and L 3 edges of 3d ferromagnetic metals, the photon mean free path 1/µ is at least ca. 10 nm, as was stated on p. 76. This means that a resonantly absorbed X-ray beam with initial intensity I 0 traversing a sample thickness t of several tens of nanometres still possesses a non-vanishing number of photons. The corresponding intensity I out carries information about magnetization encountered along the line of sight parametrized by a coordinate u according to:

I out (P) = I 0 exp - ˆt u=0 µ 0 + δµP( - → e • - → m) du , (2.16)
where P is once more the degree of circular polarization, as defined from Eq.(2.12) on p.73. µ 0 and δµ are also defined as in Eq.(2.12). If we are able to measure I out for CL and CR polarizations, then the XMCD contrast obtained after X-ray beam transmission (following the convention from Eq.(2.13)) reads:

C = I out (-1) -I out (+1) I out (-1) + I out (+1) = exp δµ ˆt u=0 ( - → e • - → m)du -exp -δµ ˆt u=0 ( - → e • - → m)du exp δµ ˆt u=0 ( - → e • - → m)du + exp -δµ ˆt u=0 ( - → e • - → m)du = tanh δµ ˆt u=0 ( - → e • - → m)du (2.17)
As a result, the XMCD contrast in such a transmission experiment has the same sign as the integral in the hyperbolic tangent's argument in Eq.(2.17). Here, if we set -→ m -→ e , we retrieve the conclusion from Eq.(2.14) p.74 that the contrast is positive. In that discussion, we had mentioned that if the magnetic contrast measurement reflects the absorption of X-rays and not their outgoing intensity, the contrast sign is opposite. Such an experiment corresponds to an electron yield measurement as described e.g. by Stöhr [96]. If X-ray PEEM is used to image the magnetic sample, then as we have seen earlier, the signal corresponds to secondary electrons emitted after excitations at most a couple nanometres from the surface. As a result, the number of photoelectrons is proportional to -dI ph /dz at the surface, i.e. µ • I [96,99]. In this case, the XMCD contrast arising from the sample surface reads:

C = µ 0 -δµ( - → e • - → m) -µ 0 -δµ( - → e • - → m) µ 0 -δµ( - → e • - → m) + µ 0 + δµ( - → e • - → m) = - δµ µ 0 ( - → e • - → m) . (2.18)
The minus sign in Eq.(2.18) illustrates the above-made point. This being said, it must be kept in mind that the information provided by the transmission and electron yield methods are fundamentally different. Indeed, Eq.(2.17) shows a contrast depending on the integrated component of magnetization along the beam, whereas the contrast in Eq.(2.18) corresponds to a local measurement of magnetization along the beam. "Local" means here that only ferromagnetic material on the surface and from a lateral area defined by the instrumental resolution [99] contributes.

To put it differently, an electron yield measurement of XMCD is limited to the sample surface but only probes magnetization at this point, whereas a transmission measurement probes the sample volume at the cost of a mathematical surjection. This is to say that a given contrast value at one point can correspond to a number of different micromagnetic configurations along the line of sight. experiment. The sample is a bimodulated cylindrical nanowire on a silicon wafer (dark gray), and lit by an X-ray beam with a 16 • incidence angle. Only a thin beam sheet is shown in bluegreen for the sake of clarity; the propagation direction, perpendicular to the wire axis, is indicated in red. The electron intensity collected from the different areas is illustrated by the orange arrows with different thicknesses. The photoemission from the substrate is largely outmatched by that on the sample's surface (therefore represented bright) owing to resonant X-ray absorption. Since the beam's incidence is low, a wide shadow is created behind the sample, similarly to everyday-life shadows at sunset. In the present case, the wire's thin section is high enough to allow its full shadow to be visible. The number of electrons in this area is all the weaker as X-ray absorption in the wire is strong.

Si wafer

With the above in mind, it is time to tackle the question: how to retrieve the photon intensities after passage through the sample? While this is readily achieved by the experimental setup itself in STXM [106], a simple solution in XMCD-PEEM consists in using the substrate on which the investigated three-dimensional sample is located. Therein lies the power of the experimental approach pioneered by Judith Kimling et al. [107]. In this work, coreshell nanowires consisting of a cylindrical nickel rod inside a SiO 2 /Fe 3 O 4 /SiO 2 trilayered shell were investigated with XMCD-PEEM after dispersion on a silicon wafer. Thanks to the incidence angle of 16 • between the beam's wave vector and the horizontal [107], the authors gained access not only to direct/electron yield XMCD-PEEM imaging of their samples, but also to a wide shadow. This shadow's origin is very simple: due to resonant X-ray absorption, the photon intensity after traversing the nanostructures is strongly reduced. Moreover, its width is easily understood by comparing the experiment with looking at, say, a tree's shadow at sunset. At that time of day, it is the sun rays' low incidence angle that geometrically leads to a shadow that can extend much further than the tree's height. Coming back to PEEM: a 16 • incidence implies that along the beam, all distances in the shadow are inflated by a factor of about 3.6 [99]. Thus, the actual resolution along the vertical direction "inside" the sample is enhanced by the same factor [99]. Since the electron intensity from the shadow reflects the photon intensity after passage through the sample, it is clear from the previous paragraphs that the shadow is a source of volume information on magnetization.

To summarize the above, Fig. 2.15 shows a schematical view of such a dual XMCD-PEEM experiment in which both the 3D sample surface and its shadow are of interest. Hereafter, such an experiment will be referred to as transmission or shadow XMCD-PEEM. In this case, the sample is a cylindrical nanowire with two diameter modulations, resembling the structures presented in section I.2.c. It is represented in light gray to account for the resonant photoemission, and lies on a bare silicon wafer. An X-ray beam is shone on the sample, with its wave vector indicated in red. For the sake of clarity, only a thin vertical beam sheet is represented in blue-green. The beam incidence is perpendicular to the wire axis. When the sample is reached, a strong photoemission of electrons (represented in orange) is triggered on the sample surface. Conversely, the beam is progressively absorbed as it traverses the cylindrical wire; for the sake of clarity, the latter is here fully opaque. In reality, there is still a (weak) number of photons after passage through the sample, and the correspondingly low photoemission in the shadow with respect to outside the shadow reflects the resonant X-ray absorption. Here, the height of the thin wire section is sufficient to allow its full shadow to be well-separated from the wire itself, as is sometimes the case in experiments (see e.g. Chapter 4, section II).

Since the seminal work by Kimling and coworkers, a number of shadow XMCD-PEEM experimental investigations of three-dimensional samples have reported [48,99,[124][125][126][127][128][129][130][131]. Still, one must keep in mind the fact that the 2D shadow image obtained from a sample's volume pertains to only one component of magnetization, and that the information results from a surjection or projection, thus, not the interpretation is not always straightforward. Yet, beyond the simple picture presented above (notably with Eqs.(2.18),(2.17)), it has been shown by DA COL, JAMET and coworkers [99,125] that micromagnetic simulations [132] combined with numerical computations of XMCD-PEEM contrast enable not only to confirm physical intuition on simple cases such as uniform magnetization or curling configurations [99] but also to predict fine features in imaging and even to identify domain wall types in soft cylindrical nanowires without ambiguity [125]. Indeed, the TVW and BPW that have been presented in Chapter 1, section III present starkly different XMCD contrast patterns, notably in the wire shadow, as shown in Fig. 2.16:

II.4 Review

The goal of section II was to introduce first the physics being X-ray Magnetic Circular Dichroism, then PhotoEmission Electron Microscopy as primarily surface-sensitive microscopy technique, and finally the shadow XMCD-PEEM approach and its marked added value in the investigation of three-dimensional samples.

As we have seen, XMCD is in principle present at any photon energy but most easily measured at electronic resonances corresponding to X-ray absorption edges. That is why in practice XMCD-based imaging is always performed at such an edge; in this work, the L 3 and L 2 edges of the relevant 3d ferromagnetic metals Fe, Ni and Co will be relevant. These edges correspond to electrons being excited from core 2p 3/2 respectively 2p 1/2 states to empty 3d states at the Fermi level. It turns out that the associated transition rates depend on the circular polarization state of the X-ray beam, which means non-zero XMCD, provided that the sample's magnetization has a projection onto the beam. The physical explanation is based on the following necessary elements:

• band structure close to the Fermi level affected by magnetization,

• spin-orbit-coupling-induced splitting of the 2p band,

• selection rules for the electronic transitions,

• varying overlaps between excited wave functions and the final states in the 3d band.

Retrieving the XMCD contrast is commonly achieved by switching the circular polarization state of X-rays. Leaving aside techniques other than PEEM, there are two main ways of measuring an asymmetry ratio or contrast representative of XMCD. Transmission or shadow-type measurements use a contrast based on photon intensities with CL and CR polarizations obtained after passage of the beam through the sample. In shadow XMCD-PEEM, the "screen" on which the attenuated beam's intensity is measured is the substrate's surface on which the sample of interest lies. While they intrinsically probe magnetization in the volume, these measurements are surjective (or projective) by nature because of integration along the photon path in matter. Hence, except in some simple cases, numerical simulations of the shadow XMCD-PEEM contrast are often needed for the interpretation. By contrast, electron-yield-type measurements use PEEM in a more conventional way since only the sample surface is probed, and not the volume. However, the interpretation is usually much more straightforward 12 than in the shadow.

In both cases, the outcome is a certain photoemission intensity collected with PEEM so as to form an image of the sample surface. Thus, both the sample's direct image and its shadow's image are acquired at the same time, at least in principle. The instrumental configuration of PEEM microscopes leads to a small incidence angle and hence to a wide shadow for 3D samples. Since the typical fields of view range from a few micrometres to about 10 µm to 20 µm, shadow XMCD-PEEM is a technique of choice for the investigation of micromagnetic configurations in cylindrical nanowires with a length on the order of a (couple) dozen micrometres. Experiments on this type of system has brought experimental evidence for the two domain wall types whose existence had been predicted in such soft ferromagnetic systems [125], and demonstrated sufficient spatial resolution to reveal the internal domain wall configuration. It is in this context that my work concerning domain wall motion in cylindrical nanowires has begun.

III Quasistatic fields for synchrotron experiments III.1 Current supply and sample holder concept

The first natural step towards influencing the micromagnetic configurations in nanostructures during synchrotron investigation consists in being able to apply a constant magnetic field. The orientations of interest are all the directions in the plane of the substrate on top of which the nanowires are located: indeed, moving domain walls in our system implies a field parallel to the structures. Similarly, nucleating domain walls from saturated nanowires is most efficiently achieved with a field along the wires. Arguably, being able to apply fields in the perpendicular direction may be useful in the prospect of sample demagnetization as described in the previous section. However, since the amplitude of magnetic induction required to perform an actual demagnetization on cylindrical systems is µ 0 M s /2, where µ 0 M s is the spontaneous magnetization of the investigated material. As opposed to domain wall depinning or even nucleation inductions that are typically in the range of a few millitesla to a few tens of millitesla, the saturation induction of Permalloy for instance is about 1 T. Even with the factor two which arises from the infinite cylinder's demagnetizing factor, magnetic inductions of such a magnitude are far from easy to produce; these are rather in the range of electromagnets. Such limitations are even more tangible in the constrained environment of a PEEM microscope.

Considering this, there is a need for a sample holder which allows the application of in-plane inductions up to, say, 100 mT so as to have a safety margin (in particular in cases where the investigated nanowires are not well aligned. Furthermore, it must match the requirements of PEEM, which are compatibility with ultra-high vacuum, and more importantly the size and shape which accommodate the PEEM cartridge.

The cartridge holds the sample holder per se thanks to a rounded depression and a threaded through-hole thanks to which a screw pushes a plate from the other side. The round basal plate has a diameter of 14 mm, and the sample holder's bulk is a 7×11×8 mm 3 parallelepiped (width, length, height). It is on top of the latter that the sample is placed. What really holds the latter still is the so-called cap. This is a round metallic plate featuring a gap in its center, as can be seen in Fig. 2.17. The latter defines the area which may be observed during the experiment, as the relative position of the cap and the sample does not change. Indeed, the X-Rays involved in X-Ray Magnetic Circular Dichroism (XMCD)-PEEM only hit the sample if they go through this gap. In fact, the latter also plays a role in the microscope's optics 13 . Furthermore, this gap ensures a proper grounding of the sample to the so-called Start Voltage or STV through direct contact of four pins, which are indicated with black dots in Fig. 2.17. The STV can be considered as the ground for any electronic connected to the sample itself; this voltage was introduced in section II.2. Although it is not the earth ground for the whole experimental set-up, it must be stressed that in a first introduction, the STV is better thought of as the reference potential for all electronics that has a role at the level of the sample but does not contribute to the PEEM imaging.

Coming back to the application of fields, the solution I chose consists in mounting a horizontal solenoid in the sample holder's bulk, through which it passes thanks to traversing holes. This design has the advantage of simplicity and total absence of remanence when no current is fed to the fieldproducing coil; furthermore, it is straightforward to ensure good spatial homogeneity of the field over the observable sample area by making the coil much larger than the latter.

The in-plane field direction is imposed by that of the solenoid, but its amplitude is directly proportional to the current fed to the coil. Therein lies the biggest weakness of the solenoid-based design: (very) large currents are required to produce enough field. This weakness is enhanced by our situation in which the solenoid is below the sample; in other words, the sample lies outside the solenoid, where the field is significantly weaker compared to the interior. Still, the use of a horizontal solenoid allows to take advantage of a pick-up loop positioned right next to the field coil's end so as to detect and inductively measure the applied field. A photograph of the sample holder fabricated at Institut Néel is featured in Fig. 2.19, as well as a schematics of its working principle.

The sample holder was machined out of PEEK, which is a plastic material compatible with ultra-high vacuum.

U loop (t) Furthermore, it features a high mechanical stiffness and resilience, a low absorption of water into the volume 14 , and a strong resistance to a wide range of chemicals [START_REF] Rao | Polyether Ketones[END_REF][START_REF] Kemmish | Update on the Technology and Applications of Polyaryletherketones, ISmithers Update Series[END_REF]. I made the design with the constraints of PEEM in mind. Holes were bored into the sample holder so that I could add and glue the field coil in red just below the level of the sample. The efficiency of the applied current is thus optimal 15 .

I pulse (t)
The source of current was a pulse generator developed by MÁRLIO BONFIM and ANDRÉ DIAS during their work at Institut Néel. Since the aim is to provide large currents, the design relies on the discharge of capacitors when the apparatus is triggered and the charged-up capacitors are plugged into the circuit featuring the field coil. As a matter of fact, two capacitors discharge simultaneously. Overall, the pulse has a shape resembling an asymmetrically-stretched first lobe of a sine function, see Fig. 2

.21.

A schematic representation of part of the generator is given in Fig. 2.20. The top part provides an output voltage of V cc =12 V that is low-pass filtered from an input 15 V (provided by an external power supply) that is also lowpass filtered. In both cases, the filtering is achieved at different frequency ranges by a pair of capacitors. The clean V cc voltage is rescaled with a voltage divider bridge which is a manually-handled potentiometer, corresponding to the amplitude-setting knob on the generator. The high-voltage converter (in blue) is powered by the controlled rescaled voltage: it serves as the base voltage for the transistor T, whose emitter electrode is held at any voltage between 0.6 V and 12 V. The high-voltage converter's output is protected by a resistor in case of a short-circuit. The output current flows through a LED whose role is to indicate that the capacitors are still loading up. The two diodes to the bottom left and bottom right of Fig. 2.20 are protections against potential transient negative 16 voltages applied to the circuit, which could lead to backwards-travelling currents and thus damage to either the voltage regulator or the high-voltage converter. The output current exits this part of the circuit to load the generator's capacitors, which are not shown in this diagram, until their voltage reaches the target U capa . Triggering is done manually, but the actual switch that closes the capacitors onto the field-producing coil is built with transistors.

It is not straightforward to estimate the typical current that can be delivered because it depends on the selfinductance of the field-producing coil and all resistances in the circuit, which are quantities I have not measured. However, it is quite enlightening to know that the capacitors can be charged up to 900 V, 1000 V being an absolute (and less safe) maximum. In this configuration, inductions up to ca. 300 mT could be delivered with the sample

High-voltage converter +15 V GND Diode IN OUT VCC=12 V GND GND GND GND GND GND GND MC78M12BDT Voltage regulator +15 V GND Diode IN OUT VCC=12 V GND GND GND GND GND GND GND

U capa

Fig. 2.20: Schematics of the circuits in MÁRLIO BONFIM's pulse generator up to the capacitors whose discharge creates a pulse of current. The top half represents the voltage regulator unit, with capacitors to low-pass filter the input 15 V voltage (from an external power supply) and the output V cc =12 V voltage that is fed to the lower half of the diagram. The latter includes the manually-handled potential behind the knob on the generator that allows to control the pulses' amplitude. The setup with the transistor T yields a controlled voltage and amplified current for the input of the high-voltage converter. It is the latter's output (potentially high) voltage U capa that is supplied to the system of capacitors. During load-up, the current passes through a LED; when the process is finished, the LED is off, which helps ascerting when the generator is armed. holder of Fig. 2.19, at sample position. Details on these estimates will be given later on in section III.2 and section III.3.

III.2 Inductive characterization of the coil

In order to estimate the fields applied to a given sample during experiments, I have first carried out an inductive characterization of the field coil. The principle was to use a small loop as magnetic field sensor, taking advantage of the link between the measurable induced electromotive force and applied field. The requirements were twofold: i) small loop size so as to make a rather local field measurement ii) distortions to the signal as low as possible.

The measurement solenoid can be seen above the sample holder in Fig. 2.21: it consists in only one loop wound around a glass disk, with a diameter d l of about 1 mm. The wiring away from the loop itself, with a total diameter of 0.25 mm, is braided so that only the loop is sensitive to applied fields 17 . Also, if the field varies linearly or close to linearly (in the active surface of the loop) as a function of altitude, then the field estimate based on the magnetic flux through the coil accurately corresponds to the field at the center of the loop. Due to its small size and number of windings, the loop works as a non-invasive 18 , reliable detector.

It must be stated that the typical substrate thickness being 0.3 mm, the center of the pick-up loop is offset with respect to the height of the substrate's top surface. Therefore, the following field estimates are to be taken as lower bounds on the field H pulse (t) actually applied to the samples. This field is obtained after numerical integration of the electromotive force induced in the pick-up loop when the pulse generator is triggered, and subsequent division by the loop's surface. A trace of the flux Φ p.u. (t) through the pick-up loop is shown in Fig. 2. 21.

In order to evaluate the error on the estimated field due to this height offset, I have carried out measurements with the pick-up loop at different altitudes z above the sample holder's top surface, z = 0 corresponding to the loop in contact with that surface. Precise displacements were achieved thanks to a micrometric positioning stage. The results are presented in Fig. 2.22: as expected, the field H M (z) decreases with height. Taking an overall linear trend, and extending it into the z < 0 region, we may estimate the field H (est) pulse that would have been determined at a lower loop position. Since the thickness of the loop's wiring is 0.25 mm, taking the substrate thickness into account, an estimate H (est) pulse can be derived from extrapolation at minus the loop's inner radius λ pick-up =0.5 mm. Based on this, the field H (est) pulse at the top surface of the substrate is estimated to be about higher than the determined H M (z = 0) by a factor 19 1.2. In terms of behaviour as a function of altitude, the fact that H M (z) is close to linear 17 As long as the applied field does not vary too fast in space, contributions to the magnetic flux through the consecutive loops formed by the braided wires cancel out. 18 While mutual inductance effects on the pick-up loop are what allow the measurement, mutual inductance effects on the field coil can be safely neglected. Indeed, one can compare the orders of magnitude of the mutual inductance between the two solenoids, and of the self-inductance of the field coil. The first one should be multiplied [START_REF] Paul | Inductance: loop and partial[END_REF] by a factor (N pick-up /N f.c. ) λ pick-up /λ f.c. , where N pick-up and N f.c. are the number of windings in the loop resp. the field coil, while λ pick-up and λ f.c. are the radii of the turns in the loop resp. the field coil. In our case, N pick-up = 1, N f.c. = 10, and λ pick-up =0.5 mm, λ f.c. ≥3.1 mm (rectangular winding with 3.1 mm width). Thus, the ratio of orders of magnitude is (N pick-up /N f.c. ) λ pick-up /λ f.c. 0.04. 19 In this approach, the deviations from linear increase with decreasing height that are expected as one draws closer to the field coil's wiring are neglected. The linear trend extracted from the data in Fig. 2.22 yields a relative increase of 13% at z = -λ pick-up . Hence, considering the above-mentioned non-linearity and spread in measurements, assuming a relative increase of 0.2 seems reasonable. supports the interpretation of the extracted field as the one right through the loop center. The second step in the calibration process relied on measurements of both the electromotive force in the small pick-up loop and that in the measurement coil to the side of the sample holder. During a pulse, both solenoids are subjected to the magnetic induction produced by the field coil. Since they have different shapes and spatial positions, they do not experience the same electromotive forces, but these must be proportional to one another because the induction has the same temporal profile for both solenoids 20 . Therefore, since the applied field can be deduced from measurements with the small pick-up loop, the magnetic flux through the measurement loop can be mapped onto the applied field. Indeed, the maximum field applied during a pulse is determined from the extremum value Φ M,p.u. of Φ pick-up (t); based on the previous argument, Φ M,p.u. is proportional to the extremum flux Φ m.c,M through the measurement coil. Although this extremum can be determined from the flux Φ m.c. (t) through the measurement coil after numerical integration, it is easier to work on the measured electromotive force U m.c. (t).

Since we know that the pulse is created by the discharge of capacitors, its temporal shape is amplitudeindependent. Therefore, there is a non-ambiguous relation between the extremal values of U m.c. (t) on one hand, and Φ M,m.c. on the other hand. In other words, one only needs to record these extremal values to estimate Φ M,m.c. and through the latter, Φ M,p.u. and therefore the maximum applied field. Fig. 2.23 displays an example trace of U m.c. (t) (in red), as well as its numerically calculated integral, -Φ loop (t). Seeing that the rise time of U m.c. (t) is below 1 µs, we have an a posteriori confirmation that the measurement coil creates negligible signal distortions. Since the surface of its single loop is much larger than that of the pick-up loop, and since the measurement coil's resistance was found lower than that of the pick-up loop, we can deduce that the measurement coil's response time is larger than that of the pick-up loop. This supports our confidence in the latter as field sensor for this application.

As can be seen to the right of Fig. 2.23, there is a spike in the electromotive force U m.c. (t). Correspondingly, the flux Φ loop (t) does not go to zero even several microseconds after the pulse has ended. This is indicative of a non-zero magnetic field going through the measurement coil; in other words, an almost steady current flows through the field coil. According to DANIEL LEPOITTEVIN from Institut Néel, such a behaviour can be explained by the mechanism that closes the circuit and allows the loaded capacitors to discharge into the field coil. This mechanism, activated by the mechanical trigger of the pulse generator, consists in power transistors that build up an effective switch. This switch is meant to open the circuit after the pulse is over, but if the components have aged and/or the circuit is opened significantly later than the pulse's end, then some current may flow from the capacitors into the field coil.

Moreover, the several sharp changes of slope that can be seen in the electromotive force U m.c. (t) are an indication that these transistors may have a certain spread in their trigger voltage. This would explain these few well-separated events: this spread causes the transistors to trigger with jitter, thus resulting in temporal offsets.

Coming back to the calibration, after these preliminary measurements, I have first checked the assumption that the recorded extremal fluxes Φ M,m.c. and Φ M,p.u. are indeed proportional. To this end, I have made several The trend is quite nicely linear as expected. Taking the error bars (calculated based on relative uncertainty estimates) into account, the best affine fit to this set of data yields an intercept Φ M,p.u. (0) on the ordinate axis such that Φ M,p.u. (0) =6 ± 2 nV•s, one order of magnitude less than the first data point. Similarly, the intercept Φ M,m.c. (0) is one order of magnitude below the first data point (in absolute value). It is to note that in this fit, the origin point was not included a priori, which supports the fit's result and thus the experimental approach.

According to the affine fit, Φ M,p.u. = κΦ M,m.c. , with κ = (1.48 ± 0.03) • 10 -2 .

With the field values at hand as well as the corresponding control parameter U capa , the load capacitor voltage in the pulse generator, the calibration per se is complete. It can be summarized by the graph in Fig. 2.25, where the estimated inductions µ 0 H M at the sample level are plotted (in red) as a function of U capa . The error bars on the field are the same as before. There are none displayed for the capacitor voltage because the latter was carefully measured with a voltmeter with a precision better than a volt. The best affine fit yields a zero-voltage offset smaller than its own uncertainty of 3 mT, and a slope k ind. =0.58 ± 0.01 mT/V. This being said, while the capacitor voltage itself can be controlled with a rather good accuracy, it is not an in-situ observable. One may well set a fixed value of U capa whatever the circuit on which the loaded capacitors are plugged, which means that the calibration in Fig. 2.25 is a priori only valid in the calibration environment, which is not that of the synchrotron experiment. The most obvious change are the several meters of microscope cable, which add a sizeable inductance to that of the field coil. Therein lies the advantage of having built in the measurement coil to the side of the sample holder.

The above calibration allows a one-to-one correspondence between the field applied at sample level and the extremal flux Φ M,m.c. through the measurement coil. This quantity, however, requires a post-processing step, namely the numerical integration of the induced electromotive force in the measurement coil. In order to avoid such post-processing, and to work only on the directly accessible voltage trace, I turned to the extremal values V 1 and V 2 (see Fig. 2.23) of U m.c. (t). Using several measurements of these voltages (for different load capacitor voltages) and comparing them to Φ M,m.c. , I was able to deduce a conversion factor from one quantity to the other, and thus to the applied field thanks to the above. For instance, I found the relation between maximum applied induction µ 0 H M and first extremal voltage V 1 to be:

µ 0 H M = V 1 ξ V 1 (2.19)
where ξ -1 V 1 =65 ± 5 mT/mV. In the end, it is the knowledge of this conversion factor that is required in experiments (t) (black) through the measurement coil and pick-up loop respectively. These were obtained by numerical integration of their corresponding induced electromotive forces. The load capacitor voltages were respectively 552 volt and 333 V. Discarding the very end of the pulses according to the previous discussion about artefacts in the electromotive forces, the differences in shape are insignificant.

to check what field was actually applied to the sample. Again, the only assumption is that the normalized pulse shape is independent of the load capacitor voltage U capa . This assumption holds quite well, as illustrated in Fig. 2.26. It features two traces of recorded magnetic fluxes, one for U capa =333 V, the other for U capa =552 V. The former was acquired with the small pick-up loop (black curve), the latter with the measurement coil (blue curve). The abscissa and ordinate axes are respectively chosen so that the beginning and maximum of both pulses coincide. Neglecting the very end of the pulses, which is affected by the above-mentioned artefacts, it is clear that the two pulse shapes match quite nicely up to a small difference in duration. Stretching the trace horizontally by a factor 1.05 (dashed red curve) leads to an even better match between the fluxes.

It appears that the first extremal voltage V 1 is more reliable in the determination of the applied field, since the difference between the two traces are smaller in the first half of the pulse. Furthermore, since |V 1 | > |V 2 |, and since both voltages are determined with a similar level of noise (see Fig. 2.23), the choice of V 1 also increases slightly the signal-to-noise ratio.

III.3 Magneto-optical characterization of the coil III.3.a Introduction

The calibration from the previous subsection relied on the knowledge of the effective surface in the pick-up loop. Albeit thin, the varnish layer of its wiring as well as the wiring itself contribute to a rather significant uncertainty on the area that must be considered in the flux-to-field conversion. Throughout subsection III.2, it was assumed that the effective surface was known with a negligible error. It was thus easier to focus on the other sources of error in the field estimates. However, the interval within which lies the true effective diameter of the pick-up loop is not so small. Indeed, a safe lower bound is given by the apparent diameter d l of 1 mm, while the higher bound would be d l plus the total thickness of the wiring (corresponding to contracting all the copper wire to an infinitely thin loop), leading to 1.25 mm. The relative difference between the corresponding surfaces (the smaller one considered as reference) is 56%. In other words, the calculated field values (with their error bars) are an upper bound that may be up to 56% higher than the actually applied field. With respect to the other sources of measurement error, this uncertainty on the surface is clearly the largest. In addition to this systematic error, there is the issue of not being able to measure the field exactly at the position where the samples are.

In order to increase the calibration accuracy, I sought another field measurement protocol that would not rely so crucially on a difficult-to-estimate effective loop surface. Since technologies such as commercial Hall probes were excluded for their lack of bandwidth 21 , the simplest approach relied on magneto-optics. More precisely, on the time-resolved measurement of a certain physical quantity that is directly proportional to the applied field via a magneto-optical effect.

Magneto-optics as a field emerged in the second half of the XIX th century [91], with the discovery of interactions between polarized visible light and ferromagnetic or paramagnetic materials. In a hand-waving manner, such interplays are expected since the electrons in a material that are responsible for the latter's magnetism, are charged particles. Light, being a time-dependent electromagnetic field, disturbs the reference state of these electrons so that magnetism is affected, in certain conditions to a measurable or even sizeable extent. Conversely, the propagation of light is influenced through the material's optical constants, which then depend on its magnetization state. One may gather the magneto-optical effects that are linear 22 in magnetization inside the material's dielectric tensor ε [93]:

ε =   ε 1 0 0 0 ε 2 0 0 0 ε 3   + Q V   0 -im 3 im 2 im 3 0 -im 1 -im 2 im 1 0   . (2.20)
Here, complex notation has been adopted for simplicity. The first term corresponds to the possibly anisotropic dielectric tensor of a non-magnetic material 23 , expressed in the basis of the eigenvalues ε 1 , ε 2 and ε 3 . In this frame, the reduced magnetization vector -

→ m = -→ M /|| -→ M || has components m 1 , m 2 and m 3 .
The second term expresses the light-magnetization interaction, whose strength is given by the material's Voigt constant Q V . Since we are dealing with plane waves that must be eigenvectors of ¯ for propagation [START_REF] Jackson | Classical electrodynamics[END_REF], the factor i implies quarter phase shifts between components of the electric field. In the end, linear polarizations are no longer eigenvectors in the magnetic material, but circularly polarized waves are. More details on the corresponding algebra can be found in the book edited by S. Sugano and N. Kojima [START_REF] Sugano | Magneto-Optics[END_REF], p.140. This is most easily seen for propagation along direction 3, with magnetization also along this direction, since ¯ then reads:

ε =   ε 1 -iQ V 0 iQ V ε 2 0 0 0 ε 3   . (2.21)
Since the propagation velocity of circularly left resp. right polarized waves are derived from ε, it follows that such electromagnetic fields travel through the magnetic material at different velocities. Thus, phase shifts between circularly polarized fields appear during propagation; this is a case of (circular) birefringence. Such shifts are interesting since they very clearly affect linearly polarized waves. Indeed, one can write a linearly polarized wave as the sum of two opposite circularly polarized waves. Applying a phase shift to only one of the circular polarizations results in a rotation of the polarization vector.

Considering that the phase shift arises from a difference in wave velocity, it is clear that it is proportional to the distance travelled inside the material. Quite naturally, one is then lead to seek to increase the travelled path, but an increased propagation distance also means a decreased wave intensity in the (frequent) case of non-transparent materials. Ferromagnetic metals such as nickel are good examples where use of an arbitrary long propagation path is forbidden by the absorption of light. Nevertheless, such samples are known to rotate the polarization of the reflected wave by a certain angle. In this case, the involved magneto-optical effect is called Kerr effect or MOKE. By contrast, the rotation of polarization experienced by a light wave that has passed through a magnetic material is said to result from the Faraday effect, which refers thus to a transmission geometry.

As a summary to the above, the symmetry breaking induced by magnetization in a magnetic material results in a rotation of the linear polarization of incident light. One might be reminded of the so-called optical activity of liquid solution containing an unbalanced mixture of the two enantiomers of a given chiral molecule. Similarly to this instance in chemistry, we are dealing with a measurable effect (the polarization rotation) that is proportional to the system's asymmetry. In our case, this asymmetry is linear in the components of magnetization 24 . Now, in cases where magnetization itself is proportional to the externally applied field, this linear magneto-optical effect allows allows for field measurements. Such a situation is by definition that of a paramagnet 25 [103]. We can therefore expect a paramagnet with significant Voigt constant to be an interesting candidate for magnetic field measurements with magneto-optics, as is the goal of section III.3. In practice, the magneto-optical efficiency of a transparent material is not presented in terms of Voigt constant but of Verdet constant V, which is defined by the ratio of the total polarization rotation angle α to the travelled distance inside the material L and the relevant 26induction J Far = µ 0 M [103]:

V = α J Far L . (2.22)
It turns out that this intrinsic quantity, which depends on the wavelength (as an optical property) and temperature, varies over (up to) four orders of magnitude depending on the material [START_REF] Weber | Handbook of Optical Materials[END_REF]. Within the family of garnets alone, a large difference was found between the Verdet constants of samarium gallium garnet at room temperature, V SGG =11.8 rad/(T•m) and that of neodymium gallium garnet at 4.2 K, V NGG =3490 rad/(T•m), at wavelengths of 633 nm and 490 nm respectively [START_REF] Weber | Handbook of Optical Materials[END_REF][START_REF] Villaverde | Terbium gallium garnet Verdet constant measurements with pulsed magnetic field[END_REF].

Aside from the suitable nature of the effect (linear in the applied field), it is noteworthy that in terms of time scale, the use of a paramagnet is considerably more favourable than, say, inductive techniques. Indeed, the relevant relaxation time scales are those of electronic states inside the considered materials, which are far below the nanosecond for instance. This approach thus delivers measurements free of temporal distortions. However, there is the constraint of using a paramagnet that is transparent in the visible range. Among the vast family of magneto-optically active materials, many are not (fully) transparent [START_REF] Sugano | Magneto-Optics[END_REF][START_REF] Weber | Handbook of Optical Materials[END_REF][START_REF] Kazei | Magnetic properties of garnets[END_REF] in the visible spectrum; one exception is gadolinium gallium garnet Gd 3 Ga 5 O 12 , hereafter denoted GGG.

A few small slabs of this material were available in the lab; these are shown in the photograph in Fig. 2.27, to the right. PIERRE MOLHO lent them to me for use in my magneto-optical calibration. For the sake of easiness, it was the largest of the slabs that I chose, with dimensions of about 1 cm×4 mm×0.50 mm. The idea of the calibration using the garnet was to place the slab vertically right atop the sample holder's top surface, so that the measurement volume would be very close to the actual sample position in experiments. The garnet used as sensor is itself not calibrated, however. In other words, its own Verdet constant is not known a priori. One may rely on measured values such as those from Valiev and coworkers [START_REF] Valiev | Faraday effect in rare-earth paramagnetic garnet Gd 3 Ga 5 O 12[END_REF] or Novotný and coworkers [START_REF] Novotný | Investigation of Gd 3 Ga 5 O 12 by Micropolarimetry[END_REF], but I chose to overcome the need to assume one value or the other. With this in mind, I devised the following experimental plan:

1. Use Helmholtz coils as a magnetic field source and first calibrate their response to a given DC current using a Hall probe.

2. Replace the Hall probe with the GGG sample and measure the Faraday rotation as a function of applied current (therefore as a function of applied field).

3. Introduce the sample holder right underneath the garnet piece and use it to applied field pulses with MÁR-LIO's generator. Measure the Faraday rotation as a function of setpoint (corresponding to the generator's load capacitor voltage).

By first calibrating the coils, one obtains a reliable field supply which can then be used to determine the garnet's response to an applied field. As a last step, the sample holder (to be calibrated) is used to send pulses of field, whose amplitude is to be measured. The additional step required by my choice not to rely on published values of the GGG's Verdet constant is the use of the coils to calibrate the garnet's field-induced rotation of light polarization. As a bonus, it allows not only to make the conversion between measured rotation and actually applied field, but it also produces a measurement of the Verdet constant. Up to the sample location, the optical path of the beam is reproduced schematically in Fig. 2.29. The presence of a magneto-optically active sample is assumed there, but the configuration without sample is simply obtained by setting the angle α to zero.

The coils were mounted on an optical rail whose brakes were used to impose a fixed separation between the coils. The green one was pressed by the first brake against the optical rail supporting the laser, while the yellow coil was held in place between the first and second brakes. Owing to the respective dimensions of these elements, the coils were not in a strict Helmholtz position, and their axis was off by about 2 mm from the laser beam path, but this should play a negligible role due to the diameter of the coils, which is roughly 10 cm. At any rate, the plane of symmetry of the coil pair still warrants a horizontal and homogeneous field 27 . In order to prevent the somewhat rigid wires to act as levers and displace the coils during the experiments, it was held in place using the pairs of columns and black parts as shown at the bottom of Fig. 2.28.

Positioning as well as holding of the Hall probe was taken care of by a triplet of micro-positioners. After wiring, the it was fixed onto the topmost positioner using Teflon screws and platelets (see white parts just above the probe's wiring on Fig. 2.28). We took advantage of the probe's situation at the tip of an elongated PCB piece to place the bulky set of positioners sufficiently far away from the coil pair. Care was taken to place the center of the Hall probe right on the beam path, and the position along the latter was adjusted to the middle of the coil pair using the scotch-taped ruler visible on the photograph of Fig. 2.28. As for feed-in current and Hall voltage read-out, a Keithley 6485 Picoammeter was used. The current provided was 5.0 mA, and the residual Hall voltage in the absence of field was 0.55 mV.

As for applying DC currents to the pair of coils, we used a MLP208 20 V-8 A power supply (from Multisources Electronique R ) provided by DANIEL LEPOITTEVIN. In practice, currents above ∼4.5 A could not be maintained with satisfactory stability or even reached easily with the apparatus, so that I restrained the study to currents ≤4 A. Furthermore, this also lessened the heating of the coil a bit. In this regard, according to the specification sheet of the probe (see appendix section A 2 .1), the output voltage varies by about 0.06 %/K. Considering this, even a 10 • C variation (which is most likely more than the typical coil heating) would induce a relative error of about 1% only. Thus, we may safely neglect the effect of potential temperature changes during the measurements.

During measurements, two signals are accessible: the difference between and the sum of the photodiodes' output voltages. The sum is a measurement of the total (transmitted) laser power, while the difference reflects the polarization rotation. Indeed, the half-wavelength plate's neutral line is positioned at 45 • with respect to the polariser, as depicted schematically in Fig. 2.30. For the sake of example, if the polariser is set vertical, then the beam's polarization becomes horizontal as it is "flipped" around the half-wavelength plate's neutral line. This -→ E is then separated into two components -→ E 1 and -→ E 2 with orthogonal polarizations by the Wollaston prism, whose pair of neutral lines is indicated in green and purple. The dots (red, green and blue) denote the points at which an interface is traversed by the laser beam. The colours of the two exiting beams corresponds denotes their polarization direction. Considering the orientation of -→ E , the amplitude of the -→ E 1 component is larger than that of -→ E 2 , as schematized after the prism. The corresponding beams impinge on the pair of photodiodes, represented in dashed blue lines, which measure their intensities. plate allows to set (without a sample in the setup) the beam's polarization in the proper direction for the Wollaston prism via tuning of the angle ψ represented in Fig. 2.30. This particular direction is 45 • with respect to its neutral lines. That way, the prism splits the beam into two components of equal power. All this corresponds to settings that are implemented prior to any magneto-optical experiment. In the presence of magneto-optically induced polarization rotation, the beam's polarization no longer has equal projections onto the two orthogonal neutral lines of the Wollaston prisms, as illustrated to the right of Fig. 2.30. Therefore, the two separate components hitting the photodiodes will have slightly different intensities. It is this difference in intensity that is a measure (as a voltage) of the polarization rotation angle α. Further details on this setting are given in the appendix section A 2 .3.2. During this calibration, the induction was also directly measured (for comparison) with a gaussmeter. However, the precision is expected to be less than with the Hall probe because the positioning was not as precise; moreover, the offset removal was not carried out in a proper shielding but only in open air, far from magnetic field sources. The estimated m gaussmeter current-to-induction conversion factor reads m gaussmeter =3.7 ± 0.2 mT/A. It is reassuring that these induction estimates are compatible with those of the Hall probe, and agree within less than 10% with these. III.3.c Calibration of the garnet's response III.3.c.(i) Experimental setup Using non-magnetic tweezers, the GGG piece was held at the position of the Hall probe as illustrated next page on Fig. 2.32. Since in the actual calibration of the sample holder, the latter's top surface would need to be right under the laser beam (see Fig. 2.36), I placed the laser spot right above the bottom edge of the GGG piece. Thus, one can rule out the effect of inhomogeneities in the magneto-optical response of the GGG sample.

α - → E ψ = π/4 - → E γ = π 4 -α -→ E 1 -→ E 2 λ/

III.3.b.(ii) Measurements

In this configuration, the current fed to the coils was no longer DC but AC, and provided by a ±20 V, ±8 A bipolar Kepco R power supply amplifier which was driven with an arbitrary waveform generator (model 33220A from Agilent R ). The control signal U c was chosen to be sinusoidal with a frequency of a few Hz so as to avoid distortions due to the coils' inductance; see appendix section A 2 .2 for further details. Before applying the AC current to the coil, its amplitude I AC was first measured as a function of control signal strength U c . It is to be noted that the current was measured with a multimeter in AC mode, hence the readout is the root-mean-square current value, not the actual amplitude I AC ; this readout was multiplied by the corresponding √ 2 factor. The result is plotted on Fig. 2.33. As expected, the agreement with an affine fit is excellent 28 (the error bars here are smaller than the markers), even though one might argue that its quality slightly decreases close to U c 5 V. This might be caused by a dropout in the Kepco R 's response, since the working frequency is f 0 =7 Hz and the delivered current amplitudes are not so small at this stage. Still, this dropout is clearly a very weak effect. Using the fitting result, one can write:

I AC = G eff • U c (2.23)
where I AC is the AC current amplitude, and G eff =0.445 ± 0.001 S is the effective conversion factor deduced from the affine fitting.

III.3.c.(ii) Measurements

As stated before, the input current for the coils was AC so that the signal of interest would be monochromatic, and therefore fully retrieved by monitoring the relevant frequency. In all the following, except otherwise specified, the signal of interest, which is the difference between the output voltages of the two photodiodes, will be referred to as the photodiode signal. The idea is the following: the photodiode signal naturally contains noise (50 Hz noise from electronics among others) of rather small amplitudes 29 at frequencies significantly higher than, say, a few Hz, but also random mean value fluctuations on a time scale of approximately 1 s. Therefore, measuring the Faraday rotation while applying a constant field is impossible because the signal to noise ratio is about unity. For the same reasons, one cannot apply a positive current, make a measurement, switch to the opposite DC current value, make a measurement and use the difference between the two signals because of the long time scale fluctuations. In this context, however, one can exploit the linearity of the physics at play to probe the garnet's response at a given frequency: provided one knows the amplitude of the applied AC induction, it is possible to focus only on the signal of interest since it should only exist at the input current frequency. Thus, unwanted fluctuations and noise sources can be circumvented provided that this working frequency lies between the low-frequency regime of large mean value fluctuations and the higher-frequency regime where several noise sources come into play.

First off, upon applying an AC current to the coils at a frequency of f 0 =7 Hz, I monitored the Fourier transform amplitude of the photodiode signal E as computed by the oscilloscope. The sampling duration was ∆t =5 s, and the sampling rate high enough to ensure having at least one data point per Hz (and of course one point at f 0 ).

As expected, there was a peak at the working frequency, and I could make sure that the background noise level around this peak was never more than one or two percents of the peak amplitude. The latter, referred to as E(f 0 ), is plotted on the following figure as a function of drive voltage U c : The method I used here is somewhat reminiscent of lock-in detection, with the multiplication by a reference signal and extraction of amplitude being taken care of by the oscilloscope Fourier transform, in a sense. Here, its power is demonstrated in the sense that amplitudes as low as a few hundred µV can be extracted with a good accuracy. Furthermore, taking the measurements' error bars into account, the affine fit to the data (performed with Igor [144]) yields an intercept with the ordinate axis (at U c = 0) of -2 × 10 -2 mV, which is below (in absolute value) its own error estimated during the fit.

III.3.d Review on the garnet's calibration

Based on the previous sections, the conversion from input current to magnetic induction produced by the coil pair is known. In addition, since I have measured the AC amplitude of this current as a function of control voltage U c (on the waveform generator), I am able to create an AC magnetic induction of known amplitude with the coil pair and the Kepco R power supply.

Using such an AC magnetic induction at a frequency of f 0 =7 Hz, a sinusoidal Faraday rotation is induced with the GGG piece. It has been recorded as the amplitude of the photodiode signal's Fourier transform at the working frequency; this quantity is E 7Hz . With the help of the previously determined G eff conversion factor, one can now plot the garnet's response E 7Hz as a function of applied induction B. Indeed, the latter reads: B = (m Hall G eff )U c . The factor m Hall G eff has a value of 1.75 ± 0.03 mT/V. Such a plot is presented on Fig. 2.35:

The data are the red dots, while the dashed blue line is the linear fit to the measurement. As of now, it is possible to directly convert variations in the photodiode signal (which is a voltage) to the corresponding magnetic induction component along the laser beam. This can be done using the conversion factor ξ GGG extracted from the fit:

ξ GGG = 0.387 ± 0.008 mV/mT.

(2.24)

Thus, the magneto-optical response of the GGG sample has been characterized, and may now be used to calibrate arbitrary applied magnetic inductions. In the following, I will detail how the sample holder was calibrated. The same was realized for a small detection loop on the back side of the sample holder (not visible here). Since it faces the field-generating coil, it responds to a pulse of magnetic field by an induced electromotive force which is recorded with the oscilloscope 30 . As can readily be seen, a part of the laser beam is scattered by the sample holder itself. This ensures that the magnetic field produced by the sample holder is probed right above the top surface of the latter, which is after all where the samples lie during actual experiments. To be more quantitative, I have estimated from the photograph the full vertical extent of the laser spot based on known dimensions (e.g. the lengths of the side of the GGG slab). The criterion I used was saturation to white on the image. The laser spot size w Laser was measured as the sum of the vertical extent δh 1 of the first spot on the vertical sample holder side (to the left of the GGG slab), and that of the spot on the first-encountered side of the GGG slab. I find w Laser =1.57 ± 0.06 mm. Therefore, the altitude h spot of the laser spot's center with respect to the sample holder's top surface reads h spot = w Laser /2δh 1 , where δh 1 =0.33 ± 0.06 mm, thus h spot =0.46 ± 0.09 mm. The uncertainties on h spot and δh 1 were added in quadrature because the two measurements are independent. Coming back to the nominal sample thickness of 300 µm, we are led to an offset in the field sensor's center of at most 0.21 mm, whereas the inductive calibration of the sample holder featured a minimal offset of 0.45 mm. In this respect, the magneto-optical calibration can be expected to be more representative 31 of the fields applied to the samples.

On the other hand, one might argue that this modifies the photodiode signal. That is why I performed a few measurements before applying pulses of field: still using the pair of coils with an AC current in the same conditions as before, I compared the GGG's response with that measured in section III.3.c. Within experimental error, the obtained values were identical to the calibration of the garnet with U c from 0 to 4 V. This supports the hypothesis that the active spot area is contained in the laser spot visible on the side of the garnet slab.

At this stage, one might also argue that the AC magnetic field should produce an unwanted electromotive force in the red coil of the sample holder. The latter would result in a bias of the field applied to the sample. Even though no change could be measured, let us estimate this electromotive force at U c =3 V, which corresponds to an induction amplitude of B AC 4 mT. The red coil has approximately N = 10 windings, each of them enclosing a surface S 1 cm 2 . Assuming that this field is homogeneous over the whole red coil, the magnetic flux amplitude ϕ reads:

ϕ = N B AC S (2.25)
Thus, the amplitude e of the electromotive induced force at the working frequency f 0 =7 Hz is such that:

e = 2πf 0 ϕ = 2πf N B AC S 2 × 10 -4 V (2.26)
This confirms that induction in the sensor loop is expected to play no role in this measurement.

III.3.e.(ii) Measurements

The sample holder cannot produce pulses with a repetition rate. Therefore, making use of AC measurements at a given frequency as was done before is no longer possible. Fortunately, the pulses are 16 µs long, which brings us to time scales where the slowly varying mean value of the photodiode signal is not an issue any more. Still, apart from low-amplitude white noise, there is a non-negligible noise component around 100 kHz, of amplitude ≤10 mV. Thus, there is a need for large amplitudes of signal in order to have a negligible influence of the latter noise, the origin of which is unknown 32 .

In this context, measurements were performed in the following way. First, the capacitor of the pulse generators were charged up to a given setpoint value (the latter was measured with a regular voltmeter). Then, pulses were applied and the oscilloscope recorded both the induced electromotive force U loop (t) from the detection loop (it was triggered on this signal) and the voltage from the photodiode system U photodiode (t) as a function of time t. The quantities of interest were the following:

• the maximum value E m reached by |U photodiode (t)| during the pulse • the local maxima V 1 and V 2 successively reached by |U loop (t)| during the pulse E m is directly linked to the maximum induction B m applied during the pulse, while V 1 and V 2 are the only two relevant quantities which can be measured during experiments and are also related to this maximum induction value. One can write:

V 1 = ξ 1,loop B m V 2 = ξ 2,loop B m
where ξ 1,loop and ξ 2,loop are two proportionality constants taking into account the geometry of the field-producing and field-detecting coils, the position at which the applied field is measured with respect to the coil, and so on. Since the pulse generator was built to deliver the same pulse shape regardless of its amplitude, one can assert that 31 This matter is complicated by the finite extent of the laser spot and its unknown intensity profile. Strictly speaking, the measured signal is a convolution of the sought induction with this laser profile. However, the laser profile can be reasonably expected to be (close to) gaussian in shape, which gives less weight to areas far from the spot center. Furthermore, the measured spot size may be artificially larger due to scattering. 32 One hypothesis would be that this is related to how exactly the Melles-Griot R He-Ne laser works/is pumped. One possible explanation is that its power supply operates in switched mode. There is, however, no means to check and no information hinting at this could be found in the apparatus' documentation.

the knowledge of V 1 and/or V 2 is sufficient to determine a posteriori the value of the induction which was applied during a given pulse.

A typical measurement of U loop (t) and U photodiode (t) is shown on Fig. 2.37. The latter voltage has been multiplied by a factor 100 for clarity. One can distinguish on this curve the aforementioned low-amplitude white noise as well as the ∼100 kHz noise. Measurements at different values of the capacitor voltage U capa were carried out, not always with the same number of repetitions. The measurement errors at each voltage were evaluated from standard deviations, to which a contribution from the periodic noise at ∼100 kHz was added in quadrature; the chosen value was the noise's amplitude of ∼10 mV. The results with their corresponding uncertainties are all represented on Fig. Compared to the other measurements, the fit is not quite as nice. The zero-voltage point was imposed to increase its quality. The main source of measurement error remains this periodic noise at ∼100 kHz: it is difficult to ascertain how much a given measurement of E m was biased, not to mention whether it was towards larger or smaller values. Nevertheless, one can still extract from this the calibration parameter k linking the maximum induction B m applied to the sample when the capacitors are charged with a voltage U capa :

B m = k Fara. • U capa (2.27)
The fit yields k Fara. =0.46 ± 0.02 mT/V. This is significantly lower than the same coefficient extracted with an inductive method, k ind. =0.58 ± 0.01 mT/V.

It is now time to make the link between the known field values and what can actually be measured during the experiments, that is to say quantify the relation between B m , V 1 , and V 2 . The measurements are summarized in the following plot:

(V) V 1 V 2 Linear fit to V 1 Linear fit to V 2
Fig. 2.39: Induction applied at the sample level B m as a function of the voltage extrema V 1 and V 2 measured on the induced electromotive force in the detection loop. The data points related to V 1 (resp. V 2 ) are in red (resp. green), while the dashed blue (resp. orange) line is the linear fit to the data.

Here, the error bars corresponding to the induction have been left out for the sake of clarity (they would be smaller than the dots). Here too, the zero-voltage, zero-induction point was added with a negligible uncertainty to the data to increase the fit's quality. The agreement is not great, especially for the V 1 data set, but the (forced) linear relation between

B m , V 1 , and V 2 is still realistic. The new slopes ξ V 1 and ξ V 2 such that in the fit ξ V 1 • B m (V 1 ) = V 1 and ξ V 2 • B m (V 2 ) = V 2 read: ξ V 1 = 25.3 ± 0.1 µV/mT ξ V 2 = 10.0 ± 0.1 µV/mT
It must be noted that this uncertainty corresponds only to the determination of the best fitting coefficient, not the interval around the induction estimated from this conversion in which the actual applied induction lies. The maximum relative discrepancy found is 8% using V 2 , 11% using V 1 . Globally, the least reliable measurement is V 1 , which is unfortunate since its larger values for a given induction (with respect to V 2 ) should make it a finer induction measurement tool. This would indicate that it is worth recording both voltage values during experiment to estimate the applied induction, instead of relying on only one extremum.

The extraction of these values concludes this calibration. I now have the relevant parameters with which to know, during a given experiment, what maximum induction was actually applied during a pulse regardless of any a priori knowledge of the capacitor voltage setpoint U capa . Relying on the detection loop is of a great advantage in cases where the loading process of the capacitor is modified. The typical example occurs when a fine tuning of the applied field is required. If one reduces the current fed to the capacitors for a given manual setting of the knob controlling U capa , then the achieved load voltage is decreased. The result is two-fold: the absolute maximum field that can be applied is reduced, and it becomes easier to accurately change the applied field by small amounts.

In practice, to that end, a current divider bridge is soldered between the DC-DC converter which supplies the capacitors with current and the capacitors themselves. For a given input in terms of rotation of the knob commanding the voltage regulator (and thus the load setpoint), such a current divider bridge reduces the current fed to the capacitors, therefore reducing the voltage U capa . This is particularly useful if low inductions, say ∼10 mT or less, are to be applied, but would render any calibration of the capacitor setpoint U capa as a function of number turns on the knob useless. However, since the electromotive force measured on the detection loop only depends on the applied induction, it remains a solid tool. Bibliography and measurement objective Considering that the above-described Faraday setup with the pair of coils allowed me to measure appreciable magneto-optical signals with accuracy, I set out to measure the garnet's Verdet constant V so as to compare my result with published values of this property. In my bibliographic searches, I was only able to find two publications on this subject: the work of Valiev and coworkers [START_REF] Valiev | Faraday effect in rare-earth paramagnetic garnet Gd 3 Ga 5 O 12[END_REF] in which the GGG's Verdet constant is reported (without experimental details) as a function of sample temperature, at different wavelengths, and an article by Novotný and coworkers [START_REF] Novotný | Investigation of Gd 3 Ga 5 O 12 by Micropolarimetry[END_REF]. In the latter, Faraday measurements were carried out using a rather large 4 mm×5 mm×10.5 mm GGG single crystal, which they put in the center of a ring of out-of-plane magnetized NdFeB. Unfortunately, no information is given regarding:

• the ring dimensions

• whether multiple reflections inside the crystal are taken into account

• how narrow the filters they use to make measurements at a given wavelength are (they do not use lasers, but filtered halogen lighting)

The ring dimensions may well play a role since they will govern the homogeneity of the field applied to the GGG crystal. In my case, inhomogeneity effects can be ruled out thanks to the (quasi-)Helmholtz setup and the sample thickness d =0.5 mm with respect to the coil radius, ∼5 cm. Furthermore, one may also argue that the total rotation seen may result not only from the direct beam, but also from the part of the beam which has been reflected twice inside the crystal before being transmitted towards the diode. Indeed, the corresponding field amplitude should be lower than that of the direct beam, but its rotation is three times higher. Along this line, one should not leave aside the potential influence of even higher orders. These effects are discussed in more detail in the appendix section A 2 .3.1.

As for the magneto-optical data from Valiev et al. [START_REF] Valiev | Faraday effect in rare-earth paramagnetic garnet Gd 3 Ga 5 O 12[END_REF], the reported measurement at room temperature and a wavelength of 630 nm indicates V (V al.) GGG (630 nm) =9.61 rad/(m•T). Novotný and coworkers report a value of the GGG's Verdet constant of V (N ov.) GGG (612 nm) = 12.5 rad/(m•T) [START_REF] Novotný | Investigation of Gd 3 Ga 5 O 12 by Micropolarimetry[END_REF]. From there, there are two ways of estimating the Verdet constant at the wavelength λ Laser = 632.8 nm that I used. The first one consists in assuming that this magneto-optical property of the materials follows a dispersion 33 of the type V -1 ∝ λ 2λ 2 0 (where λ is the working wavelength, λ 0 a constant). With this, one obtains that theoretically, the GGG's Verdet constant at 633 nm should be V (th.) GGG (633 nm) 11.6 rad/(m•T). This value is estimated from their data point at 612 nm. If now one decides to rather follow the fit to their data (V as a function of λ, Fig. 4 in their article), one arrives at 12.2 rad/(m•T). Since no error bars are given, let us keep only the value V (N ov.) GGG (633 nm) 12 rad/(m•T). This is my point of comparison: if my measurements are correct, they ought to yield an estimate of V GGG (633 nm) at least of the same order of magnitude. Interestingly, the value found in the literature [START_REF] Weber | Handbook of Optical Materials[END_REF] for the Verdet constant of diamagnetic gadolinium aluminium garnet Gd 3 Al 5 O 12 at 633 nm is 13.3 rad/(m•T), which perhaps should not be so surprising considering that gallium is right below aluminium in the periodic table.

A matter of definition It is noteworthy that a discrepancy seems to exist between the theoretical definition of a paramagnetic material's Verdet constant V involving the material's magnetization M [103] on one hand, V = α/(µ 0 M d) (with d the sample thickness and α the measured polarization rotation angle), and the convention used in practice for this physical property on the other hand. Indeed, several experimental works [START_REF] Villaverde | Terbium gallium garnet Verdet constant measurements with pulsed magnetic field[END_REF][START_REF] Novotný | Investigation of Gd 3 Ga 5 O 12 by Micropolarimetry[END_REF][START_REF] Dentz | Terbium Gallium garnet for Faraday Effect Devices[END_REF][START_REF] Weber | Magneto-optical properties of KTb 3 F 10 and LiTbF 4 crystals[END_REF] including some studies whose published Verdet constant values are considered reference in handbooks [START_REF] Weber | Handbook of Optical Materials[END_REF], use the following definition:

V = α B a d (2.29)
where B a is the applied magnetic induction. Let us try to evaluate the difference with respect to equation (2.22) by considering a finite-volume paramagnet of susceptibility χ, in vacuum. If everywhere in space a homogeneous field H 0 is created, then the vacuum around the paramagnet experiences an induction B 0 = µ 0 H 0 . By virtue of the continuity equations of electromagnetism in matter [START_REF] Jackson | Classical electrodynamics[END_REF], the induction inside the paramagnet must be the same as in vacuum. Denoting M the material's magnetization, B m and H m the magnetic induction and field inside the material, we have two equations to reconcile:

B m = µ 0 (H m + M ) M = χH m
which leads us to:

H m = 1 1 + χ H 0 M = χ 1 + χ H 0 = H 0 -H m
As can be seen above, there may be a strong discrepancy between the induction responsible for the Faraday effect and the whole induction in the paramagnetic medium. Usually, the paramagnetic susceptibilities are very small numbers, such that M χH 0 H 0 ; one may consider examples such as χ Pt 2.7 × 10 -4 , χ Pd 8.1×10 -4 , [55,103], χ GGG (300 K) 7×10 -3 [START_REF] Ranno | Magnetometry on Gadolinium Gallium Garnet[END_REF]. On the other hand, it appears significantly more convenient to consider a paramagnet's Verdet constant as the polarization rotation per unit thickness when the sample is subjected to a known induction, i.e. external applied field in vacuum times µ 0 . This convention allows to easily compare materials or measurements in different conditions without the need to known the sample's magnetic susceptibility; it is much simpler to have reliable knowledge of the induction applied to the sample than the latter's magnetization, although only the latter is responsible for the measured polarization rotation.

In the following, I will use the latest-described convention for the Verdet constant will be used: in other words, I will consider it as the polarization rotation per unit thickness for a given induction applied to the sample.

III.3.g Experimental procedure

In order to measure the garnet slab's Verdet constant, I reverted to the setup used in section III.3.c. Once more, I applied an AC current with a frequency of f 0 =7 Hz to the pair of coils while recording the amplitude of the difference signal from the photodiode at the working frequency E(f 0 ). The amplitude of this AC current was again controlled by the Agilent waveform generator output U c . In addition, this time, I also recorded the electric signal corresponding to the sum of intensities E sum . Indeed, both are needed to calculate the polarization rotation α, following the formula:

2α = E(f 0 ) E sum = ∆I I Σ (2.30)
where I sum and ∆I are respectively the total incident light power and the difference between the intensities measured by the two photodiodes. This formula is derived in the appendix section A 2 .3.2. The measuring time was 20 s, and the sampling rate high enough to ensure having one point right at the working frequency. The measurement error was estimated by repeating a measurement four times and taking the largest deviation from the mean value to a certain measurement.

During the whole series of measurements, the sum signal E sum remained constant and equal to 2.46 ± 0.01 V. The values for E(f 0 ) ranged from 0.65 ± 0.01 mV to 3.36 ± 0.01 mV. However, at this stage, it is important to take into account a detail in the setup which did not play a role earlier: actually, the E(f 0 ) signal is multiplied by a factor 10 before it is measured, whereas the E sum signal is not 34 . Thus, the measured rotation angle α is such that:

α = α 10 = 1 20
E(f 0 ) E sum (2.31)
However, one must take into account the multiple reflections of the laser beam inside the garnet. Their effect is contained in the µ factor introduced that is derived in the appendix section A 2 .3.1. The true polarization rotation α 0 for a given induction applied to the sample reads:

α 0 = α µ (2.32)
Now, we may exploit the measurements to produce an estimate V GGG (633 nm) of the sample's Verdet constant thanks to the following equation, adapted from equation (2.29): In this affine fit, the extrapolation to zero induction yields a value of α 0 (B m = 0) of -5 × 10 -8 rad, with an error of 3 × 10 -7 rad. This, as well as the fact that such an intercept was obtained without forcing the fit function to pass by the point (0 mT, 0 mrad), confirms the expected linear behaviour of the garnet under field while lending confidence in the measurement's accuracy. The slope is 6.08 ± 0.05 × 10 -3 rad/T. Using equation (2.33), one obtains:

α 0 = B m V GGG (633 nm)d ( 2
V GGG (633 nm) =12.1 ± 0.1 rad/(m•T).
There is a very good agreement with the measurements by Novotný et al. [START_REF] Novotný | Investigation of Gd 3 Ga 5 O 12 by Micropolarimetry[END_REF].

III.4 Comparison between the two calibration methods

The two calibration methods' results disagree with one another by 26%, the inductive method leading to larger fields than the magneto-optical one. This is in agreement with an under-estimate of the measurement loop's effective surface based on its inner radius: one can expect a finite extent of the wiring to contribute to this surface [START_REF] Paul | Inductance: loop and partial[END_REF], not to mention the finite thickness of the insulating sheath around the loop's copper wire. Also, these 26% are smaller than the estimated maximum relative error on the loop's effective surface, which was 56%.

Evaluating the calibration's accuracy requires to take into account not only the estimated fitting error on the capacitor voltage-to-field conversion factors k Fara. (see equation (2.27)) and k ind. (see Fig. 2.25 and the associated results), but also the discrepancy between the fit and the measured induction values. In the inductive case, the average discrepancy is 4% (not taking the uncertainty in the loop's surface into account) with a maximum of 11% and only two points out of nine above 5%, while in the magneto-optical case, the average discrepancy is 6% with a maximum for two points out of eight of 17%, all other points below 5%.

Finally, we should consider the discrepancies between measured extrema V 1 or V 2 in electromotive force induced in the in-situ measurement coil (for a given maximal applied induction B m ), and the predicted value for these voltages. Adding these relative discrepancies in quadrature with those related to the capacitor voltage-tofield conversion factors, I can now estimate the total uncertainty on the applied induction when relying on the measurement of V 1 or V 2 . In the magneto-optical case, I find 10% from V 2 and 13% from V 1 , and 8% from V 1 in the inductive case. Despite the seemingly slightly better accuracy for the inductive method, one must not forget that this overlooks the uncertainty due to the effective surface of the small measurement loop.

One unanswered question concerns the finite laser spot size. It can be argued that along the vertical direction, the laser's beam probes different values of field, and the latter was shown to have non-negligible variations over distances of several hundred microns in Fig. 2.22. Yet, around the center of the spot, all variations in induction that are linear with the altitude are averaged to zero due to the spot's symmetry. Naturally, this only holds over the distance h spot =0.46 ± 0.09 mm from the spot's center downwards until the spot is cut off (see Fig. 2.36). Thus, linear variations of induction can be safely neglected over the distance 2h spot =0.9 ± 0.2 mm, that is to say a proportion 2h spot /w Laser = 0.58 ± 0.15 of the laser spot. Though flawed by a finite sensor size as well, the measurements presented in Fig. 2.36 speak against very strong non-linearities. Therefore, remembering that over a distance δh 1 =0.33 ± 0.06 mm the laser does not go through the garnet (see Fig. 2.36), only a minority fraction of the spot (w Laser -2h spotδh 1 )/w Laser = 0.20 ± 0.19 may be affected by linear variations in induction.

Considering all aspects, it is more sensible to rely on the magneto-optical approach, as the latter provides induction estimates without a data processing that is heavily impacted by a high-uncertainty measurement, contrary to the inductive approach.

III.5 Review

In this section, I have introduced the setup I have used in this thesis to apply quasistatic fields to nanowires dispersed onto a piece of silicon wafer. The device consists in a parallelepiped made out of PEEK, with a large coil to produce the field and a sensor loop to provide quantitative, in-situ information about the applied field pulses. Though inductive and therefore indirect, this method was shown to yield electromotive signals that are large enough for relatively accurate field estimates.

The field was produced by the coil when supplied with a ∼15 µs pulse of current from a pulse generator developed by MÁRLIO BONFIM. Calibration of the device as a whole was first performed in section III.2 using an inductive method relying on a 1 mm-diameter measurement loop. While advantageous, this approach relies critically on the knowledge of the loop's effective surface to yield a field estimate. This is by far the largest source of uncertainty about 56%. Moreover, because of the device's dimensions, the sensor loop's centre could not be placed at the altitude corresponding to samples in an experiment. Such an offset I tried to correct based on the field estimates' dependence on sensor altitude, but this does not reduce the error due to the loop's surface.

In order to gain in accuracy, I have used a magneto-optical characterization method relying on a slab of paramagnetic gadolinium gallium garnet (GGG) and on the Faraday effect, i.e. the rotation of light's polarization induced by magnetization. This time, the approach used the garnet as sensor, which, in fact, I have calibrated as well. On one hand, the sample holder's calibration process involving known fields from Helmholtz coils allowed me to reliably quantify the applied field pulses. On the other hand, by measuring the garnet slab's linear response, I was able to determine its intrinsic polarization rotation per unit length per unit magnetization: the Verdet constant (at the laser's wavelength). My experiments are in very good agreement with recent measurements [START_REF] Novotný | Investigation of Gd 3 Ga 5 O 12 by Micropolarimetry[END_REF] on the same material.

Thereafter, it is necessary to take into account not only the uncertainty on the fit parameters quantifying my calibration, but also the discrepancies between the fit itself and the measurements. That is how I am able to estimate the typical relative error on the applied induction. Such an estimate is based on the in-situ inductive measurements from the loop to the side of the sample holder. From the above, this relative error on the applied quasistatic inductions lies between 10% and 13%.

All the above was done with the goal to apply fields on the order of a few microseconds. This time scale is at least comparable, larger or even much larger than typical (relaxation) time scales in magnetization dynamics [106,118,148]. It is also larger than the predicted time scale of domain wall displacement over a few to ten microns P (t) A t T /2 -T /2 0 Fig. 2.41: Triangular-shaped pulse P (t) as a function of time t. The total pulse length is T , the amplitude is A. [71,72]. Thus, this lower bound on the pulse duration represents an obstacle to the observation of domain wall dynamics, and calls for means to apply magnetic fields on shorter time scales.

In order to go from quasistatic fields to shorter i.e. nanosecond-scale fields, my approach consisted in developing antennas compatible with high-frequency signals. "High-frequency" is here to be understood as "with a spectrum extending well into the gigahertz range". Conversely, the corresponding time scales go from a few tens of nanosecond town to a few nanoseconds. The background for this excursion into this frequency domain as well as the modus operandi will be the subject of the following section.

IV Coplanar waveguide for short field pulses IV.1 Transmission lines for short time scales and choice of geometry

IV.1.a Motivation

As was highlighted in section III.5, pulses much shorter than those delivered by the combination "home-developed sample holder plus MÁRLIO BONFIM's pulse generator" are needed to properly investigate the dynamics of domain walls in cylindrical nanowires. On one hand, the expected shortest time needed for a domain wall to travel about 10 µm in a nanowire (which is the largest extent of straight sections in the investigated samples, see section I.3.d) could be on the order of 10 ns. This order of magnitude can be derived from the steady-state velocities of BPWs under field obtained in micromagnetic simulations [71,72,[START_REF] Hertel | Multiscale simulation of Bloch point dynamics in thick nanowires[END_REF], on the order of 1 km/s. This translates into an upper bound: the pulses need to be at most 10 ns long so as to not be directly limited by the length of the samples. On the other hand, one must not forget that a significant range in field amplitude is needed so that domain wall pinning can be overcome; this requirement is all the harder to fulfil as the pulse duration decreases. Indeed, for a given pulse duration, increasing the amplitude corresponds to increasing the pulse's steepness or, to put it differently, to reducing its rise and fall times while increasing its power. The difficulty is that such a goal requires not only larger and larger power but also a circuit design with sufficiently fast and clean time response, both for the pulse generation and for the antenna creating the field. That is a trade, in and of itself.

Considering this, my approach was to rely on a commercial device to create the pulse, while designing an antenna with good enough behaviour in the high-frequency regime so as to make the most of the provided pulse. In a first step, we will not focus on the practical means to apply this pulsed field to a sample, but rather on how this "good enough behaviour" can be understood in terms of bandwidth. Let us try to visualize this. A pulse is nothing but a time-dependent signal, say a voltage described by a function P (t). In the following, we will consider a given pulse shape that is to be transmitted by a certain electronics device (e.g. an antenna). As an example, a triangular-shaped pulse with duration T and amplitude A is represented in Fig. 2.41.

If we are to define requirements in terms of bandwidth so as to not distort this pulse, we need to look at it in the frequency domain. In other words, we need to have a look at the Fourier transform P (ν) of our signal P , as a function of the frequency ν. Throughout this manuscript, the convention used for Fourier transforms will be the following:

P (ν) = ˆ+∞ -∞ e -2iπνt P (t)dt.
(2.34)

This convention has the advantages of being an isometry and dealing directly with frequencies, not angular frequencies. Let us now look at the Fourier transform of our pulse. As a real, even function, its Fourier transform is an even, real function of the frequency 35 . Therefore, only the domain ν ∈ [0, +∞[ is shown in Fig. 2.42: 2.42: Fourier transform P (ν) of the pulse P (t) as a function of the frequency ν. Of course, the zero-frequency value is the area under the curve of P (t), that is to say its mean value times the pulse duration.

AT /2 1 T 2 T 3 T P (ν) ν 4 T 5 T 6 T 0 Fig.
For the chosen triangular shape, the Fourier transform is given by: P (ν) = (4A/T )[1cos (πνT )]/(2πν) 2 = (AT /2)sinc 2 (πνT /2). This function is naturally extended by a continuation to its zero-frequency limit, which reads AT /2 i.e. the area under the curve of P (t). Based on this seemingly trivial result of Fourier analysis, we learn a first important aspect of pulse handling. Whatever the bandwidth of a given device, if the latter has a poor behaviour in the low-frequency regime, then the pulse will be strongly affected. Indeed, the Fourier transform 36 of any realistic pulse possesses one or several local maxima, and an absolute maximum at this zero-frequency value. Thus, if the lobe corresponding to this maxima is distorted or suppressed, then the outcome will be more affected than if the distortions concerned features of the Fourier transform far from the null frequency.

In this instance, "far" is determined by the natural frequency scale 2/T , which by the way delimits the first lobe of P (ν). At frequencies a few times 2/T , the Fourier transform is very rapidly decreasing: above 6/T , beyond the third lobe, the amplitude can be at most 1/(3π) 2 1.13% of the function's maximum value. In other words, the (spectral) weight of higher frequencies can be considered negligible. Based on this, the notion of "requirements in terms of bandwidth" becomes clearer. As a rule of thumb, one can estimate a minimum bandwidth for a given pulse shape if that bandwidth encompasses at the very least the three first lobes of the pulse's Fourier transform 37 . If there are no lobes, then one should refer to the frequency scale given by the inverse pulse duration: the bandwidth should go as high as several times this frequency.

So far, the notion of bandwidth was dealt with like that of an ideal low-pass filter in the frequency domain, with a constant response until a cut-off frequency f cut after which this response is exactly zero for all frequencies. That is a very crude approximation of any realistic device. Even with non-linear components, the response function of an electronics circuit cannot fall to zero infinitely fast, nor remain strictly flat over a large frequency range. We may remember for instance the low-pass RC circuit, with a very gradual drop in response until its so-called cut-off frequency. The conclusion to draw from this is that while an indication of bandwidth such as a cut-off frequency is a valuable piece of information, it is not enough. The flatness of the device's response is very important, since strongly uneven response at different frequency scale may also impact drastically the pulse's shape.

After these very general considerations, it is time to move on to the case that is relevant to this manuscript. From this introduction, the three important ideas to keep in mind are: 35 While the choice of P is quite obviously meant to ease the discussion, the latter holds in a more general case if the modulus of the Fourier transform is considered. 36 Or its modulus in the case of a complex-valued function. 37 The reader is invited to pause and consider that this statement was drawn from a particular, triangular pulse shape. A square pulse with the same duration and amplitude would lead to a sinc(πνT ) Fourier transform, with similar shape but much slower decay. This being said, especially at the nanosecond scale and below, a pulse's rise and fall times are usually not (completely) negligible with respect to the pulse duration.

• The antenna's behaviour should not be only satisfactory at high frequencies, but also in the DC to lowfrequency regime, where much spectral weight lies.

• How high the bandwidth should reach is determined by the pulse shape, and notably by its duration. Over typically a few times the corresponding frequency scale, most of the spectral weight is accounted for.

• Considering that bandwidths in electronics are never infinitely sharply defined and that response functions are never perfectly flat, it is necessary to watch out for two possible issues. The first one is that strong departures from a flat response (whatever the gain) strongly affect a pulse's shape; the second one is that a realistic device's response gradually falls off with frequency. Therefore, it is safer to target a higher bandwidth than a priori required by the above criterion so as to be less influenced by this type of behaviour. The switch for the trigger mode (in green) always was turned to the "manual position". Leaving the self-explanatory elements aside, the output is SMA (capped here with a 50 Ω load for safety), and the pulse shape control knobs correspond to two distinct ways of tuning the pulse's temporal profile. In practice, the pulse can be made slightly sharper or to the contrary dulled and broader, with a sort of plateau.

The once-and-for-all set to "manual" trigger mode switch is denoted in green, and all elements that never were needed in my investigations are highlighted in red. All others, signalled in blue, have been of frequent use. The Avtech R pulse generator's output is SMA, and capped on this photograph with a 50 Ω load for safety. In the accidental event of a trigger, safety requires (notably for the Avtech R pulse generator itself) that at least the pulse is sent to the load that the Avtech R pulse generator is designed to work with. Without such a protective load, the pulse would encounter an open circuit (the output's connector end), which could result in damage due to reflections. This discussion will be easier to understand with section IV.1.c.

Aside from the amplitude control knob, the apparatus features two other knobs which allow the user to tune the pulse shape to some extent. These are referred to as the pulse shape control knobs in Fig. 2.43. They can be used to sharpen or blunt the pulse's profile to a certain extent. Moreover, experience shows that maintaining a given pulse shape over the full range of amplitude requires to adjust these knobs along the way. Effectively, these knobs set the widths of two sub-pulses whose superposition builds up the total output signal, as well as the time delay between these components.

A typical pulse produced by the Avtech R pulse generator is shown in Fig. 2.44. The measurement was taken in the apparatus' documentation; measurements I carried out showed very similar traces. The maximum amplitude is about 800 V, and the duration τ Avtech during which the signal is above 20% of its maximum amplitude (highlighted in Fig. 2.44 by the red dashed lines) is about 5.6 ns. To a first approximation, the pulse shape is triangular. Most of the deviations from this shape are located to the sides: one can see decreases of the pulse's steepness as well as bounces i.e. oscillations towards the end of the pulse. In fact, up to the noticeable changes in slope to the side, drawing straight lines to follow the pulse shape and define a (slightly asymmetric) triangular shape gives an excellent agreement with the measured trace. The amplitude is about 800 V into a 50 Ω load, which corresponds to the Avtech R pulse generator's maximum, the minimum being around 120 V. The reason behind such high voltages will be explained in section IV. As a basis, we can thus use the discussion from section IV.1.a since it was based on a (conveniently chosen) triangular signal. Here, the characteristic frequency scale is f Avtech = 1/τ Avtech 180 MHz. The discussion of Fourier amplitudes from p.106 advises to target a bandwidth no less than about 6f Avtech 1 GHz. To go just a bit further this qualitative rule of thumb: in a pessimistic estimate, if the bandwidth were to correspond to an ideal square filter in frequency with a cut-off f cut =1 GHz, then we would essentially broaden the signal in the time domain 38 by about 1 ns. While this is by far less than, say, a factor two, it is not quite negligible with respect to the pulse width.

Therefore, if we target instead a bandwidth ∆f =10 GHz, not only do we follow the last "important idea to keep in mind" of a safety margin in bandwidth from section IV.1.a, but we also expect the signal broadening to be reduced accordingly. Indeed, the signal duration would be approximately increased by only 0.1 ns τ Avtech .

With this, we may now start to consider the design for the field-producing antenna according to the targeted bandwidth.

IV.1.c From Ørsted fields to high-frequency antennas

The most straightforward way to produce magnetic induction from electrical current is by way of Ørsted field. In the context of applying such a field to a nanowire, the first idea that may come to mind is sketched in Fig. 2.45:

In the ideal case, the applied field -→ H a (in light slate blue) should be perfectly aligned with the nanowire (in light grey), and homogeneous over the latter's length. In this respect, a rather flat strip (rough, coppery texture) geometry through which to pass the current -→ J (in red) is most suitable. can be directly calculated from the current density || -→ J || via the Biot-Savard law. For a fixed total current, it is advantageous to have a narrow strip so as to enhance the field strength, as long as this width is large compared to the sample dimensions.

At this stage, we need to keep in mind that the target is not applying current signals within the realm of quasistatic electronics (also known as lumped element model), but short pulses requiring a rather large bandwidth, as was described earlier. The departure from quasistatic electronics occurs when the signal frequencies involved correspond to wavelengths on the order of or smaller than the considered circuit's dimensions. Because of this, the propagative nature of electromagnetic waves can no longer be neglected. Our target bandwidth goes up to 10 GHz, i.e. down to a wavelength (in vacuum) of 3 cm. As a result, since the circuits at play will obviously be larger than this length scale, we are dealing with so-called "high-frequency" electronics.

An introduction to high-frequency electronics is proposed in the appendix section A 2 .4, following the nice and quite detailed book by Pozar [START_REF] David | Microwave Engineering -Third Edition[END_REF]. The focus is on the notion transmission line, that is to say a reference or ground electrode facing the signal-carrying electrode. The context of the problem is the propagation of a signal from one end of the transmission line to the other end. Discretization of an infinitesimal section of a transmission line leads to a description represented in Fig. 2.46. In the general case, such a line element of length dz → 0 acts as a quadrupole through which a certain voltage v(z, t) and corresponding current i(z, t) flow, along the z direction. v is measured with respect to the bottom reference electrode (as sketched with the ground symbol). The quadrupole features a resistance per unit length R, a self-inductance per unit length L, and a shunt conductance per unit length G as well as a capacitance per unit length C from signal to ground. The current i through the signal line is also affected by the quadrupole in dz.

Physically, this modelling is meaningful because on one hand, R and G account for ohmic and dielectric losses respectively, while L and C represent the current loop and capacitance that is created by the two facing electrodes. Already at that stage, it is interesting to pause and notice that the line's geometry is already appearing in the description of signal propagation. In other words, the line design can already be expected to be much more important than in quasistatic electronics. While we are speaking of design, let us remember that for the sake of keeping the samples (i.e. the nanowires featuring domain walls) atop the field-generating transmission line, the latter should be as flat as possible, with minimal roughness. Indeed, if the wires are to be laid atop the antenna, then any surface roughness comparable or higher than the typical nanowire radius is likely to damage the sample, not to mention make investigations more difficult. This constraint rules out using commercial Printed Circuit Boards (PCB) since the conductor layers have a significant roughness, on the order of one micron (root-mean-square). Considering this, other techniques of micro-or nanofabrication must be considered for the antenna with which to apply magnetic fields to the nanowires.

Let us now come back to the matter of signal propagation along the transmission line. Even though this was not stressed beforehand, all characteristic quantities such as R, L and so on may not be constant along the line. Based on the above paragraph, changes in geometry along the propagation direction will directly impact L and C for instance. Also, changes in the materials constituting the line are also liable to impact e.g. R or G. Therefore, optimizing signal transmission requires a fine control not only over the geometry, but also the materials of the line.

The formalism relevant for signal transmission will not be detailed here, but rather in the appendix section A 2 .4. The easiest way in is to consider the propagation of a monochromatic wave v(z, t) = V 0 exp (iωtikz) (with ω = 2πν the angular frequency) along the line. Physically, the coupling between oscillating current and voltage leads to a set of wave equations that are often called the telegrapher's equation [START_REF] David | Microwave Engineering -Third Edition[END_REF]:

∂ 2 v(z, t) ∂z 2 + γ 2 v(z, t) = 0 (2.35) ∂ 2 i(z, t) ∂z 2 + γ 2 i(z, t) = 0 (2.36)
with the propagation constant γ such that γ 2 = (R + jLω)(G + jCω). Obviously, starting from a finite amplitude at z = 0, there is a requirement Re(γ) ≥ 0. Furthermore, the voltage-to-current relation is given by:

v(z, t) i(z, t) = γ(ω) G + jCω = R + jLω G + jCω ≡ Z 0 (ω) (2.37)
where the square root is an abusive (but practical) notation for the complex square root with a positive real part, for consistency. This ratio is of course time-independent 39 , with the dimension of an impedance; it is hence named characteristic impedance of the transmission line. While the concept may appear puzzling, it is easier to understand in analogy with acoustic impedances in hydrodynamics. The lower that latter impedance, the stronger the flow for a given acoustic pressure amplitude. Here, the lower |Z 0 |, the larger the current will be at fixed voltage amplitude. Since the study of wave propagation has lead us to a certain line impedance, one can expect that an abrupt change in this property should directly affect the propagating signal. Let us have a look at the situation in Fig. 2.47, in which an abrupt change of characteristic impedance from Z 0 to Z 1 occurs along a transmission line (see orange dashed line at z = 0). Both line sections are semi-infinite and further characterized by their propagation constants γ 0 and γ 1 . The wave behaviour is determined by γ 0 and Z 0 on the left hand side, and by γ 1 and Z 1 on the right hand side. Here, a forwards-propagating wave with amplitude V p at z = 0 -is being sent from -∞. At z = 0, the wave must obey to both sets of constraints, hence, there exists a backwards-propagating wave on the left hand side with amplitude V m (at z = 0 -), as well as a forwards-propagating wave on the right hand side, with amplitude V 1 (at z = 0 + ). In other words, both the current-to-voltage relations given by Z 0 and Z 1 at z = 0, and the wave equations with γ 0 and γ 1 constitute boundary conditions for the problem.

z Z 0 Z 1 V p e -iγ 0 z V m e +iγ 0 z V 1 e -iγ
It can be shown (see appendix section A 2 .4) that solving this propagation problem leads to:

V m = ΓV p V 1 = V m + V p = V p (1 + Γ)
where

Γ = Z 1 -Z 0 Z 1 + Z 0 (2.38)
is the reflection coefficient from the left line section to the right line section. In terms of modulus, |Γ| < 1; this coefficient is linked to the power transmitted to the right hand side. Indeed, the transmitted power P t at z = 0

reads P t = P i • (1 -|Γ| 2 ), with P i = |V p | 2 /(2Z 0 )
is the forward sent power (also at z = 0). By conservation, a power P r = |Γ| 2 • P i is reflected.

We are arriving at the key notion of impedance mismatch. From the above, it is easily seen that any difference between Z 0 and Z 1 leads to non-zero reflection and thus to losses in signal power further along the line. Now, seen from z = 0 towards the right, the right hand side line section is an electronic dipole, characterized only by Z 1 . We may wonder what would happen if the Z 0 -line were terminated by a discrete impedance Z 1 at z = 0. In fact, the algebra is mostly the same, and the same reflection coefficient as well as reflected and transmitted powers. The meaning of the latter power now changes to the power dissipated by the impedance or load Z 1 .

This consideration enlightens the notion of characteristic impedance. Furthermore, we can now assimilate a discrete element Z 1 terminating a transmission line to a line with characteristic impedance Z 1 and terminated with a discrete load Z 1 , that line then stretched out to infinity. The locations of termination elements or abrupt changes in characteristic impedance are usually referred to as reference planes.

At this stage, we may summarize the above notions as follows:

• The building blocks of a pulse (in Fourier space) i.e. monochromatic waves propagate along a transmission line with a propagation constant γ which is a priori frequency-dependent. Both the geometry and the materials involved in the line design play a role to determine this constant. The middle one with width w is labelled "S" as it is the signal electrode, while the two others to the sides, with respective widths b 1 and b 2 , are grounded and therefore labelled "G". The central electrode is separated from the two others by the gaps g 1 and g 2 . The dielectric slab on which the device lies has a dielectric constant ε r and a thickness h.

• Similarly to hydrodynamics in the acoustic approximation, a line viewed as transmission medium possesses a characteristic impedance, which is the voltage-to-current ratio. Changes in this impedance along the line, in other term impedance mismatches, result in reflections both in amplitude and in power.

• Terminating a line with characteristic impedance Z 0 by a discrete load of the same impedance leads to null reflection. This means that impedance matching is the optimal situation for transmitting power without losses.

Nowadays, high-frequency cables and many pulse generators are designed to be 50 Ω-matched. This is to say that e.g. BNC or SMA cables are transmission lines with minimal losses and a characteristic impedance of 50 Ω. Conversely, these pulse generators, like the Avtech R pulse generator, have their outputs impedances matched to 50 Ω.

Practically, this means that the antenna with which to apply fields to nanowires must have a characteristic impedance of 50 Ω, and be terminated with a discrete 50 Ω load for maximum efficiency, not to mention safe use of the Avtech R pulse generator. From the above, the consequences of impedance mismatches are clear: some power is lost, and furthermore reflections over a significant frequency range imply that part of the pulse bounces back. This is of great importance for the termination impedance: here, mismatches mean that part of the pulse travels backwards and generates another pulse of Ørsted field. Such a situation is a strong hindrance since it corresponds to applying not one well-defined pulse, but more than one, with alternating signs.

IV.1.d Geometrical considerations

Now that we have discussed the notion of characteristic impedance for our antenna, it is time to consider and define its geometry. Although we represented a transmission line with two metallic electrodes in Fig. 2.46 for the sake of simplicity, in the general case, there may be just one, two, or even more electrodes for a single line [START_REF] David | Microwave Engineering -Third Edition[END_REF]. From circular or rectangular waveguides to the commonly used coaxial cables through microstrips (a planar stripe separated from an infinite ground plane by a thickness of dielectric), there is a wide variety of distinct transmission line geometries. In this section, neither the whole spectrum of transmission lines nor their tunability will be described; the focus will be on the geometry that I used: the coplanar waveguide or CPW in short.

A cross-sectional view of this type of microwave line is represented in Fig. 2.48. The CPW is a so-called Ground-Signal-Ground (GSG) geometry, which is to say that the signal-carrying electrode is surrounded by two grounded electrodes. Here, the central electrode with width w is therefore labelled "S", while the two others with widths b 1 and b 2 are labelled "G". All electrodes have the same thickness t. The signal electrode is separated from the side electrodes or ground planes by gaps g 1 and g 2 . The dielectric substrate, whose thickness is t, is represented in green and labelled by its dielectric constant ε r . In practice, CPWs are designed symmetric with respect to the vertical symmetry plane of the signal electrode, leading to only one gap g, and one ground plane width b.

As a side note: the notion of capacitance and self-inductance per unit length are clearer to visualize on this example due to the vertical facing metallic planes in Fig. 2.48 on one hand, and due to the necessary loop arising from the waveguide's termination linking the signal electrode to the ground electrodes. It is also worth mentioning here that in general, a dielectric material's relative permittivity ε r is not a simple, real constant, but a complex-valued function of the frequency: ε r (ω) = ε r (ω) + iε r (ω). Without entering the details here, it is clear 40 that this dispersive behaviour will impact the capacitance per unit length C of the waveguide. As examples, the real relative permittivities 41 of alumina have negligible dispersion from 100 Hz i.e. DC to at least 8 GHz [55], but those of antimonous sulfide iodide SbSI decrease from 25 in the kHz range to 11 in the tens of GHz range [55].

Ideally, the geometry as well as the materials chosen for the waveguide should:

• minimize dispersive effects,

• allow for an easy definition of its lateral dimension,

• retain the possibility of large current densities for the applied field.

To that end, I have chosen alumina as dielectric substrate since its relative dielectric permittivities are constant over the bandwidth that I am targeting 42 . The above-mentioned measurements [55] concern the real part ε r ; as for its imaginary part (which is responsible for dielectric losses along the line), the book by Pozar [START_REF] David | Microwave Engineering -Third Edition[END_REF] reports a tan(δ) of 3 × 10 -4 at 10 GHz, Gupta et al. 2 × 10 -4 at the same frequency [START_REF] Gupta | Microstrip Lines and Slotlines[END_REF], where the loss tangent tan(δ) is defined as:

tan(δ) = ε r ε r . (2.39)
This dimensionless quantity is helpful in evaluating the performance of a dielectric material in terms of loss. There are few (if any) dielectric materials used in high-frequency applications combining such a low loss tangent [START_REF] David | Microwave Engineering -Third Edition[END_REF] and a medium relative dielectric permittivity. The latter is favourable as it allows, in short, to shrink the line's lateral dimensions: as was stated above, decreasing the central electrode's width as much as possible is beneficial for the current density and therefore the applied field.

Coming back to the geometry, CPWs feature low dispersion of their dominant transverse electromagnetic (TEM) propagative modes until frequencies on the order of 10 GHz [START_REF] Gupta | Microstrip Lines and Slotlines[END_REF]. Another dispersion-related aspect to consider are odd modes, that is to say modes for which the electric field has the same direction in both gaps, therefore having a detrimental effect on the transported current 43 . In a hand-waving approach, they constitute a channel for electromagnetic power propagation at a given frequency, but in a way that does not benefit to the applied field. These odd TEM modes are allowed in the GSG geometry but can fortunately be suppressed by short-circuiting the lateral ground planes.

Yet other modes can disturb the signal propagation along the CPW: there may exist so-called transverse electric (TE) or transverse magnetic (TM) modes, for which only one of the propagating field is perpendicular to the line. The coupling between the longitudinal field and the other field components combined with the boundary conditions arising from the transmission line's geometry results in a non-linear dispersion relation with a cut-off frequency (which might be zero). Recalling equation (2.37), we can expect the characteristic impedance of the transmission line for such a mode 44 to be strongly frequency-dependent. Therefore, the very notion of current for a given voltage amplitude is blurred for TE or TM modes; these are hence to be suppressed as much as possible. This can be achieved by using their aforementioned cut-off frequencies: pushing these frequencies above the targeted bandwidth provides a means to avoid exciting the TE and TM modes. Concerning the CPW geometry, the relevant cut-off frequencies to consider are taken as those corresponding to the surface transverse modes of a dielectric slab with a metallic plane on one side 45 , or approximations thereof [START_REF] David | Microwave Engineering -Third Edition[END_REF][START_REF] Gupta | Microstrip Lines and Slotlines[END_REF][START_REF] Hasnain | Dispersion of Picosecond Pulses in Coplanar Transmission Lines[END_REF]. The cut-off frequency f TE 1 of the first TE mode, the TE 1 , reads [START_REF] David | Microwave Engineering -Third Edition[END_REF]: 40 For further details, the reader is invited to read for instance the beginning of chapter 7 in the book by Gupta [START_REF] Gupta | Microstrip Lines and Slotlines[END_REF]. 41 There are two, since Al2O3 has hexagonal crystal structure, and is therefore not isotropic. 42 For such a bandwidth, it was shown by Gupta et al. [START_REF] Gupta | Microstrip Lines and Slotlines[END_REF] (see e.g. pp. [112][113] that alumina as a substrate is one of the best dielectric for transmission lines in this frequency range. 43 Aside from this, they considerably complicate the propagation problem since they have different propagation constants than the dominant, even mode [START_REF] Gupta | Microstrip Lines and Slotlines[END_REF]. 44 It is very important here to observe that implicitly, the characteristic impedance so far was always corresponding to a TEM mode, assumed to be the dominant line mode. 45 Surely there would be corrections due to the detailed geometry, but the following estimates should remain of the right order of magnitude. More details on such discussions can be found in the book by Pozar [START_REF] David | Microwave Engineering -Third Edition[END_REF].

f TE 1 = c 4h ε r -1
where c 3 × 10 8 m/s is the speed of light in vacuum, and h is still the dielectric slab's thickness. As for the transverse magnetic modes, the corresponding cut-off frequencies f TMn for the TM n modes (still of the grounded dielectric slab) read:

f TMn = nc 2h ε r -1
where n is a positive integer, so that the first TM mode has null cut-off frequency. Thus, only modes with index n ≥ 1 can be avoided. As can be seen from these equations, the lowest cut-off frequencies for transverse modes differ only by a factor two. An estimation of f TE 1 yields 70 GHz to 80 GHz for an alumina substrate with h =330 µm (depending on the chosen value 9.34 to 11.5 of ε r ). As it is lower than f TM 1 but much larger than the targeted bandwidth of 10 GHz, we can conclude that neither TE nor TM modes should be excited in the considered case. It should therefore be safe to reason only on the dominant, (quasi-)TEM waveguide mode.

As last few arguments in favour of choosing the coplanar waveguide over other line geometries (see Ref. [START_REF] David | Microwave Engineering -Third Edition[END_REF] for examples of other geometries), it was found that the losses of the CPW are usually lower than those of the coplanar strips (ground-signal geometry analogous to the CPW, with only two electrodes) for comparable design dimensions [START_REF] Gupta | Microstrip Lines and Slotlines[END_REF], and that the conductor losses in the CPW geometry can be made comparable or lower than those in the microstrip geometry (one full side of the dielectric slab is grounded, while a strip on the opposite side is the signal electrode). Those two other line types would have been the other most relevant ones aside from the CPW considering the goal of magnetic field application. Another argument in favour of the CPW over the microstrip is that the latter does not allow for an easy design of the line termination because the signal and ground electrodes are not on the same plane. As was explained at the end of section IV.1.c, the issue of line termination is critical in my case.

IV.2 Specifications for the coplanar waveguide and fabrication

Now that the transmission line geometry has been chosen, its dimensions remain to be determined. The remaining constraints are the following:

• featuring a characteristic impedance of 50 Ω for matching,

• being terminated with a 50 Ω load in order to suppress reflections at the CPW's end,

• having as low a central track width w as possible in order to enhance the applied field through the current density,

• possessing a surface with low enough roughness in order to not damage the nanowires to be placed upon it,

• keeping a safe margin with respect to dielectric breakdown.

The last constraint obviously results from the pulse voltages which can be delivered by the Avtech R pulse generator. This will be examined again after a first estimate for the line dimensions is given. That estimate is built on the 50 Ω characteristic impedance constraint, while searching for an optimal signal electrode width w. In this process, I first made a coarse search for relevant parameters before turning to CHRISTOPHE HOARAU from Institut Néel to refine the initial guess with electromagnetic field simulations 46 . The a priori coarse guess was obtained with the nice, light and free software TXline [153]. It only simulates CPW with infinite ground planes; it turns out that wide lateral ground planes decrease the line impedance for fixed other parameters, thus making it (in practice) easier to go down to 50 Ω. For consistency in the design then,

b/w → ∞, b/h → ∞, b/g → ∞.
Because of an instrumental constraint that the sample holders in this work all have a metallic top surface, the geometry to simulate with TXline [153] is not the simple CPW but rather the so-called conductor-backed coplanar waveguide. As the name implies, the single change with respect to the CPW is an infinite (grounded) metallic plane on the bottom side of the dielectric slab serving as substrate. This geometry is illustrated in Fig. 2.49, which presents an output from TXLine with parameters relevant for my work: alumina substrate with relative dielectric permittivity ε r =9.8 and thickness h =330 µm, copper metallization with trial thickness t =1 µm, and a width and In the software, "um" stands for "µm". Material parameters (top left) and lateral dimensions (bottom right) have been chosen so as to obtain a characteristic impedance of about 50 Ω at a working frequency of 10 GHz.

gap w respectively g leading to a target characteristic impedance close to 50 Ω at a working frequency of 10 GHz.

As for the lateral ground planes, their extent was limited by a constraint on the width of the sawed alumina substrates (see section III.1), however, the ratio (w/2 + g + b)/(w/2 + g) ranged from about 7 to 8. This is already in the range where the effective dielectric permittivity is no longer sensitive (by more than a percent) to the value of this ratio, see e.g. [START_REF] Gupta | Microstrip Lines and Slotlines[END_REF] p.389 Fig. 7.9. Therefore, we actually are in the infinite ground planes approximation.

It must be noted that varying the used parameter ε r between, say, 9.34 and 11.5 using the values reported in the Handbook of Chemistry and Physics [55] leads to variations of a few ohms in the characteristic impedance. Furthermore, there is a (low, a few ohms from 100 MHz to 10 GHz) dispersion of this property, so that seeking 50.00 Ω is pointless.

In this case, a perhaps more surprisingly low variation in characteristic impedance occurs when the gap value g is varied from 0.25 mm to 0.50 mm. Considering the dielectric material's thickness h at play, following Fig. 7.11 p.393 in the book by Gupta et al. [START_REF] Gupta | Microstrip Lines and Slotlines[END_REF], one would expect the characteristic impedance to vary47 by a little less than 10 Ω. With TXline, a variation in characteristic impedance of only about 3 Ω was found for the above-mentioned variation in gap. Using the thickness-corrected effective dielectric constant ε r,eff 6.2 calculated by TXline for a conductor thickness of t =1 µm, we may try to understand the splitting δν 4 GHz [START_REF] David | Microwave Engineering -Third Edition[END_REF]. Although quantitatively larger, this value is in agreement with the simulated behaviour, enabling us to understand that the multiple resonances seen especially well in Fig. 2.51 correspond to standing waves in the CPW, trapped between its extremities. Since both the reflection and the transmission drop (as they should) due to these standing waves, the power lost inside the device peaks at these resonance frequencies, as can be seen on the blue curve in Fig. 2.54. This graph represents the quantity 1 -|S 11 | 2 -|S 12 | 2 which is the total relative power (i.e. normalized by the input power) minus the transmitted (|S 12 | 2 ) and the reflected (|S 12 | 2 ) relative power: in other words, the power which, one sent into the device, does not come out. This power is then lost for instance through ohmic losses or radiated away.

IV.2.a Simulations for a 25 mm waveguide

From the above, it is clear that with an effective dielectric constant ε r,eff that is fixed by the constraints of geometry, materials and impedance matching, such resonances will always be present a priori. It is therefore desirable to reduce their effect, e.g. by suppressing their amplitude or by pushing their resonance frequencies out of the 10 GHz bandwidth. This is an easier way out, since it consists in controlling the length over which there is invariance by translation along the line, i.e. over which standing waves can appear. In the previous paragraph, it was L the device's length, but local breaking of translation symmetry can reduce this effective length. This can be achieved without actually changing the line geometry by implementing so-called vias or lateral contacts between the grounded electrodes to the side of the CPW and the conductor backing. Such electrical contacts only enforce an equipotential between two nominally grounded electrodes, and so they do not disturb the geometry, while breaking the translation symmetry.

Such vias were added in another set of simulations by CHRISTOPHE HOARAU. They are round holes in the dielectric substrate (with diameter about 0.2 mm) filled with a conductor joining the top and bottom ground electrodes, and spaced by 1 mm. From the above, the first resonance frequency is expected to be an order of magnitude higher, and as can be indeed seen in Fig. 2.53, no peaked minimum of transmission remains in the 10 GHz bandwidth. Furthermore, the reflection behaviour appears much less erratic, as is shown in Fig. 2.52. It is striking to see how the trend of increase in |S 11 | with frequency is suppressed; also, the maximum value reached is now -30 dB. As a conclusion regarding this comparison, it is clear that the addition of vias should greatly improve the line behaviour.

Coming back to losses, there is very clear improvement shown in Fig. 2.54 when vias are added, see red curve. The baseline is similar albeit lower with vias, and the peaked losses have been completely suppressed, as could be guessed from the improvement especially in |S

IV.2.b Simulations for an 8 mm waveguide

The other set of simulations concerned a line with the same lateral dimensions, but shorter: the length was L =8 mm. With the same approach as above, this length should correspond to a splitting between resonances of about δν 8mm (exp) = c/(2 √ ε r,eff L ) 7.5 GHz. This is larger than the position of the first dip in the transmission in Fig. 2.56, corresponding to a marked, localized dispersive feature in the reflection in Fig. 2 Once more, the optimum in terms of central electrode width w is seen in Fig. 2.56 to lie between 250 µm and 300 µm, but even more so from Fig. 2.55 where the reflection is seen to be lowest in the DC regime for 300 µm. For this width, the average value of |S 11 | over the 10 GHz bandwidth is also the lowest for the simulated widths.

Considering this width, the effect of 0.5 mm-diameter via holes separated by 1 mm along the CPW's length was also investigated, for a width of 200 µm. The results of the corresponding simulations are displayed in Figs.2.57 and 2.58.

The stark minimum present in |S 11 | for all geometries, at a width-dependent position between ca. 7 GHz and 8.5 GHz, seems almost unaffected by the presence of the vias, supporting the idea that this is the signature of a transverse or finite-width resonance. A further interpretation of this resonance goes beyond the scope of this work: it can be seen in Fig. 2.58 to correspond to high |S 12 | transmission anyway, and since it is a dip in reflection and not a local maximum, it does not contribute to losses. The latter are noticeably lower than for the longer waveguide, as is illustrated by Fig. 2.59 (note the vertical scale), which indicates that there is a role of impedance mismatch but that losses per unit length (due to conductor losses mostly [START_REF] Gupta | Microstrip Lines and Slotlines[END_REF]) are not completely negligible. Impedance mismatch alone with no losses per unit length would be seen as a constant reflection behaviour with respect to the device length, while if the impedance mismatch were zero, then the lost power should be L /L = 0.32 times smaller in the shorter device 48

IV.2.c About the metallization thickness

In all the simulations discussed in section IV.2.a and section IV.2.b, the top (copper) metallization thickness t was assumed to be uniform and equal to 1 µm. Now that we have looked at the losses inside the waveguide and seen that conductor losses are not negligible, the question of whether this parameter may help reducing them 49 . In the DC regime, the cross-section of the signal electrode increases with t, which is beneficial as it decreases the waveguide's resistance, while in the high-frequency regime, it is the skin depth δ s that matters, or rather the ratio δ s /t. thickness t with two halves (along the length), both perfectly impedance-matched to one another, with a sharp interface between two inequivalent conductors. At the working frequency ν, let us assume that the skin depth δ s1 is quite lower in the first half than that δ s2 in the second half. Now, an electromagnetic wave with frequency ν is incident on the interface. The electric and magnetic fields decay faster in the first conductor, that is why in the volume there will be a discontinuity; the latter can only be corrected with a reflected wave travelling back into the first half of the CPW. The additional fields of the reflected wave compensate the initial lack in the volume and allow for continuity. So far, however, the line thickness has not played a role. Starting from the situation in which t is much larger than both skin depths, we have a first imbalance in fields. If now t becomes comparable with the larger skin depth δ s2 , then the fields in the second line half from the top and bottom surfaces begin to add up (which previously was impossible because of the skin effect) but those in the first line half cannot do it as efficiently. Therefore, for a fixed incident wave amplitude, the reflected wave's amplitude must grow significantly. For copper, the skin depth at 1 GHz respectively 10 GHz is 5.2 µm respectively 1.7 µm. Therefore, the transition from the regime where the thickness has a strong impact to the "perfect conductor" regime (i.e. δ s /t → 0) should occur in the micrometer range.

CHRISTOPHE HOARAU has performed simulations for a constant CPW geometry, varying only the thickness. Typical results are shown in Fig. 2.60: the transmission behaviour of a CPW up to 10 GHz is represented for a copper thickness t from 1 µm down to 500 nm. As expected from the above discussion, |S 12 | decreases at any frequency when t decreases (except for the positive and negative deviations for the pink curve ↔900 nm, which is believed to be a simulation artefact). Moreover, since the difference in thickness between one curve and the next is 100 nm, is is clear that the thickness-dependence of |S 12 | over t is small at larger thickness and increases with decreasing thickness. Considering the above discussion, it is not so surprising to see that going into the micrometer range corresponds to a vanishing dependence of the line behaviour on δ s /t.

Considering this, is appears desirable to target a thickness on the order of 1 µm to 2 µm or higher. However, there is a very tedious issue associated with large thicknesses of metal: their surface roughness. Indeed, we must keep in mind that the top surface of the CPW must feature a roughness that is as low as possible, as was mentioned p.110. Also, surface roughness can complicate investigations with AFM-MFM or XMCD-PEEM, as discussed in section II. The problem is that thick film fabrication techniques lead to an increasing roughness with increasing film thickness.

IV.2.d Review

In this section, we have obtained first a coarse, then a refined set of geometrical parameters for the conductorbacked CPW, considering an alumina substrate of about 300 µm. Under the constraint of characteristic impedance matching to 50 Ω, the target width is about w =300 µm for a gap g of 250 µm. The lateral ground plane width b was set by a constraint on the total device width w + 2g + 2b which results from the available size under the PEEM sample holder cap (see Fig. 2.17 p.81).

With respect to the list of requirements from p.113, we now know that a rather small dispersion is expected in the 10 GHz bandwidth. Furthermore, the typical lateral dimensions are large enough to allow for precise definition e.g. by optical lithography: they would be an issue for machining tools, but these would only be relevant for PCB-type conductor layers, and these have been ruled out for their surface roughness.

The application of large fields depends of course on the maximum magnetic field that can be produced just over the central track of the CPW. Giving an order of magnitude based on an estimate of the current associated with a pulse is feasible, but predicting the applied field value requires precise knowledge of the propagating mode of the actual device, which is way beyond the scope of this work. Assuming perfect, dispersion-free impedance matching to 50 Ω, the current pulse corresponding to the Avtech pulse generator's voltage pulse (see Fig. 2.44) has the same shape and current values are obtained by dividing the voltage values with 50 Ω. Thus, the maximum current I max can be at most I max 16 A. Let us assume that this current density is distributed homogeneously over the whole cross-section of the signal electrode 50 , thus corresponding to a current density J max = I max /(wt). The maximum induction µ 0 H a at a height δz above the central electrode's top surface, in its vertical plane of symmetry, can be shown to be ca. µ 0 H a =33 mT for δz =10 nm to 200 nm, t =1.5 µm, w =300 µm. The key parameter is the track width w; though there is more to it than just the impact on current density. A reasonable field homogeneity requires the samples to be very close to the track, if possible directly above its center, which is very challenging for widths on the order of w =300 µm, unless the wires lie directly on the CPW.

Although realistic losses have not been included in this estimate, it must be noted than even with a loss of half the field amplitude, this field estimate remains above most, if not all, of the propagation field distribution sampled by SANDRINE DA COL [29] and co-workers [81] in wires very similar to those this manuscript is concerned with. In these MFM investigations, it was found that at least 90% of the DC propagation fields of domain walls in Permalloy or nickel nanowires (however with diameters 80 nm or slightly less) were below 15 mT. It is noteworthy that the above-discussed derivation of lateral dimensions for impedance matching, avoidance of dielectric breakdown, large field amplitude and so on was presented in a logical but not chronological order. Every geometrical parameter influences several distinct aspects of pulsed field application, with many constraints at the same time, not to mention the choice of pulsed voltage amplitude. Historically, there has been quite a lot of back-and-forth and trial-and-error before a compromise satisfying all requirements was found. A good example is the voltage amplitude: it is desirable to have it as large as possible, but then there is a risk of dielectric breakdown in air. Therefore, one could think of increasing the gaps, but then the central electrode width must be increased for the sake of impedance matching. This width in turn plays the biggest role in the maximum field amplitude through the current density, and increasing it causes the applied field to drop significantly.

At this stage, the design of the transmission line itself is complete. There remains the questions of the 50 Ω termination impedance, the line fabrication in practice, the connection to the Avtech pulse generator, and the sample holder(s).

IV.3 Ni 80 Cr 20 termination for 50 Ω loads

As was discussed previously, the typical thickness of the CPW is expected to be on the order of a micron. In order not to create additional impedance mismatches, it is desirable to create a 50 Ω load with as easy a connection to the rest of the line as possible. From this perspective, a thin stripe of conductor with adequate dimensions and resistivity appears to be the simplest design. Thus, considering the GSG geometry of interest, the termination has to feature two 100 Ω resistors in parallel from the signal electrode to the lateral ground planes, as illustrated in Fig. 2.61.

The resistance of one strip R load being equal to ρ load L load /S load , with ρ load the resistivity of the load's material, L load and S load = w load t load the strip's length and cross-section, we should note that there already is one constraint. Indeed, the length L load corresponds to the gap g, as can be seen in Fig. 2.61. Therefore, the impedance design consists in choosing a conductor with sufficiently high resistivity in order to attain a strip resistance R load =100 Ω. The high resistivity (with respect to usual metals) is necessary in order to retain lateral dimensions w load and t load sufficiently high so as to ensure good control over these.

As an example, the thickness t load of such a 100 Ω resistor with an 80 µm width would be estimated at 5 Å with the material's bulk resistivity [55]. Obviously, this value is meaningless since the bulk resistivity of copper loses relevance at such a thickness scale, but it illustrates well the difficulty of obtaining large resistance values with conventional metals.

In order to take inspiration from reliable sources, I investigated an impedance standard substrate from Cascade Microtech R (reference 101-190 C, Ground-Signal-Ground). The full arrangement of impedances on the substrate is reproduced in the appendix Fig. A 2 .3; a zoom-in on a few components is presented in Fig. 2.62.

Of particular interest is the 50 Ω load; I took it as my starting point since the dimensions (in order of magnitude) and impedance were exactly what I was targeting. I measured the length and width to get an estimate: 50 ± 10 µm and 20 ± 2 µm; the thickness was about 50 ± 10 nm. The (large and) different measurement errors are linked to the method, which was mechanical profilometry, using a Dektak R 6M apparatus. The principle is to scan the sample surface with a stylus terminated in a diamond tip, which has a radius of 12 µm. A constant force is applied, ensuring contact, and the tip height is recorded during a line scan. Due to the fact that along the length, the resistor strip is bounded by the abrupt edge of the connector pad (in yellow in Fig. 2.62), whereas across the width the sides are free, there is a more significant influence of the tip's finite radius in the first case. Thereafter, I turned to energy-dispersive X-ray analysis (EDX) to have a crude idea of the material used in the Cascade Microtech R 50 Ω load. I had beforehand an educated guess thanks to discussions with CHRISTOPHE HOARAU, according to whom nickel-chromium alloys are used in such devices. As an example, Ni 80 Cr 20 is known [START_REF]ASM International Handbook Committee, Properties and Selection: Nonferrous Alloys and Special-Purpose Materials[END_REF] to feature a rather high resistivity, about 1 µΩ•m at 20 • C, low relative variations of resistance with temperature (less than 3% for more than 100 • C temperature increase), as well as a significant resistance to oxidation. Interestingly, this alloy has been reported "non-magnetic" [START_REF]ASM International Handbook Committee, Properties and Selection: Nonferrous Alloys and Special-Purpose Materials[END_REF], which is somewhat in agreement with the expected null magnetic moment per atom as obtained from the Slater-Pauling curve [103] close to this composition 51 .

The EDX spectrum obtained on the Cascade Microtech R 50 Ω load (field emission SEM from ZEISS ultra+, ZEISS R Germany) is represented in Fig. 2.63. Several well-defined peaks, labelled with coloured straight lines and rectangles for the most intense, are visible. They can be attributed to elements present in the volume probed by the electrons, which reaches further (the order of magnitude is the micrometre) than the resistor strip. Therefore, the alumina substrate used for the impedance substrate yields oxygen and aluminium peaks. It is very clear that most if not all the other peaks are explained by the presence of nickel and chromium 52 . This strongly supports the hypothesis that the resistor strip is a nickel-chromium alloy.

The composition analysis was of course wrong since the whole volume that is probed by the electrons reaches much deeper than the resistor strip is thick. Since the analysis software assumes a homogeneous medium, the contribution from the thin resistor is volume-weighted against the much larger portion of alumina substrate that is also probed. This contributes to explaining the relatively low intensities for the Ni and Cr peaks as opposed to Al and O. The strong deviation from the sample model used in the composition analysis and the low signals also forbid any conclusions regarding the ratios of Ni and Cr contents in the resistor strip. For the sake of exhaustiveness, the computed composition is represented as inset in Fig. 2.63; it seems (logically) rather consistent with the presence of the Al 2 O 3 substrate.

Nevertheless, the confirmed presence of Ni and Cr in this commercial impedance is a strong incentive to try and use a nickel-chromium alloy in my case. While there may well be several other elements which went undetected in the EDX, for the above-mentioned reasons, the simple Ni 80 Cr 20 alloy appears to fulfil all requirements for the termination impedance of my CPW.

To that end, with the support of PHILIPPE DAVID from Institut Néel, a Ni 80 Cr 20 sputtering target with 76.2 mm diameter, 3 mm thickness and purity above 99.9% was purchased from Neyco R .

APPENDICES FOR CHAPTER 2

A 2 .1 THS118 Toshiba Hall probe specification sheet A 2 .2 Working frequency for the coils during magneto-optical calibration Considering that the input current for the coil pair is sinusoidal, the corresponding field is necessarily sinusoidal itself. One may however argue that the working frequency should be chosen as low as possible in order to benefit from an optimal current amplitude and therefore a maximum magnetic field amplitude.

It turned out that the LeCroy oscilloscope (model Waverunner 625Zi) used during the experiments had difficulties to properly trigger on AC signals with a frequency lower than 7 Hz. For this reason, I set the working frequency to be f 0 =7 Hz. In order to check that the coil system's cut-off frequency f c was significantly higher, I performed measurements of the photodiode signal amplitude at two frequencies 53 , f 1 =9 Hz and f 2 =18 Hz; the results were E(f 1 ) =3.32 ± 0.04 mV and E(f 2 ) =3.11 ± 0.03 mV. Let us call κ = 0.94 ± 0.02 the ratio E(f 2 )/E(f 1 ). Low-frequency electronics allows us to write:

1 + f 1 fc 2 1 + f 2 fc 2 = κ (A 2 .1)
Considering that f 2 = 2f 1 , this can easily be solved to yield f c :

f c = f 1 3 1 -κ 2 = 48 ± 8 Hz (A 2 .2)
This result confirms that the cut-off frequency is definitely above the chosen working frequency, yet not even one order of magnitude larger. Therefore, the choice to work just above the threshold for proper oscilloscope triggering is justified. Since the substrate is transparent at the wavelength of interest, internal reflections do occur. One expects the field amplitude to decrease fast (as a power law) with the number j of reflections, yet, the polarization rotation increases (linearly) with j, which means that this effect is possibly not negligible. What matters here is the refraction index n of the sample at the working wavelength, because it will govern the strength of the internal reflections. According to several publications [START_REF] Kazei | Magnetic properties of garnets[END_REF][START_REF] Hoefer | Thermo-optic properties of gadolinium garnet laser crystals[END_REF][START_REF] Bohm | Optical dispersion of gadolinium gallium garnet (GGG)[END_REF], the optical index of GGG at room temperature and 633 nm is n = 1.96 ± 0.01. Thus, we can now introduce the reflection coefficient r for the electric field amplitude at the GGG/air or interface. Considering a value of 1 for the optical index of air as well as our normal incidence in the experiment, its value read:

r = n -1 n + 1 = 0.324 (A 2 .3)
Since the value of the transmission coefficient t will be the same for all beams coming out of the sample, we can safely forget it in the calculation of the total field -→ E t about to be transmitted across the GGG/air interface. In summary, what needs to be described is very reminiscent of a Fabry-Perot interference device, and is sketched on the following figure:

Keeping the same conventions as on Fig. A 2 .1, let us now consider (with complex notation) an initial electric field -→ E 0 right after transmission through the air/GGG interface such that:

-

→ E 0 = E 0 e iθ 0 ( - → e x -i - → e y ) (A 2 .4)
Here, coming back to real fields, one has:

Re( -→ E 0 ) = E 0 (cos θ 0 - → e x + sin θ 0 - → e y ) (A 2 .5)
that is to say a field whose polarization is tuned by the angle θ 0 . Since the polariser is vertical, θ 0 = π/2, but for the sake of generality, it is necessary to keep a component along -→ e x for the following. Part of this field will be transmitted through the GGG/air interface and constitute the direct beam, which should account for most of the detected polarization rotation. If we denote α the Faraday rotation angle experienced after traversing once the sample thickness d, the field --→ E (1) corresponding to the direct beam reads:

--→ E (1) = -→ E 0 e iα (A 2 .6)
Indeed, if one comes back to real fields to compute intensities, one finds that:

Re --→ E (1) = E 0 [cos (θ 0 + α) - → e x + sin (θ 0 + α) - → e y ] (A 2 .7)
However, after a double bounce on the interfaces with air, a part of the beam with an amplitude r 2 times smaller is also transmitted through the GGG/air interface and therefore contributes to the polarization direction of -→ E t . Let us denote this part --→ E (3) ; it reads:

Re --→ E (3) = --→ E (1) r • e iα 2 (A 2 .8)
The real field from which to compute the intensity then reads:

Re --→ E (3) = Re --→ E (1) r • e iα 2 = r 2 E 0 [cos (θ 0 + 3α) - → e x + sin (θ 0 + 3α) - → e y ] (A 2 .9)
Based on this, the total electric field -→ E t before transmission through the GGG/air interface reads:

-→ E t = +∞ j=0 ----→ E (2j+1) = E 0 e iθ 0 ( - → e x -i - → e y ) e iα +∞ j=0 re iα 2j = E 0 ( - → e x -i - → e y ) e i(θ 0 +α) 1 1 -r 2 e 2iα (A 2 .10)
Let us work on the e i(θ 0 +α) /(1r 2 e i2α ) term. It reads:

e i(θ 0 +α) 1 -r 2 e i2α = cos (θ 0 + α)[1 -r 2 cos (2α) + ir 2 sin (2α)] + i sin (θ 0 + α)[1 -r 2 cos (2α) + ir 2 sin (2α)] [1 -r 2 cos (2α)] 2 + [r 2 sin (2α)] 2 = cos (θ 0 + α) -r 2 [cos (θ 0 + α) cos (2α) + sin (θ 0 + α) sin (2α)] 1 -2r 2 cos (2α) + r 4 cos 2 (2α) + r 4 sin 2 (2α) + i sin (θ 0 + α) + r 2 [cos (θ 0 + α) sin (2α) -sin (θ 0 + α) cos (2α)] 1 -2r 2 cos (2α) + r 4 cos 2 (2α) + r 4 sin 2 (2α) = cos (θ 0 + α) -r 2 cos (θ 0 -α) + i[sin (θ 0 + α) -r 2 sin (θ 0 -α)] 1 -2r 2 cos (2α) + r 4
Thus, we can now rewrite the total field -→ E t . Denoting E 0 = E 0 /(1 -2r 2 cos (2α) + r 4 ), it reads:

-→ E t = E 0 cos (θ 0 + α) -r 2 cos (θ 0 -α) + i[sin (θ 0 + α) -r 2 sin (θ 0 -α)] ( - → e x -i - → e y ) = E 0 [(cos (θ 0 + α) -r 2 cos (θ 0 -α)] - → e x + [sin (θ 0 + α) -r 2 sin (θ 0 -α)] - → e y ] + i[sin (θ 0 + α) -r 2 sin (θ 0 -α)] - → e x -i[cos (θ 0 + α) -r 2 cos (θ 0 -α) + i sin (θ 0 + α)] - → e y (A 2 .11)
The field of actual interest is

---→ E (Re) t = Re -→ E t . It reads: ---→ E (Re) t = E 0 [(cos (θ 0 + α) -r 2 cos (θ 0 -α)] - → e x + [sin (θ 0 + α) -r 2 sin (θ 0 -α)] - → e y (A 2 .12)
The polarization angle α with respect to -→ e y is such that:

tan α = - ---→ E (Re) t • - → e x ---→ E (Re) t • - → e y = - cos (θ 0 + α) -r 2 cos (θ 0 -α) sin (θ 0 + α) -r 2 sin (θ 0 -α)
We may now insert θ 0 = π/2. The previous equation reads:

tan α = sin α + r 2 sin α cos α -r 2 cos α = tan α 1 + r 2 1 -r 2
Taking the limit of small angles 54 , the measured polarization rotation angle α is larger than the Faraday rotation α for one passage through the sample by a factor µ such that:

µ = α α = 1 + r 2 1 -r 2 = (1 + n) 2 + (1 -n) 2 (1 + n) 2 -(1 -n) 2 = 2 1 + n 2 2 • 2n = 1 + n 2 2n = 1 2 n + 1 n (A 2 .13)
The numerical value of this enhancement is 1.24 ± 0.01 for n = 1.96 ± 0.01. This formula is interesting because of its interpretation in limiting cases:

• If n = 1, then there is no reflection at the interface, therefore, µ = 1 because only the direct beam matters.

• µ increases with n and goes to infinity if n → +∞. Indeed, the Faraday rotation could become arbitrarily large since |r| → 1 as n increases, which means that the reflected beams contribute more and more to the global rotation. However, the case |r| = 1 means that the transmission coefficient (in terms of power, not amplitude) goes to zero! This is normal because in that case, the whole beam is trapped inside the sample.

This correction of the effect of interfaces is important to extract the real rotation per unit induction per unit length, whose meaning is only material-dependent and should not be affected by the medium (here, air) surrounding the finite-size sample.

A 2 .

Polarization rotation extraction

So far, we have not delved into how the polarization rotation α is measured in practice. It is far less straightforward (if at all possible) to try and extract polarization components of the electric field vector than measuring their respective intensities. Let us take the field ---→ E (Re) t from equation (A 2 .12), and assume r = 0 and θ 0 = π/2 for the sake of simplicity. One can write:

---→ E (Re) t = -E 0 sin α - → e x + E 0 cos α - → e y (A 2 .14)
Orthogonal polarizations can be separated using a Wollaston prism; let us therefore try to extract α from two such intensities I 1 and I 2 . Since the -→ e x and -→ e y directions may not correspond to the optimal directions, let us keep a degree of freedom ψ: the angle between the directions -→ e x ( -→ e y ) and -→ e 1 ( -→ e 2 ). Experimentally, this corresponds to the angle at which the neutral line of the half-wavelength plate is set in the plane perpendicular to the beam (see 2.30 in section III.3.b). We can form the difference ∆I and sum I Σ such that:

∆I = I 1 -I 2 = ---→ E (Re) t • - → e 1 2 - ---→ E (Re) t • - → e 2 2 = E 2 0 (-sin α cos ψ + cos α sin ψ) 2 -E 2 0 (sin α sin ψ + cos α cos ψ) 2 = E 2 0 • [sin 2 (α -ψ) -cos 2 (α -ψ)] = -E 2 0 cos [2(α -ψ)] I Σ = I 1 + I 2 = ---→ E (Re) t • - → e 1 2 + ---→ E (Re) t • - → e 2 2 = E 2 0 (-sin α cos ψ + cos α sin ψ) 2 + E 2 0 (sin α sin ψ + cos α cos ψ) 2 = E 2 0 • [sin 2 (α -ψ) + cos 2 (α -ψ)] = E 2 0
54 This is reasonable since the polarization rotations are on the order of a few mrad. This (educated guess) is beginning to look promising, since these two quantities are accessible as they are proportional to the output voltages of the photodiodes. Their ratio reads:

∆I I Σ = -cos [2(α -ψ)] (A 2 .15)
At this point, the degree of freedom ψ can be put to use. For ψ = 0, we retrieve Malus's law, and the measured magneto-optical effect follows a squared cosine law. Since we want to extract the rotation that is expected to be a very small angle, this functional behaviour is highly unfavourable. In other words, the sensitivity to small α's is low. We can quantify this sensitivity S with the derivative of the above equation:

S = ∂ ∂ψ ∆I I Σ = 2 sin 2(α -ψ) (A 2 .16)
Since we have seen that ψ = 0 is not favourable, and since α 1, let us look for the maximum sensitivity S M = max(|S|) by neglecting α with respect to ψ. We thus find S M = 2 for ψ = π/4. This corresponds to an actually intuitive configuration: the one which is farthest from the crossed polariser-analyser positions. In the latter, defined by ψ ≡ 0 [π/2], the sensitivity to small rotation angles is the smallest, so that the optimum must be in the symmetric middle position. This is the explanation for the use of a half-wavelength plate: it allows to ensure that at rest, without sample, the laser's polarization arrives at 45 • with respect to the neutral lines of the Wollaston prism. The latter then splits the orthogonal polarizations whose respective intensities are then measured by the pair of photodiodes.

Assuming ψ = π/4, we now have:

∆I I Σ = -cos [2(α -π/4)] = -cos (2α -π/2) = sin 2α (A 2 .17)
In the small-angle limit, we now have:

∆I I Σ = ∆V V Σ 2α (A 2 .18)
where ∆V and V Σ are respectively the difference and the sum of the output voltages of the photodiodes used to measure the intensity of one of the orthogonal polarizations.

I Introduction

This chapter focuses on an original investigation led jointly by colleagues from the University of Zaragoza (Spain), synchrotron Soleil (France), the University of Erlangen (Germany), and Institut Néel. The sample fabrication was performed at the University of Zaragoza, and most of the experimental observations and measurements were then carried out at synchrotron Soleil. I have thereafter taken the lead in the interpretation of the obtained data [] pertaining to X-ray spectroscopy and XMCD-PEEM. Though primarily, or better said originally, a surface technique, XMCD-PEEM has been shown to be an efficient microscopy for three-dimensional ferromagnetic samples when used in the so-called shadow mode, which has been presented in Chapter 2, section II.3. A few works in the literature report investigations of ferromagnetic structures with a sizeable extent perpendicular to their substrate. From ca. 250 nm-diameter core-shell nanowires [1] lying on a Si wafer to vertical nanorods piercing a superconducting film [2] through micrometre-wide rolledup membranes aligned with [3] or perpendicular to the substrate [4], or yet hemispherical caps [5], a variety of systems for which transmission microscopy is relevant have been studied so far.

In these works however, the magnetic samples possess either aspect ratios (significantly) below ten or close to one [2,5], or aspect ratios much larger than one but are lying on the substrate [1,3,6,7], which prevents the observation of the full structure's shadow. The investigation of tilted, rolled-up Ni membranes by Robert Streubel [4] is an exception, but the imaging of only a third of the structure's length was reported. This raises the question: can the shadow of a whole upright magnetic sample with high aspect ratio be imaged with XMCD-PEEM? Furthermore, while a tomography-like analysis [4] allows to retrieve even complex domain patterns, the capabilities of shadow XMCD-PEEM suggest that simpler systems than exchange-decoupled multiple layers in a rolled-up tubular geometry can be investigated and well understood without the need for complex data processing.

As for a (relatively) simple ferromagnetic system featuring not completely trivial configurations as well as possessing both a high-aspect ratio and a naturally three-dimensional character, one may think of a soft cylindrical ferromagnetic nanowire. The like of such systems have been introduced in Chapter 2, section I; however, their fabrication with electrochemistry does not allow to keep the nanostructures vertical while singling out only one for transmission microscopy, because of the template. Another fabrication technique permits to grow single, vertical and high-aspect-ratio cylindrical nanowires with a control over both shape and material, namely Focused Electron-Beam-Induced Deposition (FEBID in short).

II Sample fabrication and experimental strategy

FEBID, as the name implies, consists in depositing material thanks to a focused electron beam which acts as trigger [8]. The key principle, which is illustrated in Fig. 3.1, is the dissociation of precursor molecules in gas phase by the primary electron beam and also to some extent by secondary electrons [9]. The latter process, as will be illustrated later on, is sometimes far from negligible. The gas precursor molecules are often carbonaceous [10][11][12], and contain the material to be deposited. The latter can be quite a number of elements1 from the periodic table [14]. The dissociation process does not have a 100% yield, which often prompts the search for deposition parameter optimization [15], post-growth purification strategies [16], or structure coating methods in order to prevent oxidation [17].

In this latter case, the structure's core material is (mainly) cobalt obtained from the complex dicobalt octacarbonyl Co 2 (CO) 8 , and the coating is produced with a platinum-rich shell from trimethylmethylcyclopentadienyl platinum2 CH 3 CpPt(CH 3 ) 3 . In the first case, contamination from either carbon leading to the formation of carbides or oxygen can occur in the case of partial CO ligand dissociation [13]. Due to the large C content, carbon contamination is prominent in the case of the Pt deposition [13]. However, vertical nanowires fabricated with a Co core and a Pt-rich shell, hereafter denoted Co@Pt core-shell nanowires, were shown to feature slightly smaller volume C and O content as well as reduced surface oxidation/contamination with respect to the same Co nanowires fabricated without a Pt-rich shell [17]. This can be understood as a protective effect of the shell preventing oxygen from the ambient atmosphere to further oxidize the Co core. Aside from the EELS spectra, electron-holography-based induction measurements support this interpretation [17] since it was found that Co@Pt core-shell nanowires possess a larger magnetic induction than Co nanowires fabricated in the same conditions, but without the Pt-rich shell. The measurements were facilitated by a uniform, monodomain micromagnetic configuration following the straight nanowires' long axis, as expected for a soft magnetic material.

Despite the fact that the sample's magnetization only originates from cobalt, the Co-rich material produced with FEBID is indeed soft, as was shown in other studies [11,19]. This results from the textureless, nanocrystalline nature of the material [20], which consists of nanocrystals embedded in a carbonaceous matrix. Thus, thanks to an effective growth spot size which may reach 30 nm [21], thin, magnetically soft [19], vertical or tilted nanowires [15] can be grown with a homogeneous composition [22], a tunability of the wire's shape at rest [23] and a protective coating [17] thanks to FEBID.

It is based on these capabilities of FEBID that upright Co@Pt core-shell nanowires featuring sharp bends by JAVIER PABLO-NAVARRO, JOSÉ MARÍA DE TERESA and CÉSAR MAGÉN at the Universidad de Zaragoza (Zaragoza, Spain). As in their previous work [17], they proceeded in a two-step fashion: deposition of the Co core, then coating from two opposite sides with a tilted ad-hoc SEM stage to fabricate the Pt-rich shell. More details on the fabrication can be found in the publication reporting on this investigation: [24]. The first step's result can be seen in Fig. 3.2.a), and the addition of the Pt-rich shell is featured in Fig. 3.2.b). The indicated diameters and height were measured by JAVIER PABLO-NAVARRO, the height being corrected for the 52 • SEM stage tilt. The scale bar corresponds to the horizontal direction only. Obviously, the deposition of the shell has increased the structure segments' diameter, by an amount of ca. 10 nm to 13 nm. The nanowire represented in Fig. 3.2 is the only one among many structures fabricated by JAVIER PABLO-NAVARRO et al. that has been investigated with synchrotron radiation. In the following, it will be referred to as F11 1 .

An overview of the possibilities of FEBID growth in terms of geometries is displayed in the appendix section A 3 .1, Fig .A 3 .1. In terms of composition, my Spanish colleagues indicated that the core's cobalt content should be about 70% for the batch of samples that had been sent for the beam time at SOLEIL. Although the wires are supposedly all very similar, strictly speaking, not all deposition parameters were kept exactly constant during the growth of all ca. 200 nanostructures.

The suitability of F11 1 's geometry for the investigation of an upright ferromagnetic nanostructure with a nontrivial magnetic configuration is illustrated in Fig. 3.3. The scheme to create a stable remanent state featuring domain walls follows the idea reported in the work of Da Col and coworkers [25] and takes advantage of the nanowire's bends. Such bends have been shown to act as pinning sites for domain walls, both in simulations [26] and in experiments [27][28][29][30]. This behaviour also emerges in a simple 1D approach to a bent geometry featuring a domain wall, see Chapter 1, appendix section A 1 .2, p.34. When applying a sufficiently strong horizontal field -→ H a in the bends' plane, -→ M is saturated. Its trajectory to remanence is sketched in red. Right: owing to the bends [25], three domain walls are expected to be nucleated. This is dictated by the soft magnetic nature of the material; shape anisotropy leads to alignment of magnetization with the local long axis. For a given field value, magnetization tends to realign faster (due to stronger demagnetizing effects) in the vertical segments, but the crucial question is: up or down? This is imposed by the tilted segments [e.g. segments ( 5) and ( 6) in Fig. 3.2] which still have a magnetization component along the applied field. For any finite field, magnetization in these parts will be slightly tilted towards the nearest wire axis orientation; in the case of Fig. 3.3, downwards for segment (6) and upwards for segment (5). That is how degeneracy is lifted for e.g. segment ( 7): exchange interaction imposes that magnetization in this segment rather follows the downwards tilt, instead of re-aligning upwards and increasing the exchange energy.

µ 0 -→ H a = - → 0 -→ M µ 0 -→ H a
As a consequence, two domain walls are nucleated in the 90 • bends of the structure. They share the same polarity, which is head-to-head in the case of Fig. 3.3, and are represented there as red-rimmed black dots with a "+". These two identical domain walls in turn imply the presence of a third domain wall in-between. The latter is represented with the red-rimmed black dot with a "-" in Fig. 3.3. In the idealized case with a perfect wire, and where the applied field is perfectly horizontal, the bends either exactly vertical or tilted by exactly 45 • , one expects this latter domain wall in the middle of vertical segment (4). However, considering realistic misalignments as well as the pinning site nature of bends, there is a marked possibility that this domain wall be found at a 135 • bend.

III PEEM imaging and nanowire vibrations

III.1 Application of field F11 1 (and many other wires) were fabricated on a silicon wafer with Au/Ti alignment marks fabricated at Institut Néel with my designed mask (see section I.4 in Chapter 2, in particular Fig. 2.6 p.67). The structures' bends were produced along principal directions of the alignment marks so as to facilitate the sample orientation. Thanks to this design, it was possible to apply a horizontal magnetic field along the desired axis at the HERMES beamline [31] at synchrotron SOLEIL. An induction of µ 0 H a =0.73 T was applied in the plane of the bends [24] with an electromagnet. Considering the ease of orientation granted by the marks visible to the naked eye (see Fig. 2.6 p.67), I estimate that the misalignment in the horizontal plane between field and bends cannot have been larger than 10 • . However, since only line marks were used for the horizontal alignment, the field may have had the orientation depicted in Fig. 3.3 or its opposite. In the second case, the domain wall polarities are opposite as well. A priori, misalignments in the vertical plane were of the same order of magnitude, but nanowire F11 1 was shown after the synchrotron experiments [24] to be tilted with respect to the normal to its substrate 4 . Hence, the following discussion's conclusions are to be taken with some caution. The requirement from section II to have domain walls was a "sufficiently strong" horizontal field to saturate magnetization. To be cautious, it is reasonable to assume dealing with polycrystalline pure cobalt (albeit our FEBID wire material is not pure) with a saturation induction µ 0 M s,Co =1.81 T [32]. The natural induction scale is thus the shape anisotropy induction for a cylindrical infinite cobalt rod µ 0 H cyl,Co = µ 0 M s,Co /2 =0.91 T. Considering the above, the reader may rightly wonder: is µ 0 H a =0.73 T actually sufficient? Let us examine the problem in more detail. As was argued previously, what primarily matters is that magnetization points along the field in the 45 • -tilted segments [e.g. segments ( 5) and ( 6) from Fig. 3.2]. In other words, the applied field must be higher than the switching field. Along this line of thought, let us evaluate this in an approach à la Stoner-Wohlfarth, assimilating a segment to an infinite cylinder. Obviously, we neglect here the finite length and connection to neighbouring segments, however:

• the typical segment aspect ratio is larger than 3, so that the infinite cylinder limit should not be too far off,

• the behaviour of the vertical segments either follows that of the tilted segments, in which case three domain walls are expected, or it differs from it, which would result in more domain walls. Only in the case of very large field misalignment should one expect fewer domain walls, if some segments switch in the wrong direction.

On this basis, if only switching of our 45 • -tilted segments is targeted, then the required horizontal induction is in our Stoner-Wohlfarth modelling: µ 0 H sw = 1/2 • µ 0 H cyl,Co =0.46 T< µ 0 H a . What's more, we may estimate an angular "safety margin" from the Stoner-Wohlfarth astroid [33] from the angles furthest from 45 • for which the switching induction µ 0 H sw (θ a ) is still lower than the applied induction. Here, θ a is the angle between the segment's long axis and the applied field -→ H a ; thus, 45 • corresponds to our horizontally applied field and the tilted nanowire segments. For the sake of clarity, the astroid is represented in Fig. 3.4 (in green). Its maximum is of course the shape anisotropy induction µ 0 H cyl,Co . The applied induction µ 0 H a =0.73 T (red horizontal line) sets the maximum switching field and therefore the possible angular range, which is coloured in light brown. From this, the estimated safety margin is about ±40 • around 45 • . Such a high value is only natural, considering that the applied field is not so far below the shape anisotropy field. It is noteworthy that in the low θ a range, an isolated finite-length segment would behave very differently from this astroid owing to domain wall nucleation processes.

As a conclusion, we have derived an estimate of the field needed in experiments to obtain domain walls at the 90 • bends of our nanowire. This field is significantly lower than the applied field, with a comfortable margin on its direction. Furthermore, the above discussion does not even rely on a reduced saturation induction with respect to pure, bulk cobalt, even though we know that such a reduction is to be expected [17]. This gives us confidence in the outcome of the field application with the electromagnet at synchrotron SOLEIL.

III.2 PEEM imaging

3 µm After introduction into the microscope of the HERMES beamline's XPEEM branch, the sample was investigated at and around the Co L 3 edge, which is associated with large magnetic dichroism [34]. Care was taken in the sample orientation so as to have the beam in the plane defined by the nanostructure's bends; this was enabled by the alignment marks on one hand and by the FEBID design oriented with respect to them on the other hand. A first PEEM image of nanowire F11 1 and its surroundings is displayed in Fig. 3.5. This is an uncropped image obtained in the following way. The elementary step is that four PEEM 1 s-snapshots are averaged into one. One hundred such snapshots are acquired for both circular beam polarizations: circular right (CR) and circular left (CL). All these are drift-corrected so that it is possible to average them. For the purpose of PEEM imaging without magnetic information, the CL and CR images are averaged. The end result is the image in Fig. 3.5. Quite a few different features can be spotted in this image:

X-ray beam

• the golden circle arc (added for clarity) corresponds to the microchannel plate's round opening before the CCD camera screen; anything outside that arc is not (exclusively) the sample image,

• the better part of the actual sample image (left half and bottom right) has rather low intensity except for a large number of bright dots, whose size is seen to increase towards the top right of the image,

• a bright halo is located to the top right of the image; it contains a strong dip more or less at its center (see light blue cross), where the intensity falls to almost zero,

• from the dark dip out a straight dark line extends towards the bottom left of the image; its tip is indicated with the red arrow.

From here on, we will first examine these latter three features, starting with the last in section III.2.a, then moving to the other two in section III.2.b. It will thereafter be easier to have a first global view on the observed situation, as described in section III.2.c. Then, a longer section follows, namely section III.2.d, in which I will provide an explanation for unexpected and surprising aspects of the PEEM image, pertaining to the dark oblique line. For the sake of completeness, section III.2.e and section III.2.f review the state of nanowire F11 1 after the experiments with SEM imaging, both for damage checking and consistency with what has been observed in PEEM. Finally, section III.2.g sums up all of section III.2 so as to provide a concise view on the PEEM imaging of our core-shell Co@Pt sample.

III.2.a Wire shadow and field emission

The aforementioned line is actually F11 1 's shadow, which is projected onto the silicon substrate thanks to the ψ =18 • X-Ray beam incidence angle (as measured with respect to the sample surface, no the normal to it). Thus, the dip at the cross corresponds to the immediate surroundings of the nanostructure's foot. One may wonder: why is the intensity there so much lower (ca. 6% of the maximum intensity from the halo) than in the nearby halo? The answer lies in field emission from the nanowire's tip. Remembering that the sample is brought to 20 kV, and that its tip 2.7 µm above the substrate has a radius of curvature of about 40 nm, the likeliness to a lightning rod is striking. Actually, the field emission was confirmed using Low-Energy Electron Microscopy (LEEM): in this mode, even with a negative start voltage (STV) i.e. with a sample completely repelling the illuminating electron beam, a strong intensity could be collected from this area 5 . By contrast, its surroundings were completely dark. This means that a process only depending on the voltage difference between the sample and the imaging system (20 kV+STV) led to the generation of a sizeable current. In this context, the only plausible culprit is field emission, which is also consistent with the measured current from sample to microscope: the latter was slightly but definitely higher than in "usual" experiments in this microscope (always on the order of a few dozen nA), for instance clean metallic thin films without topographic relief .

Coming back to the PEEM image, it now seems surprising to see a low intensity at the wire foot compared to the neighbouring halo. Let us first remember that the intensity is not zero, and that in PEEM, the focus was on the sample surface, not the wire tip more than 2 µm away. Furthermore, while in LEEM everything was done to suppress intensity from the rest of the sample, here photoemission from the substrate was optimized; hence, intensity at the wire foot must be compared not with zero as in LEEM but with a finite contribution. Along this line, there is a good reason for photoelectrons coming from the wire foot's surroundings to not be collected by the microscope: these electrons are nominally accelerated to 20 keV on the microscope's optical axis, but are here disturbed by the field emission. The electric field is locally strongly affected by F11 1 's tip, and possibly by the emitted current. As a result, the photoelectrons are deflected (and the image thus distorted in this area), and potentially stopped in the imaging column by the contrast aperture, which selects only electrons emitted parallel to the optical axis.

Due to the incidence angle of ψ =18 • , the total shadow length is increased with respect to the structure height by a factor 1/ sin (ψ) 3.2. Therefore, one way to retrieve the nanowire's height is to measure the extent of its shadow: it can easily be traced on both sides since the center of the very dark area on the PEEM images corresponds to the location of the structure. From the images, the shadow length is about 8.1 ± 0.3 µm, which results in a height of 2.5 ± 0.1 µm, while SEM measurements performed before shipping of the samples indicated 2.7 µm. SEM investigation after the synchrotron experiments indicated that the nanostructure was bent with respect to its initial state, as is illustrated in Fig. 3.9. The measured height was 2.3 ± 0.1 µm.

III.2.b Halo and dots

The previous section provided explanations for the low intensity at the wire foot compared to the close surroundings. Concerning the strong photoemission in the halo, it is natural to suspect the presence of cobalt for two reasons. The first one is spectroscopy: the image was acquired at the Co L 3 edge, hence a much stronger photoemission with respect to the substrate hints at this element's presence. Qualitatively, this is confirmed by images taken at energies a couple eV below/above the Co L 3 edge: on these (not shown in this section, but in section IV.1, p.161), the halo intensity is comparable to that of the dark areas in Fig. 3.5, which are attributed to the Si substrate. This resonant behaviour is the signature of cobalt as element. Such an in-situ proof is quite definitive, yet, in fact, we have already briefly touched the other reason to expect cobalt around the FEBID nanostructure. Indeed, at the beginning of section II, the dissociation of precursor molecules by secondary electrons i.e. electrons scattered away from the SEM's beam spot was mentioned. This results in material deposition away from the target, thus constituting a halo as described in many works [12,15,18,20,35]. It was therefore quite logical to foresee the halo as well as the dots lying farther away from F11 1 .

Both features suggest caution a priori, as long as any influence from their cobalt content on our nanostructure's magnetism has not yet been ruled out. Fortunately, these parasitic deposits have been investigated in several instances: it was shown for instance that they are insulating 6 [36], and more importantly that the halo surrounding FEBID deposits is not ferromagnetic [37]. In the case of vertical pillar depositions, Córdoba et al. report a low cobalt content in the halo as plausible cause for the non-ferromagnetic behaviour [15], due to a likely complete oxidation 7 . Therefore, one can safely consider nanowire F11 1 as an isolated magnetic object.

III.2.c Illustration

As a simplistic but more comprehensible illustration of F11 1 's situation, I propose a view that I created with POV-Ray [39] in Fig. 3.6. It features an imagined representation of nanowire F11 1 standing on the Si substrate (in uniform gray) above which is only vacuum (in black). The cobalt core of the structure as well as the grains mimicking the FEBID halo are rendered bright, consistently with the imaging technique. The thin platinumcontaining shell is translucent gray. Trailing behind the wire, the shadow produced by the X-Ray beam extends far according to the 18 • incidence angle.

III.2.d Wire vibrations

III.2.d.(i)

A closer view on the shadow Now that we have described almost all the image in detail, let us come back to the shadow. The strong intensity collected on the halo at the Co L 3 is now understood as the signature of its cobalt content. Yet, this does not explain why the contrast between the shadow and the surrounding halo area is so low, in contradiction with the expectation of an opaque Co core as represented in Fig. 3.6. The instrumental resolution in PEEM being below 40 nm [40], the shadow center should actually be orders of magnitude darker than the halo. Indeed, X-Rays falling there have travelled a path inside the wire that is at least F11 1 's cobalt core diameter 8 . Considering the indicated FEBID cobalt purity of 70%, the expected attenuation corresponds to a 93% decrease in intensity, whereas the intensity ratios ("shadow divided by halo") from the image in Fig. 3.5 are as high as 70%-80%! In addition to this disturbing element, it is noteworthy that even close to the wire tip, the intensity in the shadow remains barely lower than that from areas that are attributed to the Si substrate. The intensity ratios ("shadow divided by substrate") are on the order of 80% or even higher. That is shocking in comparison with several shadow PEEM works on magnetic nanostructures [1,6,7], all of which display a much more visible shadow. The 6 Experimentally, no significant charging was observed. Among the possible reasons for this, one can consider the fact that a FEBID platinum deposition was undertaken after the cobalt-rich core fabrication, thus possibly spreading a more conducting material around the nanostructure. Also, as will be shown with X-ray spectroscopic imaging (see Fig. 3.12 on p.161), a significant portion of substrate does not exhibit resonant photoemission at the Co L3 edge; therefore, these areas could be dominated by the silicon wafer's behaviour, mitigating charging. I do not have more insight into this matter. 7 Along this line, one may note for instance that CoO is an antiferromagnet with a Néel temperature TN=291 K [32] and CoCO3 is a canted antiferromagnet with TN=18.1 K [38]. Many other compounds may be present however. 8 The absorption through the platinum shell, which corresponds to a thickness of about 13 nm with a linear absorption coefficient only about an order of magnitude smaller than that of cobalt [41] (taking into account a reasonable purity for the Pt-rich shell [14]), can be estimated to be at most 5% intensity decrease. Oxygen and carbon are much lighter elements, thus interact less with X-Rays; an educated guess at their contribution's order of magnitude places it way below that of platinum. high shadow-to-surroundings intensity ratios not only close to the wire foot but also far from it indicate that this intriguing behaviour has a priori little to do with the sample surface. Therefore, we are looking for an unexpected circumstance that affects the whole wire but not its substrate, and which seems to cause the X-Ray beam to only "partially" feel the presence of F11 1 . Here, I wish to mention that I have tried to account for this behaviour firstly by a strong background intensity such as described by Jamet and coworkers [40], to no avail. Even the addition of the X-Ray beam's finite divergence could not explain the observed too faint shadow. What's more: even if this refined modelling predicted a certain shadow widening, considering the FEBID structure's diameters, the predicted shadow widths (full-widths at half maximum) were on the order of 150 nm, whereas the measured widths are rather close to 200 nm. This is illustrated in Fig. 3.7, which features the same image as in Fig. 3.5 in a) and b) with different contrasts so as to highlight different parts of the shadow. The X-Ray beam direction is represented in Fig. 3.7.a) with a green arrow, the shadow tip still with a red arrow and the wire foot with a light blue cross in Fig. 3.7.a-b). On each, a vertical line is drawn: intensity profiles were extracted from both lines and represented in Fig. 3.7.c): the correspondence is shown by the labels as well as the colour (red/black) and style (dashed/full) of the curves/lines. The profiles have been horizontally offset so that the shadow's dips match, and rescaled so that the full shadow amplitudes also match. The criterion to delimit the shadow that I used was to encounter (roughly) the first location where the derivative of the intensity versus abscissa vanishes. In the case close to the shadow's tip, this is justified towards negative abscissa by the presence of almost uniform background (i.e. the substrate). In the other case, close to the wire foot, the local maximum at positive abscissa is clearly located outside the shadow [see Fig. 3.7.a)] so that the latter cannot extend beyond it.

Comparison between the two profiles in Fig. 3.7.c) clearly indicates that the profile close to shadow tip is broader than that close to the wire location. To be more quantitative and to be less sensitive to the marked asymmetry of the shadow profiles, I estimated the curvature around zero abscissa by fitting both profiles to a polynomial f (x) = a + b • (xx 0 ) 2 . Thanks to the normalization to the same shadow amplitude, the ratio of curvatures i.e. the ratio of the shadow widths is directly that of the two b coefficients from the fit 9 . From this I could deduce that the shadow close to the tip was 44% wider than close to the wire location, despite the adverse ratio of cobalt core diameters between wire tip and foot, see Fig. Both have been offset so the dip's center (corresponding to the wire shadow) are at zero abscissa, and scaled so that the shadow's amplitudes match. The criterion to not be biased by the non-uniform background was to reach the first vanishing derivative of the PEEM intensity with respect to the abscissa. This procedure allows to compare the profile shapes, in particular their widths, irrespective of their amplitudes.

III.2.d.(ii) Review of the investigation of the shadow

As a reminder, starting from the observation of an unexpectedly weak shadow, I have gone one step further and showed that the latter was not only wider than suggested by the wire diameters, but also with a larger width close to the tip than close to the wire location. Furthermore, I was able to rule out a predominant role of the X-Ray beam's divergence and/or of a (realistic) background intensity. Historically, another element regarding the XMCD-PEEM shadow contrast profiles was helpful in my interpretation; it can however be included here by looking at the profiles in Fig. 3.7.c): the dip does not have abrupt borders, which should actually be the case. Indeed, as shown in Fig. 3.8, the theoretical X-Ray transparency corresponding to F11 1 is very flat over, say, half of the profile. This transparency has been computed for normal incidence on a perfectly cylindrical coreshell structure with cobalt and platinum purities of 70% and 20% respectively. I calculated the linear absorption coefficients from the literature [34,41] taking these purities into account and neglecting the contributions of oxygen and carbon. The abscissa ranges where only the Pt-containing shell contributes are shaded in grey, while the inner part where both shell and Co core absorb the beam is shaded in blue. The platinum contribution appears stronger than previously mentioned on page 152. What is misleading is that the travelled distance inside the shell is maximum not at zero abscissa but exactly at the Co core radius, here 36 nm. There, the travelled length is about 45 nm, about three times the diameter increase caused by the shell, here 13 nm.

It is striking that the profile is very flat, with rather sharp flanks. This is rather logical considering that the mean free path of X-Rays in pure cobalt at the Co L 3 edge is about 18 nm [34]: except close to the core's edge, the beam always travels through a length significantly higher than this mean free path. Hence the low, flat minimum in transparency. These features are in stark contradiction with the shadow measured in PEEM. It appears as though the shadow is smeared out so as to be less strong and wider than expected, with less steep flanks than expected. Moreover, as we have seen, this smearing is larger close to the shadow's tip. Here, incidence perpendicular to the wire axis has been assumed for simplicity. The assumed purities of cobalt and platinum are 70% and 20% respectively, the absorption coefficients relevant to the Co L 3 edge were derived from the works of Nakajima [34] and Henke [41]. The wire is a perfectly cylindrical cobalt core with 36 nm radius, and a 6.5 nm thick platinum-containing shell. Both parts are indicated with shaded areas. The shell's influence appears misleadingly strong because just outside the cobalt core, the beam traverses only platinum, but over a distance much larger (here ca. 45 nm) than the 13 nm diameter increase that was mentioned on page 147.

III.2.d.(iii) Field-emission-induced vibrations

lution with the density of probability of positions in the horizontal direction perpendicular to the X-ray beam 11 . Moreover, the reason for the shadow's weakness would be very clear: at a given point behind the wire on the substrate, the time-averaged incoming X-ray intensity would be increased compared to the immobile wire case because a fraction of the time, the nanostructure would not absorb the photons impinging at this location! Thus, in the shadow, the measured intensity would contain a contribution from the naked substrate: this appears all the more plausible considering the high intensities in the shadow inside the halo. This hypothesis being put forward, it now needs justification. Why should F11 1 oscillate inside a microscope whose mechanical stability is key in its operation? In fact, since the X-Ray beam itself can safely be discarded (neither radiation pressure nor beam time stability could explain oscillations), as can the vacuum, the only stimulus applied to the nanostructure is the microscope's high voltage. At first sight, this might appear an unlikely candidate as the applied high voltage is constant and stable (for the sake of proper microscope operation); the situation seems far from oscillating nanostructures under time-dependent electric fields as reported by e.g. Wang and coworkers [43]. However, it has been shown by Lazarus and coworkers [44] that an elongated nanostructure in front of an anode could undergo a mechanical instability and enter a self-sustained oscillating regime under sufficiently strong DC voltage, if the latter can trigger field emission.

The key ingredient is the strong, position-dependent non-linearity of the field-emission. Physically, the sample's tip position has an enhanced impact on the field-emission current due to this non-linearity [44]; consistently, the voltage difference between the nanostructure's tip is changed in a non-linear way. The instability mechanism now comes full circle: the electrostatic force applied to the tip is proportional to the AC voltage deviation from its mean value. Thus, position fluctuations induce fluctuations in the force that can drive the tip away from its equilibrium positions. Of course, there is a threshold DC voltage below which the self-oscillations cannot exist; Lazarus et al. [44] have worked out this condition as well as the other requirements for the dynamical system involving the tip position, applied voltage and intensity to become unstable. This is to say that the static equilibrium tip position goes from stable to unstable equilibrium: any small perturbation 12 will drive the nanostructure into oscillations.

Comparison of parameters with the publication by Lazarus and coworkers so as to quantitatively determine the self-oscillation threshold and so on goes significantly beyond this work. Such comparison is hindered in the first place since the expected nanowire resistance in my case is a few kΩ, based on the structure dimensions and core-shell structure as well as the indicated material purities. The corresponding mean resistivity in the µΩ•m is consistent with what has been measured by Begun and coworkers [11] on an as-deposited FEBID cobalt strip. Taking into account the protection from oxidation that results in our case from the Pt-containing shell may even bring our nanowire's resistivity slightly lower. In any case, it is already too low to try and evaluate the unstability criterion presented in the study by Lazarus et al. [44], who investigate a regime of much higher resistance.

In view of the above, I cannot present definite proof of the field emission's role, since the alleged wire vibrations never were directly evidenced 13 . However, if it were to be fully ruled out, then to my knowledge, the only other possible cause for oscillations would be mechanical vibrations of the microscope. However, the corresponding amplitudes should be a few tens of nanometres at least 14 ; this is quite inconsistent with the reported typical resolution in PEEM [45][46][47]. One would then expect a resolution on the order of 100 nm or more, whereas experiments with the same instrument only a few days after the experiments on FEBID structures indicated a resolution of at most 40 nm [48]. To put it briefly: the hypothesis of vibrations (leading to the afore-mentioned weak shadow etc.) caused entirely independently of the field emission appears to require an inconsistent culprit, since all others have to my knowledge been discarded. This is not to say that a contribution from resonant amplification of microscope vibrations is ruled out. This will be discussed below, after estimation of the nanowire's vibration frequency's order of magnitude.

Considering this, and assuming that these field-emission-induced vibrations are present with or without a contribution from the microscope mechanics, we arrive at a rather consistent picture of the PEEM experiment. The broadening and loss of contrast in the shadow can be explained as in the beginning of this section. Furthermore, the fact that this broadening becomes stronger towards nanowire F11 1 's top does suggest a motion reminiscent of the first transverse mode of the rod clamped at one end and whose other end is free [49], in resemblance with the experiments of Wang et al. [43], see e.g. Fig. 1 or Fig. 2.b. As a side note, since the wire tip can move in two orthogonal directions, two modes of vibrations (as can be seen in [43], Fig. 2.b-c) can be excited [50]. The modes can be (close-to-)degenerate [43] or well-separated [51] depending notably on the wire symmetry. In the present case, considering the marked anisotropy of F11 1 with two obvious special directions, one can expect the two fundamental modes to be well apart from one another.

Since the beam is in the plane defined by the wire bends (as was explained at the beginning of section III.2), the dominating effect is probably that of the fundamental mode in the direction perpendicular to the bends. This being said, it is reasonable 15 to expect that the mode in the bends' plane is excited as well.

As for orders of magnitude, we now must at any rate estimate the order of magnitude of these resonance frequencies: they have to be (much) higher than 1 Hz for consistency with the hypothesis of the "normal shadow" being convoluted with the density of probability of positions in the direction perpendicular to the wire and the beam. Taking into account the Young's moduli and volume mass densities of cobalt and platinum [52], as well as the core-shell geometry, but neglecting the role of carbon and oxygen, the order of magnitude I estimate 16 is 6 MHz to 40 MHz. The consistency with the hypothesis of vibrations on time scales significantly shorter than the image integration time is thus ensured. Even with a much reduced effective Young's modulus E, due to the square root dependence of this frequency on E, the order of magnitude remains much higher than the inverse image acquisition time. Interestingly, tungsten nanowires grown with a similar technique, namely Focused Ion-Beam-Induced Deposition (FIBID), have been shown [53] to resonate in the megahertz range. Differences are notably due to the differences in length and composition, of course. The authors found a strongly reduced Young's modulus with respect to bulk W; this was attributed to the inhomogeneous, core-shell-like composition of the nanostructure 13 Historically, the suspicion of their role came long after the end of the experiments. 14 This seems reasonable as long as the mechanical vibrations are far away (most likely below) the nanostructure's resonance frequency. As will be detailed later on, neglecting resonant couplings is legitimate; schematically, all of the nanowire's environment is so much heavier than it that vibrations at the wire's resonance frequency should be too high for the microscope's inertia. 15 Under the present conditions of rather small amplitudes and hence of linearity, which ensures independence of the two orthogonal modes. 16 The model is the crude one of a perfect, straight, ca. 2 µm-long cylindrical structure with a homogeneous composition-averaged volume mass density, and a Young's modulus between 75 GPa and 400 GPa [52]. The calculated frequency ν1 is that of the first flexural mode of the bent rod [49], reading 2πν1 = a 1 L 2 πER 4 4m . a1

3.5160 is the first root of the equation cos ( √ x) cosh ( √ x) + 1 = 0, L is the wire length, E its Young's modulus, R its radius, and m the mass per unit length.

with an impact of the co-deposited gallium, which is softer.

Let us estimate the order of magnitude of vibrations caused by microscope vibrations. Considering the abovementioned upper bound of 40 nm on the resolution [48], it is safe to assume that the total microscope vibrations extend over at most ∆x micros. =40 nm. Even though the experimental resolution of X-ray PEEM is influenced by many other factors [54], let us reason on this value for the sake of safety. Disregarding the spectral features in the instrument's mechanical response, we shall assume that these vibrations are homogeneously spread over a frequency domain ∆f , resulting in a spectral amplitude density ρ x = ∆x micros. /∆f . For self-consistency, ∆f > ν 1 in this approach. Now, likening the core-shell nanowire to a harmonic oscillator with a resonance frequency ν 1 and a quality factor Q, it is straightforward to show that the peak-to-peak oscillation amplitude ∆x tip in the resonance's bandwidth ν 1 /Q is (at most) ∆x tip = Q∆x mic.,band. where ∆x mic.,band. is the spectral peakto-peak amplitude of microscope vibrations integrated over the resonance bandwidth. In our white-noise-type description, ∆x mic.,band. = (ν 1 /Q) × ρ x = (∆x micros. /Q) × (ν 1 /∆f ). As a result, the oscillation peak-to-peak amplitude reads ∆x tip = Q∆x mic.,band. = ∆x micros. × (ν 1 /∆f ) < ∆x micros. . The above treatment is crude, but if we consider the profile width close to the shadow tip in Fig. 3.7 and its width of about 200 nm, it is clear that subtracting the instrumental resolution of 40 nm, the wire diameter of 80 nm and the estimated maximal broadening due to microscope vibrations ∆x tip =40 nm, about 40 nm width remain unaccounted for. It must be kept in mind that this follows from the most pessimistic estimate in terms of bandwidth ∆f = ν 1 and portion of instrumental resolution due to microscope vibrations. As a conclusion: many details such as the spectral mechanical response of the instrument or the harmonics ν n of nanowire resonance have been left out, yet the estimated contribution from microscope vibrations does not appear to be sufficient to explain all of the observed shadow broadening.

III.2.d.(iv) Review on the wire vibrations

Here is a short review on the whole section III.2.d. Starting from the anomalously strong intensity collected from the wire shadow, we have noticed that intensity profiles extracted perpendicular to the shadow were not only broader than expected, but also wider towards the nanostructure's tip. The profile shape was drastically different from a theoretically modelled wire transparency. These features could be explained neither by a parasitic background intensity nor the small but finite beam divergence.

In the experiment, only very few stimuli were applied to the sample. Considering the above, there must be one or several whose effect is to provoke mechanical oscillations on a time scale below the smallest image integration time (on the order of the second). Microscope vibrations can be safely ruled out for the sake of consistency with a realistic instrumental resolution. The exposure to X-Rays is expected to contribute to a negligible radiation pressure. The last and plausible hypothesis is that of a field-emission-associated mechanical instability (under the experimental high voltage) which brings nanowire F11 1 into a self-sustained oscillatory regime. The latter accounts for the widened, weakened and less steep shadow. As a check of consistency, the resonance frequency was crudely estimated to lie well in the megahertz range, so that the various positions assumed by the wire are indeed averaged during the acquisition of one image, thus spreading and decreasing the shadow amplitude. The frequency's order of magnitude seems supported by experiments carried out on W-rich FIBID nanowires [53] where mechanical resonances were imaged at a few MHz.

III.2.e Shadow length

III.2.e.(i) Shadow extent and post-experiment SEM imaging Section III.2.d suggested a strong influence of the AC, high-voltage-induced forces applied to nanowire F11 1 on the imaging of the latter's shadow. The focus was then on the transverse direction, yet, there is also a surprising feature along the length which we had noted earlier. Indeed, the shadow length is supposedly increased by a well-known factor due to the X-Ray beam's incidence angle, as discussed in most publications on shadow XMCD-PEEM experiments [1,3,5,6,40,55]. This enhancement factor reads 1/ sin (ψ), where ψ is the (smallest) angle between the X-Ray beam and the horizontal direction. As mentioned previously, ψ =18 • so that 1/ sin (ψ) 3.2; however, the measured shadow length of 8.1 ± 0.3 µm [24] corresponds to a nanowire height of 2.5 ± 0.1 nm, whereas SEM measurements before the experiment indicated 2.7 µm (see Fig. 3.2). This small but clear difference prompted me to perform SEM measurements after the synchrotron beamtime, with the decisive help of SÉBASTIEN PAIRIS. With a special stage adapted to tilting, the images presented in Fig. 3.9.a) and c) [24] could be obtained; the image in Fig. 3.9.b) I took on my own. There is a clear tilt of the nanowire's main axis compared to the status prior to the synchrotron experiments. A plausible cause may be transportation to the facility and back. Another possibility I think is worth mentioning involves the static or time-averaged distribution of electrostatic forces applied to the nanostructure. Section III.2.d discussed in detail the likely effects of the time-dependent forces, but their "mean" might have played a role in the observed plastic deformations.

Thanks to the known orientation of the FEBID design with respect to the substrate's alignment marks and the observed direction of the shadow with respect to these, we know a priori the direction from which the X-Ray beam was falling onto the nanowire during the synchrotron experiments. It turns out that the bends were facing the beam i.e. the X-Rays propagation direction was from the right to the left if reported on Figs.3.2.a-b) or Fig. 3.9.a). Based on this, one can easily imagine how the shadow's length was decreased in PEEM. Furthermore, the height as measured with SEM after the beamtime was 2.3 ± 0.1 µm, in better agreement with the PEEM measurement. This suggests that some plastic deformations may well have taken place before the PEEM experiments started, or right when they did start.

III.2.e.(ii)

Changes in the FEBID halo Aside from the wire itself, the reader may have had an intrigued look at the surroundings of the FEBID nanostructure. The relief in the close vicinity of the foot was quite clearly absent from the imaging that had followed the sample fabrication, as illustrated in Fig. 3.2. While reminiscent of the bright dots from the PEEM imaging (cf. Fig. III.2) and the actual reason behind the representation in Fig. 3.6, these granular features suggest a halo more complex than a smooth surface as imaged by Córdoba and coworkers [20], see Fig. 5.a). It is worth remembering that the FEBID halo does not under originate a priori from the exact same decomposition process as the main sample, hence, it is a different material. I do not have an explanation for these scattered grains around the wire.

For the sake of completeness, Fig. 3.10 displays an SEM image SÉBASTIEN PAIRIS and I acquired with secondary electrons and a wider field of view (here about 15 µm). The granular features from Fig. 3.9 are still visible, but their apparent size and density decreases away from the wire foot, until wider round objects take over. Considering the scales, this transition corresponds to that seen in PEEM, see Fig. 3.5. Thus, we are led to thinking that the FEBID halo which appeared continuous in the previous sections possesses marked topographical features. Further away, the SEM image strongly suggests that the bright dots observed with PEEM correspond to other, seemingly flatter cobalt-containing objects.

III.2.f Damages to the sample?

Considering how the nanostructure has been bent with respect to its initial orientation and how the halo has changed aspect, it is legitimate to wonder whether changes for instance in the sample's microstructure have occurred. After the beamtime, my collaborators from Zaragoza suggested that they try to pluck nanowire F11 1 from its substrate thanks to micromanipulation and perform transmission electron microscopy. The goal was to gather definite evidence showing that the polycrystalline nature of the cobalt core had been preserved and that hence the 1 µm Fig. 3.10: Wide field of view (about 15 µm) SEM image with secondary electrons featuring the surroundings of nanowire F11 1 after the beamtime. The well-defined granular objects from Fig. 3.9 are easily recognised close to the wire foot. The corresponding amount of material seems to decrease radially away from the nanostructure. When the grains vanish, other features take over: these are reminiscent of the bright dots seen in PEEM, and the distance to the wire matches that of Fig. 3.5. The faint, somewhat square feature encompassing the nanostructure at about 2 µm distance might be due to repeated SEM observations with small fields of view.

interpretation of XMCD data under the assumption of a soft magnetic material was justified. Unfortunately, the nanowire was broken and lost after sample shipping to Spain. The SEM imaging presented above does not show much visible damage. There is of course the plastic deformation associated to the tilt: the two "90 • " bends appear to have a larger angle than in Fig. 3.2, and there are two clots of matter stuck to the wire, see the features within dashed lines in Fig. 3.11. From the outside at least, there does not appear to be worrying changes in the sample aspect. In the experience of JAVIER PABLO-NAVARRO, JOSÉ MARÍA DE TERESA and CÉSAR MAGÉN, strong distortion or destruction of such FEBID nanowires usually result from significant temperature increase (e.g. during exposition to a focused laser beam in magneto-optical experiments [23]).

Yet, this does not answer the question of internal changes. Apart from chemical processes and/or matter removal/diffusion promoted by the PEEM environment as mentioned in the previous section, i.e. possibilities I cannot rule out, there is the question of the direct influence of the field-emission-associated current. Indeed, under sufficient electrical current through the nanostructure, one may expect annealing-type processes to occur: this would lead to crystallite growth as suggested by several reports on heat-treated FEBID deposits [10,11,16].

Thus, I have tried to give a realistic upper bound on the temperature increase that may have occurred during the synchrotron experiments. It must first be noted that in this respect, the beam's influence is negligible: experiments on samples with high temperature sensitivity on the HERMES beamline [56] indicated that even the resonant absorption of the X-Ray beam on a thin film (thus with a much larger absorbing surface than nanowire F11 1 ) did not increase the sample temperature by more than 10 K. Refined calculations [57] led to a total power Q beam transferred to the wire of about 0.3 µW [57].

In order to translate this thermal power into a temperature increase, I modelled the foot of the nanowire as described in appendix section A 3 .2. Based on the fact that from the bottom surface, the Si wafer was thermalised (thanks to a metallic sampled holder) to room temperature, I could estimate the temperature increase at the wire foot as a function of the thermal power Q received by the wire 17 . Then, using a simple thermal conduction model taking into account the bulk thermal conductivities of cobalt, platinum and amorphous carbon as well as the coreshell wire structure, I solved numerically the heat diffusion equation [58] with Mathematica [59] including Joule heating due to the field-emission-induced current. Even with a significant overestimation of the latter, a realistic resistivity based on the measurements performed by Begun and coworkers [11] on FEBID Co stripes, and an assumed reduction by a factor ten of the estimated thermal conductivity due to the reduced dimensionality (this is inspired by the work of Ou et al. [60], with an added margin), the output temperature increases are below 1 K. Therefore, I consider the claim that temperature increases the during synchrotron beamtime were negligible to stand on solid ground. This is a very important step: in order to safely interpret the results of shadow XMCD-PEEM imaging in the frame of a soft magnetic material, the hypothesis of nanocrystalline material is crucial. That is why the above-mentioned large margin over which current density and resistivity (thermal conductivity) can be increased (decreased) is key in ruling out significant temperature increases which would have induced annealing or other changes in the sample's microstructure.

nm

III.2.g PEEM imaging: section review

In the above, we have started from the full-field PEEM image of nanowire F11 1 in Fig. 3.5 and described one feature after another. All bright features on the image, from the bright dots away from the wire foot to the more dense area closer to it, have been identified as well-known FEBID halo deposition. Its aspect is not only quite different from the usually continuous layer, but also from what had been imaged by JAVIER PABLO-NAVARRO before the beamtime. I have proposed a couple of leads in this respect, though a convincing explanation is beyond my reach.

In the close vicinity of the nanostructure, the low intensity originates in the field emission that is caused by the high voltage and wire aspect ratio. From this area out the shadow of F11 1 extends quite far, but not exactly as far as expected considering the beam's incidence angle. While this was explained by post-beamtime SEM imaging, the very low contrast between the shadow and the neighbouring area was not.

This element, in combination with an increasing broadening of the shadow towards its tip as well as an unexpected lateral intensity profile shape, has led me to consider several aspects of imaging. Neither parasitic background intensity nor beam divergence could explain these unexpected features. Conceptually, however, fast (with respect to the image acquisition) wire oscillations could. Such oscillations could not result directly from the beam nor from mechanical vibrations of the instrument.

An explanation that appears much more plausible to me is that the field emission came with a mechanical instability, the crossing of which led to a self-sustained oscillation regime. To put it shortly, fluctuations in position have a strong impact on the emitted current through the nonlinearity of the field emission process. This current in turn is linked to the voltage difference between the microscope's objective and the sample, thus to the electrostatic force applied to the wire tip. This feedback loop causes the static mechanical equilibrium to be unstable. In this frame, the interpretation for the weak and inhomogeneously broadened shadow is that many oscillations of the wire, allegedly at its fundamental frequencies, are averaged in one image, resulting in a spread-out shadow. I have followed a simplified approach to estimate the lowest mechanical resonance frequency of such a core-shell FEBID wire, and found an order of magnitude in the (tens of) MHz range. Not only is this more than sufficient to explain the above-mentioned time average of many different wire positions in one image, but such an estimate is also close to an experimental measurement of fundamental mechanical resonance carried out by Córdoba and coworkers [53] on FIBID tungsten nanowires.

With this better knowledge of the sample and the imaging conditions in mind, we shall now move on to further characterizations of the nanowire: first with spectroscopy in section IV, then with magnetic imaging in section V.

IV X-Ray Absorption Spectroscopy (XAS) IV.1 Introduction and experimental measurements

As the name implies, XAS is performed by varying the beam's photon energy and recording changes in intensity. This is typically done by averaging the intensity over a region of interest on a PEEM image while the energy is scanned. In this process, one image is acquired per value of photon energy. The principle is first illustrated in Fig. 3.12, where 11 images at different energies spanning the Co L 3 edge are gathered. Each image/energy has a total integration time of 1 s only. Fig. 3.12.a) is just at the upturn associated with the cobalt's absorption edge, hence the halo and dots are already discernable against the silicon surface.

a) b) c) d) e) f ) g) h) i) j) k)
For the same reason, a part of the shadow is barely visible against the halo. In Fig. 3.12.b), the halo's intensity has considerably risen. Between Fig. 3.12.b) and Fig. 3.12.c), the shadow becomes more slightly visible against the dots because while the latter's emission is approximately constant, the shadow's absorption becomes larger as the photon energy crosses the L 3 edge's sharp slope. In Fig. 3.12.d), the emission from the dots and the halo is about maximum. The small difference between Fig. 3.12.d) and Fig. 3.12.e) is again attributed to the increase in absorption from the shadow.

Figs.3.12.f-h) correspond to the L 3 edge's downturn; in Figs.3.12.i-j), the shadow is more visible because of the drop in halo emission while the shadow's absorption is still significant. Fig. 3.12.k), far beyond the edge's peak, highlights the persistent "baseline" post-edge absorption from the nanowire while the halo and dots continue to fade.

These qualitative features can be retrieved from absorption spectra extracted from such XAS series of images. The post-processing involves drift correction 18 , then extraction of intensity from a selected region of interest. In order to process intensive data, i.e. to be independent from the photon flux which varies with the X-Ray energy, the normal procedure would be to normalize each image with a measure of the corresponding photon flux. This procedure will be discussed in Chapter 4, section II.2. In the present case, no adverse effect of discrete energy steps was observed on spectra extracted from large areas, indicating that the photon flux smoothly followed the beamline's spectrum. Based on the HERMES beamline description [31], I estimated that over a 10 eV wide window around the Co L 3 edge (nominally at 778.1 eV [61]), the photon flux varies by ca. 7 . Therefore, as will be clearer in a moment, the slight improvement of a proper normalization would be far below experimental noise: that is why I have not performed this operation on the XAS data.

It must be noted that the photon energies in this chapter are the target or input X-Ray energies: there can be an offset between these and the actual energy of photons in the beam. In the beamline configuration pertaining to this beamtime, the existence of such an offset is known [62]; if corrected, the cobalt L 3 peak appears at the reported energy [61]. Furthermore, this is the reason why the precision on the energy appears better than allowed by the instrumental energy resolution of about 0.1 eV.

As a complement to Fig. 3.12, XAS data from different regions of interest is presented in Fig. 3.13. The main frame's graph displays the measured absorption spectrum of the nanowire's shadow. The two curves correspond to CL (blue) and CR (red) beam polarizations. The pre-edge values have been set to zero so as to have a baseline, and the CL data was scaled to ensure matching post-edge tails. The data comes from two distinct image series, which is why their respective energy ranges differ. Here, the intensities are negative, corresponding to a reduced photoemission in the shadow area compared to its immediate surroundings, which were taken as reference. There are three main reasons for the quite high noise:

• the drift correction is most effective when supported by objects with sharp boundaries and strong contrast; in the XAS series, no such features are present, so that "jumps" can occur and easily impact the small shadow area even for small drift (see e.g. 770.00 eV, 771.00 eV),

• while the noise is small compared to the typical PEEM intensities from the surroundings, it is far from negligible with respect to the difference between the intensity in the shadow and around it; we have discussed this in section III.2.d,

• since the drift correction relies on image correlation, in the case of a XAS series, it is mandatory to use image features that are known to not display dispersive behaviour; since no such feature is present, the procedure's stability is impaired.

The labels "a)" through "k)" have been added to help linking the spectrum to the XAS series from Fig. 3.12. For the sake of clarity, only purple-rimmed dots were added in the inset. The latter shows spectra obtained by average over different regions of the image: the continuous halo (green), one single bright dot (thick blue), and many such features (red). A rescaling of the halo data was performed to match pre-edge intensity 19 , followed an offset to bring the pre-edge part to zero. This processing was done so that the intensity scale numerically corresponds to that of the main frame, in order to highlight the shadow's weakness. The reader is invited to follow in parallel the evolution of the behaviour in the shadow and on the halo/dots to get a better feeling for the description of Fig. 3.12.a-k).

IV.2 Interpretation

In the light of section III.2.d, one must keep in mind that the XAS data from the shadow is contaminated by the FEBID parasitic deposit because of the wire's vibrations. As a result, spectroscopic information from the bulk of the wire obtained by transmission is mixed with information from the surface of the sample 20 .

Let us therefore start with the graph in the inset of Fig. 3.13. Like in the PEEM image from Fig. 3.5 in section III.2, the halo dominates the dots in terms of photoemission, possibly as a consequence of the difference in material amount. The dots are apparently very similar, as suggested by the comparison between one single dot and the average of many such features. Overall, the spectra from the halo and the dots share the same shape, albeit with more pronounced relief for the halo. The dots' spectrum features a pre-edge and a post-edge shoulder, with a flat

Fig. 3.13: X-Ray Absorption Spectra pertaining to the wire shadow (main frame) and other parts of the PEEM image (inset). The shadow's absorption corresponds to the negative sign: the spectra associated with CL (blue) and CR (red) were obtained by difference with the averaged background in the vicinity. Both spectra are offset so that the baseline pre-edge absorption corresponds to zero. The labels a), b) etc. refer to the images in Fig. 3.12. Their counterparts in the inset are purple-rimmed white dots on the XAS data from the FEBID halo (green). For comparison, spectra from an ensemble of bright dots (red) as well as an isolated dot (thick blue) are added. The intensity scale in the inset corresponds to that of the main frame; the spectra from the inset were rescaled to have the same pre-edge levels, which has also been offset to zero for simplicity.

top spanning about 1 eV; this is incompatible with pure metallic cobalt. Instead of a single, well-defined peak [34], the L 3 edge has a very clear multi-peak structure in the halo XAS. Unsurprisingly, this suggests the presence of cobalt oxide and/or carbide around the nanowire: a similar shape was reported for CoO [63] and cobalt carbide [64]. Both materials are likely to be present considering the previously discussed FEBID halo. By comparison, the shadow's spectrum seems to exhibit a slightly less pronounced shoulder at 767.50 eV photon energy, even if the strong noise makes a detailed analysis of the peak shape difficult. However, considering that both the cobalt-containing background as well as the outer rim of the wire's Co core (due to vibrations) contribute and thus necessarily decrease the apparent spectrum quality, the XAS data suggests a better metallicity of cobalt in the FEBID wire than in the surrounding halo/dots. As a manner of example, Fig. 13.19 p.622 in the book by Stöhr [65] gives an example of XAS data taken on a partially oxidized thin cobalt film. While conclusions should not be drawn just from comparison with these spectra, their experimental Co L 3 edge shape bears interesting similarity to the data in Fig. 3

.13.

A more quantitative argument is provided by the post-edge behaviour of the recorded spectra. Looking at the intensities at, say, 775.00 eV [or at Fig. 3.12.k)] shown in Fig. 3.13 and comparing them with those at 768.50 eV or 769.00 eV [corresponding to Fig. 3.12.d-e)], the photoemission from the dots/halo clearly is strongly. In other words, the typical "edge tail" intensity value I (halo/dots) tail from the halo/dots is negligible with respect to the maximum value I (halo/dots) M in this spectrum It must be noted that these quantities are measured as an increase with respect to the pre-edge value (taken as baseline). Their behaviour corresponds to the description made of Fig. 3.12.k) (see p.161).

Such a comparison cannot be extended to the shadow XAS, since it does not correspond to an electron yield measurement. Actually, such data could be impacted by saturation effects [34]; the latter would lead to a flattened spectrum. However, I have observed saturation as well as its absence in shadow XAS obtained on Ni 60 Co 40 nanowires with diameters at least twice larger than nanowire F11 1 (see section II.2 p.195). Thanks to these data, I can claim that if saturation effect are involved in the present case, they are not strong enough to jeopardize the following.

Physically, the bulk of the post-edge photoemission at the L 3 edge of a ferromagnetic transition metal com-

Photoemission

Photoemission

Tab. 3.1: Computed tail-to-peak ratios from literature reports (left of double vertical line) and same ratios corresponding to my XAS data (right thereof).

pounds corresponds to transitions from the core 2p 3/2 electrons to unoccupied states that are lying further and further above the Fermi level. In metallic cobalt, these levels are free-electron-like s-type states [32] in addition to the so-called "atomic" [66] contribution of truly free electrons. In the case of an insulating transition metal oxide, schematically, the picture is first changed by the fact that s-like conduction states are not available just above the absorption edge because they lie significantly higher [65]. Furthermore, the d-type bands have been shown [67,68] to be more localized (compared to the corresponding metals), which results in less broad features above the edge. As a result, in this case, little help i.e. fewer available states are granted to excited electrons by the material's electronic structure. This hand-waving comparison certainly simplifies/neglects many aspects of the complex physics pertaining to transition metal oxides [67] and their spectroscopy, but gives an intuitive view of the reduced ratio of post-edge X-Ray absorption over L 3 edge peak absorption in transition metal oxides, with respect to their ferromagnetic metal [68].

The starkest changes in XAS upon oxidation of a ferrromagnetic transition metal are the appearance of the above-mentioned multi-peak structure or shoulders as in the case of Fe and FeO [68], but as discussed in the previous paragraph, the "edge tail" is expected to fade with respect to the edge peak. In order to further try and investigate the state of cobalt in the FEBID wire core, I have calculated the above-mentioned ratios I (halo/dots) tail /I (halo/dots) M corresponding to my measurements. The tail values were taken at 775.00 eV. Then, I have extracted the corresponding ratios from different reports in literature [34,63,68] pertaining to Co and CoO so as to compare their values with mine. In all cases, I took care to subtract the reported spectra's pre-edge baseline values (if it had not already been done) to the quantities of interest before calculating the ratios. Only then may comparison make sense, considering my own data processing.

All the computed ratios are presented in the "Tail-to-peak ratio" line in Tab.3.1, the processed literature data to the left and my results (labelled as "This work") to the right. Quite clearly, my estimates of Tail-to-peak ratios concerning the dots and halo match rather well with the CoO values calculated from the works of Regan et al. and de Groot et al. [63,68], while the ratios for cobalt are in much better agreement with my estimate from the shadow XAS. The comparison's outcome for the halo and dots is unsurprising considering their nature as well as the edge shape and measurement method. As was stated above, the comparison of the shadow XAS with the electron yield measurements is not direct, yet It must be noted that I have extracted the tail values from literature at the energy corresponding to the lowest local minimum between L 3 and L 2 edge. In the case of bulk Co, the latter lies a bit more than 12 eV higher than the L 3 peak [68], in stark difference with the ca. 6 eV difference between the shadow XAS minimum and the aforementioned 775.00 eV. Considering the monotonic behaviour of the spectra between the L 3 edge's maximum and the minimum I evaluated (disregarding the multiplet structure in the oxide's case), evaluating the ratios with the 6 eV offset would increase them: those from Regan et al. would increase from 4.2% to 5.7% and for CoO and from 15.4% to 19.9% for instance. Similarly, the ratios from Nakajima et al. and de Groot et al. would increase to 24% and 7.8% respectively.

In other words, the evaluation of tail-to-edge ratios I first did at the lowest local minimum between L 3 and L 2 edge (for the sake of caution) gives a reasonable agreement between the behaviour of the background and dots, and the reported bulk CoO.

IV.3 Review on XAS

As a summary, the powerful coupling of microscopy and spectroscopy allowed in PEEM has allowed me to further investigate the nature of the FEBID core and the nanostructure's surroundings, notably by comparing their absorption/photoemission across the Co L 3 edge.

The behaviour of the numerous bright dots was found to be quite homogeneous yet slightly different from the halo. A difference in amount of material between these features might be part of the explanation, but the multiplet peak structure that is marked for the halo and faint for the dots does suggest a stronger presence of cobalt oxide and/or carbide. In any case, the fact that both spectra display vanishing intensities between the Co L 3 and L 2 edges further supports their oxide and/or carbide nature. This spectroscopic investigation is thus in agreement with the existing literature about the well-known FEBID halo.

Furthermore, this vanishing edge tail intensity comes in stark contrast with the persistent shadow absorption above the Co L 3 edge. As was shown by a comparison with CoO spectra from the literature, the edge tail XAS from the dots and halos divided by the maximum photoemission is consistent with what has been reported for pure CoO. On the other hand, comparison with other shadow XAS spectra in NiCo nanowires allow on one hand to estimate that saturation effects, if present, were small, and on the other hand to have a further indication of the material's metallicity. Of course, trying to quantify the proportion of oxidized/carbidized cobalt atoms in the wire core is far out of reach. In this regard, it must be reminded that due to the previously discussed nanowire vibrations, the shadow XAS mixes transmission through the bulk of the cobalt core with transmission through the outer, more oxidized shell. Despite this adverse effect and the noise, XAS has allowed to confirm the good material quality that had been measured by JAVIER PABLO-NAVARRO et al. [17], albeit with less quantitative results.

In the light of this investigation, XAS appears a promising tool for future studies of such three-dimensional FEBID structures. Thanks to the element specificity of the technique, the metallic/oxidized state of complex e.g. core-shell samples with several metals of interest can be assessed. This would however require to solve the problem caused by field emission on one hand so as to gain access to more of the sample's shadow, and that of the strong FEBID halo. Among the leads I proposed in these regards [24], the prospect of micromanipulating the nanostructure after fabrication to a place devoid of halo is perhaps the most likely to overcome the halo issue. As for the field emission, it could be interesting to go for a more horizontal orientation, such as the desired engineered nanowire hanging between two smooth, elevated FEBID posts. Incidentally, sample vibrations should also be reduced. The height would allow to offset the shadow (even at the wire's tips) away from the posts themselves, and thus grant more safety with respect to field emission. Provided the distance is sufficient, the project shadow image on the substrate would be entirely accessible, contrary to this experiment where only a portion of the shadow could be used for XAS.

Speaking of the wire shadow, the reader will probably have noticed that the shadow XAS data was acquired with circular X-Ray beam polarizations. Since the shadow is expected to display magnetic dichroism, one could wonder whether such a signal could affect the shadow spectra. Indeed, dichroism can be present in spectroscopic features such as absorption edge shoulders, as is demonstrated e.g. by the XAS obtained by Chi Võ Vȃn [64] (see Fig. 2.5 p.34). The reason why the two curves in Fig. 3.13 for CL and CR polarization are essentially the same will become clearer in the next section, in which the magnetic configuration of the wire will be discussed. In essence, whatever the dichroism in the XAS images, its amplitude was too small so that with the short integration time, the noise-to-signal ratio was far too high to actually observe it.

V Magnetic state of the wire V.1 Full-field magnetic image and contrast level As a reminder, a horizontal field was applied to the sample in the plane of the bends with an electromagnet so as to saturate the whole nanowire before mounting the sample into the microscope. The applied induction was µ 0 H a = 0.73 T. We have discussed in section III.1 the expected outcome, i.e. three domain walls in the nanowire, two of them in the 90 • bends. So far in this chapter, we have tackled the many features from the PEEM imaging yet without magnetism, as well as discussed the spectroscopic data that has been recorded. Keeping these discussions in mind, it is time to move on to magnetic imaging.

An example of a full-field XMCD-PEEM image is featured in Fig. 3.14. This image corresponds to the normalized difference between one CL image and a CR one. Both are the average of 100 1 s-snapshots that have been drift-corrected beforehand. Thus, the quantity displayed in grey scale at each pixel (x, y) is the contrast C(x, y) defined as: Here, I CL (x, y) [resp. I CR (x, y)] is the photoelectron intensity on the CL (resp. CR) image at the same point. By definition, |C| ≤ 1. Considering the working energy set at the Co L 3 edge, in the absence of noise, C is nonzero only where there is magnetic circular dichroism associated to cobalt 21 . At first glance, the contrast in Fig. 3.14 seems to display quite a lot of features. However, the linear gray scale of contrast goes from -2.5% (black) to +2.5% (white), and the better part of the field of view is dominated by noise. For instance, the coarse-looking feature to the top right of the image corresponds to the field emission area i.e. the wire foot location (see Fig. 3.5 for comparison). In most of the image, the contrast is zero (except for the noise), indicating that the dots and the halo are not ferromagnetic 22 , as is suggested by the work of Fernández-Pacheco et al. [37].

C(x, y) = I CL (x,
It must be noted that even with this narrow contrast range, the main features in this image that are the one bright and two dark strips (from the center towards the bottom left) appear quite weakly. These correspond to magnetic contrast from the wire shadow. Such low contrast levels would be surprising in comparison to published reports [40,46] if not for the fact that we know the shadow to be very weak in our case. To explicit the link between this weak absorption and the low contrast, let us work on Eq.(3.1) and introduce the X-Ray beam's intensity I 0 , as well as σ(x, y) the proportionality factor between photon intensity I ϕ (x, y) impinging on the substrate and the number of detected photoelectrons per unit time at location (x, y). We may first write:

I CL (x, y) = σ(x, y) • I (CL) ϕ (x, y) = σ(x, y) u • I 0 exp - ˆwire µ L dl + (1 -u)I 0 I CR (x, y) = σ(x, y) • I (CR) ϕ (x, y) = σ(x, y) u • I 0 exp - ˆwire µ R dl + (1 -u)I 0
where u is the fraction of time during which the wire blocks the X-Ray beam on average. During this time, the incoming beam's intensity I 0 is decreased by the absorption through the wire core (its details being regrouped in the "wire" integration limit), and the rest of the time it directly hits the substrate surface. For the sake of simplicity, magnetization is assumed to be exactly along the beam; this also corresponds to the most optimistic contrast estimate. Dichroism is contained in the absorption coefficient per unit length for CL (resp. CR) polarization µ L (resp. µ R ). This is a crude "on/off" model 23 but the essential features are there. With this, the contrast [as defined in Eq.(2.13), p.73] reads:

C(x, y) = u • I 0 exp - ˆwire µ L dl + (1 -u)I 0 -u • I 0 exp - ˆwire µ R dl + (1 -u)I 0 u • I 0 exp - ˆwire µ L dl + (1 -u)I 0 + u • I 0 exp - ˆwire µ R dl + (1 -u)I 0 = exp - ˆwire µ L dl -exp - ˆwire µ R dl exp - ˆwire µ L dl + exp - ˆwire µ R dl + 2(1 -u) u (3.2)
To have an order of magnitude, let us recall that the values of µ R and µ L (for magnetization aligned along the beam direction) are 0.047 nm -1 and 0.080 nm -1 [34] and the wire diameter (not taking the projection angle into account) is about 70 nm. The effect of the different from one cobalt content is omitted here. Overall, the exponentials are at most on the order of 3%. Considering the previously discussed wire vibrations and the pronounced shadow broadening and low average absorption, it is reasonable to expect the factor (1u) to be more than 50%, taking the high intensity in the shadow from PEEM imaging (see section III.2.d) into account. This is why the u-term in (3.2) is in bold: it dominates the numerator of C. Based on this, the contrast cannot be expected to be higher than, say a couple percent. Therein lies the answer to the question related to CL/CR polarization for the shadow XAS that had been raised at the end of section IV.3 p.165: whether the shadow XAS was acquired with linear or circular polarization, the expected difference is so small that it could only be completely below noise level; it is worth remembering that the XAS images have 1 s integration time (see section IV.1), as opposed to 100 s in total for the XMCD-PEEM CL/CR images.

The above approach does not take into account noise on the photoelectron counts. As could be seen on the PEEM image from Fig. 3.5 or even in the halo/dots X-Ray absorption spectra from Fig. 3.13, the signal-to-noise ratio in areas outside the shadow is quite high, but this small noise is non-negligible with respect to the intensity differences corresponding to the shadow. Based on this, the coarse aspect of Fig. 3.14 is better understood. A more quantitative representation of the dichroic signal is given in Fig. 3.15, which features two line profiles extracted at the same position from XMCD-PEEM images. The latter were obtained at target photon energies of 768.20 eV (blue curve) 768.65 eV (red curve).

V.2 Interpretation of the dichroic contrast

The best shadow XMCD-PEEM image obtained being that in Fig. 3.14, we will now focus on it to try and understand the magnetic configuration of nanowire F11 1 . As a first step, it is sensible to delimit the shadow segments on the XMCD-PEEM image before interpreting the contrast and notably its visible change of sign. With the knowledge of the wire geometry and the beam's incidence angle, it should be possible to obtain a consistent picture of the whole, featuring the positions of the different wire segments on one hand, and the position of the wire foot on the other hand. ). The rays passing through junctions between segments delimit the shadow sections corresponding to segments (4) through (7). b) Shadow XMCD-PEEM image of the nanowire with dashed red lines delimiting these segments according to a). The domains with uniform contrast correspond to the segments (4), ( 5) and ( 7) from Fig. 3.2.b). The two black markers above segments ( 4) and ( 5) indicate the lateral positions of the contrast profiles in Fig. 3.16.c). The transition region between segments (4) and ( 5) is where a domain wall was pinned. The dashed blue line indicates the structure's foot position according to a). c) Contrast profiles across the shadow in domains (5) (in purple) and (4) (green), along with their best gaussian fits in dashed lines. The profiles were averaged laterally over ca. 400 nm for the sake of statistics. The linear gray scale goes from -2.5% to +2.5%.

To the right, the noise is slightly stronger where the structure stands because of the field emission.

To that end, a schematic but true-to-scale view of the X-ray beam falling onto the structure is featured in Fig. 3.16.a): it illustrates how the subsequent nanowire segments (4) through (7) are projected onto the substrate to form their respective shadow sections. One should note that the nanostructure's possible tilt is not featured; this will be explained shortly. Correspondingly, the XMCD-PEEM image of the structure's shadow shown in Fig. 3.16.b), with a red (resp. blue) arrow indicating its tip (resp. the nanowire's position) reproduces these sections, which are vertically aligned with their counterparts in Fig. 3.16.a). The structure itself is located on the right on the image, where the noise is stronger due to the field emission causing a strong local reduction of intensity. Fig. 3.16 was obtained first from the constraints that the red and blue arrows correspond to the shadow tip and wire foot respectively. Such conclusions were drawn by comparison with the PEEM imaging. Then, using the 18 • incidence angle, vertically above the wire foot, I obtained its tip. Then, drawing the sketch in reasonable resemblance to the known wire geometry 24 , I constituted the blue representation of nanowire F11 1 . It must be noted that since the wire foot's position is not known with great accuracy, many such sketches are plausible if some tilt is given to the nanostructure. This being said, the reader is invited to consider the SEM imaging presented in Fig. 3.9 and judge whether the agreement after correcting for the more precise geometry is acceptable. Two further elements convinces me that it is: the first one being the clot of matter on the lower wire bend highlighted in Fig. 3.11 can be expected to locally broaden the shadow due to the wire vibrations. It turns out that at the position on the shadow that can be predicted from the height of this clot, the shadow is broader, see ca. 0.5 µm to the left of the red dashed line in Fig. 3.7.a). Also, the position of the transition from (4) to (5), about 5.5 ± 0.3 µm away from the structure, is consistent with its SEM-measured height [see Fig. 3.2.a)] of 1.5 ± 0.1 µm:

(5.5 ± 0.3 µm)× sin (ψ) =1.7 ± 0.1 µm, where ψ =18 • is the beam incidence angle. Thanks to these "anchor points" as well as small corrections to the exact wire geometry, I have confidence in the correspondence between projected wire segments in Fig. 3.16.a) and labelled areas in Fig. 3.16.b).

We may now come back to the three identifiable domains in Fig. 3.16.b) : a faint dark domain at the tip corresponding to segment (7), and further away from the latter another dark domain in segment (5) followed by a bright one in segment (4). Even though the background emission is strong, the close to 1% contrast amplitude is clearly above noise in the contrast profiles of Fig. 3.16.c), see full purple and green lines. These have been extracted perpendicular to the shadow, and their position along the latter is indicated with the black triangles in Fig. 3.16.b). The dashed lines are gaussian fits to these XMCD contrast profiles, with the fit function f defined by

f (x) = A exp [-(x/∆) 2 ]
. The agreement between the data and the fits is surprisingly good, with fitting results ∆ (exp) 120 nm in both cases. It is worth remembering that the actual diameter of the Co rods is about 70 nm.

Actually, the reader may well be far less surprised than I was when I realized this unexpected agreement: we have discussed extensively the likely vibrations experienced by the wire during the PEEM investigations as well as their effect on the shadow (see for instance section III.2.d and Fig. 3.8). Historically, these fits which should not have worked were the starting point for the reflections which lead to the idea of vibrations. In the absence of vibration i.e. u = 0, the translation of Eq.3.2 for the XMCD contrast C(x) along the x direction (x = 0 being the wire center) reads:

C(x) = tanh κ 1 - x 2 d 2 (3.4)
where κ is proportional to the difference δµ = µ Lµ R in linear absorption coefficients of X-rays in Co (for magnetization parallel and anti-parallel to the beam at the Co L 3 edge) and the local nanowire diameter d. This is the basis for the wire transparency profile that was shown in Fig.

k -→ M k k X -r a y s k 3.8. a) b) c) d) α
Fig. 3.17: Schematic view of the four possible magnetic configurations [a) through d)] at the bend between segments (4) and (5) (in blue). The X-ray beam's wave vector k (in red) determines the shadow XMCD-PEEM contrast with the same sign as -→ M • k. Considering the orientations at play, our experiment corresponds to b), indicating the presence of a domain wall (in gray) in the bend.

As a side note, wave optics should be mentioned here: the nanowire diffracts the incoming beam, which leads to an image onto the substrate with different lateral dimensions. Fringe patterns due to Fresnel diffraction on 3D objects have already been reported [40] in XMCD-PEEM experiments. In our case, we are dealing with Fresnel diffraction as well, and a rough numerical computation of the diffraction pattern (not shown here) leads to a satisfactory agreement in terms of profile shape. However, taking into account the instrumental resolution of ca. 30 nm to 40 nm as well as the beam's divergence of ca. 0.008 mrad leads to an estimate of the width ∆ (theo) 80 nm to 90 nm, in rather poor agreement with ∆ (exp) . In the present case, the effect of wave optics is probably second order at most. To give an order of magnitude: the peak-to-peak amplitude of Fresnel fringes observed with PEEM by JAMET et al. was about a fifth of the difference between the intensity on the substrate and in the shadow of a ca. 300 nm-thick FeNi nanowire [40].

Let us now come back to the magnetic state of the observable parts of the nanowire's shadow. Denoting ϕ the angle between the beam's wave vector and magnetization, the XMCD contribution to absorption is proportional to cos ϕ [70]. Now, the absence of contrast in segment ( 6) is due to both the beam's incidence angle (ϕ closest to 90 • among all visible segments) and the presence of a bright dot in this area, as can be seen in Fig. 3.5. This dot does not contribute to the dichroic signal because of its non-ferromagnetic nature, but does enhance the background level following the above-described mechanism [see Eq.(3.3)], and therefore reduces the signal-to-noise ratio [40]. Similarly, to the right of segment (4), the halo emission is too strong and no magnetic contrast can be retrieved from this part.

The soft nature of the deposited cobalt, along with the constant contrast sign within segments ( 7), ( 5) and ( 4), suggests that magnetization is uniform inside them and aligned with the local segment axis. As for the change in contrast sign from segment (5) to segment (4), we must be careful because its interpretation heavily depends on the angle α between them, see Fig. 3.16.a).

Remembering that the contrast has the same sign as -→ M • k, where -→ M is magnetization and k the X-ray wave vector, we must consider the direction of the segments (4) and ( 5), since magnetization will be aligned to them. Fig. 3.17 illustrates the four possible scenarios at this bend, taking into account all relative orientations in our experiment. As another way of considering these, the reader is invited to note that going downwards from segment (5) to segment (4), the local tangent to the wire axis keeps the same sign of projection onto k. Therefore, one can actually forget the bend and reason about magnetization as if both segments had the axis of segment (5).

It can be seen that the only situation corresponding to our contrast pattern is that of Fig. 3.17.b). The key feature is a change in contrast sign; thanks to the latter, we can conclude that magnetization reverses from segment (5) to segment (4): in other words, a domain wall is pinned in this bend. This interpretation remains even in the event of a wrong association of contrast sign to magnetization direction (C → -C) since only the domain wall polarity would be changed, corresponding thus to Fig. 3.17.c). This being said, the application of the in-plane field carried out as depicted in Fig. 3.3 should have led to a negative domain wall polarity, provided the field direction was indeed the same. Let us therefore try and go one step further.

As was stated by Joachim Stöhr [71], "The hande[d]ness of circularly polarized light is not uniquely defined". This suggests. . . caution. Although the L 2 and L 3 absorption edges of ferromagnetic transition metals are known to possess XMCD with opposite sign [65,69] (as was mentioned in Chapter 2, section II.1), linking experimental measurements to a magnetization direction with certainty is not that straightforward. There is the question of beam helicity (positive/negative?) or conversely the sense of rotation for circular polarization, and then a further source of uncertainty is the alignment of photon spin either "with the majority spins" of the studied ferromagnet or "with its magnetization". Also, one should be careful with i.e. "regular" XMCD-PEEM intensity measurements as opposed to transmission data from e.g. STXM, as they behave in a complementary way. Last but not least, one should mind the matter of convention or to put it differently: is the XMCD contrast defined and computed as in Eq.(3.1), or with the opposite convention?

Based on the comprehensive (especially in terms of conventions) introduction to XMCD by Baudelet [72] on the theoretical side and for instance the experimental confirmation by Chen et al. [73], the contrast as defined in Eq.(3.1) in conventional XMCD-PEEM imaging (e.g. of a thin film) is positive at the L 3 edge of cobalt and negative at the L 2 edge if magnetization is along the beam's propagation direction. Therefore, it must be negative (resp. positive) at the Co L 3 (resp. L 2 ) edge in shadow imaging for the same alignment. I have checked that the image processing I used to obtain the shadow XMCD-PEEM views followed the contrast definition of Eq.(3.1). Thus, the contrast pattern remains that of Fig. 3.17.b), however the orientations of magnetization in the domains must be flipped.

It is worth remembering the uncertainty mentioned at the beginning of section III.1 concerning the actual direction of application of the horizontal field. Considering the above, it appears more probable that the field was applied with an orientation opposite to that depicted in Fig. 3.3, since the observed domain wall is head-to-head. It may well correspond to the positive-polarity counterpart of the tail-to-tail domain wall depicted in segment (4) in Fig. 3.3, and have moved to the junction between segments (4) and (5).

Due to the lack of contrast from segment (6), it is unfortunately impossible to assess the presence or absence of the other expected domain wall in this area. Taking into account the weak but negative contrast from segment (7), there are only two possibilities: either segment (6) has magnetization antiparallel to both segments (5) and ( 7), in which case there are two additional domain walls, or segments (5) through (7) share the same magnetization direction. In the latter case, there may have been domain wall motion and subsequent expulsion at the wire tip, or additional domain wall nucleation and annihilation of a pair. No interpretation can be definitive, and furthermore there is the question of the magnetic state of segments (1) through (3). However, looking back on the adverse experimental circumstances faced, namely strong background and significant nanowire vibrations, three sevenths of the global magnetic configuration could be retrieved despite low signal-to-noise ratio.

VI Prospectives

This chapter's object has been the synchrotron investigation of a ferromagnetic nanostructure with a marked threedimensional character. Not many techniques apart from FEBID allow the fabrication of such an elaborate sample, featuring both a significant height and a large aspect ratio. Of particular interest in this regard are novel two-photon lithography techniques, either with negative resists and physical deposition methods onto the 3D scaffold [74] or positive resists and electrodeposition into the 3D channels [75], and additive fabrication via (micro-)plasmaenhanced chemical vapour deposition [76]. Taking the considerable versatility of FEBID regarding the shape and composition owing to its "3D-nanoprinter-like" ability, there appears to be a rich variety of subjects in 3D nanomagnetism where this fabrication technique may be a powerful tool. Since the nanowire under study was a core-shell structure, an immediate example would be a similar sample with exchanged non-and ferromagnetic materials. Individual ferromagnetic nanotubes have raised significant interest [48] notably on the side of fabrication and theory/simulations [77][78][79][80] and recently imaging [1,55] Contrary to the work by Streubel and coworkers [3,4] on much larger and bulkier structures, this investigation did not consider only regions significantly far away (about 150 µm [4] in the "standing" configuration) but attempt to recover magnetic information from all of the sample's shadow. I have explained the different obstacles to imaging in the immediate vicinity of the wire and highlighted the importance of the well-known FEBID halo as well as field emission and the vibrations it induced [24]; such effects had not been encountered in previous shadow XMCD-PEEM experiments to my knowledge. Among the solutions that could be implemented, there is notably the possibility of micromanipulation so as to (at least) remove the nanostructure from the halo-contaminated area. Provided that it does not damage the sample itself, the deposition of a uniform metallic layer via physical methods could also be beneficial since it could prevent photoelectrons from the contaminated substrate surface to escape into vacuum. On one hand, the larger linear absorption coefficients of, say, gold and platinum [µ Au (Co L 3 ) µ Pt (Co L 3 ) 0.015 nm -1 as opposed to µ Si (Co L 3 ) 6.9 × 10 -4 nm -1 [41,52]] slightly reduce the number of photons interacting with the contaminated substrate and more importantly lead to higher, spectroscopically flat background, hence a more contrasted shadow. On the other hand, this metal thickness prevents whatever phototelectrons were created by the beam in the contaminated surface to reach vacuum, provided it is larger than the photoelectron mean free path (about 10 nm in this energy range [69]).

Provided the sample design allows it, another way to study geometrically engineered wire-like structures with shadow XMCD-PEEM could be to "attach" these to two elevated posts on both sides. Not only should this reduce the influence of vibrations, but the choice of post height allows to project the sample shadow far away enough so as to not be bothered by field emission any longer. In principle, the posts' fabrication can definitely be independent from that of the actual sample, and need not take place at the same location if micromanipulation is an option.

Such a tool has the drawback of potential damages to the fabricated sample. The reader is referred to Chapter 4 section I.5 and appendix section A 4 .1 and. Nevertheless, as an element of context, tungsten nanowires fabricated with FIBID have been shown to possess higher yield strength (i.e. the threshold stress before the plastic deformation regime) than bulk W, albeit their stiffness was lower. Furthermore, micromanipulating an object that is held to its substrate only by a thin foot as nanowire F11 1 is much easier to do without damage than micromanipulating a wire lying horizontally on the surface. Along this line, the majority of those wires which we tried to image during the beamtime was still standing. Aside from shocks during transportation, the structures may have been damaged during sample mounting and also by the imaging technique itself. The wire counts have been summarized in Tab.3.2. Although they are far too low to speak of statistics or draw conclusions, it is interesting to see that going from coated to uncoated nanostructures, the tendency to survival appears to be drastically suppressed. 1.h), significantly different core diameters can be achieved, and besides, the tuning of this geometrical parameter as well as the interplay with the core composition has been investigated by JAVIER PABLO-NAVARRO and coworkers [81]. Aside from the diameters, the composition and the presence/absence of coating, the large extent of the accessible ranges in structure height, bend angles as well as structure tilt [see Fig .A 3 .1.f)] is evident. Coming back to magnetism, it was shown that these structure possess ca. 65%-70% Co in terms of composition. Quite naturally, the spontaneous induction µ 0 M s of the deposited cobalt-rich material was shown [81] to increase with the Co purity, and with the presence of coating [17].

A 3 .2 Modelling of the wire's temperature increase This section's goal is to estimate first the temperature increase at the foot of nanowire F11 1 as a function of the total received thermal power Q. It is this analysis which I have briefly described in the published report on this experiment [24].To that end, I assume a thermal equilibrium reached, featuring:

• a well-defined temperature T ∞ far away from the wire foot, corresponding to room temperature (the silicon substrate being on a metallic sample holder in this experiment),

• a starting temperature T 0 at the foot of the wire, the quantity T 0 -T ∞ being the sought temperature increase,

• spherical symmetry in the half space corresponding to the silicon substrate.

The temperature T 0 is assumed to be reached on a hemispherical cap with diameter 2r 0 corresponding to the nanowire's diameter close to its foot. This cap is represented in blue in Fig. A 3 .2. A certain heat flow, represented by the -→ j th (r) = j th (r) -→ e r thermal flux density current (in green), carries away the heat received by the wire. r and -→ e r are the usual spherical radius and radial unit vector respectively. On the basis of small thermal power received even by a full-film cobalt sample under the X-Ray beam at resonance [56,57], heat transfer to the Si substrate 25by the beam is neglected. The beam and nominal photoelectron emission direction are represented for ease of representation, since the vertical in Fig. A 3 .2 goes from the Si wafer's top surface into the substrate.

For a start, we shall assume that the silicon's thermal conductivity λ Si is that of bulk undoped Si at room temperature. If large temperature increases are found, we will have to depart from this simplification. Considering that the sample is in ultra-high vacuum, only thermal conduction through the substrate is considered for heat evacuation: the wire is thermally insulated everywhere except at its foot.

The required thermal equilibrium requires first [82]:

T = 0. (A 3 .1)
Using the expression of the Laplace operator in spherical coordinates [32] and the spherical symmetry inside the Si substrate, we obtain after integration: As can be seen by comparing these different structures, quite a variety of segment lengths, diameters and orientations can be achieved. It must be reminded however that the FEBID growth process depends on many different parameters, the local temperature not being the least of these. As a result, fine and reproducible geometry tuning is not as straightforward as it may seem at first glance.

r 2 dT dr = A (A 3 .2)
where A is a constant. The boundary condition at infinity now allows us to write:

T (r) = T ∞ - A r . (A 3 .3)
The boundary condition at r = r 0 , corresponding to T 0 , results in:

T (r) = T ∞ + (T 0 -T ∞ ) • r 0 r . (A 3 .4)
Determining T 0 requires to plug in the power evacuated in the steady-state regime, that is to say Q. As a first step, the thermal flux density current reads [82]: surface is in pink, the vertical z goes inward into the Si substrate, who is assumed to be at thermal equilibrium with temperature T ∞ sufficiently far from the center of the wire foot (see black cross in the center). The wire radius is here denoted r 0 , and its foot is assumed to have a temperature T 0 . For the sake of modle simplicity, this temperature is assumed reached on the hemispherical blue cap touching the rim of the wire foot. Considering perfect insulation from the sides, all the thermal power Q received by the wire must be evacuated through its foot, resulting in a temperature increase T 0 -T ∞ > 0, and therefore a temperature profile T (r) and thermal current flux density -→ j th (r) (in white). Spherical symmetry is assumed in the half-space corresponding to the Si wafer.

T 0 T ∞ T(r) X -R
j th (r) = -λ Si dT dr = λ Si (T 0 -T ∞ ). r 0 r 2 (A 3 .5)
On the other hand, we can write:

ˆ2π ϕ=0 ˆπ 2 θ=0 j th (r) sin (θ)r 2 dθdϕ ≡ Q. (A 3 .6)
Thus, combining equations (A 3 .5) and (A 3 .6), we get:

2πr 0 λ Si (T 0 -T ∞ ) = Q. (A 3 .7)
As a result, the temperature increase reads:

T 0 -T ∞ = Q 2πr 0 λ Si . (A 3 .8)
Numerical values for a thermal power Q beam due to X-Ray absorption (on the order of a few tens of microwatts, as explained in section III.2.f) alone are about 8 mK to 10 mK. To claim that the beam played little role in elevating the wire temperature is thus an understatement.

Actually, this is the case provided the thermal conductivity of the nanowire is not too low. One has reasons to expect a decrease in thermal conductivity with respect to bulk values: from the report of Ou et al. [60] for instance, a significantly reduced thermal conductivity of nickel was measured in a nanostrip. Though this nickel strip's overall electrical behaviour was metallic, there may be however an influence of some oxide layer due to the lithography and etching process.

The approach I chose consisted in estimating a composition-weighted, core-shell-structure-and temperaturedependent 26 average conductivity first based on bulk values available for cobalt, platinum [52], and amorphous carbon [58]. These values are represented in Fig. A 3 .3. The purities were 0.7 for cobalt in the core and 0.3 for amorphous carbon, and 0.2 for platinum in the shell and 0.8 for amorphous carbon in this part. neous in a cross-section. q(x) is the thermal power received by the wire per unit volume at abscissa x.

The simplest approach was to fit 27 the conductivities with second-order polynomials in the temperature; thus, the structure-and composition-average λ avg also is a second-order polynomial of the form:

λ avg (T ) = λ 0 1 + T -T min T scale 2 (A 3 .9)
With this, it is easier to seek the stationary solution of the heat diffusion equation [82] inside the nanowire. With the above-described material parameters, I found T min =1172 K, T scale =1098 K, and λ 0 28 W/(m•K). I have chosen to write it in a 1D approach for the sake of simplicity: the wire is assimilated to a straight rod parametrized with an abscissa x, with length L =2.7 µm and uniform temperature in a cross-section. The cylinder radius is the outer radius r T =47.5 nm. This is represented in Fig. A 3 .4. One can write:

λ avg T (x) • d 2 T dx 2 = -q(x) (A 3 .10)
where q(x) is the thermal power per unit volume transferred to the wire. The simplified evaluation of λ avg is helpful because the problem is non-linear by nature. The boundary conditions are that T (x = 0) = T 0 at the wire foot, and that dT dx (x = L) = 0 at the wire tip, since the thermal flux density current must vanish there. Introducing the variable Tmin = T min /T scale and the function T (x) = T (x)/T scale , the problem becomes:

d 2 T dx 2 = - q(x) λ 0 T scale 1 + [ T (x) -Tmin ] 2 T (x = 0) = T 0 /T min d T dx (x = L) = 0 T 0 = T ∞ + πr 2 T ˆL 0 q(x)dx 1 2πr T λ Si (A 3 .11)
The last line in system (A 3 .11) is the translation of equation (A 3 .8), for consistency. In the following, q(x) is constant and attributed to Joule heating only, q = ρJ 2 . I assumed a resistivity on the order of 1 µΩ•m as measured by Begun et al. [11] on as-deposited uncovered FEBID cobalt stripes. Concerning the current density J, I relied on the indication from RACHID BELKHOU that in PEEM experiments without electrical discharges, the current I sample flowing from the sample to the microscope is a few tens of nA. For the sake of safety, let us assume I sample =100 nA, and that all of it flows only through nanowire F11 1 . Taking into account the wire cross-section, this results in an average current density of about J =2 × 10 8 A/m 2 .

System (A 3 .11) I solved using Mathematica 9.0 [59]. The maximum temperature increase was of course at the wire tip; using the previously mentioned parameters, I found about 16 µK. This was reassuring, but to gain more confidence, I increased the material's resistivity from 1 µΩ•m to 100 µΩ•m and the current from 0.1 µA to 10 µA (which is actually unrealistic, PEEM being impossible in these conditions); the temperature increase gained correspondingly 6 orders of magnitude, but 16 K temperature increase is still benign.

To summarize: even with a very largely overestimated current density and a safety margin on the material's resistivity based on experimental reports, the estimated maximum temperature increases are clearly below a (few) hundred kelvin, for which annealing processes could be expected. Also, even under the assumption that the thermal conductivity is much closer to that of amorphous carbon e.g. λ 0 28 W/(m•K) → 2.8 W/(m•K) due to the reduced dimensionality, keeping the more reasonable (but still overestimated) current I sample =100 nA and upper bound on the resistivity ρ =100 µΩ•m, the estimated temperature increases reaches at most 16 mK. Therefore, the estimates provided by this modelling are still well within the latter's limits of validity: as a reminder, I had assumed that the silicon's resistivity could be taken as the room temperature one, and obviously changes in the material's resistivity can be safely neglected, as is done in my model.

It thus seems safe to claim that the FEBID nanowire cannot have experienced significant heating during the synchrotron experiments.

DOMAIN WALL TRANSFORMATIONS IN CYLINDRICAL NANOWIRES UNDER FIELD

The work presented in the previous chapter was focused on the shadow XMCD-PEEM investigation of a novel type of sample, namely an upright ferromagnetic nanowire with a heavy-metal-based coating and featuring (at least) one domain wall. The latter was evidenced at rest, and apart from the initial magnetic field, no stimulus was applied to the nanostructure. Therefore, the behaviour of domain walls in this system remains so far an open question. In particular, it would have been interesting to compare the domain wall pinning sites constituted by the FEBID nanowire bends in the light of simulations [1] and experiments [2][3][4][5] related to bends in quasi-1D structures as pinning sites.

In the present chapter, ferromagnetic nanowires will be studied with (shadow) XMCD-PEEM as well, but both the samples and goals will be different. The focus will indeed be on electrodeposited structures out of Ni 1-x Fe x and Ni 1-y Co y alloys; these systems will be the hosts for domain walls as well as conduits in which to move the latter under magnetic fields. As was discussed in Chapter 1, section III (p.20), only two different types of domain walls are expected from simulations and have been experimentally evidenced in such systems: the transverse-vortex wall, hereafter denoted TVW, and the Bloch point wall, hereafter denoted BPW. While several predictions from the side of theory [6] or simulations [6][7][8][9] have been published so far regarding the behaviour of these domain walls under field or current (for the TVW) [10], no experimental reports have yet shown evidence supporting/disproving these expectations.

Beyond the stark difference in mobility under field, what should distinguish the two domain wall types is the fact that the BPW is expected to possibly change configuration depending on both its initial state and the applied field. As was discussed in Chapter I, section III. 3 (p.25), it has been predicted that motion lifts the degeneracy between the two possible senses of magnetization curling around the Bloch point, leading to the selection of one, namely right-handed with respect to the direction of motion [8]. During motion, the velocity v therefore defines a right-/left-handed helicity; assigning a positive (negative) sign to right-handed (left-handed) helicity, then it can be defined as h v with the following equation:

h v [ - → m] = ˛Cd/2 → × - → m • v dl ˛Cd/2 → × - → m • v dl (4.1)
where -→ m = -→ M /M s is the usual unit vector with the direction of magnetization, and C d/2 is the circle with radius d/2, d being the wire diameter, that lies in the plane of antisymmetry for the longitudinal component of -→ m. To put it differently, this is the plane normal to the wire axis containing the Bloch point itself. In this plane, -→ m ⊥ v, and irrespective of the coordinate system, → × -→ m points along v for a right-handed rotation of -→ m around v. The normalization by the integrated absolute value of ( → × -→ m) • v ensures that irrespective of deviations with respect to an orthoradial circulation of magnetization 1 , h v = ±1. Thus, this quantity 2 does reflect the binary notion of right/left-handed with respect to the direction of motion. Also, it can be evaluated "by eye" provided that the BPW configuration is available, as is the case with shadow XMCD-PEEM.

The existence of such a helicity selection has not been demonstrated beyond doubt yet, however the extensive and rigorous experimental work by Aurelio Hierro-Rodriguez and coworkers [14] with STXM on magnetic multi-1 Such deviations are expected, both from theory of the Bloch point's immediate surroundings [11] and simulations in soft nanowires with square [6,12] or cylindrical cross-sections [12]. 2 Inspiration originated from the helicity defined in particle physics [13]. The counterparts of particle, spin and momentum would here be the BPW, the integral of the curl of -→ m, and v. By the way, it is noteworthy that the convention in this manuscript regarding left/right circularly polarized light (sense of rotation of the vector field as viewed in the direction opposite to the propagation direction) is the opposite of that used in e.g. the book by Schwartz [13] (for example "left" means there left-handed vector rotation around the propagation direction). This illustrates the comment by Joachim Stöhr made in Chapter 3, p.171. . . layers stands out as a noteworthy step in this direction. First, the immediate vicinity of the evidenced Bloch points bears close resemblance to the BPW in cylindrical nanowires. In this study, the middle NdCo layer supports the equivalent of the longitudinal domains from wires, while around them in the NdCo and above or below them in the NiFe or the GdCo layer, and the flux closure pattern corresponds to the longitudinal curling. The observations by Hierro-Rodriguez et al. indicate (at least in one of their samples) a clear majority (out of dozens of events) involving propagation of BPWs with positive h v helicity, as expected from theoretical arguments.

The synchrotron investigations reported in this chapter aimed at shedding light on this predicted selection of BPW helicity. The samples hosting domain walls will be presented first, in section I. The focus will then move on to fine features of XPEEM imaging and X-ray spectroscopy in section II. I will thereafter discuss my results pertaining to domain wall motion, this topic being split into two sections. First, section III will tackle the preparation of nanowires hosting domain walls, as well as their identification. Then, section IV will tackle the search for evidences of BPW helicity selection across events of domain wall motion between pinning sites, nucleation and subsequent pinning, and unexpected changes in domain wall type. The latter findings will not be discussed in detail in that section but in the following one, section V. After more details on reproducible observations of this kind, I will present a comparison between the experimental results and theory as well as simulations. Finally, an implementation of the high-frequency developments that have been presented in Chapter 2, section IV will be reported in section VI: an attempt at domain wall motion driven by a static magnetic field and assisted by a nanosecond-scale pulse.

I Sample preparation

As a general remark: all substrates holding nanowires were demagnetized using the setup introduced in Chapter 2, section I.5.a. The durations over which the field amplitude was ramped down to zero ranged from one to three days. In all cases, the starting induction value was on the order of 1.3 T to 1.4 T i.e. higher than the saturation induction of Ni 60 Co 40 (about 1.08 T) or Ni 80 Fe 20 (1.04 T), even Ni 70 Fe 30 according to the Slater-Pauling curve [15]. Since the requirement for transverse saturation of an infinite cylinder is the application of half the material's saturation induction (as was shown by the cylinder's demagnetizing factors, see Chapter 1, section section I.1.c ), there is a large safety margin in all considered cases.

I.1 NiFe nanowires

The investigated NiFe nanowires were produced at Institut Néel by RAJA AFID and LAURENT CAGNON by electrodeposition into nanoporous aluminum oxide membranes received from the University of Erlangen. These templates, designed and etched by SEBASTIAN BOCHMANN and JULIEN BACHMANN, featured pores with two values of diameters, such as was described in Chapter 2 (see section I.2.c). The larger diameter D thick of the two pore extremities was about 250 nm while the smaller diameter D thin of the middle section was about 150 nm. In terms of length, all three sections were designed to be about 10 µm. Details of the fabrication process lead to asymmetric diameter modulations, with one being steeper than the other one.

This asymmetry is expected to play a role in the interaction between diameter modulation and domain wall. Simulations performed by Judith Kimling [16] in modulated wires whose diameters still maintain quasi-uniform magnetization (see Fig. 6.16 p.129) already suggest that the magnetostatic repulsion induced by the diameter modulation changes as a function of the latter's extent. This is readily seen in Fig. 6.17.(b) especially for the -30 mT simulation points: when pushed towards the modulation with fixed strength, a domain wall actually gets closer and closer to the modulation center as the length "l" of the modulation increases. In an hand-waving approach, this can be attributed to the spreading of magnetostatic charges over a larger extent, while conserving the total charge. Fig. 4.1 shows an SEM image of the abrupt modulations in an AAO template produced at the University of Erlangen, while Fig. 4.2 displays the other type of modulation. As can be seen from comparing both, the longitudinal extent of the transition in diameter is comparable; the measured ca. 300 nm in Fig. 4.2 stand out as an exception rather than the rule and may well originate from a cut in the alumina template with some angle. The principal difference lies in the above-described overhang of the steep modulation.

Into these pores, an NiFe alloy with target composition Ni 80 Fe 20 i.e. that of permalloy with vanishing magnetocrystalline anisotropy and magnetostriction [17] was deposited by RAJA AFID and LAURENT CAGNON. The electrodeposition was performed in potentiostatic conditions i.e. the deposition potential was kept constant. In the opinion of my electrochemists colleagues, the actual sample composition of the first batch probably was offset towards lower content of iron with respect to the target; EDX measurements indicated a composition closer to Ni 90 Fe 10 than Ni 80 Fe 20 . The second batch of such samples also had an offset of ten percentage points in iron content, but towards nickel-poorer composition. This second batch was electrodeposited into AAO templates with the pore geometry described in section I.3.

I.2 Silica sheath

In all the previous paragraph, it has been assumed that the investigated nanostructures contain no other element than iron, nickel, and oxygen on the surface. However, as was mentioned in section I.1, the etching of the pores serving as mould for the nanowires involves the deposition of a silica shell by ALD. Since this material is rather resilient to the etching agents used for the dissolution of the alumina templates, the absence of these SiO 2 sheaths in the synchrotron experiments is not warranted.

This being said, even if the ALD layer is still present on the nanowires, it may only impact (XMCD-)PEEM imaging or XAS acquisition on the wire surface. Indeed, the linear X-ray absorption coefficient of silicon (in silicon dioxide) at the Fe L 3 edge µ Si (Fe L 3 ) is less than 8 × 10 -4 nm -1 , and that of oxygen is less than 2 × 10 -3 nm -1 [18], corresponding to mean free paths over 1.2 µm and 0.5 µm respectively. Therefore, the attenuation (in the shadow) that may result from this shell is weak.

The effect on the wire surface would on the other hand be more severe since the nominal ALD layer thickness of about 10 nm is larger than the photoelectrons' escape depth [19]. In other words, photoelectrons created in nickel or iron close to the wire surface cannot leave the nanostructure. This implies that the behaviour of the photoelectron counts in this case are dominated not by the actual wire but by silica, whose photoemission is weak, as was stated above.

While I have no clear evidence of this occurring with e.g. spectroscopic studies supporting the above scenario, it may have been a cause for weak XMCD-PEEM contrast on the surface of some of the investigated nanowires.

I.3 NiCo nanowires

After the review of NiFe nanowires in section I.1, the present section tackles a different type of samples, both in geometry and in composition. While the previous nanowires were out of NiFe (deposited at Institut Néel into AAO templates from the University of Erlangen) with an amplitude of diameter modulation D thick -D thin 100 nm, the present structures were out of a Ni 60 Co 40 alloy deposited into templates with diameter modulations D thick -D thin 50 nm. Both template etching and alloy electrodeposition were undertaken at the University of Erlangen.

The addition of a NiCo alloy to the studied NiFe was notably motivated by the effort put into the development of this material by SEBASTIAN BOCHMANN and JULIEN BACHMANN in Erlangen, as well as magnetometry measurements performed on Ni 1-y Co y nanowires with variable cobalt content y [20]. These indicated a minimum in coercive field and remanence for the Ni 60 Co 40 composition (as measured on wires still in the AAO template) and a more abrupt magnetization switching from comparison between magneto-optical Kerr effect (MOKE) on a single Ni 60 Co 40 nanowire and a Ni 82 Co 18 nanowire. The former measurement's interpretation was that the Ni 60 Co 40 was softest, and the latter's interpretation that domain wall pinning and stochastic nucleation are absent for this composition.

It should however be noted that on one hand, an alloy's soft nature i.e. the strength of its magnetocrystalline anisotropy constant(s) cannot be directly inferred from a coercive field measurement on an array of interacting nanostructures. Coercivity in these systems primarily reflects domain wall nucleation followed by wire switching after domain wall propagation [21]. Nucleation is in turn heavily dependent on dipolar interactions which depend on the spontaneous magnetization M s (thus on alloy composition) as well as the array geometry, so that magnetocrystalline anisotropies, if present, do not solely determine the array's coercive field. Furthermore, concerning the "soft" magnetic nature of the alloys, measurements on bulk NiCo alloys [22,23] show that alloys with composition close to Ni 80 Co 20 not only have a (2 nd order) magnetocrystalline anisotropy constant more than one order of magnitude lower than Ni 60 Co 40 alloys, but also comparable or weaker saturation magnetostriction coefficients. While transferring these facts from macroscopic samples to nanostructures is not obvious considering their fabrication process, even if one follows the criterion of BOCHMANN et al., the difference in remanence and coercive fields for Ni 60 Co 40 and Ni 82 Co 18 are not starkly different anyway 3 .

On the other hand, one might argue that depending on the position of the laser spot used for MOKE magnetometry on single wires in reference [20] may play a critical role in the aspect of hysteresis loops. Unfortunately, this issue is discussed neither in this work nor in the investigation by Vega and coworkers [24]. Furthermore, irrespective of the actual laser spot position, the lack of statistics for these magneto-optical measurements seems insufficient to be able to make statements about stochastic processes such as domain wall nucleation and pinning.

Leaving the issue of material aside, the change in geometry was motivated by the fact that the too strong diameter modulations appeared to build mechanical weak points. During the dissolution of the nanoporous alumina templates and subsequent purification/rinsing steps (prior to dispersion of wires onto a substrate), some stirring is mandatory, and in many instances nanowires were found (after dispersion) broken precisely at the diameter modulation. Along a different line, the absence of XMCD-PEEM observations of domain walls being stopped by the diameter modulations suggested that they geometry might be unsuitable.

An SEM view of these new samples is presented in Fig. 4.3. Here, a cluster of nanowires was formed probably before dispersion onto the silicon substrate. It is worth mentioning that the dissolution (by RAJA AFID and MICHAL STA ŇO) of the alumina membranes containing these nanowires was found more difficult than that of previous, similar templates. Even after this problem was solved with ion beam etching (IBE) prior to dissolution, the dispersed wires were still found to exhibit a certain surface roughness. . With these images, is clear that even with larger fields of views, the surface roughness remains visible. Considering that this roughness was observed on a very large majority of dispersed Ni 60 Co 40 nanowires and (to the best of my knowledge) not on the majority of the investigated NiFe nanowires, there appears to be a statistically significant difference in surface state between the two batches of nanostructures.

I do not have a decisive explanation for this surface roughness. Let us however recall that before the template dissolution, aside the metallic samples themselves, the only materials present are (presumably) aluminium oxide and silicon dioxide 4 . Let us also consider that nanowires liberated from the membrane without prior IBE and after two days in sodium hydroxyde had significantly rougher surfaces than the ones shown in Fig. 4.5, who had necessitated only one day in NaOH after the IBE of both membrane sides. In the former case, the peak-to-peak roughness reached 20 nm to 30 nm, and in the latter about 10 nm or less. One final fact to consider is that the etchant-resistant SiO 2 layer is about 10 nm thick in both cases and should not cover the whole nanowire as it is deposited before the pores are fully etched; yet, SEM imaging by MICHAL STA ŇO (not shown here) reveals that this surface roughness extends over all of the concerned nanostructures.

Based on this, one plausible explanation is as follows. The dissolution of alumina may not have been as efficient as in previous cases and the rough features are template remnants; such a hypothesis might have been checked with PEEM imaging at Elettra. Indeed, a flux over 1 × 10 12 ph/s is available5 at the Nanospectroscopy beamline "from 50 to 600 eV" [25], which encompasses the L 1 , L 2 and L 3 edges of Al at 117.8 eV, 72.95 eV, and 72.55 eV respectively [26]. This, unfortunately, was not done. However, Transmission Electron Microscopy imaging with Energy-Dispersive X-ray spectroscopy (EDX) performed by MICHAL STA ŇO indicates the presence of aluminium in the rough "shell" around the nanowires. Silicon might be present but the signals (not shown here) are generally significantly weaker. These findings suggest that the rougher nanowires result from insufficient template etching. This being said, one should keep in mind that the etching time leading to the wires in Fig. 4.5 still was longer than usually needed for template dissolution. Taking into account the spiky to lamellar aspect of the surfaces, it seems legitimate to wonder whether the removal of alumina might not have also lead to significant (localized) oxidation. In this respect, X-ray Absorption Spectroscopy (XAS) revealed (see section II.2) pronounced oxidation on some wire surfaces from the Co L 3 peak shape and (sometimes) substructure on one hand, suggesting an absence of protection from oxidation (in other words, remnants of alumina apparently do not constitute a passivating layer) and from the shadow PEEM imaging of a cobalt oxide rim at a photon energy where the wire's bulk was still transparent.

As a conclusion, there appears to be a few elements suggesting that the Ni 60 Co 40 nanowires' surface may have been degraded by way of potentially non-uniform oxidation. I wish to emphasize, however, that I am only presenting the scenario which seems most likely to me, and not claiming to have the correct explanation. Furthermore, as stated above, a number of facts elude me, such as why these particular templates were more difficult to etch.

I.4 Sample orientation, introduction and PEEM imaging

The nanostructures had been tentatively aligned with the help of a permanent magnet during dispersion onto the patterned silicon substrates. The outcome was a slightly preferred orientation, whose mean was observed with optical microscopy. In order to benefit from the maximum field amplitudes, the investigated wafers were positioned on the PEEM sample holder (see Chapter 2, section III.1) so that the mean long axis of the nanostructures was aligned with the direction of the field produced with the coil I wound. Then, the system was put on the PEEM cartridge so that the X-ray beam was more or less perpendicular to the mean wires' long axis. This configuration is summarized on the annotated optical microscope view in Fig. 4.6. The density of wires is rather large yet a number of isolated structures can be seen as thin elongated dark streaks. Thanks to light scattering, it is possible to see even individual wires although their diameter is much lower than the wavelengths of visible light. It must be pointed out, however, that even the thinnest structures seen may actually be a pair of adjacent wires. In other words, optical microscopy alone is not sufficient to determine where single, isolated nanowires lie. Still, mapping out samples using a large number of images such as presented in Fig. 4.6 is useful as it allows to pre-select single-wire candidates. As a last remark on this imaging: considering that the structure length was expected to be 30 µm (see section I.1), the majority of nanowires observed here are incomplete: either broken or more likely shorter than expected due to insufficient electroplating time.

I.5 Micromanipulation of nanowires

Since one goal of the beamtimes I took part in was to apply nanosecond-scale pulses of magnetic field, I initiated a campaign of micromanipulation in order to place nanowires onto a coplanar waveguide (CPW) I had fabricated on an alumina substrate following the simulations results from Chapter 2, section IV.2. The copper CPW tracks were fabricated with triode sputtering, the termination impedance with magnetron sputtering using the Ni 80 Cr 20 target mentioned in Chapter 2, section IV.3. The micromanipulation procedure was expected to allow both the choice of the most suitable structure to pick from the silicon wafer and an optimal alignment i.e. perpendicular to the CPW's central track. As a reminder of Chapter 2, the purpose of the operation was to benefit from the optimal applied field amplitude by disposing the nanowires as close as possible to the waveguide's top surface. Moreover, the design with impedance matching (as target) was meant to apply a single well-defined pulse without echoes. Due to misstep during fabrication, the NiCr strips constituting the termination impedance had to be bonded back to the CPW with silver paint; CHRISTOPHE THIRION took care of this owing to his longer experience in such delicate procedures. In the end, the DC termination impedance was about 62 Ω.

Micromanipulation of micro-/nano-objects is performed inside an SEM chamber equipped with Focused Ion Beam (FIB) capabilities and a stage holding the manipulating tip. In this case, all the operation was performed by JEAN-FRANÇOIS MOTTE. The principle is to mechanically remove the object of interest from the substrate's surface and gluing it to the tip so that displacement over large distances may be reliably performed. Particularly adverse are short-range adhesion forces resulting e.g. from van der Waals interactions; significant forces are needed to overcome them and lift off the object, but such forces applied locally by the tip may damage it. In a manner of example, Fig. 4.7 shows this mechanical removal process in the making. Starting from a nanowire oriented as outlined by the orange dashed lines, the strongly adhesive contact between nano-object and substrate is broken by pushing the former aside with the tip. Since this process is local, the tip is moved along the full wire length to completely unglue this nanostructure. Here, the tip travel direction is indicated in green. The induced deformations are very often plastic. Once the bulk of the adhesion forces have been overcome, the micromanipulating tip is glued to the object by deposition of material (thanks to the injection of gas and decomposition under the electron/ion beam) localized on the interface between the two items. In the present case, it was carbon. During the micromanipulation campaign, it was clear in many instances that solution residues still present around the nanowires (potentially due to insufficient rinsing) may dramatically hinder the separation from the substrate.

µm

This complication is illustrated in Fig. 4.8, which features SEM images of a nanowire before and after an attempt to pluck it off. In this instance, the halo-like feature surrounding it, which is associated to solution residues, was gluing the wire strongly enough to prevent its removal. Indeed, when part of the wire was finally detached under sufficient pressure from the tungsten tip, it actually resulted in mechanical failure. This specific case was that of an ill-rinsed substrate with a dispersion of nanowires, and as such not representative of what can be obtained with dispersions. A bit more insight on the mechanical resilience of nanowires under micromanipulation can be found in section A 4 .1.

Overall, the process of nanowire micromanipulation was found to be much more time-consuming and difficult than anticipated. This issue of adhesion to the substrate was one complication, but even in the few successful cases, dropping the nanowires on the target substrate (i.e. the CPW) was also challenging. One reason for this is the large aspect ratio of the wires: it was not straightforward to bring the wire close to the surface while keeping it horizontal 6 and landing it safely. Furthermore, the structures proved to be very sensitive to residual electrostatic forces when approaching the CPW. More often than not, this resulted in wires standing almost perpendicular to the substrate. This is more easily understood when considering a typical successful lift-off situation as presented in Fig. 4.9. Here, a nanowire was plucked-off without overwhelming damage and the tip was retracted significantly with respect to the substrate; the latter therefore appears blurred. The nanowire has been glued to the tip at a point in one of its thick sections. Since this is the only contact point, any force distributed over the whole wire length, such as electrostatic forces, easily exerts a large torque. Several dark areas can be seen along the wire (see blue arrows): they correspond to patches of carbon used to attach the nanostructure to the surface. These were useful not only when detaching the wire from the tip as they provide anchor points, but also in the correction of the wire's vertical and horizontal alignment. Quite clearly, some significant plastic deformations were induced by the micromanipulation. This can be seen as a further drawback: the process is too difficult to allow control over the amount of mechanical stress induced in the manipulated wires.

µm

µm

II Synchrotron imaging and spectroscopy

After sample introduction into the microscope and setting of the latter, navigation with PEEM over the silicon wafer and imaging of nanowires can begin. Throughout the experiments, the energy of the incident photons is set (close) to an L-edge (L 2 or L 3 ) of one of the elements constituting the nanowires, i.e. Ni or Fe. As a first example, Fig. 4.11 shows a nanowire with both diameter modulations. Its case is interesting as the middle section is fully lifted, as demonstrated by the absence of contact between the top of the direct wire image and the bottom of the thin section's shadow. The top and bottom view are the same image, only with different intensity ranges: the top one is saturated but allows to see the shadow more easily than the unsaturated bottom one. The latter reflects better the wire's geometry. Here, however, thanks to the beam's incidence angle, one can have a sense of depth in the direction perpendicular to the substrate surface owing to the shadow. The pronounced separation between the wire itself and most parts of its shadow reflect their height offset. Although the focus is on the nanostructure rather than the shadow, a closer look seems to indicate that the latter's intensity might not be homogeneous over the thin wire's section. This will be tackled in more detail in section II.4.

X-ray beam 1 µm

II.2 Spectroscopy

The above-mentioned resonant imaging requires of course to set the beam's energy to one absorption edge of nickel or iron, and therefore to perform X-ray Absorption Spectroscopy (XAS), as was described in the eponymous section in Chapter 3,section IV. Since a typical (XMCD -) PEEM image contains three distinct types of area, that is to say the direct wire image, the wire's shadow, and the silicon substrate, three different spectra can be acquired:

• on the wire, corresponding to total electron yield spectra [27] and thus reflecting the dispersion in X-ray absorption cross-section of the elements constituting its surface,

• in the wire shadow, in other words using the substrate as screen on which the sample's transmission is projected and measured,

• on the Si wafer, whose absorption cross-section varies little over the typical XAS energy ranges; yet, these variations are not negligible, say, from an L 3 edge to an L 2 edge (of Fe or Ni).

In practice, determining the position of an absorption edge from shadow XAS does not require to take the energy-dependent X-ray absorption of silicon into account since the latter varies very little over the width of an absorption edge (less than 4 eV over the range covering the L 3 and L 2 edges of metallic iron, cobalt and nickel [27]). However, an accurate shadow spectrum independent of the substrate must be corrected by normalization to XAS data acquired on that substrate. It must be noted that since such a spectrum is proportional to the silicon photoemission cross-section and the photon flux, this normalization ensures that the corrected spectrum no longer depends on the X-ray intensity that varies with the energy.

The case of XAS acquired on the surface of a wire is different as the absorption cross-section is directly the one of interest. The only normalization needed (and always used in the following) originates from variations in the photon flux as the X-ray energy is changed. A measure of the flux is provided by a gold grid along the beamline, ahead of the microscope onto which a part of the X-ray beam falls [28]; the photoelectric current (itself normalized by the photoemission cross-section of gold) reflects the number of incident photons.

Let us illustrate the above with experimental examples. Fig. 4.13 presents two spectra acquired with the same nanowire: the graph to the left corresponds to the Fe L 3 edge, while the one to the right corresponds to the Fe L 2 edge. On both graphs, the red curve represents XAS data acquired as described above on the wire surface, and the black one to shadow XAS scaled by an arbitrary factor for the sake of representation. Additionally, the blue dashed curve is the opposite of shadow XAS, rescaled and offset so that its pre-edge values match those from the wire surface. The interest is to emphasize the offset in photon energy between the maximum of absorption in the shadow and the maximum of emission on the wire7 on both edges. Both graphs feature data acquired from the wire surface (red) and from the shadow (black). An equivalent representation of the latter (dashed blue curve) is a vertically mirrored then offset spectrum that allows more direct comparison with the wire XAS. At both edges, the maximum of absorption in the shadow lies 1.9 ± 0.4 eV lower in energy than the maximum of photoemission on the wire, suggesting surface oxidation. This is a strong indication of surface oxidation. Indeed, the spectra of several iron oxides feature a maximum of absorption at the Fe L 3 shifted with respect to the single peak of metallic iron [29]. Also, the Fe L 2 tail for these oxides starts higher in energy than it does for metallic iron [29]. Such a behaviour is not surprising as the nanowires spent some time in solution before dispersion, and were stored (in vacuum) for days/weeks before the beamtime. As a result, shadow XAS is a more accurate tool in the determination of the working photon energy for the metal of interest: since shadow XAS probes the nanowire's bulk, it is less affected by its surface state which is expected to be oxidized. This conclusion had been reached by JAMET et al. [30].

A representation similar to Fig. 4.13 but at the Ni L 3 edge is given in Fig. 4.14. XAS data from the wire surface and the shadow are featured on the same graph, as well as a vertically mirrored shadow spectrum for easier comparison with the photoemission spectrum. Contrary to the previous case, the shift in peak position is much smaller, about 0.2 eV from the graph. Furthermore, the shadow peak lies above the surface peak, which is unexpected in the "oxidized surface/metallic bulk" picture. Still, the beamline's resolving power E/∆E in this energy range is about 3000 [31], thus the energy resolution at the Ni L 3 edge should be ca. 0.3 eV. As a result, since the spacing between metallic Ni peak and NiO peak at the L 3 edge is about 0.24 eV [29], one should not expect to resolve the wire surface and shadow peaks in this experiment.

XAS was of course also acquired on NiCo nanowires. An example of spectrum at the Co L 3 edge is shown in Fig. 4.15 with the same conventions as previously. In this case, the somewhat flat peak from the wire XAS and the tail emission levels suggest surface oxidation as well. It is noteworthy that this data was extracted from a very large portion of the wire surface (over ca. 70% of its length).

Although the discussion from section I.3 concerning the surface state of the NiCo nanowire may come back to mind, the above neither means that all NiCo nanowires were similarly oxidized nor proves the point related to possible over-etching during alumina template dissolution. As a counter-example, Fig. 4. 16 shows XAS acquired on another wire, this time with a less flat L 3 edge and less apparent peak substructure. Along with a higher tail photoemission (with respect to pre-edge values) than in the previous case, one is led to conclude on a better metallicity in this case. Likewise, the graph features data acquired from the wire surface (red) and from the shadow (black). An equivalent representation of the shadow XAS (dashed blue curve) is a vertically mirrored then offset spectrum that allows more direct comparison with the wire XAS. This time, the maximum of absorption in the shadow and the maximum of photoemission on the wire are quite close. Conclusions about oxidation are harder to draw because of specific spectrum shape corresponding to nickel [29]. The apparent spacing is about 0.2 eV. Like in the previous spectra, the graph features data acquired from the wire surface (red) and from the shadow (black). An equivalent representation of the shadow XAS (dashed blue curve) is a vertically mirrored then offset spectrum that allows more direct comparison with the wire XAS. The close-to-flat edge peak (over 2 eV), the plausible shoulder at 783 eV and to some extent the photoemission levels around 785 eV compared to pre-edge levels suggest that cobalt on the wire surface may be partially oxidized [29].

This being said, a spectacular evidence of surface oxidation was provided by the NiCo nanowire shown in Fig. 4.17. While a PEEM image at an energy corresponding to the Co L 3 edge's maximum of shadow absorption (top image) reveals nothing peculiar except for pronounced topographical features on the wire surface, another one acquired at a pre-L 3 edge energy E rim shows a transparent shadow with a dark rim. The former is expected but not the latter. Considering the wire XAS shown in Fig. 4.17, the nanostructure's surface is quite clearly oxidized [29], and the photon energy E rim is seen to correspond to a pre-L 3 edge shoulder.

As a result, this rim is only apparent because of the difference in absorption between bulk metallic and surface Like in the previous spectra, the graph features data acquired from the wire surface (red) and from the shadow (black). An equivalent representation of the shadow XAS (dashed blue curve) is a vertically mirrored then offset spectrum that allows more direct comparison with the wire XAS. In this case, the less flat (with respect to Fig. 4.15) peak from wire XAS and slightly higher edge tail photoemission compared to the pre-edge values speak for a less oxidized material [29]. oxidized materials at that energy. Intensity profiles across this rim yielded ca. 100 ± 15 nm in terms of rim width 8 . Taking into account an instrumental broadening due to the microscope resolution assumed to be 40 ± 10 nm as well as the beam's incidence angle [30], I estimate this rim to correspond to a layer on the wire surface with thickness 17 ± 5 nm. A few more details need be given. The bright features on the bottom side of the nanowire are brighter than the bulk of its surface, well above the microscope's resolution, and yet the shadow's top rim displays no relief. Moreover, they do not correlate with variations in intensity on the aforementioned rim; thus, these features appear no to contribute to the latter.

To check further that oxidation was predominantly on the wires' surface and to improve the signal-to-noise ratio there, we have performed some sputtering with argon ions in a separate microscope chamber. The goal is to strip the alleged oxide layer and reveal the more metallic bulk nanowire material. During this beamtime, two sputtering rounds were performed, the first one being 2 min long, the second one 5 min long. XAS data before, in-between and after these rounds are compared in Fig. 4. 18. In order to meaningfully compare three separate series, an additional normalization to the same peak amplitude was performed. That way, changes between the full blue, green and red curves (before the first round, after the first round, and after both rounds respectively) reflect changes in the ratio of metallic iron to iron oxide 9 . The blue, green and red full curves correspond to the virgin sample state, the state after two minutes of Ar sputtering, and the state after an additional five minutes of sputtering. This wire surface XAS is accompanied by shadow XAS processed as described previously: these data are the full black and dashed brown (vertically mirrored, then offset) curves. The surface wire spectra have been normalized to the same peak amplitude, which corresponds to iron oxide as demonstrated by the offset of this peak with respect to the maximum in shadow absorption. The latter's position hints at a shoulder in the wire surface XAS, whose relative amplitude (with respect to the peak value) appears to increase with the sputtering time. Here, the shadow spectrum corresponds to the state after one sputtering round. The two wire XAS after that round were acquired on the same wire.

The energy at which to look for such changes is indicated by the full black and dashed brown curves in Fig. 4.18. Similarly to the previous figures, the full line is the photoemission from the shadow and the dashed line is an offset, vertically mirrored duplicate of this data. Its maximum corresponds to a shoulder in the measured wire surface spectra, at about 706.2 eV. Therefore, the evolution of the intensity at this energy as a function of sputtering time is an measurement of the changing surface metallicity. Admittedly, the spectra I extracted have non-negligible noise and only display a weak difference, still, the observed evolution is in agreement with the scenario of an oxide layer being partly etched thanks to the argon sputtering.

Coming back to the shape of the shadow XAS, one may be surprised to observe for instance a rather flattened L 3 edge compared to e.g. the sharper peak for metallic iron measured by Regan et al. [29] in total electron yield. In fact, as pointed out by JAMET et al. [30], in the infinitely thick wire limit 10 , no photon should traverse the sample and impinge on the substrate. Thus, the absorption should saturate: close to the L 3 edge, the peak shape may therefore be flattened. To be more precise: the wire XAS is directly comparable with a total electron yield measurement, while the shadow XAS is a measure of exp [-µ(E) × t], where µ is the linear X-ray absorption coefficient, E the photon energy and t the traversed material thickness 11 . The exponential is responsible for the seemingly flattened spectra at high absorption. It is therefore neither directly comparable with a total electron yield spectrum, nor with the behaviour of µ(E). These saturation effects in transmission measurements have been described e.g. by Nakajima and coworkers [32].

The determination of the distance travelled inside the sample for which this saturation should occur is difficult as it is heavily influenced by the wire composition and the beam incidence angle. Just to give an order of magnitude, the proportion of linearly polarized photons at the Fe L 3 that are unabsorbed after 250 nm distance in a Ni 90 Fe 10 at 45 • incidence (between X-ray beam and wire long axis) is 12% [29].

As a further emphasis, Fig. 4.19 displays two shadow spectra encompassing both the L 3 and the L 2 edges of iron. They were acquired on the same wire, processed in the same way, and are represented without offsets or further rescalings. The vertical scale reflects the ratio of photoelectron counts in the shadow to the average counts on the silicon surface (due to the normalization to the background). While the data from the thin section shows rather well-defined absorption peaks, the thick section is clearly seen to saturate at low intensities. This results in weaker, seemingly broader peaks. To ease comparison, the data from the thick section's shadow has been added after multiplication by a factor 2.15. If saturation did not play a role, transmission XAS from both sections should be identical up to a simple multiplication. Clearly, this is not the case here. The spectrum from the thick section's shadow clearly shows saturation and appears therefore flatter than its counterpart from the thin wire section. To better emphasize this, the dashed purple curve shows the XAS from the thick section's shadow after multiplication by a factor 2.15. In the absence of saturation, the intensities transmitted through wire sections of different diameter should collapse onto one another with a simple multiplicative factor, which is not the case at all energies.

II.3 Wave optics

As a conclusion to this first part on spectroscopy, I present a feature that had not been anticipated at the time of these experiments, namely X-ray interference fringes close to the rim of a nanowire's shadow. These are understood as Fresnel fringes [30], considering the distance from the top of the wire's surface to the "screen", i.e. the silicon substrate. Since they are not visible away from the absorption edges of the relevant metals, they may be a direct consequence of resonant elastic scattering and/or the rapid change in transparency caused by the wire.

It is interesting to note that this occurrence of wave optics had not been reported by DA COL et al. [33] during the investigation of thinner NiFe nanowires, nor by Bran et al. [34], also on thinner wires. One possible explanation is be that the larger diameters involved in this work induce a faster attenuation and possible accumulation of phase across the shadow rim, allowing the latter to diffract. Two of the best examples of interference fringes seen during this 2014 beamtime are shown in Fig. 4.20. Both nanostructures are here imaged at the Fe L 3 edge. It must be noted that the wire on the right hand side is not visible partly as a result of intensity range on the image but primarily due to the microscope settings at the time. Further details in this respect can be found in reference [30]. The top row comprises PEEM images that are each the sum of two images: one acquired with circularly left polarized photons, the other with circularly right polarized photons. The markers/red line indicate the direction along which I extracted intensity profiles so as to examine the fringes. Excerpts of these profiles are featured in the middle and bottom row. Barely more than a period of oscillation can be discerned even on such profiles. However, as can be seen especially on the top left image in Fig. 4.20, the fringes follow closely the shadow rim's curve. Due to the fast dispersion and large peak values of the real part of iron's resonant X-ray scattering factor close to the L 3 edge [27] (see Fig. 9.6.), the role of elastic scattering per se is hard to estimate. On the other hand, the absorption is strong (resulting in the marked shadow) and is likely to be a non-negligible cause of the Fresnel diffraction.

X-ray beam X-ray beam

The fringes' amplitude is seen to be much lower above the thin middle section's shadow. This is in agreement with the above discussion of diameters: an increased wire radius R means a longer distance √ R 2x 2 travelled through matter for X-rays at fixed depth x. Therefore, scattering is more pronounced since the change in sample transparency is more abrupt. Fringes could also be observed in one case at the Ni L 3 edge: this even more striking observation is reported in Fig. 4

II.4 Composition

The NiFe nanowires under study during the 2014 had been electrodeposited by RAJA AFID and LAURENT CAGNON in potentiostatic conditions, i.e. the reduction reaction leading to the alloy deposition was conducted under constant voltage. As was described in Chapter 2 (see section I.3.c), the electrodeposited alloy's composition is heavily influenced by the local conditions of temperature, pH, species concentration in solution. A further issue is that of agitation [35] as it becomes critical in assuring equal, homogeneous concentrations in the AAO template pores. Investigation pertaining to the electrodeposition of NiFe alloys (into nanoporous membranes) close to the composition of permalloy by SANDRINE DA COL [36] indicate an increase in ca. 10 percentage points in nickel content by a change of only 5% in deposition voltage, all other conditions being fixed. This sensitivity with respect to one parameter, associated with the non-linearity in the kinetics of electrodeposition [35] and the diffusion into the long, narrow AAO pores, suggests that deviations from the nominal alloy composition are likely to occur in the potentiostatic deposition mode. The latter indeed drags the electrochemical system into a steady state where the conditions (notably concentrations) at the deposition site may depart from the initial conditions.

The relevance of these considerations, so far only theoretical, can in principle be checked with the spatially resolved spectroscopic capabilities of XPEEM in a synchrotron. Since the transmission through the investigated nanostructures depends on the thickness of material traversed by the beam and, for a given photon energy, the path-averaged composition, monitoring the composition along the wire should be possible provided the diameter does not change.

Following this idea, I have considered the nanowire presented in Fig. 4.22; it is the same wire as in the righthand side of Fig. 4.20. In the following, this structure will be referred to as Cicero. The thick and thin diameters are estimated from the shadow to be D thick 0.5 µm and D thick 0.2 µm, respectively. To the left-hand side is a image with a broad field of view (10 µm) with microscope focus on the substrate (thus the shadow), and the right-hand side is a zoom-in on the thin section, the microscope being focused on it. From these images, no clear change in the thin section's diameter is visible neither from the shadow nor from the wire itself, although the instrumental resolution on the order of 30 nm to 40 nm [37] must be kept in mind for the direct wire image. There are, however, several structural features such as a break (about one-third of the thin section's length, starting from the right) and a notch on the top side, as revealed by the shadow. are the silicon surface and the wire's top surface, respectively. While undesirable features such as a break in the middle section and at least one notch (revealed by the shadow) on the top side of this thin section, no significant changes in wire diameter can be seen here.

Making use of an image of this wire with a total long integration time (40 images, each with an acquisition time of 10 s), I obtained the PEEM images in Fig. 4.23. These images were corrected for the dark field i.e. the electron counts on the microscope's CCD camera even in absence of signal. It must be added that the X-ray beam was circularly polarized at the time, but the wire's magnetic state was single-domain in the thin section. As a result, changes in intensity along this section cannot be attributed to changes in magnetization: in other words, XMCD is uniform here. The top image's intensity range is chosen to closely match the spread in intensity within the middle section's shadow. That is why everything else is saturated towards low intensity (thick section's shadow) or high intensity (e.g. the substrate).

While at first sight nothing is amiss, this image reveals non-negligible variations in intensity along this part of the shadow. To get a better view on this, I have extracted an intensity profile following the red broken line, from the left to the right of the image. The lateral width over which the intensity (for a given point) was averaged was ca. 40 nm.

The profile shows an overall increase in intensity starting right after the abrupt diameter modulation. Since from the profile start point to the end point, the intensity (as measured on the Si surface above the shadow) is uniform within 3%, curves (in black) representing this window around the extracted intensities are drawn. The scale bars are 1 µm, and the X-ray beam's direction is shown by the orange arrow. In order to quantify these changes, an intensity profile was extracted along the broken red line shown in a). This data is shown on the graph in c). The red curve is the extracted intensity, and the black curves define a 3% window around the profile based on the variations of intensity on the Si wafer just behind the shadow's rim.

Even with this tentative error estimate, the above-mentioned upturn remains clear. Let us now turn to the interpretation of such a profile. As was discussed previously, since no change in wire diameter is visible in PEEM (nor expected since the thin pore sections in the AAO templates are defined well into the self-ordered regime, see Chapter 2 section I.2.c), the only other explanation for changes in shadow intensity are inhomogeneities in composition. Indeed, the beam's incidence angle with respect to the wire changes negligibly, and no changes in magnetization can affect the shadow intensity. The latter reads:

I shadow (x) = κ Si • I 0 exp -µ C d cos θ i n Fe (x) + I bckgnd (4.2)
where κ Si is the photon-to-photoelectron conversion coefficient pertaining to the silicon substrate 12 and I 0 is the X-ray beam's intensity before interacting with the nanowire. µ C is the linear absorption coefficient of X-rays for the relevant circular polarization, d is the wire diameter, and θ i is the incidence angle of the beam with respect to the wire's main axis; θ i =90 • would mean perpendicular incidence. n Fe (x) is the atomic percentage of iron in the wire, as a function of the abscissa x along the profile. I bckgnd corresponds to the background intensity described by JAMET et al. [30], after subtraction of the CCD camera's dark counts. As argued in Ref. [30], the measured intensity in the shadow of very thick wires should be vanishing. Yet, even at the Ni L 3 edge (nickel being largely predominant in the samples), some intensity could still be measured in the middle of a 250 nm thick nanowire (not shown here), about 21% of the intensity measured on the bare silicon surface! This is striking if one considers that the ratio predicted from the absorption coefficients of nickel [32] is on the order of 10 -4 at most. This effect is less pronounced at the iron edges because of the smaller average fraction of iron in the samples. Still, taking into account the thick diameter of about 380 nm of wire Cicero as well as a 10% atomic fraction of iron and the beam incidence angle, the expected shadow's intensity (in the thick part) should be about 1% of that measured on the substrate, whereas in the experiments it reaches 7%.

Coming back to the issue of composition, it is clear that the intensity profile from Fig. 4.23 must be corrected for this background contribution. After this, the difference δn(x 1 , x 2 ) in laterally-averaged composition between two abscissae reads:

δn(x 1 , x 2 ) = cos θ i dµ C .ln I shadow (x 1 ) I shadow (x 2 ) (4.3)
where the intensities I shadow (x 1 ) and I shadow (x 2 ) have been corrected for I bckgnd as described above.

Assuming an absorption coefficient µ C between the extremal values for iron at the Fe L 3 edge [32] to estimate an error on such a composition difference, I arrive at a maximum difference of 1.8 ± 0.4%. This is not large, but as was shown in the intensity profiles, some variations may be quite localized close to the diameter modulations. Just to give an order of magnitude in terms of magnetization: based on the Slater-Pauling curve only [15], such a variation should lead to a change in spontaneous induction of about 30 mT to 40 mT. Therefore, their actual influence on domain wall motion in experiments might be rendered more complex with respect to the "homogeneous material" assumption, if the variations are on the scale of the domain wall width. In the present wire's thin sections, domain wall widths on the order of 1 µm and below have been observed in over twenty instances, and a significant part of the intensity variations in Fig. 4.23.c) occur over less than 1 µm.

A more spectacular example of composition inhomogeneity was provided by the micromanipulated NiFe nanowire D12 shown e.g. in Fig. 4.12 p.195. As was mentioned in section II.1, this nanowire's shadow seemed to present intensity inhomogeneities along the thin wire's section. These inhomogeneities are shown in Fig. 4.24.a), which includes an even stronger emphasis thanks to a contrast image in Fig. 4.24.b). This second image was obtained using the same image processing as an XMCD-PEEM image but using as sources a pair of images with the same linear polarization state, but acquired at two different photon energies. These were the positions of the Fe L 3 edge and the Ni L 3 edge, respectively. The idea, originally from MICHAEL FOERSTER, was to enhance the composition inhomogeneities in the same way as XMCD is enhanced by the familiar normalized difference of images corresponding to different photon beam polarizations.

In that case, the resonant absorptions at different energies is used to highlight these spatial variations. The computed contrast is a function of the distance travelled by the photons through the nanowire, the linear absorption coefficients of iron and nickel at the Fe L 3 and Ni L 3 edges, and of course the local nickel content n Ni or conversely the local iron content n Fe = 1n Ni . More quantitative details can be found in the appendix section A 4 .3.

Since the contrast in Fig. 4.24.b) is proportional to the intensity measured at the Fe L 3 edge minus the intensity at the Ni L 3 edge, in the shadow, a brighter contrast means less absorption by nickel; in other words, the brighter areas are richer in nickel. It must be noted that the numerous gold impurities on the CPW copper substrate are still visible on the contrast image in Fig. 4.24.b) simply because the background intensities on copper have been adjusted to the same value (so that there is no bias in terms of shadow contrast). Due to the different evolution of X-ray absorption by gold as a function of photon energy, there necessarily is an imbalance in the shadow of these gold particles.

To go beyond this very clear but so far only qualitative representation, I have derived in appendix section A 4 .3 an expression relating the measured shadow contrast in Fig. 4.24 to the sample composition. This calculation assumes null parasitic beam intensity, which is probably not the case; as a result, the measured contrast is likely reduced (similarly to shadow XMCD-PEEM contrast [30]) and therefore the estimated composition variations must be considered as lower bounds. From the left hand side to the right hand side of the blue arrow in Fig. 4.24, I estimate the nickel content to increase by 4±1%, and from the left hand side to the right hand side of the red arrow, it decreases by 2.4 ± 0.7%. With respect to the first example, not only are the (lower bounds on the) composition variations larger, but they are much more abrupt. Here ends my analysis of composition homogeneities.

The above discussion has dealt with the background intensity I bckgnd in order to extract meaningful composition variations irrespective of this background's origin. Let us examine it in more detail. Since the dark field electron counts (from the CCD camera) were subtracted, this excess intensity corresponds to real photoelectrons [30]. In other words, there appears to be an additional channel for photoemission that evidently depends on the beam energy but not much on microscope settings [30]. This channel cannot be a phenomenon arising in the shadow only because its "screen" is the same silicon surface as outside the shadow. The only reason why this effect is measurable in the shadow and not outside is its weakness. As a result, there seems to be a small, unanticipated number of photons which somehow are not or less affected by the presence of the wire on their path.

After discussions with ANDREA LOCATELLI, ONUR MENTE Ş and MICHAEL FOERSTER, I have found a plausible explanation for these facts: it relies on the X-ray beam harmonics, in the sense "beam with twice the working photon energy". Although these photons are filtered out by the beamline monochromator 13 , realistic experimental situations may involve a second order contamination of the primary beam at least 0.2% and reaching at most 4% [41] 14 . This may not seem much and usually is not, but this may be enough to explain part of the 13 A common interference-based wavelength selection is performed using so-called Variable Line Spacing (VLS) gratings [38], as is the case at synchrotrons Soleil [39] and Elettra [40]; additionally, further instrumental methods such as Variable Groove Depth (VGD) can reduce the contribution of higher diffraction orders [39] i.e. photons with an energy that is an integer multiple of the working energy. 14 These numbers is representative of the PEEM on the Circe beamline at synchrotron Alba. Considering that it is more recent than the one from the Nanospectroscopy beamline at Elettra, the order of magnitude at least should be similar.

above-mentioned 7%.

The first crucial aspect of a photon energy of about 1410 eV i.e. twice the Fe L 3 energy is that on one hand iron and nickel feature absorption coefficients close to pre-L 3 edge levels [18]. This means that the beam harmonics is barely attenuated by traversing the wire. Now, the linear absorption coefficient of silicon at this energy is just a factor ca. four times smaller than at the Fe L 3 edge [18]. In other words, not taking changes in electronic mean free path in Si from 705 eV to 1410 eV into account 15 the number of detected electrons per incident photons at 1410 eV is about four times lower than at 705 eV. The combination of wire quasi-transparency and still significant absorption from silicon allows to maintain a photoelectron counts associated to the beam harmonics still comparable with the shadow intensity associated to the primary beam.

This reasoning is semi-quantitative only and much of the quantitative details depends on the exact quantification of this contamination. Furthermore, I do not mean to claim I have fully understood the background tackled by JAMET et al. [30]: for instance, the above-mentioned very large shadow intensity at the Ni L 3 edge does not fit into this frame, or then unreasonably large contaminations must be invoked. Still, in my view this explanation cannot be discarded off-hand as it matches some of the experimental observations while several other causes have been ruled out [30].

III Domain wall nucleation, pinning, and identification III.1 Requirements and modus operandi for domain wall injection into nanowires

Assessing whether the field-induced dynamics of the Bloch point wall (BPW) lead to a favoured helicity [8] h as defined in Eq.(4.1) first necessitates an easy triggering of BPW motion, thus the presence of domain walls in a large number of wires. The engineered diameter modulations featured by the nanowires I have investigated (see section I) are meant to assist in this by confining the domain wall in the nanowire's thin section. It is worth noting that this case is optimal insofar as the same micromagnetic object is moved but not annihilated at a wire's extremity. For this situation to occur, there is a need for at least two pinning sites with asymmetric behaviour: on one hand they must prevent a domain wall from escaping the segment they define and therefore have a large outward pinning field, but on the other hand they must not impede domain wall motion inside this segment; as a result, their inward pinning field must be low.

When domain walls have been observed in one or several wires, the application of magnetic field starts. During beamtimes, the applied fields are always in the substrate's plane, with a component along the wire axis. In the following, the reported induction values correspond to this longitudinal component. As for the actual orientation of the applied field i.e. whether a positive or negative current must be fed to the coil or electromagnet, the practice consists in deducing it empirically. This is information typically obtained from the first wire that is subjected to magnetic fields. Of course, the latter affect all nanowires on the substrate and therefore domain walls may be lost during the period of focus on a specific structure. Considering that equipment equivalent to the demagnetizing stage from Chapter 2, section I.5.a (and following sections) is not available during the synchrotron beamtimes and that the demagnetizing process would cost too much time, the demagnetized state is irremediably lost after the first field application.

Quite often, in the early stages, it happens that the applied fields exceed the nanowire's pinning fields by far. Thus, domain walls which were present after the transverse demagnetization at Institut Néel are lost and the nanowire is single-domain. It is therefore useful to target the nucleation of a domain wall followed by its pinning somewhere in the wire. The principal drawback here is that two additional, poorly controlled and/or understood phenomena come into play with respect to the above case, namely the nucleation itself and the passage through one diameter modulation. Nevertheless, such a scenario only requires the presence of one pinning site with a strength (slightly) above the nucleation field: this is a significantly less stringent requirement. Furthermore, the farther away from the diameter modulations this pinning site lies, the smaller the influence of the nucleation process and the passage through a modulation should be. This configuration is illustrated in Fig. 4.25, presenting a tri-segmented nanowire with only one pinning site in the thin section; the diameter modulation's effect is negligible. Starting from the single-domain state at zero applied field H a , the field strength is increased until the nucleation field value H nucl is reached. In this figure, nucleation occurs on the left hand side wire tip. Depending on the pinning field H p associated to the aforementioned site, two outcomes are possible. In the top row, where H p,1 < H nucl , the nucleated domain wall propagating under H a = H nucl is not stopped by the pinning site. Thus, its propagation to the right hand side wire tip leads to magnetization reversal of the full wire. In the second row, H p,2 > H nucl = H a , so that the domain wall is pinned and can be observed when the applied field is ramped down to zero. the diameter modulations have negligible pinning strengths here. The top (bottom) row corresponds to the case where the pinning site's strength H p,1 (H p,2 ) is lower (higher) than the nucleation field H nucl . Initially (left column), the applied field H a is zero and the wire is uniformly magnetized to the left (red arrows). When a field (in green) pointing to the right is applied and reaches the nucleation field H nucl (right column), a domain wall is nucleated, here from the left, and propagates. If the pinning field is lower than H nucl = H a (top row), the domain wall propagates on and annihilates at the opposite wire tip. To the contrary, if the pinning field is higher than H nucl = H a (bottom row), the domain wall is pinned and remains so as the applied field is ramped down to zero.

Pinning site, µ

0 H p,1 < µ 0 H nucl µ 0 H a = 0 Pinning site, µ 0 H p,2 > µ 0 H nucl µ 0 H a = 0 Pinning site, µ 0 H p,1 < µ 0 H nucl µ 0 H a = µ 0 H nucl Pinning site, µ 0 H p,2 > µ 0 H nucl µ 0 H a = µ 0 H nucl
In practice, the majority of domain wall motion instances I encountered were of the latter type. To be brief: in the cases where domain walls were pinned in the thin wire section, the fields needed to depin them were so strong that whatever barrier the diameter modulations constituted was (almost) systematically overcome. Although they have been studied with simulations [16] and experimentally [42], the actual domain wall retention strength of diameter modulations remains so far difficult to assess, possibly invisible because buried beneath the too high pinning fields encountered in the wires. This is one reason why the observation of a domain wall far away from the diameter modulation in the demagnetized sample state is not always good news: it might (and in practice often does) mean that the current nanowire's pinning fields are rather high.

During the investigation of a given structure, the procedure is in essence always the same. If a domain wall is present, then depending on the outcome of the first application of field, a different step is made. Either the nanowire's micromagnetic state has not changed, in which case the applied field strength is increased, or the domain wall has moved. In the latter case, if the domain wall is still inside the wire 16 , then the field is adjusted either to pursue motion in the same direction or to push the domain wall back. If the domain wall has annihilated at a wire's extremity, then either this structure is no longer studied, or the investigation is continued with the "nucleation configuration" in mind.

This situation is that of a wire devoid of domain walls. Typically, the attempts to nucleate a domain wall start with rather large fields. If the wire's magnetic state is unchanged, the field is increased by a certain increment. In the beginning, the steps in applied field are coarse to limit the number of steps needed before the applied field is above the wire's nucleation field and a domain wall is nucleated. At this stage, the latest applied field value and the one applied before are upper and lower bounds on the nucleation field. Then, using dichotomy, the window within which the nucleation field lies is narrowed. By applying fields just above the latter, the goal is to nucleate a domain wall and have it pinned somewhere inside the wire.

It is along these lines that domain walls are nucleated or depinned in order to investigate the configuration of BPWs. In practice, the nanowires' shadow is the preferred source of information because it reflects the sample volume on one hand, and is associated to an enhanced spatial resolution on the other hand thanks to the beam incidence angle [33,43]. Features that may be unclear on the narrow wire surface are spread over a larger area in the shadow, the latter being supported by a flat surface onto which it is easier to focus.

III.2 Distributions of nucleation or depinning fields

As mentioned above, a large number of cases involving domain wall motion that I encountered were nucleation events followed by pinning. Usually, the number of domain wall nucleations per studied nanowire was not high enough to be considered statistically meaningful, and neither was the number of depinning events. Nevertheless, one NiFe nanowire with thick diameter D thick 0.5 µm and thin diameter D thick 0.2 µm (nanowire Cicero, see section IV.1) was investigated long enough to yield a not so small number of observations of failed/successful nucleation as well as depinning. This nanowire's geometry is reminded on the PEEM image in Fig. 4.26. The data analysis I conducted as follows. Each interpretable17 pair of subsequent XMCD-PEEM observations between which a field has been applied was attributed a success estimate s as well as two field values. The success estimate was simply 0 in case of failure i.e. absence of nucleation or depinning, and 1 in the opposite case. The two afore-mentioned field values were the lower and upper bounds (as derived from the errors resulting from my field calibrations) on the field18 applied to the nanostructure, both being corrected for the angle between wire axis and applied field. The two series of XMCD-PEEM image in Fig. 4.27 display nucleation events with s = 0 for lower and upper bounds 13 mT and 18 mT, and s = 1, lower and upper bounds 14 mT and 19 mT respectively. Examination of the available sequences of images led to a three-column table for nucleation in this nanowire, and two similar tables for each observed pinning site: one table for leftward domain wall depinning and the other for rightward depinning. There is indeed no reason a priori to assume a symmetric behaviour.

µm

Then, these raw data were browsed and binned by a C program I wrote in order to build an estimate of the probabilities of nucleation and depinning as a function of applied field. Each event k was attributed a statistical weight 1/(δB) k where (δB) k is its span of induction values. The values s k were sorted and used to build the probability of successful event within each bin. Assuming that the the numbers n 0 k and n 1 k associated to s = 0, 1 should follow a certain binomial distribution with unknown probabilities, I considered that the error on each probabilities should be 1/ n 0 k + n 1 k [44]. The results of this post-processing are presented in Figs.4.28 and 4.29, for a chosen bin size of 1 mT. In practice, only the fact that very small increments of field could be reliably performed in experiments despite the rather large (several mT) systematic error on the applied field allows the resolution seen there. That is why one single event always contributes (with a low weight) to several bins. In total, there have been 65 events pertaining to nucleation and 39 to depinning (at the considered site, for rightward propagation). The applied induction µ 0 < H nucl > such that the associated nucleation probability is 1/2 is comparable or lower than the switching inductions 19 of 22 mT to 100 mT measured on individual nickel nanowires by Pitzschel et al. (with diameter modulations) [42] or Vilanova Vidal et al. [45]. The wire dimensions in these studies are 17 µm to 20.5 µm length with 80 nm to 160 nm diameter, and 12 µm length with 160 nm diameter, respectively. It must be noted that all these systems have diameters above 7∆ d , where the dipolar-exchange length of nickel is ∆ d 7 nm [15]. Also, µ 0 < H nucl > is significantly lower than the coercive induction measured on arrays of thin (diameters of 50 nm and below) Fe, Ni or Co nanowires [46]. This quantity is expected to decrease with increasing wire diameter above the onset of the curling reversal mode [47] and this prediction has been confirmed in experiments [46,48].

As for pinning, it is well known from experiments and simulations pertaining to soft flat strips [2,49,50] that there can be a large dependence of pinning fields on the domain wall configuration. The details of the pinning site itself can influence the domain wall structure at rest or under applied field [2,49,50]. Here, unfortunately, the relevant pinning site's nature is not clear.

Even when considering the largest total error on the applied induction, on the order of ca. 6 mT, the above graphs display a transition from null to unity probability whose width cannot be explained by the error on applied field alone. Although the errors on the estimated probabilities are large, it appears that even the sharpest curve possibility based on the estimates and their errors features a width larger than this upper bound on the spread in inductions. This being said, extrapolating the energy barriers obtained by Paulus et al. [46] in Ni and Fe nanowires (above diameters of 7∆ d ) to the range of diameters which is relevant here, as a guideline, leads to very large barriers, on the order of 10 4 k B T and above at room temperature. While the behaviour of energy barriers associated to nucleation may be much more complex, this order of magnitude seems incompatible with a measurable broadening of the nucleation field distribution thanks to the applied pulsed fields' time scale of ca. In this wire, about 15 nucleations at an applied induction of 28 ± 1 mT followed by pinning at one composition transition point [see red arrow in Fig. 4.30.b)] could be triggered. Nucleation always proceeded from the right hand side towards the left in Fig. 4.30. What is striking is that in 15 events out of 17, domain wall depinning in the opposite direction (to the right, see yellow arrow) was achieved by inductions of (11 ± 1) mT to (15 ± 1) mT, thus displaying a difference in depinning field larger than ca. 10 mT at the very worst. Admittedly, the diameter modulation at this pinning site (see Fig. 4.24 p.206) may well have played a role, yet it must be noted that within the experimental resolution, the domain walls' extremity closest to this diameter modulation appeared exactly level with the position associated to an abrupt composition variation.

III.3 Domain wall identification

As a preliminary note, it must not be forgotten, that in many instances the full extent of a nanowire's shadow is not accessible to XMCD-PEEM imaging. As a result, when the magnetic contrast pattern on the wire surface is unclear and too much of the shadow is hidden, the domain wall type cannot be identified. An example of this situation is shown in Fig. 4.31. The top and bottom images are XMCD-PEEM views at the Fe L 3 edge of the same NiFe nanowire before and after the application of a magnetic induction along the wire of 5 mT to 10 mT. The change of contrast both on the wire surface and in the shadow indicates the injection of a domain wall. However, the corresponding contrast patterns do not allow unambiguous identification. The more plausible hypothesis is a BPW considering the seemingly uniform contrast on the wire. Due to the presence of contrast from the neighbouring domains on one hand and the absence of information from the bottom of the shadow on the other hand, it is not possible to conclude unambiguously on the domain wall type. 20 With respect to the domain wall width. From this pinning site, domain wall depinning towards the right (see yellow arrow) occurred under inductions of (11 ± 1) mT to (15 ± 1) mT. By contrast, nucleation from the right side towards the left, followed by pinning at this site, was repeatedly obtained under applied inductions of 28 ± 1 mT. There is therefore an easy depinning direction, shown by the yellow arrow. Away from the domain wall itself and especially to the left of the blue arrow, an orthoradial component of magnetization in a single domain is revealed by the bipolar contrast pattern. If magnetization were fully along the wire axis, no contrast would be seen due to the orthogonality between wire axis and beam.

µm

Fig. 4.31: XMCD-PEEM views at the Fe L 3 edge of a NiFe nanowire before (top) and after (bottom) the application of a magnetic field from 5 mT to 10 mT (its direction being shown in red). The X-ray beam's propagation direction is indicated by the green arrow. The linear gray contrast scale goes from -0.07% to 0.07%. It is noteworthy that the contrast on the rim of the left hand side's thick section shadow (see area encompassed with an orange dashed line) also has changed sign, while in the bulk (see area encompassed with a purple dashed line), the uniform positive contrast has not. On the rim (inside the area encompassed in orange), the contrast goes from -(1.2 ± 0.6)% (close to the diameter modulation) to 2.4 ± 0.7%, while in the bulk (inside the area encompassed in purple) it goes from 1.7 ± 1% to 1.8 ± 0.8%. The errors are the standard deviation evaluated on areas representative of the shadow's rim or bulk.

By contrast, the micromanipulated NiFe nanowire D12 section II.4 allowed an easy domain wall identification thanks to its altitude offset above the sample. Due to the marked differences in sample height along the wire length, magnetic imaging was only performed in the shadow for the sake of focussing on only one plane. Initially, nanowire D12 was imaged with shadow XMCD-PEEM in a uniformly magnetized state. Thanks to an electromagnet underneath the CPW, it was however possible to first switch the nanowire completely and then to reliably nucleate domain walls, under inductions21 of 29 ± 1 mT. The electromagnet's field was aligned with the long wire axis, and in the substrate plane. Two particular pinning sites were evidenced: the one at which newly nucleated domain walls were always found, in the thin wire's section and another one close to the diameter modulation. The first one is illustrated with the two shadow XMCD-PEEM views in Fig. 4.32.

In these images, two domain walls differing in both type and polarity are shown in the shadow, while the wire itself is not visible due to the microscope settings. The BPW is easily recognised thanks to its bipolar contrast pattern [30,33]. By comparison, the more complex pattern is associated "by eye" to the TVW; this can be confirmed by simulations (performed by JEAN-CHRISTOPHE TOUSSAINT) of the expected contrast [30,33], as illustrated in Fig. 4.33.

While there are quantitative differences, the qualitative features in Fig. 4.32's top image, namely the marked perpendicular border of the dark contrast area and the more or less triangular-shaped inclusion featuring positive contrast are reproduced by the simulations. These were based on the relaxed equilibrium micromagnetic configuration a of TVW obtained with the finite-elements-based code FeeLLGood [51,52]. The simulated nanowire had a diameter of 140 nm and material parameters corresponding to Permalloy.

With the above in mind, it is possible to identify the domain wall type and even some details of its internal structure. In the BPW's case, the most interesting feature is of course its winding in the prospect of determining helicities after motion. However, one important experimental detail must be considered before this can be done. It is indeed crucial to note that raw images obtained with XMCD-PEEM have undergone a vertical flip (top becomes bottom) because of a mirror in the imaging system [53]. While most experiments are unaffected by such a flip, here this is of utmost importance as the direction of motion is unchanged, but the apparent winding is reversed. The meaning of positive/negative contrast indeed remains the same, and the direction of motion is unaffected by respectively. For the sake of comparison, Fig. 4.34.f) is a PEEM image of the same structure, with indication of the beam direction. This orientation provides contrast both from the domains in the thin wire's section and from the domain wall themselves. The events presented here show that a number of different configurations can be obtained, as well as three distinct pinning sites, labelled A, B and C in see Fig. 4.34.a1): A corresponds to a position where the wire appears wholesome, B to the notch on the bottom of the thin section [as visible in Fig. 4.34.f)], and C to the position where the wire is broken. Nothing from the PEEM image in Fig. 4.34.f) seems striking around pinning site A, except for a locally strong intensity.

- It is mostly on the former pinning site (A) that conclusions could be drawn regarding the selection of BPW helicity. Indeed, as is clear from Fig. 4.34.e2), some configurations can be significantly influenced by the notch, which is reminiscent of the case of strips to some extent [2,49,50]. Still, in total 12 events could be associated unambiguously to BPW, leading to the statistics in Tab.4.1. In all these events, the direction of motion was from the left to the right. The reported helicities h v are defined according to Eq.(4.1) (p.185), that is to say positive (negative) for right-handed (left-handed) magnetization winding around the direction of BPW motion (given by v). As a reminder, micromagnetic simulations on Permalloy square-cross-section wires (40 nm-side) [6] and on cylindrical Permalloy tubes 22 (outer diameter 60 nm/34 nm, thickness 10 nm/6 nm) [54,55] predict that positive helicity should prevail. Therefore, the observations should reflect a majority of cases where the selection of winding has led to h v = +1.

→ m a1) f ) 2 µm -→ m -→ m a2) b1) b2) c1) c2) d1) d2) e1) e2) A B C
It was possible in one instance to move a domain wall from pinning site A from to B, i.e. from left to the right. Motion in the opposite direction was impossible: either nucleation took place before depinning or pinning site A did not stop the domain wall. This conclusion is drawn from the comparison between the values of nucleation field (starting from a saturated state) and the fields applied in cases which could be interpreted as depinning to the left.

Final pinning site (origin)

Head-to-head polarity Tail-to-tail polarity

h v = +1 h v = -1 h v = +1 h v = -1 A (nucleation) 3 0 5 0 B (site A) 0 0 0 1 B (nucleation) 0 0 4 3 Total 3 0 9 4
Tab. 4.1: Summary of the nucleation and pinning events in nanowire Cicero.

As for pinning at site A, XAS and PEEM images at different foci (not shown here) indicate that the strong intensity at this position shows resonant behaviour at the Fe L 3 edge and that its source's altitude is below the thin wire's section. Since the shadow itself does not mirror this behaviour i.e. does not show a localized, marked dip as the photon energy is swept through the Fe L 3 edge, it seems that this intensity does not reflect a defect on/in the wire itself. Since no shadow extends just behind the direct wire image, the explanation I find most plausible is that this unexpected intensity originates from an impurity on the sample substrate. It probably has relief, which is why it is most visible with a focus above the silicon surface, but not so much as to feature a shadow reaching beyond the direct wire image.

Although the domain walls involved are pinned on a site whose nature is not clear, the results from Tab.4.1 clearly indicate a majority of selection of winding in agreement with simulations [8]. This majority of positive helicities irrespective of domain wall polarity is even unanimity if one discards the observed configurations pinned at site B (wire notch), whose influence is unknown. In all cases involving nucleation (i.e. the majority of events), the domain wall propagates over a distance in excess of 10 µm; by comparison, the travelled distance before BPW winding switch obtained by Thiaville and Nakatani [6] (in a square, 40 nm-side Permalloy wire) was less than 2 µm. Furthermore, the selection of helicity has been observed in these simulations under an applied induction of 3.5 mT, and Yan and coworkers [54] as well as Otálora et al. [55] have observed the same phenomenon in soft tubes under applied induction lower than 8 mT (after travelled distances also on the order of 2 µm). Comparison with these simulated systems is not straightforward, yet in the present work all events of BPW motion occurred under inductions always in excess of 11 mT.

IV.2 Nanowire D12

In this micromanipulated nanowire, a number of BPW nucleation events could be triggered, ten in total. With respect to the experiments from previous section, the pinning site at which domain walls stopped after nucleation was different as it allowed motion in the opposite direction without direct domain wall annihilation. This pinning site is illustrated in Fig. 4.35 for two BPW of identical polarity (here, tail-to-tail) but opposite winding. These two images were obtained at the Fe L 3 edge, with a microscope focus on the shadow. Due to the significant height of the wire above the substrate (see Fig. 4.12 p.195) as well as start voltage, the wire itself is invisible.

Considering that both domain walls propagated in the same upwards direction (indicated by the vector v in Fig. 4.35), the event associated to positive helicity is featured on the right hand side image. Overall, both nucleation experiments in this wire and domain wall motion between pinning sites led to an equal proportion of positive and negative helicities h v . These are detailed in Tab.4.2.

It must be noted that out of the 21 events of motion between pinnning sites featuring a BPW in the initial and final state, there have been only two cases such that the winding switched. In both cases, this resulted in negative helicity. In the 19 remaining cases, with the exception of one unclear event, the BPW winding did not switch. Out of this majority of motion events without winding switch, there have been 10 events such that the BPW was expected to switch so as to select positive helicity but it did not happen. In the remaining 8 cases in which no switching was observed, the helicity during motion was positive. This distribution of events (not taking the unclear event into account) is reported in Tab.4.3.

An example of motion with winding switch resulting in a negative BPW helicity is shown in Fig. 4.36.

In the cases of motion between pinning sites, it is noteworthy that the distance travelled by BPW is ca. 2 µm; this is a significant difference with respect to the experiments mainly relying on nucleation from section IV.1.

Initial wall type Initial position Final position

Head-to-head polarity Tail-to-tail polarity

h v = +1 h v = -1 h v = +1 h v = -1 BPW D U 0 5 3 3
TVW D U 0 0 0 0 Nucleation (NA) (NA) U 2 0 2 4 BPW U D 3 0 2 5 TVW U D 4 0 1 0 Nucleation (NA) (NA) D 0 0 0 0 Total 9 5 8 12
Tab. 4.2: Summary of all domain wall motion events leading to a BPW in nanowire D12.

One possible explanation for the absence of switching in the above-mentioned 10 cases is that although the initial winding corresponds to negative helicity, the distance travelled by the BPW is too short to allow switching. In the 8 motion events such that the BPW's initial winding corresponded to positive helicity, the predicted selection of helicity should contribute to maintaining this initial winding. Finally, the five events involving transformations of TVWs into BPWs will not be tackled here but later on in section V.

IV.3 NiCo nanowires

In Ni 60 Co 40 nanowires, a much smaller number of events involving domain wall motion could be triggered. In a large majority of instances, domain wall motion was followed by annihilation at one wire's extremity. There

Final helicity Initial position Final position

No winding switch Winding switch

h v = +1 D U 4 0 h v = -1 D U 6 0 h v = +1 U D 4 2
h v = -1 U D 4 0 Total 18 2
Tab. 4.3: Summary of the events with BPW motion between pinning sites in nanowire D12. have been, however, a few cases in which the final domain wall configuration was that of a BPW. Out of these, three were associated with nucleation, and one with a transformation from TVW to BPW. In the former series, one nucleation yielded a positive helicity, while the two others yielded a negative helicity. The transformation into a BPW corresponded to a positive helicity. In NiCo nanowires, all events of BPW with their associated helicities are summarized in Tab.4.4.

As can be seen here, the majority of events is contrary to the predicted selection of helicity. This being said, the number of events is quite small.

Type of event \ Helicity

h v +1 -1 Nucleation 1 2
TVW-to-BPW transformations during motion 1 0

Motion between pinning sites 0 1

Both (total) 2 3

Tab. 4.4: Summary of all domain wall motion events leading to a BPW in NiCo nanowires.

IV.4 Review

Let us now have a look at the whole ensemble of application of fields leading to the observation of BPWs. I have sorted these events with dynamics in Tab.4. Here, the square root of the total number of events corresponding to positive/negative helicity is presented as least poor estimate of the uncertainty on the event counts [44]. This estimate is based on the assumption of a binomial distribution (with unknown probabilities) describing the probabilities of observing a positive helicity: a pessimistic evaluation of the standard deviation amounts to simply taking the square root of the total number of events in each category [44].

Tab.4.5 thus captures an important result I have obtained from all the afore-mentioned XMCD-PEEM observations. It appears that BPW motion under magnetic induction (with values all above ca. 10 mT) does not lead to a very clear selection of helicity positive i.e. right with the direction of motion, contrary to what has been predicted by simulations [8]. The majority established in Tab.4.5 is 55 ± 10%; of course, the number of events is not very high. In terms of comparison, the careful and thorough investigation by Hierro-Rodriguez et al. [14] on trilayers shows (without direct imaging of the Bloch point's vicinity) a majority of positive helicities for one sample with a comparable number of events, and a much more mitigated or even adverse result for their other sample, with an order of magnitude more events. This was attributed by the authors to stronger, non-local interactions between the stripe domains thanks to which the presence of Bloch points was induced in the samples.

The present work has the advantage of dealing with isolated individual objects, both the nanowires and the investigated domain walls. This being said, there is yet much to learn for instance with respect to the observed pinning sites. Furthermore, there is the question of the distance travelled by the BPW: micromagnetic simulations indicate that winding switch only occurs after a minimum travelled path that may be as large as 2 µm [6]. As was stated before, this distance is very close to the one involved in the many motions between pinning sites from section IV.2; therefore, these events may bias the ensemble. If we consider only nucleation events, which involve travelled distances in excess of 10 µm, there are 17 motions with positive helicity and 9 with negative helicity. Thus, a somewhat clearer majority of 65 ± 13% emerges.

Beyond this question of distance required for the helicity selection, one should keep in mind that micromagnetic simulations have predicted finite energy barriers both for the winding switch associated to helicity selection and for the additional, energetically unfavourable winding switch [6]. In other words, it is likely that dynamically, a field-driven BPW moving with positive helicity may undergo a winding switch leading to negative helicity provided that the field is large enough. This behaviour has not been observed for the BPW in simulations so far [6,8,56,57], which rather hint at plateaus in BPW velocity without further configuration changes. However, simulations on soft tubes [55] involving vortex-like domain walls (the analogues of BPW without the Bloch point singularity) have shown that these walls can undergo this unfavourable winding switch at high enough applied fields. An analogy can be drawn with the Walker breakdown [58] in soft strips at that stage. Indeed, all energy barriers opposing rotation of magnetization around the propagation direction have been overcome, so that periodic winding switches follow [58]. In the present experiments, the picture of helicity selection would clearly be blurred as a result of this behaviour.

Another important and striking topic is that of the observed domain wall transformations, a few occurrences of which have been mentioned previously. In next section, these processes will be reviewed in more detail. In the above-described NiCo nanowires, there has been one event not involving domain wall motion but still a transformation from TVW to BPW. While the domain wall displacement is non-existent or below the detection threshold, if one considers the direction in which the domain wall would have moved if not pinned, then the final BPW configuration has a positive helicity. Upon re-application of the same field, this domain wall moved and retained a positive helicity. This transformation is illustrated in Fig. 4.38. Here, the magnetic signal-to-noise ratios are appreciably higher than in Fig. 4.37. The reason is that although the XMCD is weaker at the Co L 2 edge [32] than at the Co L 3 edge, the smaller mean X-ray absorption allows the transmitted X-ray intensities to be larger with respect to the parasitic background level discussed previously. As a result, their differences are less buried by this background.

V Domain wall transformations V.1 Transformations in NiCo nanowires

A transformation in the opposite sense, i. Then, significant depinning followed the application of a larger field and the domain wall was pinned close to the other, abrupt diameter modulation (not shown here): it is one of the very few cases in which (the vicinity of) a diameter modulation stopped domain wall motion. At that stage, the domain wall's configuration was unclear. Nevertheless, it could be brought back close to its initial position from Fig. 4.39. There, it could be imaged and seen to have transformed into a TVW, as is illustrated in Fig. 4.40.a-b). These two XMCD-PEEM images were acquired at different foci, first with focus on the wires' surface, and then in the shadow. The rich contrast pattern is incompatible with that of a BPW [30]. In

V.2 Reproducible evidence for transformation between transverse-vortex and Bloch point domain walls

We have observed that the final state depended on the strength of the applied induction. ) can be unambiguously attributed to a BPW and a TVW, respectively. Since the initial state was the same, this means that only the difference in applied magnetic field could be the cause for the different outcome. In other words: there appears to be a threshold field above which a transformation from BPW to TVW occurs. From the point of view of topology, this experimental result shows that the injection of a micromagnetic topological defect is possible and does not require extremely high field amplitudes. In other words, the distinct topological natures of the BPW and the TVW do not prevent these DWs to transform into one another.

A further confirmation of this fact is shown in Fig. 4.42. The sequence of shadow XMCD-PEEM images is exactly of the same nature as in Fig. 4.41, but this time with opposite domain wall polarity. The same sequence of DC fields leads to the same outcome: at low fields, the BPW does not transform, but does upon the application of higher fields.

I have gathered all events of domain wall motion in order to highlight the influence of the applied field. They are sorted in the following table, Tab.4.6, as a function of final and initial domain wall types according to the applied induction values. Here, "NA" means that no experiments with the corresponding starting configuration and applied field were conducted.

The sum of these experimental observations show that the transformation from TVW to BPW and vice-versa is possible, corresponding to the injection or expulsion of a topological defect. Moreover, a stark difference appears in the BPW behaviour as a function of applied field, suggesting a certain threshold above which this domain wall structure is prone to transformation. Therefore, the so-called protection that is sometimes associated with the topologically non-trivial character of the BPW configuration is shown to not hold at sufficiently high magnetic fields.

V.3 Comparison with theory and simulations

No matter how many instances of transformations are observed, our imaging before and after the application of field does not reveal their detailed process. Still, it is worth remembering that the TVW features no topologically non-trivial textures per se but two well defined micromagnetic objects whose real-space configuration on the nanowire's surface covers half the unit sphere, namely the vortex and antivortex at the intersection between domain wall core and wire surface that have been described in Chapter 1, section III.2 p.22. This is schematically illustrated in Fig. 4.43, which gathers in a first row schematic representations of a vortex and an antivortex defined on a flat sample as well as an illustration of a Bloch point's spherical vicinity in a BPW. The latter illustration is reproduced from the 2016 publication by Riccardo Hertel [8]. When reported on the unit sphere, these magnetization configurations defined on surfaces cover only half of it in the first two cases, and the whole 4π surface in the third case. Since the TVW possesses these two objects, each corresponding to a "half unit sphere" and the BPW does not, one TVW-to-BPW transformation mechanism that comes to mind is that vortex and antivortex somehow fuse on the wire surface during motion and yield a Bloch point. When motion stops, the latter eventually reaches its equilibrium position on the wire axis, leading to the final BPW configuration. Conversely, starting from a BPW at rest, it would seem that if the micromagnetic singularity manages to reach to wire surface, then there might be a "decomposition" path into vortex and antivortex thanks to the locally strong spatial variations of magnetization resulting from the Bloch point's presence. I must add that the above discussion and Fig. 4.43 is slightly simplistic in the sense that the polarity of vortex or antivortex does not mean that the surface they cover on the unit sphere goes "to the South pole" or "to the North pole". The key element is here that these algebraic surfaces have opposite signs such that these "halves" cancel out (otherwise the TVW would be topologically non trivial).

The above-described mechanism is plausible from the mathematical point of view, but its physical validity must be checked with micromagnetic simulations of a TVW or a BPW under field. In the following, the focus will be on the simulated field-driven dynamics of a TVW, starting from a configuration relaxed to equilibrium under zero magnetic field. The sample, simulated with FeeLLGood [51,52], is a 70 nm-diameter wire with 1 µm length. The material parameters are close to those of permalloy: saturation induction µ 0 M s = 1 T, α = 0.05 for the Gilbert damping parameter during dynamics (α = 1 was used for the configuration relaxation only), no magnetocrystalline anisotropy, and the typical cell size was 2.5 nm. During motion, the simulated domain wall is moved backwards In order to follow the transformation, an algorithm has been developed by CHRISTOPHE THIRION based on the definition of the whole micromagnetic configuration's S 2 winding number [59] N to first determine whether a given configuration (either from micromagnetic simulations or from our theoretical construction) contains a Bloch point. As a reminder: this quantity amounts to the algebraic surface covered on the unit sphere divided by 4π. If a Bloch point is present, the algorithm locates the singularity in the three dimensions up to an arbitrary precision. In micromagnetic simulations, the latter's lower bound is given by the mesh cell size.

A field of 8.2 mT is applied in a stepwise manner at t = 0. At this stage, the estimate of the S 2 winding number N sim (t = 0) is 0, as expected. Fig. 4.45.b) illustrates that at t 1 =0.45 ns the vortex and antivortex are no longer diametrically opposed but rather very close to one another, as shown by the neighbouring extrema of opposite signs in the colour map of -→ m • n. The reason why initially the vortex and antivortex draw closer is the following. If one looks at either object alone (without considering the other one) in, say, "half a wire", without changing the reference frame, one expects a given applied field to deflect the antivortex as well as the vortex in a direction perpendicular to the wire axis due to the gyrotropic effect. Now, the key issue is that the deflection is in the same direction for both objects. This is a consequence of the gyrovectors that could be associated to them [6] as surface objects, and which are parallel. The easiest way to see this is probably to look only at the wire surface [see Fig. Physically, the gyrovector arises from magnetization's precessional dynamics. In a hand-waving approach, for textures such as vortices and antivortices, the fact that magnetization precesses counterclockwise around the effective field from the LLG equation breaks a symmetry in the core's vicinity. Let us consider for instance magnetization ahead of the core of, say, a vortex at rest. One sees that forward motion, once started, imposes a certain time evolution, for a given configuration of the effective field. Now, independently, if we change the sense of magnetization precession, the effective field caused by the vortex is the same, but magnetization's time evolution must be different since it must precess the other way. On a broader level, when the object is taken as a whole, the asymmetry we just touched results in the object's core being pushed in the direction perpendicular to motion, the sense being determined by the texture's configuration [6].

Let us come back to the domain wall's further evolution. The surface magnetization still clearly displays the vortex and antivortex features at t 1 , and the calculated isolines (in green) of m z = -0.5, 0 and 0.5 (z being the direction of the wire axis) on the surface are very pinched. At t 2 =0.62 ns, these curves have become much more separated, while the vortex and antivortex have both disappeared, as revealed both by the surface magnetization and by the colour map of -→ m • n. This striking transformation corresponds to the injection of a Bloch point, which is the result of the recombination (into the volume) of the vortex-antivortex pair as it is annihilated on the surface. The estimated winding number N sim (t 2 ) = 1 + , with < 3 • 10 -3 , confirms this visual interpretation. Since the latter involves the motion and eventually the vanishing of the vortex and antivortex, monitoring these e.g. with the normal surface magnetization -→ m • n is key. Its local extrema, determined using level curves of -→ m • n, allow us to use a bracketing algorithm to track them during the simulation. With both their positions, we can compute the difference in angular position ∆φ(t) = φ V (t)φ AV (t) ("V" stands for vortex) and in abscissa on the wire axis ∆z(t) = z V (t)z AV (t). ∆ϕ(0) = π and ∆z(0) = 0, corresponding to the equilibrium configuration of the TVW. As can be seen in Fig. 4.46, these quantities' behaviour defines two regimes. In the first one (up to ca. 0.34 ns), ∆ϕ decreases monotonously while ∆z increases in absolute value then decreases. In the second regime, these quantities vary less though ∆ϕ still decreases. This behaviour accelerates until the critical time t c at which ∆ϕ is close to zero, while ∆z is still about as large as four mesh cells. It must be noted that just before the merging, the maximum values of | -→ m • n| starkly decrease and -→ m • n ( r) spreads much, so that this feature does not provide a reliable position estimator. Nevertheless, the behaviour that can be observed on visualizations of surface magnetization is reproduced by these estimators. Furthermore, even though there appears to be a finite distance between the vortex and the antivortex at the point where they vanish from the wire surface, we have a further element indicating that these object merge to yield a Bloch point in the volume. Indeed, using the first time step t B > t c such that the S 2 winding number associated to the micromagnetic configuration is 1 + , we find that the angular and longitudinal positions φ BP and z BP of the injected Bloch point are very close to the estimated positions of the vortex and antivortex. In Fig. 4.46, the red and green crosses indicate ∆z BP = z BPz AV and ∆φ BP = φ BPφ AV at this time step t B , respectively. It is noteworthy that ∆φ BP -11 • is only slightly larger in absolute value than ∆φ(t B ) -6 • , and |∆z BP | < ∆z(t B ). The radial position on the other hand is ρ BP 19 nm, i.e. significantly smaller than the wire radius of 35 nm. This is possibly due to a rapid Bloch point "fall" towards the wire axis due to a strong repulsion induced by the surface. With this in mind, this investigation of the Bloch point's initial position strongly suggests that it results indeed from the merging of the vortex and antivortex. This supports our proposed mechanism along which the vortex and antivortex draw closer and ultimately merge as a Bloch point is injected as the domain wall type changes. The latter is also reflected in the curves of the domain wall velocity and or position versus time in Fig. 4.47: the contrast between the initial and slow regime and the regime after Bloch point injection goes in the same direction as the existing theoretical predictions regarding BPWs [6,57,60], here, the estimated velocity is 0.39 ± 0.50 km/s. However, these micromagnetic simulations leave out imperfections in the sample (roughness, grain boundaries) and temperature, and do not handle properly the singular nature of the Bloch point, around which the modulus of magnetization progressively falls to zero [11,61]. Therefore, no quantitative conclusions should be drawn from these position and velocity curves.

By contrast, simulations starting with exactly the same conditions but at an applied induction of 8.1 mT led to a markedly different outcome. To put it briefly: no transformation from TVW to BPW was observed even at much larger time scales. This is apparent from visualizations as well as the fact that the computed S 2 winding number remains zero within a typical error less than a percent. A view representative of the steady state reached in this simulation is shown in Fig. 4.48, at a simulation time t steady =14.85 ns.

As can be seen from the surface magnetization and especially from -→ m • n (still as colour code), both vortex and antivortex are still present after a time more than one order of magnitude larger than t c in the previous simulation. At this stage, the motion perceptible from such visualization is a precession of the object pair which appears rigid, as the domain wall slowly propagates under the applied field. This behaviour is illustrated in Fig. 4.49, notably with its first graph representing the averaged components of magnetization that are perpendicular to the wire axis. After a transitory regime, they are seen to oscillate with π/2 phase difference at a frequency ca. 0.22 GHz, as obtained from Fourier transformation. Aside from this clear precession of a magnetic moment transverse to the wire axis, the middle and bottom graph in Fig. 4.49 reveal the dynamics of the TVW under 8.1 mT. The end of the transitory regime corresponds to a well-defined mean velocity (with some oscillations) of 3 ± 1 m/s, i.e. two orders of magnitude lower than the estimated velocity in the 8.2 mT simulation after the Bloch point injection.

The observed oscillations on the velocity (see inset) with a time period of ca. 0.5 ns were not expected. Over this time scale, the domain wall has moved along the z-axis by about 1.5 nm, which is over a factor three smaller than the mesh cell size of 5 nm. In the same amount of time, the precessional motion with the frequency of 0.22 GHz corresponds to a rotation of the vortex-antivortex pair of ca. 24 nm. The influence of the mesh cell size cannot be ruled out based only on these considerations, but it nevertheless does not seem that the velocity oscillations' time scale correspond to motion of magnetic textures over the cell size. It must be noted that the spike in domain wall velocity at ca. 19.5 ns results from domain wall re-centring. Due to the finite length of the simulated wire, eventually the propagating domain wall should reach the cylinder's end. This is avoided by longitudinal offset of the whole micromagnetic configuration, resulting in this type of short artefacts. If one computes the expected TVW velocity within the 1D model's frame [6] of domain wall motion, using the domain wall width calculated by JAMET et al. [62], one finds 11 m/s. The fact that this prediction is an overestimate by only a factor ∼ 4 is actually surprising, considering the complex three-dimensional domain wall configuration. The proper domain wall width to use for quantitative comparison, however, is the Thiele width [6], which has not been calculated for these simulations.

In fact, all the expected qualitative features of the field-driven TVW motion in a perfectly cylindrical nanowire [6,7] are found, even though the domain wall configuration is quite distorted with respect to the static equilibrium. Thus, there really appears to be a threshold field at which the TVW dynamics bifurcates. As a last emphasis on this threshold behaviour, Fig. 4.50 shows a zoom-in on the transitory regime of both simulations, in terms of the -→ m• n viewing the whole sequence of micromagnetic configurations indicates that after a transitory regime, the state displayed here is representative of all others except of the angular position around the wire axis. The pair of object appears to precess as a whole around that axis at a well-defined angular velocity while the (distorted) TVW propagates on.

domain wall position. The top graph pertains to the 8.1 mT simulation, the bottom one to the 8.2 mT simulation. What has been surprising during this simulation campaign, however, is that even at large applied inductions (over 40 mT) no transformation from BPW to TVW could be observed. Even a slight misalignment of the applied field (a couple degrees with respect to the wire axis) did not suffice to induce this transformation, which has been evidenced experimentally. While this raises further questions, it must not be forgotten that by nature, these micromagnetic simulations cannot accurately describe the Bloch point's behaviour.

As a conclusion, the dynamics of domain walls in magnetically soft cylindrical nanowires appears to be richer than what simulations in nanowires of either disk [56,60] or square cross-section [6] had suggested so far. One possible explanation lies in the investigated diameters in literature [10,63] that were often below the threshold for absolute stability of the BPW of about 7∆ d [62] (∆ d being the dipolar-exchange length), as opposed to those in our simulations or experiments. Both indicate that cylindrical wires can host domain wall transformations from BPW to TVW and vice-versa, similarly to soft flat strips where transitions between transverse and vortex domain walls are possible [6,64,65], but do not involve a topological defect. As illustrated by our simulations, its injection is a consequence of precessional magnetization dynamics, namely the gyrotropic effect bringing the vortex and antivortex features of the TVW closer together until their merging.

V.4 Review

In this section, I have gathered experimental evidence that the two domain wall types found in soft cylindrical nanowires [62] may evolve into one another when subjected to magnetic fields, despite their topologically different natures. It is useful to remember that experiments in other research areas in condensed matter physics involving topologically non-trivial states have demonstrated their resilience [66,67] but the latter is finite. The existence of these transformations is perhaps not so surprising because contrary to what misconceptions regarding topology may lead to think, there is no "topological protection" preventing the injection or annihilation of a Bloch point. Indeed, there has to be an energy barrier since either discontinuous transformations or the sample's edges are needed for such a transformation (as we have shown), but predicting its height lies outside the realm of topology. Experimental and theoretical studies in other fields [68,69] have shown how topological transitions can be induced by the continuous sweep of a physical parameter, similarly to our case.

Furthermore, the rather low field amplitudes needed experimentally to trigger the transformations imply that care must be taken in the interpretation of any domain wall velocity measurements (e.g. with magnetic force microscopy) in which the domain wall type cannot be assessed. Indeed, even if velocity measurements yield distinct orders of magnitude, the attribution of one measurement to only one domain wall type on the basis of this wall's expected mobility overlooks the possibility of domain wall transformations. Concerning the not observed They are seen to oscillate with π/2 phase difference, as is expected for a magnetic moment perpendicular to the wire rotating at a constant angular velocity. Fourier analysis on these functions yields a very-well defined peak with an oscillation frequency ca. 0.22 GHz. The middle graph shows the estimated domain wall velocity versus time, and correspondingly the bottom one displays the estimated domain wall position z DW (t). In this case, although there is also a plateau in position, the dynamics does not lead to a disappearance of vortex and antivortex. Intead, the TVW conserves its type and propagates slowly (with respect to the estimated "steady-state" velocity from Fig. 4.47. All these features are in qualitative agreement with the known behaviour of this type of domain wall under field [6,7], although it is here markerdly distorted.

BPW-to-TVW transformation, there is a need for a better understanding of the Bloch point's field-driven dynamics, its interaction with defects, and the role of temperature . Such insight would be valuable in the study of the BPW-to- TVW transition, which we have evidenced experimentally and has been seen in simulated cone-shaped nanowires [7], but has not yet been seen in simulated straight nanowires [7,56]. In all the observations, the trigger was the application of magnetic field, yet not all transformations occurred after domain wall motion over large distances (with respect to the experimentally measurable domain wall width). Some cases of TVW-to-BPW (and vice-versa) conversions were triggered while the domain wall was still on the same pinning site. Taking this into account as well as the fact that in most cases, the DC applied fields were ramped to their maximum value over time scales on the order of a second, it appears that domain wall pinning sites may play a role in the observations. However, this role so far only seems theoretically plausible, and the static XMCD-PEEM observations before and after application of magnetic field cannot rule out the scenario of transformations occurring during domain wall motion. Furthermore, the above-mentioned finite-elements simulations do indicate the existence of a threshold in terms of applied field above which the TVW-to-BPW transformation takes place. While the absence of the reverse transformation in this simulation campaign (even when the problem's cylindrical symmetry was broken) is intriguing, the latter proves nevertheless that there exists a dynamical path for the TVW to transform into the topologically non-equivalent BPW.

VI Application of a nanosecond-scale pulse of field

As was described in section I. 5 (p.191), the strategy to apply short pulses of field during synchrotron experiments was to micromanipulate nanowires and place them atop a coplanar waveguide (CPW) of my design. Both the CPW's alumina substrate and its tracks were adapted in terms of geometry to fit the stringent requirements of the PEEM sample holder. During the beamtime at synchrotron Alba where this strategy could be implemented, the Avtech R pulse generator was stationed inside the microscope rack and connected to the sample holder via the microscope cable. The final contact to the coplanar waveguide itself was achieved between the dedicated sample holder from the beamline and the CPW track pads with microbonding, performed by JORDI PRATT from the Circe beamline.

The schematic experimental setup as well as this special sample holder are shown in Fig. 4.51. In order for the sample holder's cap not to cause unfavourable short-circuits by direct contact with microbonding wires, the Fig. 4.52: Shadow XMCD-PEEM images at the Fe L 3 edge of nanowire D12 featuring a BPW. The wire itself is invisible on these images. Initially (top image), the domain wall is observed at the pinning site associated to nucleation in this nanowire. The (immobile) dark dot on the images encompassed by a red curve is related to an impurity on the sample substrate and is not related to the micromagnetics of nanowire D12. The linear contrast gray scale goes from -10% to +10%. The scale bar is 1 µm, and the X-ray beam direction is given by the green arrow. After a first application of static field (with the sample holder's electromagnet) of 11 ± 1 mT, the domain wall has slightly moved to the right but is still pinned (middle image). After re-application of the same field combined with a pulse from the Avtech R pulse generator, depinning is achieved (bottom image). On the bottom image, the dark ring is an artifact related to the substrate surface.

APPENDICES FOR CHAPTER 4

A 4 .1 Mechanical resilience A preliminary micromanipulation test was conducted by JEAN-FRANÇOIS MOTTE prior to the campaign mentioned in Chap.4 (see section I.5). The objective was to assess the overall difficulty of the process. It turned out that the corresponding NiFe nanowire geometry was that featuring the larger (ca. 100 nm) diameter modulations and thus also the larger diameters; later experiments indicated that this was a significant help in ungluing the wires without breaking them. In addition, the silicon substrate on which nanowires had been dispersed was cleaner than the substrate holding NiCo nanowires mentioned in section I.5. This being said, JEAN-FRANÇOIS MOTTE could perform an interesting test of the mechanical resilience of these NiFe nanowires. This test is illustrated in Fig. A 4 .1. From the top downwards, the tungsten micromanipulating tip is visible, with a nanowire attached to it. The nanostructure had been successfully lifted off beforehand, and its axis has a significant component along the normal to the substrate on the left image. To the right, the tip has been pushed towards the image's left hand side and brought closer to the silicon surface. As a result, the bottom wire tip being in contact with this surface, the nano-object starts to bend. Although a quantitative measurement is difficult due to the 3D orientations of the wire sections, the bend angle is easily over 35 • "by eye". From the image alone and without any corrections, the apparent bend is between 65 • and 70 • . On the latter image is featured a bend angle close to the "record": JEAN-FRANÇOIS MOTTE pushed the tip further but almost immediately after this image, the wire detached from the tip. Owing to the large amount of elastic energy stored 23 , the nanostructure "sprang" far outside the field of view and was irremediably lost.

General conclusion

During my thesis, significant endeavours have been undertaken towards the application of fields to nanowires during XMCD-PEEM studies. Field pulses with about 20 µs width could be applied and measured in-situ with a sample holder I designed and calibrated specifically for this experimental configuration, in combination with a pulse generator provided by MÁRLIO BONFIM. In addition, I have designed the fabrication of impedance-matched coplanar waveguides with a low-loss target, so as to be able to apply nanosecond-scale pulses of field to nanowires. Although the corresponding setup could not be finalized and put to use during this thesis, most technical issues have been overcome. There are two remaining problems: the placing of nanowires in the proper orientation with a high throughput without damaging the waveguide itself, and the waveguide's electrical connection to the pulse generator. The former cannot be fully addressed with micromanipulation, which was unfortunately shown to be too time-consuming and harmful to the samples to an unknown extent. A solution to the latter may be found with micro-bonding, instead of using a dedicated probe.

Outside the realm of field-driven domain wall motion, I have made use of the spatially resolved spectroscopic capabilities of PEEM with synchrotron radiation. This has notably allowed the evaluation of an original sample, namely an upright, core-shell, soft ferromagnetic cylindrical nanowire with a geometry featuring bends for domain wall pinning [1]. X-ray spectroscopy allowed to assess the sample core's metallicity, confirming independent investigations using Transmission Electron Microscopy and Electron Energy Loss Spectroscopy [2]. Additionally, I have been able to provide a frame of understanding for the observed XMCD shadow contrast pattern, revealing a domain wall, as well as the difficulties in imaging due to unanticipated wire vibrations. In the electrodeposited samples I have investigated the most, I have used X-ray spectroscopy to try and quantify the composition homogeneity thanks to the volume probing intrinsic to shadow PEEM imaging.

As for field-induced domain wall motion, I have first sought to observe the orthoradial winding of Bloch point walls after application of magnetic field so as to determine whether the predicted selection of helicity [3] occurs in experiments. Uncontrolled pinning in wires as well as the absence of domain wall containment from the nanowires' diameter modulation have hindered the observation of a large number of displacements. Considering all cases of domain wall motion leading to a BPW, only a very weak majority of events in agreement with this prediction emerges. Although the total number of events is not very high, my observations under inductions a priori large enough (exceeding all predicted thresholds from literature) to lead to helicity selection rather seem to imply a more complex picture. It must be noted that on one hand, a significant proportion of motion events were associated with a travelled distance on the order of the distance over which the helicity selection has been predicted to take place in wires [3] and in tubes [4] for the BPW's counterpart. In my experiments, motion over this short distance almost always proceeded with constant BPW winding, even when the latter was expected to switch. By contrast, a more solid majority of motion associated with the expected helicity emerges if only nucleation events followed by BPW pinning are considered. The difference to be noted is that the travelled path is several times higher than the aforementioned minimal distance for helicity selection.

Apart from investigating these configuration changes pertaining only to the BPW, I have observed field-induced transformations from transverse-vortex wall to BPW and vice-versa. These have occurred in both investigated materials (NiFe and NiCo), and could be triggered several times in one nanowire. There is evidence for a threshold in applied magnetic field below which the Bloch point wall retain its structure, and above which it transforms into a TVW. Such a transformation had been unanticipated in theory and simulations, rather, forward motion with retained BPW character had been predicted even under applied inductions on the order of 200 mT [5]. Micromagnetic simulations of BPWs under field performed by JEAN-CHRISTOPHE TOUSSAINT did not lead to the BPW-to-TVW evolution, however, the inverse transformation was observed above an applied field threshold. Detailed study of this simulated dynamical process could be done with the help of tools from topology, which were derived and implemented by CHRISTOPHE THIRION. It has been shown that the vortex and antivortex features of the TVW move closer and closer until they merge; at this stage, the simulated configuration becomes topologically non-trivial. Moreover, the first detection of the corresponding Bloch point singularity allowed to place it quite close to the position where the vortex and antivortex vanished as they merge. Thus, the TVW-to-BPW transformation proceeds thanks to these two objects and their merging, which results in the injection of a Bloch point into the nanowire's volume.

The experimental evidence for these transformations illustrates the fact that the topological non-equivalence between two micromagnetic states does not imply a large energy barrier separating them, as demonstrated by the applied field values. Moreover, it invites caution in future experiments on domain walls in soft cylindrical nanowires: without the knowledge of the domain wall type during field-driven motion, the attribution of an estimated velocity to the TVW or BPW is not straightforward. This being said, my experiments cannot rule out or quantify the role of defects in the sample. Considering that the quite reproducible transformation events were triggered by slowly ramped up magnetic fields (i.e. rise times on the order of a second), it is quite possible that the site at which the domain wall was eventually pinned played a major part in the transformation.

Nevertheless, micromagnetic simulations have shown that in nanowires with sufficient diameter, a threshold in field exists above which the transformation from TVW to BPW can occur. Previously, only variations in nanowire diameter had led to domain wall type transformations under field [6,7]. This consideration, in addition to the fact that most simulations of soft nanowires under field usually involved diameters below the threshold diameter ∼ 7∆ d identified by JAMET and coworkers [8], are likely to explain partly why this behaviour had not been reported earlier. While these simulations have provided insight into the dynamical transformation process, the reason behind the absence of BPW to TVW transformation even with rotational symmetry breaking remains to be determined. This is of course a topic closely linked to the actual transformation process in the shadow XMCD-PEEM experiments, which do not provide time resolution.

Prospectives Samples

As was previously mentioned, domain wall pinning is still a strong obstacle in experiments under field. Despite the endeavours concerning the fabrication of nanowires, it is presently difficult to estimate how far the samples are (in terms of material) from the lowest achievable pinning strengths and density of pinning sites. Several distinct topics are here entangled:

• the absence or presence of magneto-crystalline anisotropy, as well as its homogeneity in the latter case,

• the homogeneity of alloy composition along the wire length,

• the exact role of diameter modulations with respect to domain wall pinning,

• the influence of the surface state, in terms of both roughness and oxidation.

Quite evidently, local deviations from a soft alloy composition can lead to non-uniform anisotropies, and mechanical strains originating from the liberation of wires from their templates may in turn lead to magnetostrictive contributions. As another example of entanglement, some Transmission Electron Microscopy experiments performed by MICHAL STA ŇO on nanowires with diameter modulations suggest that these geometrical features may impact the electrochemical deposition process, thereby inducing localized composition variations. The latter are not controlled and may therefore have a non-negligible influence on the modulations' pinning strength.

The above list is but a summary of the topics I have grounds to believe might be linked to the difficulty of domain wall motion experiments so far. I do not claim to provide an exhaustive list nor to assess which list item is the most relevant. These grounds are as follows. Even defects causing variations of anisotropy with a size smaller than the domain wall width can contribute to pinning [9]. Since the preparation of a wire dispersion requires mechanical agitation, not to mention the dispersion itself, unknown amounts of stress are involved in the sample preparation. Although the quantification of composition variation in a single wire is difficult, at least in one experiment I have demonstrated abrupt variations of composition reaching two to five percentage points. In experiments, only few cases of domain wall pinning close to diameter modulations could be observed, whereas in many instances pinning sites in the wires' thin section have been evidenced. This shows that in a majority of cases, whatever pinning field is associated with the diameter modulations is lower than the depinning fields. Finally, it has been experimentally shown that pinning at sample edges, as well as other phenomena, can significantly affect domain wall motion even in samples that are significantly wider than the corresponding domain walls [10,11].

The question of composition homogeneity suggests taking a step back and focussing on the deposition process. As mentioned in the manuscript, pulsed electroplating might help reducing deviations from the nominal sample composition, perhaps even at diameter modulations. Admittedly, assessing the corresponding progress is difficult as for instance EDX is poorly suited to spatially resolved investigations of nanowires. Yet, using not single wires but bundles (as are easier to obtain with smaller dilution etc.), this issue might be alleviated.

After the uncertainties about composition have been lifted, one could furthermore think of tuning a wire's composition on purpose along the wire. Unintentional and abrupt composition changes may have played a role in the marked asymmetry observed on one domain wall pinning site, with a difference in pinning field in excess of 10 mT. Recent work has shown that strong, abrupt composition changes in nanowires with uniform diameter could lead to domain wall pinning [12][13][14], with some degree of reproducibility even [14].

The main disadvantage of such a strategy is that it requires a fine control over the electroplating parameters so as to reproducibly tune the nanowire composition. Moreover, both the amplitude and extent of the composition modulation can be expected to play a role; therefore, there are at least two independent parameters to explore. The advantage is that no diameter modulations are in principle required, so that their unknown role in domain wall pinning can be left out.

Further work at the interface between material science (on the side of nanowire fabrication) and nanomagnetism has recently begun with the PhD thesis of MICHAEL SCHÖBITZ. One of the goals is to strengthen the control over the electroplated materials and reduce the influence of pinning in nanowires, so that their domain wall physics can be more easily addressed.

Domain walls

Despite their apparent simplicity, soft ferromagnetic cylindrical nanowires harbour a rich physics, some facets of which yet to be understood. In particular, the response of domain walls in these systems to applied stimuli still requires further experimental investigation. In this respect, although this thesis has yielded elements of confirmation concerning the Bloch point wall's (BPW) helicity selection, it also has raised the question of domain wall type stability. An important unknown at this stage is the distribution of pinning sites along an average nanowire, or to put it differently the energy landscape seen by a domain wall. Since all experiments, be it MFM or XMCD-PEEM, provide only static images before and after motion, it is so far not clear whether the observed domain wall displacements occur in a creep regime [15] or in a linear flow regime. The predictions from micromagnetic simulations regarding notably the Bloch point wall's field-driven dynamics have been obtained in perfect nanowires. The influence of potentially strong pinning remains to be determined, in particular if the pinning fields are above the threshold in field for the BPW's helicity selection.

Aside from domain wall pinning, the difficulties of applying sufficiently short but strong pulses of field to nanowires so as to trigger domain wall dynamics must be overcome. Several technical issues remain in this regard, such as the dispersion of nanowires on a waveguide and reproducible electrical connection of the latter to a pulse generator. However, recent experiments by BEATRIX TRAPP on nanowires dispersed on very thin substrates seem to show that the pulsed induction amplitudes are too weak in this configuration, a couple millitesla at most. This reduction of pulse strength is due to the separation between waveguide track and sample corresponding to the substrate thickness of several tens of micrometres at least. It would seem that domain wall motion triggered solely by a magnetic field short pulse requires the nanowires to be directly atop the waveguide.

Provided that the issues of pinning and of short, strong pulses of field can be solved, the questions of helicity selection on one hand and of domain wall transformations on the other hand can be tackled in a simpler context. Furthermore, only such experiments can reliably explore the high-field regime of BPW propagation. The predicted drop in mobility associated with the excitation of spin waves [16,17] is a remarkable dynamical behaviour permitted by the expected high domain wall velocities. If confirmed, this property could render the control over domain wall displacement (at fixed pulse length) easier due to the corresponding drop in the velocity's sensitivity to the applied field. Indeed, if the pulse strength is well above the threshold for spin wave excitation, a small error on this pulse amplitude leads to a reduced offset in domain wall velocity, since the latter no longer varies as abruptly with field as before the threshold.

The BPW has yet to reveal another interesting dynamical property, one that is not related to longitudinal motion. When viewed at rest in the plane perpendicular to the wire axis and containing the Bloch point, the domain wall configuration quite resembles a vortex except for the absence of a core. Still, the Bloch point's central position corresponds to a minimum of energy, and therefore a gyrotropic motion around the centre is expected. Already in simulations of longitudinal field-driven motion [16], this type of Bloch point excitation has been observed with well-defined resonance frequencies with 10 GHz as order of magnitude. Using waveguides in the proper geometry and alignment, it should be possible to excite this gyrotropic motion similarly to the vortex core motion in soft ferromagnetic dots [18,19]. Not only could these Bloch point oscillations be observed with time resolution thanks to the stroboscopic pump-probe capabilities of XMCD-PEEM (with the beam perpendicular to the wire), but also the emission of spin waves along the wire axis [17] could be investigated. In particular: if the domain wall stands still, whether the spin wave tails become symmetric (in terms of wavelength) is an open question.

Finally, an exciting prospective is that of current-driven BPW motion. Indeed, the TVW's motion under current is not expected to be very different from the field-driven one [20] if one considers only spin transfer torque, whatever the non-adiabaticity parameter usually denoted β. Only differences in precessional angular velocity as a function of βα (α being the Gilbert damping) have been predicted [20]. On the other hand, strong deviations from the hypothesis of adiabatic spin transfer torque can be expected in the Bloch point wall's case. Since spin transfer torque is all the stronger as the spatial variations of magnetization are large (in the adiabatic picture [21]), the interaction between Bloch point and spin-polarized current may be significantly larger than the interaction between Bloch point and magnetic fields [5]. As a result, the BPW's current-driven motion may not feature the Bloch point's "lagging behind" observed under large magnetic fields [5].

In this respect, the efforts dedicated to electrical contacting of nanowires by SYLVAIN Y. MARTIN are a first, significant step. They have notably resulted in MFM measurements of domain wall velocities under a pulse of current on the scale of ten nanosecond, yielding lower bounds on the order of 300 m/s. During his PhD thesis, MICHAEL SCHÖBITZ will build on these pioneering experiments so as to further investigated the current-driven motion of domain walls in soft nanowires.

Fig. 1 . 1 :

 11 Fig. 1.1: Schematic representation of a tetragonal lattice system hosting ferromagnetism (blue arrows) under the influence of a magnetocrystalline anisotropy. On each site (red ball), the system's magnetic moment points preferentially along the vertical bonds, as suggested by the elongated halos.

E 12 ,Fig. 1 . 2 :

 1212 Fig. 1.2: Schematic representation of two aligned point magnetic dipoles µ (at the origin) and µ 2 at a variable position with respect to µ, see the dashed yellow lines.If the angle θ is such that µ 2 is in the blue-shaded area, the parallel alignment is energetically favourable; if cos 2 (θ) > cos 2 (θ c ) as in the red-shaded area, the alignment becomes unfavourable.

Fig. 1 . 3 :

 13 Fig. 1.3: Photographs of a self-assembled ring of hard magnetic beads before (left), during (middle) and after (right) manual application of pressure on the sides.

Fig. 1 . 4 :

 14 Fig. 1.4: Schematic representation of a pair of hard magnetic beads in their equilibrium configuration (left) and after rotation of one with respect to the other (right).

Fig. 1 . 5 :

 15 Fig. 1.5: Schematic representation of two possible equilibrium configurations for a soft ferromagnetic disk of radius R and thickness t. a) Uniform magnetization under the dominant influence of exchange interaction. b) Non-uniform pattern in which magnetization follows the sample edges, leading to null surface magnetostatic charges.

Fig. 1 . 6 :

 16 Fig. 1.6: Schematic representation of our boundary conditions. This problem's unknown is the variation of -→ m with the coordinate y.

Fig. 1 . 7 :

 17 Fig. 1.7: Illustration of the Bloch wall profile determined by equation Eq.(1.23). Magnetization, represented as cones, features m x as colour code going from red (-1) to ochre (+1).

Fig. 1 . 8 :

 18 Fig. 1.8: Schematics of the Stoner-Wohlfarth particle, with an easy axis (horizontal dashed line) of magnetization from which the system's magnetization -→ M is pulled away by an angle θ under the application of an external field -→ H a at an angle θ H .

d 2 e dθ 2 θ 2 =Fig. 1 . 9 :

 219 Fig. 1.9: Hysteresis loops of the Stoner-Wohlfarth particle (adapted from [29]) at different angles θ H , indicated in degrees on the graph for each curves. The (reduced) remanence M r and coercive field H c of the 45 • and 80 • cases are illustrated in blue and brown, respectively.

Fig. 1 . 10 :

 110 Fig. 1.10: Schematic representation of the curling reversal mode [37] in a ferromagnetic cylinder (transparent gray). Magnetization progressively cants away from the wire axis under the influence of an applied field.

Fig. 1 . 11 :

 111 Fig. 1.11: Illustration of the damped precession of a magnetic moment µ in an externally applied field -→ H a . The length || µ|| is conserved, and so µ evolves on a sphere (in gray). Starting from µ along -→ e x , relaxation occurs after ca. ten revolutions, as shown by the magnetic moment's trajectory (black curve). Here, the Gilbert damping is α = 0.08.

Fig. 1 . 12 :

 112 Fig. 1.12: Micromagnetic simulations of a soft flat strip with Permalloy material parameters, with width 240 nm and thickness 10 nm. The orientation of magnetization is shown by the colour wheel and the black arrows. a) Initial vortex wall configuration at rest. b) Snapshots at the indicated simulation times, under an applied induction of 1.5 mT. The Gilbert damping is α = 0.01. In this case, the field is above the Walker threshold and the domain wall configuration periodically transforms via the ejection and injection of vortices (see image at 24 ns) followed by antivortices (see image at 12 ns). Adapted from Ref.[63].

Fig. 1 . 13 :

 113 Fig. 1.13: Schematic representation of a head-to-head (left) and a tail-to-tail (right) domain wall in cylindrical nanowires.Their respective magnetostatic charges are shown at the domain wall position.

Fig. 1 . 15 :

 115 Fig. 1.15: Equilibrium configurations of transverse walls in soft nanowires with square cross-sections obtained by finitedifferences simulations. a-b) Cuts containing the wire axis and perpendicular to the wire side, the latter being 30 nm; material parameters are those of Permalloy, exchange constant A =10 pJ/m and µ 0 M s =1.0053 T, hence ∆ d 5 nm [67]. Cell size 1 nm × 1 nm × 2 nm. The y component of magnetization is highlighted as colour code, showing a distinctly triangular core shape for this head-to-head domain wall. c) Similar cut for a wire with 35 nm side and same material parameters [63], featuring a tail-to-tail domain wall. As expected ( -→ m → --→ m), the core

Fig. 1 . 16 :

 116 Fig. 1.16: Views of a tail-to-tail TVW's equilibrium micromagnetic configuration simulated with FeeLLGood [70] in a cylindrical nanowire with Permalloy material parameters. The wire diameter is 70 nm, the scale bars 50 nm. The mesh size was 5 nm, and the seed configuration derived from a Bloch-wall-like profile. a-c) Cuts in the plane containing the wire axis and domain wall core. The m x , m z and m y components are successively shown as colour code on magnetization (cones). a), b) and c) respectively highlight the marked curling features described by JAMET et al. [67], the still quasi-triangular wall shape and the narrow wall core. d-f) Representations of magnetization on the wire surface (displayed as green cylinder), with -→ m • n as colour code: n is the local, outward-pointing unit vector normal to the wire surface. d) and e) focus on the domain wall's vortex and antivortex surface features, while f) shows the TVW at the same angle as a-c. Simulation credit: JEAN-CHRISTOPHE TOUSSAINT.

Fig. 1 . 18 :

 118 Fig.1.18: Schematic views of a head-to-head BPW's cross section in the plane perpendicular to the wire axis and containing the Bloch point. Both windings are shown (left column, black arrows) at the initial stage of application of a magnetic field -→ H a along the wire axis, towards the reader, and above the selection field H sel . The brown arrows represent the corresponding torques on the winding. In addition to forward motion, the applied field promotes inequivalent distortions for the windings. Based on Refs.[63, 81], the right column shows the steady-states. The magnetization from the winding right-handed with the direction of motion (top) has canted but the winding remains the same. On the other hand, the opposite winding has switched (dashed purple arrows) to become right-handed with respect to the BPW velocity -→ v DW .

Fig. 1 . 19 :

 119 Fig. 1.19: Simulated BPW velocities in a square crosssection nanowire with Permalloy material parameters, as a function of initial winding. The wire side was 40 nm. Full dots correspond to a BPW winding right-handed with the direction of motion, hollow dots to a winding initially left-handed with the direction of motion. The difference in mobility (see straight lines starting at origin) as well as the left-handed winding's velocity plateau are clear. Figure adapted from [63].

Fig. 1 . 20 :

 120 Fig. 1.20: Simple objects, a stick and a torus. We shall try and distinguish them without resorting to geometry.

Fig. 1 . 21 :

 121 Fig. 1.21: Simple objects, a stick and a torus. The former defines only one set of closed loops, which can be slid around or next to it. The latter defines by contrast two distinct sets of closed loops: those encompassing it (e.g. to the right) and all others.

Fig. 1 . 22 :

 122 Fig. 1.22: Magnetization configurations (left column) sampled only on circular closed loops, on which magnetization (black arrow) is drawn. The images of these loops in the order parameter space (the unit circle for planar-only magnetization) are shown in blue (right column).

Fig. 1 . 23 :

 123 Fig. 1.23: Magnetization configuration corresponding to a Bloch point within a BPW (left column, image adapted from Ref.[81]) and representation of the spherical surface encompassing the singularity on the unit sphere i.e. magnetization's order parameter space (right column).

Fig. A 1 . 1 :

 11 Fig. A 1 .1: Schematics of the considered system. -→ m is the local magnetization orientation defined by its angle θ(s) with respect to the z direction. s is the curvilinear abscissa along the wire, and 2θ 0 is the bend's full opening angle.

A 1 . 3 . 2

 132 Frame of work and definition of the force III.2.a Hypotheses

A 1 . 3 . 4

 134 Geometrical interpretation of the gyrovector III.4.a 2D case

Fig. 2 . 1 :

 21 Fig. 2.1: SEM views of nanoporous alumina templates, taken from Ref. [26]. a) Side view of a membrane partially filled with electrodeposited nanowires. b) Top view of well-ordered, empty pores.

Etching

Fig. 2 . 2 :

 22 Fig. 2.2: Schematic representation of a AAO membrane with its pores still closed (left) and after pore opening (right). The opening is performed either with wet etching methods or e.g. RIE. Adapted from Ref.[29].

Fig. 2 . 3 :

 23 Fig. 2.3: Scanning electron micrograph of a nanoporous alumina template featuring repeated diameter modulations along the pores. Both the membrane and the above image were produced by SmartMembranes GmbH R .

Ni 60 Co 40 :

 40 Electrodeposited by Sebastian Bochmann and Julien Bachmann in Erlangen[50]. Ni 1-x Fe x : Electrodeposited at Institut Néel by Raja Afid, Beatrix Trapp and Laurent Cagnon with the target composition of Permalloy Ni 80 Fe 20 . Deviations in composition will be discussed later on (see Chapter 4, section II.4).

Fig. 2 . 5 :

 25 Fig. 2.5: Schematic representation of a nanowire featuring two outer thick sections of the same diameter D thick and a middle section of diameter D thin < D thick . The magnetostatic charges corresponding to the diameter modulations as well as the domain wall are highlighted in red and their corresponding (positive) signs.

Fig. 2 . 6 :

 26 Fig. 2.6: Left: detail of the alignment mark pattern within a unit cell in the original design. Here, the view shows only the (abstract) pattern as edited. The red circle represents a 30 µm field of view, as is used during navigation in PEEM.The coloured areas are developped i.e. free of resist after the lithography, hence they correspond to the regions eventually covered with metal. Right: portion of an optical map of a silicon wafer with this mark pattern and a dispersion of nanowires. The latter appear as black rods on the image owing to light scattering. As can be seen from the sharp transition in the middle of the image, the map is constituted by stitching together frames with some overlap between one another

Fig. 2 . 7 :

 27 Fig. 2.7: Left: Detail of the mark pattern within a unit cell in the final design. The red circle represents a 30 µm field of view as is often used for navigation with PEEM, just like in Fig.2.6 and the area is the same for comparison. Right: Overview of the mark pattern in the final design.

Fig. 2 . 10 :

 210 Fig. 2.10: Schematics of a demagnetization sequence, displaying the applied field B as a function of time t, both normalized by the maximum value B M respectively the time scale of the envelope's decay (here exponential for the sake of example) T .

Fig. 2 . 11 :

 211 Fig. 2.11: Demagnetization stage featuring the two electromagnets as well as a pair of pole pieces, here shown unmounted.

Fig. 2 . 13 :

 213 Fig. 2.13: Schematic view of the electronic transitions at the L 2 and L 3 absorption edges of a ferromagnetic 3d transition metal, with emphasis on the L 3 edge (see turquoises arrows) at a photon energy ω = E L3 . Here, CL light is shone on a sample with magnetization along the beam, i.e. with a majority of down spins. The spin and orbital momenta s and l are respectively represented in red and blue for the 2p bands, and s is also represented on the 3d bands.The corresponding densities of states (calculated with Density Functional Theory[104]) in dark gray were taken from[105]; the light gray and dark shaded areas correspond to the densities of state associated to p and s electrons.Along the electron energy axis E, the Fermi level at E F is indicated with a vertical line. The asymmetry in density of empty states at that energy is illustrated with the down and up spin densities ρ ← and ρ → in green.

Fig. 2 . 14 :

 214 Fig. 2.14: Schematical representation of a PEEM.Image taken from Ref.[96].

  Electrons S h a d o w X -r a y b e a m

Fig. 2 .

 2 Fig.2.15: Schematical POV-Ray[123] view of a shadow XMCD-PEEM experiment. The sample is a bimodulated cylindrical nanowire on a silicon wafer (dark gray), and lit by an X-ray beam with a 16 • incidence angle. Only a thin beam sheet is shown in bluegreen for the sake of clarity; the propagation direction, perpendicular to the wire axis, is indicated in red. The electron intensity collected from the different areas is illustrated by the orange arrows with different thicknesses. The photoemission from the substrate is largely outmatched by that on the sample's surface (therefore represented bright) owing to resonant X-ray absorption. Since the beam's incidence is low, a wide shadow is created behind the sample, similarly to everyday-life shadows at sunset. In the present case, the wire's thin section is high enough to allow its full shadow to be visible. The number of electrons in this area is all the weaker as X-ray absorption in the wire is strong.

Fig. 2 .

 2 Fig. 2.16: XMCD-PEEM views of NiFe nanowires featuring a BPW (left) and a TVW (right) [125]. The wires are outlined in red, the shadows in blue.

Fig. 2 . 18 :

 218 Fig. 2.17: Schematic plans of the cap holding the samples in place in PEEM experiments [133]; all dimensions are in millimetres. Left: side view with a partial cross-section (to the left of the curvy line indicated with a blue arrow). The 5 • mention refers to the tilt of the cap towards the center. Right: top view where the four tangs with through holes that are used for screwing the cap (and sample) into place are visible. The aforementioned tilt allows four pins (black dots, see black arrow to the bottom) to hold the sample in place and ensure electrical contact. The central disk corresponds to an hole in the cap delimiting the sample area which can be imaged.

Fig. 2 .

 2 Fig. 2.19: a) Schematics of the sample holder (grey), featuring the field coil (red), the pick-up loop (ochre), and a sample (dark grey) on top of it. The current I pulse (t) flowing through the field coil generates the field -→ H pulse (t) applied to the sample. This field in turn creates a voltage U loop (t) at the terminals of the pick-up loop. This measured voltage is defined with respect to the STV, which is the refence potential in the experiment, and to which the field coil and sample are grounded as well. b) Photograph of the sample holder. While the field coil (red) stands out, the pick-up loop (orange) is harder to see because of the glue holding it in place.

Fig. 2 .

 2 Fig. 2.21: Left: photograph (picture credit: MÁRLIO BONFIM) of the sample holder with a small measurement loop above its top surface. Right: inductive measurement of the pulse sent with MÁRLIO BONFIM's pulse generator. The flux Φ p.u. (t) through the loop can be obtained by integration of the electromotive force at the loop's terminals. This flux in turns is proportional to the applied field H pulse (t) at the location of the loop.

Fig. 2 . 22 :

 222 Fig. 2.22: Magnetic induction µ 0 H M (z) versus the altitude z of the pick-up loop, as measured with respect to contact with the sample holder. The error bars on the field are calculated from a 4% uncertainty on each data point. These 4% were estimated from the maximum spread in a series of 10 successive measurements in the same configuration.

1 F

 1 M,m.c.

Fig

  Fig.2.23: Measured electromotive force U loop (t) (red) induced in the pick-up loop during a pulse sent with MÁRLIO BON-FIM's pulse generator, the load voltage being 552 V. Numerical integration of this signal leads to the opposite of the magnetic flux through the loop, Φ loop (t). -Φ loop (t), represented in blue, is proportional to the applied field H pulse (t). As can be seen, it does not fall to zero after the pulse, meaning that a low current is still present after the pulse. The appearance of this current follows the spike on U loop (t), indicated with an arrow. The extrema V 1 and V 2 of U loop (t), as well as the extremum Φ M,m.c. of Φ m.c. (t), are also indicated.

FFig. 2 . 25 :

 225 Fig.2.24: Extremal fluxes Φ M,p.u. versus Φ M,m.c. (red), as determined after numerical integration of their corresponding electromotive forces induced in the pick-up loop and the measuring coil, respectively. The sweeped parameter was the capacitor load voltage U capa . The error bars were calculated following several estimates of relative uncertainties on the fluxes (a series of ten identical measurements for one relative uncertainty). The best affine fit to the data (dashed blue line) taking the error bars into account is also represented.

FFig. 2 . 26 :

 226 Fig. 2.26: Magnetic fluxes Φ M,m.c. (t) (blue) and Φ M,p.u. (t) (black) through the measurement coil and pick-up loop respec-tively. These were obtained by numerical integration of their corresponding induced electromotive forces. The load capacitor voltages were respectively 552 volt and 333 V. Discarding the very end of the pulses according to the previous discussion about artefacts in the electromotive forces, the differences in shape are insignificant.

Fig. 2 .

 2 Fig. 2.27: GGG slabs provided by PIERRE MOLHO. In these experiments, only the bigger one on the forefront was used for easiness of handling.

  Photodiodes

Fig. 2 . 28 :Fig. 2 . 29 :

 228229 Fig. 2.28: Setup used to calibrate the field created by the Helmholtz coils (in green and yellow). A Hall probe at the tip of a PCB piece is placed in the laser beam path, in the plane of symmetry of the coil pair.

Fig. 2 . 30 :

 230 Photodiodes

  From the specifications of the Hall probe, given the control current value of 5 mA, we have the following voltage-to-induction conversion factor: ξ Hall : 1.19 ± 0.01 mT/mV. The voltage U Hall acquired while applying a DC current I coil to the two coils is plotted on Fig.2.31. Clearly, there is a zero-induction offset, but the quantity of interest is the slope R of this curve. The affine fit (see dashed blue line) to the data (red markers) yields a slope R =3.31 ± 0.01 mV/A. Thus, for a given applied current I coil , the magnetic induction B felt at the location of the Hall probe reads B = m Hall •I coil where m Hall is such that m Hall = ξ Hall •R =3.94 ± 0.05 mT/A.

Fig. 2 . 31 :

 231 Fig. 2.31: Measured Hall voltage U Hall as a function of the DC current I coil applied to the pair of coils. The red markers correspond to the data, while the dashed blue line is the affine fit to the measurement. The error bars, on the order of 10 mA for I coil and 1 × 10 -2 mV for U Hall , are smaller than the markers except for the highest-current data point.

Fig. 2 . 32 :

 232 Fig. 2.32: Photograph of the GGG piece held by non-magnetic tweezers on the laser beam path. The latter's incidence point (visible by scattering) close to the straight bottom edge was chosen thinking ahead of the setup involving the sample holder, see Fig.2.36.

Fig. 2 . 33 :

 233 Fig. 2.33: AC current amplitude I AC as a function of control voltage setpoint U c . The data are the red markers, and the dashed blue curve is the affine fit to the measurements. The error bars on the current, on the order of a few mA, are smaller than the markers.

Fig. 2 . 34 :

 234 Fig. 2.34: Peak amplitude E(f 0 ) of the photodiode signal's Fourier transform at the AC current input frequency of f 0 , as a function of control voltage U c . The data are the red markers, and the dashed blue curve is the affine fit to the measurements

Fig. 2 . 35 :Fig. 2 . 36 :

 235236 Fig. 2.35: Peak amplitude E(f 0 ) of the photodiode signal's Fourier transform at the AC current input frequency of f 0 =7 Hz, as a function of applied induction amplitude B.

Fig. 2 . 37 :

 237 Fig. 2.37: Measured induced electromotive force in the detection loop (red) for a pulse delivered with a 552 V capacitor voltage, and corresponding photodiode signal (blue). The latter has been multiplied by 100 for clarity. The recorder extremum E m (which must be evaluated despite the noise) is indicated with an arrow.

Fig. 2 . 38 :

 238 Fig. 2.38: Maximum voltage E m measured on the photodiode signal as a function of capacitor voltage setpoint U capa . The data points are in red, the affine fit to them is the dashed blue curve.

III. 3 .

 3 f Determination of the garnet's Verdet constant III.3.f.(i) Context and bibliography

. 33 )Fig. 2 . 40 :

 33240 Fig. 2.40: Faraday rotation α 0 as a function of the amplitude B m of the applied AC magnetic induction. The data points are in red, the affine fit to them is the dashed blue curve.

IV. 1 .Fig. 2 . 43 :

 1243 Fig. 2.43: Photograph of the front end of the Avtech R pulse generator. The parts that I have not used are signalled in red. All the parts that have been of frequent use are highlighted in blue.The switch for the trigger mode (in green) always was turned to the "manual position". Leaving the self-explanatory elements aside, the output is SMA (capped here with a 50 Ω load for safety), and the pulse shape control knobs correspond to two distinct ways of tuning the pulse's temporal profile. In practice, the pulse can be made slightly sharper or to the contrary dulled and broader, with a sort of plateau.

Fig. 2 . 44 :

 244 Fig. 2.44: Pulse trace provided by Avtech R . The dashed red lines delimit the top 80% of the pulse's amplitude. The corresponding time during which the signal is above 20% of its maximum value is about 5.6 ns. Overall, the pulse can be thought of as a close-to-triangular signal, with low-amplitude corrections to the sides: less steep flanks and also bounces, especially towards the pulse's end. I have measured the pulses from my Avtech R pulse generator and found little difference; much of it depends on the fine shape settings.

Fig. 2 . 45 :

 245 Fig. 2.45: Schematic and not true-to-scale representation of a diameter-modulated nanowire (light grey) on a metallic strip with rectangular cross-section (rough, coppery texture). A current -→ J (in red) is being passed through the strip, generating the Ørsted field -→ H a (in light slate blue) which winds around the strip's cross section.

1 z 0 Fig. 2 . 47 :

 0247 Fig. 2.47: Sketch of a transmission line with characteristic impedance Z 0 . An abrupt change of characteristic impedance from Z 0 to Z 1 at z = 0 is denoted by the dashed orange line. A voltage wave with amplitude V p (at z = 0) propagates from the left to the right with the left hand side line segment's propagation constant γ 0 . The impedance mismatch then leads to a transmitted wave V 1 propagating into the right hand side line segment and a reflected wave V m in the left hand side segment. These correspond to propagation constants γ 1 and γ 0 .

Fig. 2 . 48 :

 248 Fig.2.48: Schematic representation of a CPW's cross section. The conductor (in red) has a thickness t for the three electrodes.The middle one with width w is labelled "S" as it is the signal electrode, while the two others to the sides, with respective widths b 1 and b 2 , are grounded and therefore labelled "G". The central electrode is separated from the two others by the gaps g 1 and g 2 . The dielectric slab on which the device lies has a dielectric constant ε r and a thickness h.

Fig. 2 . 49 :

 249 Fig. 2.49: Screenshot from TXLine illustrating a simulation of a conductor-backed CPW (see cross-section to the top right).In the software, "um" stands for "µm". Material parameters (top left) and lateral dimensions (bottom right) have been chosen so as to obtain a characteristic impedance of about 50 Ω at a working frequency of 10 GHz.

Following

  this first estimate, simulations were performed by Christophe Hoarau with Advanced Design System by Keysight R . The first set of simulations is represented in Figs.2.50 and 2.51. It features the high-frequency behaviour of a 2.5 cm-long conductor-backed CPW without termination, both in reflection for the |S 11 | parameters in Fig.2.50 and in transmission for the |S 12 | parameter in Fig.2.51.

  96 ± 0.01 GHz between the successive peaked maxima in |S 11 | from Fig.2.50, which are at the same position as the minima in Fig.2.51. One finds an electric length associated with the line length L equal to Λ = L/ √ ε r,eff 4.0 mm, then an expected splitting δν (exp) 25mm = c/(2 √ ε r,eff L) 2.

Fig. 2 . 56 :

 256 Fig. 2.56: Simulations by CHRISTOPHE HOARAU on the same geometries as Fig.2.55, with the different curve colours corresponding to the same widths. The graph displays the transmission |S 12 | as a function of frequency. There is almost no influence of the signal electrode width w, apart from very slight resonance frequency shifts and amplitude increases. This resonance corresponds to the dispersive feature in Fig.2.55, while the dip in reflection appears to have no counterpart.

Fig. 2 . 57 :Fig. 2 . 58 :

 257258 Fig. 2.57: Simulations by CHRISTOPHE HOARAU on the conductor-backed geometry from Fig.2.55, with a signal electrode width w of 200 µm. The reflection |S 11 | has been calculated for this simple geometry (blue curve), and with vias (red curve), here 0.5 mm-diameter, copper-filled holes separated by 1 mm along the CPW's length. Their impact here appears small; perhaps the clearest feature is that the dispersive feature at 5.7 GHz seems suppressed by the vias.

5 →Fig. 2 . 60 :

 5260 Fig. 2.60: Simulations by CHRISTOPHE HOARAU on a CPW geometry (without conductor backing) as a function of metallization thickness t. The latter is varied from 1 µm (top curve, in blue) down to 0.5 µm (bottom curve, in dark green) by steps of 100 nm (pink, then cyan, purple, and light green curves, from second-to-topmost curve to forelast one). The behaviour of |S 11 | as a function of frequency is seen to be marginally affected by the thickness. The CPW has a width 650 µm, gap 250 µm, ground plane width 2.5 mm, alumina substrate thickness 1.5 mm with a dielectric permittivity 9.8. The waveguide length is 2.5 cm.

Fig. 2 . 61 :

 261 Fig. 2.61: Schematic true-to-scale representation of a CPW with lateral aspect ratio corresponding to the geometry defined in sections IV.2.a and IV.2.b.Here, the gray bar with width w load to the right is the conductor strip serving as termination impedance to the line.

Fig. 2 . 62 :

 262 Fig. 2.62: Zoom-in on a few impedances from the 101-190 C impedance standard substrate from Cascade Microtech R , adapted from the technical sheet reproduced in Fig.A 2 .3. From left to right: thru line, a pair of short-circuits, and a pair of 50 Ω load.

Fig. 2 . 63 :

 263 Photon energy (keV)

A 2 . 3 Fig. A 2 . 1 :

 2321 Fig. A 2 .1: Schematics of the Faraday rotation

Fig. A 2 . 2 :

 22 Fig. A 2 .2: Schematics of the beam and its multiple reflections inside the GGG sample of thickness d. The successive reflections decrease the electric field amplitude but the Faraday rotation increases in the meantime.

A 2 . 4

 24 High-frequency electronics calibration substrate Impedance Standard Substrate (Pitch: 100 -250 um, Configuration: Ground-Signal-Ground)

Fig. A 2 . 3 :

 23 Fig. A 2 .3: First page of the technical sheet for the 101-190 C impedance standard substrate by Cascade Microtech R . Several precision components for GSG lines are present on the substrate: there are short-circuit, 50 Ω and open loads, as well as thru GSG lines of different lengths. The black A, B, C etc. marks are used for probe calibration.

Fig. 3 . 1 :

 31 Fig. 3.1: Illustration of the principle of FEBID: a flow of precursor gas is injected towards the deposition location by the gas injection system or GIS. Through the action of the focused electron beam, the precursor molecules undergo a chemical reaction which results in the deposition of the desired material (in grey) while the volatile residues of the precursor (in red and yellow) are pumped away. Figure from [18].

  in the plane constituted by the planes of the nanowire, the latter's magnetization -→ M is saturated 3 . Then, upon decrease of the applied field µ 0 -→ H a towards -→ 0 , -→ M falls back onto the local wire axis, following the trajectory shown by the red arrows in the left part of Fig.3.3.

Fig. 3 . 2 :

 32 Fig. 3.2: SEM images of an upright FEBID nanostructure, hereafter denoted F11 1 with bends grown on a Si wafer; the diameters of its different segments along the height are indicated in yellow. The scale bars correspond to the horizontal direction, after correction for the 52 • SEM stage tilt. a) The first image is of the cobalt core only after its growth; its (tilt-corrected) height is about 2.7 µm. b) After the addition of the Pt-rich shell, the structure segments' diameters have increased. They have been labelled in growth order, and those which will be of interest in the XMCD-PEEM imaging are highlighted.

Fig. 3 . 3 :

 33 Fig. 3.3: Schematic side view of nanowire F11 1featuring the nanostructure's magnetization -→ M (blue arrows). Left: upon the application of a horizontal field -→ H a in the bends' plane, -→ M is saturated. Its trajectory to remanence is sketched in red. Right: owing to the bends[25], three domain walls are expected to be nucleated.

µ

  Fig. 3.4: Stoner-Wohlfarth astroid (for the switching field, in green) corresponding to a FEBID nanowire segment. The maximum switching induction is naturally the shape anisotropy induction µ 0 H cyl,Co =0.91 T. The angular range for which the switching induction µ 0 H sw (θ a ) (θ a being the angle between the segment's long axis and the applied field -→ H a ) is below the applied induction µ 0 H a determines the safety margin for vertical misalignment between sample and applied field. This range, coloured in light brown, is significantly larger than the expected experimental errors, and covers almost 90 • because the applied field is about 80% of the shape anisotropy field.

Fig. 3 . 5 :

 35 Fig. 3.5: PEEM view of nanowire F11 1 . The actual image only is what's inside of the golden arc. The structure's foot is denoted by the light blue cross; its shadow (formed by the low X-Ray beam angle) extends until the red arrow. The X-ray beam is represented by a green arrow. From the dark-appearing surroundings of F11 1 out, the shadow first crosses a bright, rounded halo, then an area where the substrate (dark) is covered by a large number of bright dots.

Fig. 3 . 6 :

 36 Fig.3.6: POV-Ray rendering of a scene representing nanowire F11 1 on its substrate (smooth gray), surrounded by the FEBID halo (bright dots). The sample is in vacuum but lit up by the X-Ray beam falling at the experimental incidence angle of 18 • . Hence, the structure's shadow extends far behind it. For consistency, the cobalt core is opaque and bright, while the thin platinum shell is translucent.

Fig. 3

 3 Fig. 3.7: a) Subpart of the PEEM image in Fig.3.7 with a focus on the wire shadow. Its tip is still indicated with a red arrow and the wire foot's approximate position with a light blue cross, while the X-Ray beam direction is denoted by the green arrow. b) Same image as a) but with a different contrast, allowing the shadow region close to the tip to be more visible. c) Intensity profiles (averaged laterally over ca. 400 nm) corresponding to the dashed red line in a) (dashed red curve) and the full black line in b) (full black curve). Both have been offset so the dip's center (corresponding to the wire shadow) are at zero abscissa, and scaled so that the shadow's amplitudes match. The criterion to not be biased by the non-uniform background was to reach the first vanishing derivative of the PEEM intensity with respect to the abscissa. This procedure allows to compare the profile shapes, in particular their widths, irrespective of their amplitudes.

Fig. 3 . 8 :

 38 Fig. 3.8: Computed transparency right behind the core-shell nanowire.Here, incidence perpendicular to the wire axis has been assumed for simplicity. The assumed purities of cobalt and platinum are 70% and 20% respectively, the absorption coefficients relevant to the Co L 3 edge were derived from the works of Nakajima[34] and Henke[41]. The wire is a perfectly cylindrical cobalt core with 36 nm radius, and a 6.5 nm thick platinum-containing shell. Both parts are indicated with shaded areas. The shell's influence appears misleadingly strong because just outside the cobalt core, the beam traverses only platinum, but over a distance much larger (here ca. 45 nm) than the 13 nm diameter increase that was mentioned on page 147.

Fig. 3 . 9 :

 39 Fig. 3.9: SEM images of nanowire F11 1 after the beamtime, with red scale bars corresponding to the horizontal direction after correction for the stage tilt, if present. a) Image with the in-lens detector at 45 • stage tilt, line of sight "perpendicular" to the bends. The nanostructure clearly appears bent with respect to the initial images presented in Fig.3.2. b) In-lens view from the top, with untilted stage. c) Secondary electrons image with line of sight along the bends and 45 • tilted stage.

Fig. 3 . 11 :

 311 Fig. 3.11: SEM image (with in-lens detector) of nanowire F11 1 after the beamtime. Clots of matter which were absent before the beamtime are highlighted with red dashed lines.

Fig. 3 . 12 :

 312 Fig. 3.12: PEEM images of nanowire F11 1 at different energies, with a field of view of 15 µm. All images have the same intensity range so as to allow for comparison. The target photon energies, corresponding to the labels in Fig.3.13 are a) 767.00 eV b) 767.50 eV c) 768.00 eV d) 768.50 eV e) 769.00 eV f) 769.50 eV g) 770.00 eV h) 770.50 eV i) 771.00 eV j) 772.00 eV k) 775.00 eV

  y) -I CR (x, y) I CL (x, y) + I CR (x, y) .

Fig. 3 . 14 :

 314 Fig. 3.14: Full-field XMCD-PEEM view of nanowire F11 1 , with a target photon energy of 768.65 eV. The coarse, noisy area to the top right is again the vicinity of the nanowire foot. The shadow extends once more towards the bottom left, yet only two distinct parts of it are visible: a stripe that is first bright then dark between the two blue crosses, and just beyond the second one there is a faint dark plume. The crosses indicate the extremities of the segment from which the contrast profiles in Fig.3.15 were extracted.

Fig. 3 .

 3 Fig. 3.16: a) True-to-scale schematic representation of the X-ray beam (in red) falling onto the nanowire; the scale length and axes are as in Fig.3.16.b). The rays passing through junctions between segments delimit the shadow sections corresponding to segments (4) through(7). b) Shadow XMCD-PEEM image of the nanowire with dashed red lines delimiting these segments according to a). The domains with uniform contrast correspond to the segments (4), (5) and (7) from Fig.3.2.b). The two black markers above segments (4) and (5) indicate the lateral positions of the contrast profiles in Fig.3.16.c). The transition region between segments (4) and (5) is where a domain wall was pinned. The dashed blue line indicates the structure's foot position according to a). c) Contrast profiles across the shadow in domains (5) (in purple) and (4) (green), along with their best gaussian fits in dashed lines. The profiles were averaged laterally over ca. 400 nm for the sake of statistics. The linear gray scale goes from -2.5% to +2.5%.To the right, the noise is slightly stronger where the structure stands because of the field emission.

Tab. 3 . 2 :APPENDICES FOR CHAPTER 3 A 3 . 1

 32331 Pt coating \ Counts of. . . # Surviving structures # Numbers of FEBID nanostructures which were still/no longer standing when the sample was first imaged with PEEM. The presence/absence of FEBID platinum coating[17] is indicated. Upright hook FEBID nanowires For the November 2015 beam time at synchrotron SOLEIL, JAVIER PABLO-NAVARRO, JOSÉ MARÍA DE TERESA and CÉSAR MAGÉN fabricated over 200 upright nanostructures, an overview of which is presented in Fig.A 3 .1. In these SEM images, only the wires in Fig.A 3 .1.a) and c) are platinum-coated Co-core nanowires; all others are uncoated. The Pt coating represents a diameter increase of about 13 nm. As can be seen by comparing e.g. Fig.A 3 .1.a) and Fig.A 3 .

Fig. A 3 . 1 :

 31 Fig. A 3 .1: Overview of the upright FEBID nanowires which have been grown by JAVIER PABLO-NAVARRO, JOSÉ MARÍA DE TERESA and CÉSAR MAGÉN at the Universidad de Zaragoza (Zaragoza, Spain) as potential samples for the beam time at SOLEIL. These SEM images are their courtesy's. The height indications have been corrected for the SEM stage tilt, and the scale bars correspond to the horizontal direction. Nanowires a) and c) are Co@Pt coreshell structures, all others are constituted of a FEBID Co core only.As can be seen by comparing these different structures, quite a variety of segment lengths, diameters and orientations can be achieved. It must be reminded however that the FEBID growth process depends on many different parameters, the local temperature not being the least of these. As a result, fine and reproducible geometry tuning is not as straightforward as it may seem at first glance.

Fig. A 3 . 2 :

 32 Photoelectrons

Fig. A 3 . 3 :Fig. A 3 . 4 :

 3334 Fig. A 3 .3: Graph representing the bulk thermal diffusivities of cobalt and platinum[52], as well as amorphous carbon[58].

Fig. 4 . 1 :Fig. 4 . 2 :

 4142 Fig. 4.1: SEM image (in-lens detector) of the steep diameter modulation produced in a nanoporous alumina template by the Erlangen process. Target diameters were D thin =150 nm and D thick =250 nm. The indicated error is derived from the finite-thickness display of the available scalebar in the original image. It must be kept in mind that the template cutting process leads to some pores being only partially imaged (see Fig.2.4 p.61). Therefore, the indicated diameter values are only lower bounds on the actual dimensions. Image credit: RAJA AFID.

Fig. 4 . 3 :

 43 Fig. 4.3: SEM image (in-lens detector) of dispersed Ni 60 Co 40 nanowires produced at the University of Erlangen in the above-mentioned nanoporous templates with reduced amplitude of diameter modulations. Target diameters were D thin =150 nm and D thick =200 nm. The indicated error is derived from the finite-thickness display of the available scalebar in the original image as well as the image's resolution. Image credit: RAJA AFID.

Fig. 4 . 4 :

 44 Fig. 4.4: SEM images of nanowires electrodeposited into nanoporous AAO templates from Erlangen. a), b), d) are images of the same NiFe nanowire; it lies on a copper substrate, and the objects around the nanostructure are gold clusters. a) and b) were acquired with in-lens detector, d) with secondary electrons detector. Image credit for these: JEAN-FRANÇOIS MOTTE. c) is a high-resolution, in-lens image by MICHAL STA ŇO of a NiCo nanowire such as were investigated in the 2015 beamtime at Elettra. Except for d), the SEM views share the same scale for comparison. Although the resolution in a), b) and d) is arguably lower than in c), features as small as ca. 13 nm in b) respectively ca. 29 nm in a) can be distinguished, while the lateral scale of the rough features in c) is 20 nm to 30 nm.

Fig. 4 . 5 :

 45 Fig. 4.5: SEM views (in-lens detector) of Ni 60 Co 40 nanowires from the 2015 beamtime. They feature a surface roughness present on most structures from this sample batch.

Fig. 4 . 6 :

 46 Fig. 4.6: Optical microscope view of a dispersion of nanowires before sample mounting on the PEEM cartridge. The yellow and blue arrow represent the directions of applied field and X-ray beam propagation respectively. To the right, two gold alignment marks can be spotted.

Fig. 4 . 7 :

 47 Fig. 4.7: SEM image (in-lens detector) of a NiFe nanowire on a silicon substrate and a tungsten micromanipulating tip in the process of ungluing the nanostructure. The green arrow indicates the direction of tip motion, and the orange dashed lines the outline of this thin wire's section before the beginning of the micromanpulation. This mechanical pushing step is necessary because the contact area between substrate and wire is initially too large to allow lifting off the full object. Image credit: JEAN-FRANÇOIS MOTTE.

Fig. 4 . 8 :

 48 Fig. 4.8: SEM images (in-lens detector) of a Ni 60 Co 40 nanowire on a silicon substrate (bottom) and a tungsten micromanipulating tip nearby, hovering just over the substrate surface. Image credit: JEAN-FRANÇOIS MOTTE. a) Approaching stage, nanowire so far untouched. b) After a failed attempt to detach the wire from the substrate, the wire is broken. Remains of the "gangue" surrounding it are quite visible.

Fig. 4 . 9 :

 49 Fig. 4.9: SEM view (in-lens detector) of a successfully lifted-off NiFe nanowire, glued to the tungsten micromanipulating tip.The latter has been sufficiently retracted with respect to the position close to the substrate that the silicon wafer's surface is quite blurred in the background. Image credit: JEAN-FRANÇOIS MOTTE

Fig. 4 . 10 :

 410 Fig. 4.10: SEM view (in-lens detector) of a Ni 1-x Fe x (x ∼ 0.3) nanowire that has been successfully transferred to the CPW surface. Part of the micromanipulating tip is visible to the top left corner, and the numerous, rather bright round features are gold clusters. In order to maintain the wire more or less horizontal, it was necessary to glue it by way of several carbon deposits (indicated by the blue arrows) extending onto the substrate. Image credit: JEAN-FRANÇOIS MOTTE.

Fig. 4 . 11 :

 411 Fig. 4.11: PEEM images at the Fe L 3 edge of a trisegmented NiFe nanowire from the first batch i.e. with estimated composition Ni 90 Fe 10 . Top: saturated view (toward high photoelectron counts) allowing to see the full shadow of the thin wire's section. Bottom: unsaturated view with the middle section's shadow outlined in orange. The beam's almost perpendicular incidence (with respect to the wire axis, in blue) reveals the fact that the middle section does not touch the substrate.

Fig. 4 . 12 :

 412 Fig. 4.12: Composite PEEM view of the NiFe nanowire D12 from Fig.4.10, at the Fe L 3 edge. Most of the shadow is well-separated from the direct wire image, indicating a significant height offset with respect to the CPW's copper surface. The numerous small features correspond to the above-mentioned gold clusters (see Fig.4.10). Dead pixels from the camera as well as image parts outside the actual field of view are encompassed in red. The positions of the wire's two diameter modulations are indicated with the green arrows on the structure itself and in the shadow. In certain instances, the carbon patches maintaining the wire in position feature a dark halo corresponding to insulating material deposited away from the electron beam spot: this is in essence the same type of halo as discussed with FEBID in Chapter 3, see section III.2.b.

Fig. 4 . 13 :

 413 Fig. 4.13: XAS from the same nanowire within two energy windows: around the Fe L 3 (left) and Fe L 2 (right) absorptionedges. Both graphs feature data acquired from the wire surface (red) and from the shadow (black). An equivalent representation of the latter (dashed blue curve) is a vertically mirrored then offset spectrum that allows more direct comparison with the wire XAS. At both edges, the maximum of absorption in the shadow lies 1.9 ± 0.4 eV lower in energy than the maximum of photoemission on the wire, suggesting surface oxidation.

Fig. 4 . 14 :

 414 Fig. 4.14: XAS around the Ni L 3 absorption edge from the same nanowire as Fig.4.13. Likewise, the graph features dataacquired from the wire surface (red) and from the shadow (black). An equivalent representation of the shadow XAS (dashed blue curve) is a vertically mirrored then offset spectrum that allows more direct comparison with the wire XAS. This time, the maximum of absorption in the shadow and the maximum of photoemission on the wire are quite close. Conclusions about oxidation are harder to draw because of specific spectrum shape corresponding to nickel[29]. The apparent spacing is about 0.2 eV.

Fig. 4 . 15 :

 415 Fig. 4.15: XAS around the Co L 3 absorption edge from a Ni 60 Co 40 nanowire. Like in the previous spectra, the graph featuresdata acquired from the wire surface (red) and from the shadow (black). An equivalent representation of the shadow XAS (dashed blue curve) is a vertically mirrored then offset spectrum that allows more direct comparison with the wire XAS. The close-to-flat edge peak (over 2 eV), the plausible shoulder at 783 eV and to some extent the photoemission levels around 785 eV compared to pre-edge levels suggest that cobalt on the wire surface may be partially oxidized[29].

Fig. 4 . 16 :

 416 Fig. 4.16: XAS around the Co L 3 absorption edge from another Ni 60 Co 40 nanowire. Like in the previous spectra, the graphfeatures data acquired from the wire surface (red) and from the shadow (black). An equivalent representation of the shadow XAS (dashed blue curve) is a vertically mirrored then offset spectrum that allows more direct comparison with the wire XAS. In this case, the less flat (with respect to Fig.4.15) peak from wire XAS and slightly higher edge tail photoemission compared to the pre-edge values speak for a less oxidized material[29].

Fig. 4 . 17 :

 417 Fig. 4.17: Left: PEEM images at different energies around the Co L 3 absorption edge on a Ni 60 Co 40 nanowire. The top onewas acquired at the maximum of shadow absorption, while the bottom one was taken (with much lower integration time) at a pre-edge photon energy E rim such that there is a faint but visible dark rim at the shadow's border, see purple arrows. Right: wire XAS acquired on the thin middle section. The photon energy E rim is indicated; the associated spectral feature is reminiscent of cobalt oxide XAS[29] although it lies much lower in energy than the shoulder known in CoO. This suggests that the rim seen in PEEM corresponds to localized oxidation.

Fig. 4 . 18 :

 418 Fig.4.18: XAS around the Fe L 3 absorption edge at different stages. The blue, green and red full curves correspond to the virgin sample state, the state after two minutes of Ar sputtering, and the state after an additional five minutes of sputtering. This wire surface XAS is accompanied by shadow XAS processed as described previously: these data are the full black and dashed brown (vertically mirrored, then offset) curves. The surface wire spectra have been normalized to the same peak amplitude, which corresponds to iron oxide as demonstrated by the offset of this peak with respect to the maximum in shadow absorption. The latter's position hints at a shoulder in the wire surface XAS, whose relative amplitude (with respect to the peak value) appears to increase with the sputtering time. Here, the shadow spectrum corresponds to the state after one sputtering round. The two wire XAS after that round were acquired on the same wire.

Fig. 4 . 19 :

 419 Fig.4.19: Shadow XAS recorded in the shadow of the thin (red curve) and thick (black curve) sections of a nanowire. No offset or rescaling were performed, only the mandatory normalizations. The spectrum from the thick section's shadow clearly shows saturation and appears therefore flatter than its counterpart from the thin wire section. To better emphasize this, the dashed purple curve shows the XAS from the thick section's shadow after multiplication by a factor 2.15. In the absence of saturation, the intensities transmitted through wire sections of different diameter should collapse onto one another with a simple multiplicative factor, which is not the case at all energies.

Fig. 4 . 20 :

 420 Fig.4.20: Top row: PEEM images at the Fe L 3 edge of two nanowires, each the sum of two images acquired with circularly left resp. right polarized light. Low-amplitude fringes can be seen below the bottom of especially the thick wire part. NB: on the right hand side, the microscope's settings and the image intensity range are such that the wire itself is not visible. The field of view is 6 µm wide to the left, and 10 µm wide to the right. The beam direction is indicated in orange. Middle row: excerpts of intensity profiles extracted along the segment between red markers (left) or along the full then dashed line (right). After the first maximum, the oscillations are very weak compared to the shadow's amplitude. Bottom row: zoom-in on the fringes themselves. The second maximum (outside the shadow) at most can be discerned.

Fig. 4 .

 4 Fig. 4.21: a-b): PEEM images at the Ni L 3 edge of two nanowires, each the sum of two images acquired with circularly left resp. right polarized light. These images only differ in their intensity ranges. Low-amplitude fringes can be seen in b). The scale bars are 1 µm, and the beam propagation direction is perpendicular to the nanowire. More interference orders than in Fig.4.20 can be seen there, as further shown in c-d), which show intensity profiles extracted perpendicular to these fringes above the thin section in c) and the thick section in d).

Fig. 4 . 22 :

 422 Fig. 4.22: PEEM images of nanowire Cicero a) at the Fe L 3 edge and b) at the Fe L 2 edge. The microscope foci in a) and b)are the silicon surface and the wire's top surface, respectively. While undesirable features such as a break in the middle section and at least one notch (revealed by the shadow) on the top side of this thin section, no significant changes in wire diameter can be seen here.

Fig. 4 .

 4 Fig. 4.23: a-b) PEEM images of nanowire Cicero at the Fe L 3 edge, obtained as sum of a CL and CR image. They are both the same, only the intensity ranges are different. In a), both nanostructure and shadow are visible, while in b) only part of the thin section's shadow is visible. A slight increase in intensity can be seen from the left to the right.The scale bars are 1 µm, and the X-ray beam's direction is shown by the orange arrow. In order to quantify these changes, an intensity profile was extracted along the broken red line shown in a). This data is shown on the graph in c). The red curve is the extracted intensity, and the black curves define a 3% window around the profile based on the variations of intensity on the Si wafer just behind the shadow's rim.

Fig. 4 .

 4 Fig. 4.24: a) PEEM image at the Ni L 3 edge of the micromanipulated NiFe nanowire D12. Due to its height above the substrate, the nanostructure itself is not clearly visible. Its out-of-focus image is outlined in green. The X-ray beam is perpendicular to the wire axis, see yellow arrow. Aside from numerous 3D gold impurities on the substrate, one can easily see a very abrupt change in shadow intensity (see blue arrow). In order to gain better insight on these changes attributed to composition variations, the image in Fig.4.24.b) was acquired. b) Difference of a PEEM image at the Fe L 3 edge and another one at the Ni L 3 edge, both acquired with linear polarization. This difference has been normalized by the sum of these images, just like a regular XMCD image. Here, the varying contrasts reflect not magnetism but sample composition. The linear contrast scale goes from -0.5 to 0.15. The red arrows in a-b) indicate the position of this wire's pinning site associated to domain wall nucleation.

Fig. 4 . 25 :

 425 Fig. 4.25: Schematic representation of a nanowire with two diameter modulations and only one pinning site in its thin section;the diameter modulations have negligible pinning strengths here. The top (bottom) row corresponds to the case where the pinning site's strength H p,1 (H p,2 ) is lower (higher) than the nucleation field H nucl . Initially (left column), the applied field H a is zero and the wire is uniformly magnetized to the left (red arrows). When a field (in green) pointing to the right is applied and reaches the nucleation field H nucl (right column), a domain wall is nucleated, here from the left, and propagates. If the pinning field is lower than H nucl = H a (top row), the domain wall propagates on and annihilates at the opposite wire tip. To the contrary, if the pinning field is higher than H nucl = H a (bottom row), the domain wall is pinned and remains so as the applied field is ramped down to zero.

Fig. 4 . 26 :

 426 Fig. 4.26: PEEM image of nanowire Cicero at the Fe L 3 edge. The microscope focus is on the silicon surface. The beam direction is given by the blue arrow.

  Fig.4.27 illustrates the image analysis on the example of failure to nucleate a domain wall [Fig.4.27.a-b)] then successful nucleation [Fig.4.27.a-b)] in nanowire Cicero with shadow XMCD-PEEM images at the Fe L 3 edge. Thanks to the beam's partial alignment with the wire axis, magnetic contrast is observed in the shadow. In this configuration, the domain wall is also visible thanks to the beam component that is transverse to the wire axis.

Fig. 4 . 27 :Fig. 4 . 28 :

 427428 Fig. 4.27: Shadow XMCD-PEEM images of nanowire Cicero at the Fe L 3 edge. The focus is the silicon surface; with the start voltage (STV) settings, the wire itself is close to invisible. The X-ray beam direction is indicated in green. Thanks to its component along the wire axis, XMCD contrast from the domains is retrieved, in addition to domain wall contrast associated to the beam component transverse to the wire. Left column: images before [a)] and after [b)] the application of an induction with longitudinal component µ 0 H a =13 mT to 18 mT. Since the shadow contrast is unchanged, no domain wall nucleation occurred, so the value of s associated to this application of field is zero. Right column: images before [c)] and after [d)] the application of an induction with longitudinal component µ 0 H a =14 mT to 19 mT. A domain wall has been nucleated, therefore the s associated to this application of field is one. 1.0 0.8 0.6 0.4 0.2 0.0 Nucleation proba.

Fig. 4 . 29 :

 429 Fig. 4.29: Estimated depinning probalities in nanowire Cicero for a given pinning site (referred to as wholesome later on) a rightward domain wall propagation, as a function of maximum applied field. The bin size used was 1 mT.

  30 µs. As a side note, the composition variations in NiFe nanowire D12 dealt with in section II.4, albeit unplanned and uncontrolled, may have demonstrated an interesting potential in terms of asymmetric domain wall pinning. Images of this nanowire are shown in Fig.4.30 to highlight this. Fig.4.30.a) is a PEEM image without magnetic contrast at the Ni L 3 edge, Fig.4.30.b) is the composition contrast image from Fig.4.24 p.206, and Fig.4.30.c) is a shadow XMCD-PEEM image showing a domain wall (encompassed in orange) pinned at the level of the abrupt 20 composition change.

Fig. 4 . 30 :

 430 Fig. 4.30: Images of the micromanipulated NiFe nanowire D12 with the X-ray beam (purple arrow) perpendicular to the wire axis. a) PEEM image at the Ni L 3 edge. Due to its height above the substrate, the nanostructure itself is not clearly visible. Its out-of-focus image is outline in green. Aside from numerous 3D gold impurities on the substrate, one can easily see a very abrupt change in shadow intensity (see blue arrow). In order to gain better insight on these changes attributed to composition variations, the image in Fig.4.24.b) was acquired. b) Difference of a PEEM image at the Fe L 3 edge and another one at the Ni L 3 edge, both acquired with linear polarization. This difference has been normalized by the sum of these images, just like a regular XMCD image. Here, the varying contrasts reflect not magnetism but sample composition. The linear contrast scale goes from -0.5 to 0.15. The red arrows in a-b) indicate the position of this wire's pinning site associated to domain wall nucleation. This pinning site is highlighted in c), which is a shadow XMCD-PEEM image featuring a domain wall (encompassed in green).From this pinning site, domain wall depinning towards the right (see yellow arrow) occurred under inductions of (11 ± 1) mT to (15 ± 1) mT. By contrast, nucleation from the right side towards the left, followed by pinning at this site, was repeatedly obtained under applied inductions of 28 ± 1 mT. There is therefore an easy depinning direction, shown by the yellow arrow. Away from the domain wall itself and especially to the left of the blue arrow, an orthoradial component of magnetization in a single domain is revealed by the bipolar contrast pattern. If magnetization were fully along the wire axis, no contrast would be seen due to the orthogonality between wire axis and beam.

Fig. 4 .

 4 Fig. 4.34: a1-e2) XMCD-PEEM views at the Fe L 3 edge of NiFe nanowire Cicero before [a1), b1) etc.] and after [a2), b2)etc.] the application of a magnetic field. The image parts lying outside the actual field of view are encompassed in red. The X-ray beam's propagation direction is indicated by the purple arrows. The microscope focus is on the wire shadow and the start voltage is such that the nanostructure is itself invisible. These images illustrate the variety of nucleation and pinning events observed on this nanowire: both domain wall polarities as well as distinct internal configurations are displayed. The three pinning sites observed in this wire are labelled A, B, and C (green crosses), see a1). In a2), b2), and c2), the direction of magnetization as well as the orientation of the BPW's winding are indicated in red. Regions of the image lying outside the actual field of view are encompassed in red. f) PEEM view of nanowire Cicero at the Fe L 3 edge.

Fig. 4 . 35 :

 435 Fig. 4.35: Shadow XMCD-PEEM images of nanowire D12 at the Fe L 3 edge. Due to the microscope settings and wire height above the substrate, only the shadow is visible. The linear contrast gray scale goes from -7.5% to 7.5%. The beam incidence angle is indicated with the green arrow. The configuration of domains and BPW winding are indicated in red. Both domain walls were nucleated from the top of the image downwards, as indicated by the vector v in blue. On the right hand side image, the abrupt diameter modulation is highlighted in green. Also, this wire's two pinning sites labelled D and U are indicated in yellow.

Fig. 4 . 36 :

 436 Fig. 4.36: Shadow XMCD-PEEM views of nanowire D12 at the Fe L 3 edge before (top) and after (middle) the application of a magnetic field. The linear contrast gray scale goes from -12% to 12%, and the red scale bar corresponds to 1 µm. The PEEM image (bottom, sum of CL and CR images) illustrates the shadow without magnetic contrast and reveals better the abrupt diameter modulation, outlined with an orange dashed line. In this case, the beam is perpendicular to the wire axis.

Fig. 4 .Fig. 4 . 37 :

 4437 Fig. 4.37: Shadow XMCD-PEEM views of a Ni 60 Co 40 nanowire at the Co L 3 edge before (a) and after (b-d) the application of a magnetic field. The PEEM image (c, sum of a CL and a CR image) illustrates both wire and shadow without magnetic contrast. The full blue line in a-c) outlines the shadow close and the abrupt diameter modulation. The X-ray beam direction is indicated in yellow. Thanks to this incidence, contrast from the domains and the domain walls is retrieved. The domain wall contrast is weak and poorly resolved, but even with a smaller field of view and perpendicular beam incidence as in d), the BPW's bipolar contrast pattern (highlighted in red and purple) in the shadow (outlined in dark blue) is weak, a couple percents in amplitude only. d) was obtained after b) without any application of field in-between.

Fig. 4 . 38 :

 438 Fig. 4.38: Shadow XMCD-PEEM views of a Ni 60 Co 40 nanowire at the Co L 2 edge before (top) and after (middle) the application of a magnetic field. The PEEM image (bottom, sum of a CL and a CR image) illustrates both wire and shadow without magnetic contrast. A pair of orange dashed lines outline the shadow close to the (abrupt) diameter modulation. The red scale bar corresponds to 1 µm, and the X-ray beam direction is indicated in yellow. Thanks to this incidence, contrast from the domains and the domain walls is retrieved. The red and green dashed lines encompass the areas in the vicinity of the domain walls with negative and positive contrast, respectively.

  e. from BPW to TVW has also been observed in a Ni 60 Co 40 nanowire. The first part of this domain wall's evolution is shown in Fig.4.39. Motion over a short distance [Fig.4.39.a-b)] under field larger than 20 mT was first achieved; it yielded a negative helicity. The PEEM image in Fig.Fig.4.39.c) shows that the observed sample actually consists in two neighbouring nanowires; it is in the top one that a domain wall was observed (see orange arrows). Further imaging at different foci also revealed that this wire of interest was bent close to the smooth diameter modulation; the bend point is indicated by the blue arrows.

Fig. 4 .Fig. 4 .Fig. 4 . 41 :

 44441 Fig. 4.39: (Shadow XMCD-)PEEM views of a pair of adjacent Ni 60 Co 40 nanowires at the Co L 3 edge before (a) and after (b) the application of a magnetic field. The PEEM image [c), sum of a CL and a CR image] illustrates both the wires and shadows without magnetic contrast. The red scale bars correspond to 2 µm, and the X-ray beam is here perpendicular to the nanowires. It turned out that the top wire featuring a BPW was bent upwards at the location indicated by the blue arrows, before the smooth diameter modulation.

Fig. 4 .

 4 41 displays four consecutive shadow XMCD-PEEM images of nanowire D12 illustrating this. Similarly to Fig.4.32, in all images of Fig.4.41, only the shadow (and not the wire itself) is visible. The respective magnetization orientations in the domains are highlighted by black arrows, while red markers indicate the position of the diameter modulation close to which DWs could be reproducibly pinned. In Fig.4.41, between images a) and b) on one hand and between images c) and d) on the other hand, quasistatic magnetic fields were applied with the above-mentioned electromagnet for domain wall displacement. In both cases, the domain wall was pushed to the pinning site associated to nucleation, however, the equilibrium configuration is starkly different. The contrast patterns in Fig.4.41.b) and Fig.4.41.d

Fig. 4 . 42 :Tab. 4 . 6 :

 44246 Fig. 4.42: Sequence of shadow XMCD-PEEM views of the NiFe nanowire D12 featuring a domain wall in its thin section; both subsequences a)-b) and c)-d) start with a BPW close to the diameter modulation (highlighted with a red marker) and end after the application of a quasistatic field. The X-ray beam direction is indicated in blue. a) The field strength is µ 0 H m = 11 ± 2 mT and b) the BPW is only displaced. c) The field strength is µ 0 H t = 24 ± 1 mT and d) the BPW has transformed into a TVW.

Fig. 4 . 43 :

 443 Fig. 4.43: Table representing a vortex with upward polarity, an antivortex with downward polarity, and a Bloch point's vicinity(as seen on a sphere surrounding the singularity) in real space in the first row, and correspondingly on the unit sphere in the second row. The Bloch point illustration was taken from[8] as the associated configuration is representative of a Bloch point in a BPW.

  1.16.d-e) p.24] on both sides subsequently. It is important to note that this separation into two distinct objects is tricky since the gyrovector is normally defined on the whole configuration. Still, the above holds if one considers only the vicinity of the vortex and antivortex cores. This being said, these considerations only explain the behaviour of the vortex and antivortex at the very beginning of the simulation: what follows cannot be grasped by such simple arguments.

Fig. 4 . 44 :

 444 Fig. 4.44: Simulated tail-to-tail TVW configuration in a 70 nmdiameter wire, magnetization being represented as cones (scale bar 50 nm). a) TVW's surface magnetization around the wire (green), with the outwardpointing normal magnetization -→ m • n as colour code. b-c) Slices through the TVW configuration, with m z resp. m x as colour code.

Fig. 4 . 45 :

 445 Fig. 4.45: Views of the surface magnetization of the aforementioned TVW subjected to a field of 8.2 mT along its axis. The scale bars are 50 nm, and the -→ m • n contrast scale goes from -1 to +1. The three green lines are the level curves of magnetization's longitudinal component corresponding to m z = -0.5, 0 and 0.5. With respect to Fig.4.44, the camera has been rotated around the z-axis by 45 • . a) At t = 0, the field has been applied but the domain wall has not yet moved. b) At t 1 =0.45 ns, vortex and antivortex are now very close to one another c) At t 2 =0.65 ns, the two objects have vanished and the extremal values of -→ m • n have significantly dropped. Moreover, the m z level curves have drawn significantly farther apart from one another.

Fig. 4 . 46 :

 446 Fig. 4.46: Time evolution of the differences in longitudinal and angular positions of the vortex ("V")and antivortex ("AV"), ∆z(t) = z V (t)z AV (t) and ∆φ(t) = φ V (t)φ AV (t). Their values at the time step t c such that the Bloch point is injected in the nanowire are indicated. The red and black crosses indicate the Bloch point's first detected position at time step t B > t c , shown here with respect to the antivortex' position. As can be seen, both indicate a clear proximity to the coordinates (on the wire surface) at which the vortex and antivortex have merged. The first detected Bloch point position with respect to the antivortex is shown with the red (∆z BP ) and green crosses (∆φ BP ).

Fig. 4 . 47 :

 447 Fig. 4.47: Graphs pertaining to the TVW's evolution under an induction of 8.2 mT. The top graph is a trace of the winding number N sim (t) with the aforementioned jump around t c , the middle one shows the estimated domain wall velocity versus time, and correspondingly the bottom one displays the estimated domain wall position z DW (t). The plateau on this graph corresponds more or less to the time interval during which vortex and antivortex are very close to one another.

Fig. 4 . 48 :

 448 Fig. 4.48: Snapshot of the simulated TVW under 8.1 mT at t steady =14.85 ns. The vortex and antivortex are still whole;viewing the whole sequence of micromagnetic configurations indicates that after a transitory regime, the state displayed here is representative of all others except of the angular position around the wire axis. The pair of object appears to precess as a whole around that axis at a well-defined angular velocity while the (distorted) TVW propagates on.

Fig. 4 . 49 :

 449 Fig.4.49: Graphs pertaining to the TVW's evolution under an induction of 8.1 mT. The top graph is a trace of the magnetization's components perpendicular to the wire axis m x (in red), m y (in blue) averaged over the whole simulated volume. They are seen to oscillate with π/2 phase difference, as is expected for a magnetic moment perpendicular to the wire rotating at a constant angular velocity. Fourier analysis on these functions yields a very-well defined peak with an oscillation frequency ca. 0.22 GHz. The middle graph shows the estimated domain wall velocity versus time, and correspondingly the bottom one displays the estimated domain wall position z DW (t). In this case, although there is also a plateau in position, the dynamics does not lead to a disappearance of vortex and antivortex. Intead, the TVW conserves its type and propagates slowly (with respect to the estimated "steady-state" velocity from Fig.4.47. All these features are in qualitative agreement with the known behaviour of this type of domain wall under field[6, 7], although it is here markerdly distorted.

Fig. 4 . 50 :

 450 Fig.4.50: Graphs pertaining to the TVW's evolution under an induction of 8.1 mT (top) and 8.2 mT (bottom). The abrupt threshold is illustrated here by the stark difference in time evolution of the position, which in one case proceeds at a steady mean velocity with some oscillations, and in the other case abruptly accelerates.

Fig. A 4 . 1 :

 41 Fig. A 4 .1: Subsequent SEM views featuring the micromanipulation of a NiFe nanowire. Left: lifted-off nanowire with one of its tips brought back into contact with the silicon substrate. Right: a tip displacement to the left and towards the surface has induced a significant bend in the wire owing to the two anchor points on the wafer and on the tungsten tip. Image credit: JEAN-FRANÇOIS MOTTE.

A 4 . 2

 42 Vortex and antivortex at rest in the TVW

Fig.A 4

 4 Fig.A 4 .2 displays the equilibrium micromagnetic configuration of a TVW in a 70 nm-diameter wire with Permalloy material parameters. The first view in Fig.A 4 .2.a) is a cut containing the TVW's transverse moment, highlighted by the colour scale. In Figs.A 4 .2.b-c), only the surface magnetization is shown, with -→ m • n as colour code. n is the local, outward-pointing unit vector normal to the wire surface.

Fig. A 4 . 2 :

 42 Fig. A 4 .2: Views of the simulated equilibrium TVW configuration in a 70 nm-diameter wire with Permalloy material parameters. n is the local outward-pointing unit vector normal to the wire surface. The scale bar is 50 nm. a) Vertical cut highlighting the downward-pointing domain wall core (in blue). b-c) Views of magnetization on the wire surface, with the normal magnetization component -→ m • n as colour code. The two views are of the same object, only along opposite lines of sight.

A 4 . 3

 43 Composition contrastThe contrast in Fig.4.24.b) p.206 reads:C(x) = I 1 e -µ(1)Ni (x)+µ

( 1 )

 1 Fe (x) L -I 2 e -µ (2) Ni (x)+µ

( 1 )

 1 Fe (x) L + I 2 e -µ (2) Ni (x)+µ

( 2 )

 2 Fe (x) L (A 4 .1)where µ(i) Ni (x) = n Ni (x)µ 0 Ni (E i ) and µ (i) Fe (x) = n Fe (x)µ 0Fe (E i ) are the linear absorption coefficients of iron and nickel in the nanowire at position x for a photon energyE i ; E 1 is Fe L 3 , E 2 is Ni L 3 . They depend on

  . 2.23: Measured electromotive force U loop (t) (red) induced in the pick-up loop during a pulse sent with MÁRLIO BON-FIM's pulse generator, the load voltage being 552 V. Numerical integration of this signal leads to the opposite of the magnetic flux through the loop, Φ loop (t). -Φ loop (t), represented in blue, is proportional to the applied field H pulse (t). As can be seen, it does not fall to zero after the pulse, meaning that a low current is still present after the pulse. The appearance of this current follows the spike on U

loop (t), indicated with an arrow. The extrema V 1 and V 2 of U loop (t), as well as the extremum Φ M,m.c. of Φ m.c. (t), are also indicated. measurements of the electromotive forces in both the pick-up loop and the measuring coil, sweeping the capacitor load voltage U capa . After numerical integration and extraction of the extremal fluxes, I have plotted Φ M,p.u. versus Φ M,m.c. in Fig.

2

.24.

  Schematic representation of an infinitesimal transmission line slice, of length dz. One electrode is grounded and serves as reference for the voltage v on the signal electrode. The voltage is different after the line slice because of the several elements defining the slice's properties: a resistance per unit length R, a self-inductance per unit length L, and a shunt conductance per unit length G as well as a capacitance per unit length C from signal to ground. The current i through the signal line is also affected by the quadrupole in dz.

			dz
	i(z, t)	Rdz	Ldz	i(z + dz, t)
	v(z, t)	Gdz	Cdz	v(z + dz, t)
			z
	Fig. 2.46:

  12 |. Simulations by CHRISTOPHE HOARAU on a conductor-backed copper CPW with 25 mm length, gap g =0.250 mm, dielectric slab thickness h =300 mm, as a function of the frequency. The different curves correspond to different signal electrode widths w. Here, the plotted parameter |S 11 | characterizes the reflection behaviour of the transmission line. The simulation assumes perfect 50 Ω-matched connections at both line ends, so that the |S 11 | truly reflects the intrinsic line properties. In terms of impedance matching, w =300 µm appears to be the optimal width: it corresponds to the lowest |S 11 | towards low frequencies, and the lowest average value (-20.9 dB vs. -20.1 dB for w =250 µm) over the whole simulated frequency range. Simulations by CHRISTOPHE HOARAU on the same geometries as Fig.2.50, with the different curve colours corresponding to the same widths. The graph displays the transmission |S 12 | as a function of frequency. There is almost no influence of the signal electrode width w, apart from very slight resonance frequency shifts and amplitude increases (see inset). Simulations by CHRISTOPHE HOARAU on the conductor-backed geometry from Fig.2.50, with a signal electrode width w of 300 µm. The reflection |S 11 | has been calculated for this simple geometry (blue curve), and with vias (red curve). These are round holes in the dielectric substrate (with diameter about 0.2 mm) filled with a conductor joining the top and bottom ground electrodes, and spaced by 1 mm. Quite clearly, they strongly suppress the overall line reflection and lead to a much more simple behaviour as a function of frequency. The latter now appears to be a "simple" resonator with modes whose spacing corresponds to the device's electrical length. Simulations by CHRISTOPHE HOARAU on the same geometries as Fig.2.52. The focus now is on the transmission |S 12 | as a function of frequency. The vias are seen (on the red curve) to completely suppress the strong resonance dips which were ocurring in the case of the geometry without vias (see blue curve). With them, the transmission is nicely flat, with a typical value of |S 12 | from ca. -0.3 to -0.4.

		0
		0
		-10 -10			Without vias With vias
	|S 11 | (dB) |S 11 | (dB)	-40 -30 -20 -40 -30 -20						w=200 µm	w=300 µm
		-50						w=250 µm	w=350 µm
		-50
		-60
	1 Fig. 2.50: -12 -10 -8 -6 -4 -2 0 |S 12 | (dB) 1 1 Fig. 2.52: -10 -8 -6 -4 -2 0 |S 12 | (dB) 1 Fig. 2.51: -60 Fig. 2.53:	|S 12 | (dB)	-2.8 -2.6 -2.4 -2.2 -2.0 1.98	3 3 2.01 Frequency (GHz) 2 2 2.00 1.99 3 2 3 2 Without vias With vias	2.02	4 4 4 4	7 7 w=200 µm Frequency (GHz) 6 5 6 5 Frequency (GHz) w=250 µm 7 6 5 Frequency (GHz) 7 6 5 Frequency (GHz)	9 9 w=300 µm 8 8 w=350 µm 9 8 9 8	10 10 10 10

  .

		-10
		-20
	11 | (dB) |S	-40 -30 -50		w=200 µm w=250 µm		w=300 µm w=350 µm
		-60
		-70
		1	2	3	4	5	6	7	8	9	10
						Frequency (GHz)

Fig. 2.55: Simulations by CHRISTOPHE HOARAU on a conductor-backed copperg CPW with 8 mm length, gap g =0.250 mm, dielectric slab thickness h =300 mm, as a function of the frequency. The different curves correspond to different signal electrode widths w. Here, the plotted parameter |S 11 | characterizes the reflection behaviour of the transmission line. The simulation assumes perfect 50 Ω-matched connections at both line ends, so that the |S 11 | truly reflects the intrinsic line properties.

In terms of impedance matching, w =300 µm appears to be the optimal width: it corresponds to the lowest |S 11 | towards low frequencies, and the lowest average value (-33.9 dB vs. -30.3 dB for w =250 µm) over the whole simulated frequency range. There is one noticeable dispersive feature at about 5.7 GHz, with a width-independent position, and a rather marked dip above 7 GHz which is significantly influenced by w.

  Computed losses from the simulations by CHRISTOPHE HOARAU in the geometries of Figs.2.57 and 2.58. Note the vertical scale: mostly due to the device's reduced length, the losses are lowered with respect to Fig.2.54. Here, |S 12 | |S 11 |, hence the losses are dominated by the transmission behaviour of the line, and almost unaffected by the resonance in the reflection at about 7 GHz.

		0.2
	Relative lost power	0.1		Without vias With vias
		0.0
		1	2	3	4	5	6	7	8	9	10
						Frequency (GHz)
	Fig. 2.59:

This can be understood qualitatively with the following hand-waving argument. Let us consider a CPW of

  5 as a function of their type: nucleation, change in domain wall type or switch of BPW winding without further transformation. In this table, "simple motion" denotes BPW motion without switch of its winding.

	Type of event \ Helicity h v	+1	-1
	Nucleation or simple motion	25	24
	TVW-to-BPW transformations
	during motion	6	0
	Switch of winding	0	2
	Total √ Total	31 5.58	26 5.10

Tab. 4.5: Summary of all events leading to a BPW in NiFe and NiCo nanowires.

Other details such as post-deposition cooling in the case of sputtering can be included.

This gain is defined with respect to null energy when the dipoles are separated by an infinite distance, || r -r0|| → ∞.

Just a bit more math will be necessary. . .

As a reminder, the presented picture is static, and we are not considering electric charge currents, so that Maxwell's equations[20]

The reader is invited to compare this with the ring of magnetic beads presented in section I.1.c .

Nucleation[41], (anti-)ferromagnetic resonance[42, 43], domain wall creep[44] to name a few.

For the historical introduction of inertial mass associated to localized micromagnetic textures, the reader is referred to the pioneering publication by Werner Döring[62], of which no translation into English exist to the best of my knowledge.

For the dipolar field only, obviously not for the magnetic induction -→ B .

Unless otherwise specified, the convention is that adapted from Jakubovics[26] and used by JAMET et al.[67], the integral of sin 2 (θ) with the present convention.

Such an interpretation is retrieved within Thiele formalism, see appendix section A1.3 Eq.(A1.50), p.41.

The name originates from the context of so-called "Bloch lines", which are transition regions between two Néel-type domain wall segments with opposite rotation. Such a line is effectively a "thin" Bloch wall. A Bloch line may itself be divided in two segments with opposite rotation: the point junction can be seen (with an argument akin to Feldtkeller's) to be a Bloch point.

The reader is referred to the reference book by Malozemoff and Slonczewski[88] for a detailed description of bubbles, which seem to be close cousins of or identical to skyrmions (in some cases)[89].

Punching holes, cutting or stitching parts of an object together are of course forbidden operations.

The reader is invited to demonstrate this with a simple proof by contradiction (reductio ad absurdum), contracting the loop while assuming that no discontinuous point exists.

Implicitly, it is assumed that no other singular points exist in the present case.

These dimensions need not be equal, and confusion in this respect is dangerous[76].

This leads for instance to the fact that topological defects should appear in pairs of opposite objects a priori. Such a consideration, however, neglects sample edges, which add considerable complexity, notably by the possibility of topological defect injection.

This is not a cubic cell of volume a 3 , but rather a Wigner-Seitz cell.

Since the interpretation of this mass is related to how much the magnetization profile is distorted when its velocity changes, one should expect the inertia to depend on the strength of the variations in external drives. For instance, step function-like applied fields would drive magnetization far from its equilibrium configuration as opposed to quasistatic field ramps, and the profile's mass should reflect this difference.

That's a relief, after all, our very hypotheses rely on the latter's existence!

The sum of the eigenvalues of D is strictly positive, and the determinant can be shown to be positive or zero. Hence, the eigenvalues must have the same sign and therefore be positive or null.

Indeed, such a configuration covers only a half unit circle on the unit sphere, that is to say zero surface.

The sign is actually determined by the product of the chirality of the vortex and the polarity of its core. For a vortex pointing along z and a counter-clockwise chirality (on a counter-clockwise path around the core), there is a minus sign.

The case of nanoparticles on the scale of a few nanometers will not be tackled here for instance. The physics of such objects is very rich owing to the small (with respect to samples with larger dimensions, in which the materials are closer to the bulk) number of atoms at play as well as the strongly enhanced contribution of surfaces with respect to the volume. However, it goes well beyond the scope of this document, and so do the fabrication techniques relevant to such nanoparticles.

Mild anodization is actually the voltage regime in which the electrochemical process is not limited by species diffusion to the interface[29], as opposed to hard anodization, for which the reactions proceeds fast enough to locally deplete the solution of reagents.

The role of the standard hydrogen electrode as reference for the definition of standard electrode potential must not be confused with that of the reference electrode during an electrochemical process. In this case, the relevant voltage difference is measured or applied between the electrodes of the two redox pairs at stake, while the reference electrodes provides the ground so to speak, that is to say the electrode at constant potential (regardless of the ongoing reaction) with respect to which the potential of the other electrodes is defined.

In fact, departure from the constant or DC regime was already mentioned in the context of modulated diameters for nanoporous alumina templates. In that case, the anodization voltage was indeed varied in order to trigger an increase or decrease in the pore diameter.

It is assumed here that the amount of deposited material is very low compared to the amount of dissolved metal ions, so that the electrodeposition has a negligible influence on the overall metal ion concentration in the bulk of the electrolyte.

The latter authors' conclusion is still that the degree of solution buffering i.e. the presence of hydroxide ions was key rather than

Since the ALD process covers only the pore walls and occurs before the deposition of nanowires, neither of their tips is actually protected by the silicon oxide layer. However, considering that this represents only a negligible fraction of a nanowire's surface, it is safe to expect any damage to be confined to the nanowires' tips in this configuration.

More precisely, P is calculated as difference between normalized squared moduli of the electric field amplitudes associated to circular left and right polarizations.

One could also add here XMCD obtained from Scanning Transmission X-ray Microscopy (STXM) as the underlying physical principle is exactly the same. For a concise presentation of STXM, the reader is referred to the book by Als-Nielsen and McMorrow[94], and for a

A known limitation being the contrast reversal described by JAMET et al.[99].

Because of this, imaging becomes more and more difficult towards the gap's edges. These exert a stronger and stronger influence on the electrons extracted from the sample, thus disturbing their path and therefore image formation on the detector.

This is not to say that water e.g. on the surface or in the holes for coil winding cannot be an issue. Significant degassing has been observed with this sample holder during pumping towards working PEEM vacuum.

Strictly speaking, it could be further enhanced by bringing the wiring even closer to the top surface of the sample holder, but it would then have been too difficult to precisely machine the traversing holes and also to preserve a flat top surface. The spacing from the latter to the top of a coil hole is 0.5 mm

It is to note that all voltages here are positive.

Similarly to the above argument on mutual inductances, one can expect the measurement loop and field coil to have an inductance smaller than the field coil's self-inductance by a factor slightly higher than 10.

That contribution cannot be quantified because it depends intricately on the component's architecture.22 There exist second-order effects, but they are left out here for the sake of simplicity. The reader is referred to e.g. Ref.[93] for an introductory description of these.23 Provided that no symmetries are broken (which is the case if magnetism is left out) other than those broken by lattice asymmetries, the dielectric tensor must be symmetric and real: indeed, it is a response tensor. Therefore, it can be diagonalized in a certain orthonormal basis.24 Other effects going beyond this first approximation exist, but magneto-optical effects quadratic in magnetization are usually much weaker[93] and will be neglected here.

If it is an insulator, then it is easily shown that the material's magnetization is directly proportional to the (supposedly homogeneous) induction that is applied to and around it.

As we have seen, the Faraday rotation originates from the magnetized medium, so that the magnetic field H does not play a role.

What matters here is the ratio of the lateral extent q of the field sensor to, say, the coil radius R. Since in our case q is given by the Hall probe or the laser spot, q ≤1 mm while R 5 cm, therefore ensuring homogeneity.

There is a zero-voltage offset, but its value is 3 mA, which can be considered negligible and within the error on the current.

Typically, 1 mV to 10 mV.

This prefactor was already included in the magneto-optical calibration of the garnet, using a known AC field produced by the pair of coils. Therefore, since no intrinsic ratio such as α has been calculated until now, it was safe to ignore this experimental feature.

Quantitatively, this filtering corresponds to a convolution of the time-domain shape with a cardinal sine with a central lobe width (from first negative root to first positive root) of δtcut = 1/fcut =1 ns.

The relative dielectric permittivity ε r (13 in[START_REF] Gupta | Microstrip Lines and Slotlines[END_REF]) is different in my case, but the characteristic impedance varies with ε r only through a square root with this property in its argument. Hence, no large differences are expected.

This hand-waving approach completely overlooks dispersive effects and resonances, and should therefore only be considered for, say, the baseline of the curves in Figs.2.54 and 2.59

There also is the question of impedance matching. However, considering that in all analytical calculations which assume infinitely thin electrodes, the respective lateral dimensions appear in ratios, one may expect the first corrections for finite thickness t to be on the order of, say, t/w, t/(w/2 + g), or t/b. Assuming that the thickness remains on the order of a micron and all other dimensions are at least several hundred microns, only negligible corrections are expected.

One can show that at least in our regime where w 200t, the inverse limit where the current density is confined only to the perimeter of this cross-section (corresponding to an infinitely thin skin depth) leads to results that are identical to within a percent. Therefore, the real current density being somewhere in-between these two limiting cases, one or the other should provide a reasonable field estimate.

While perhaps trivial at first sight, this is an important fact in the sense that magnetoresistive effects such as AMR should not affect the impedance's behaviour.

In the case of oxygen, note that the peak close to the energy of the so-called O Kα1 line (nominally at 524.9 eV[100]) is so much brighter than e.g. the neighbouring Ni peak that it cannot be explained by the L1, Lα1 and Lβ1 lines of Cr alone (nominally at 500.3 eV, 572.8 eV, and 582.8 eV respectively[100]).

Instrumental developments for domain wall motion in cylindrical nanowires

The acquisition time was 5 s, the sampling dense enough to ensure having one data point at the working frequency (on the signal's Fourier transform).

FEBID is not restricted to the deposition of metals: oxides for instance can be deposited[13], but this topic goes beyond the scope of this work.

The molecule is essentially a tetravalent platinum atom with bonds to three methyl groups on one hand, and a bond to the methylcyclopentadiene (five C-atoms ring with a methyl group and two double bonds) ring on the other hand.

A more precise criterion than "sufficiently strong" will be proposed later on.

No torsion around that axis was visible, so that the discussion about horizontal misalignments holds.

This measurement was performed with a focal plane very close to the expected altitude of the wire tip, not to the substrate surface altitude.

The fit was done with Igor Pro 6.37 by WaveMetrics R[42] using 12 data points in both cases; the number was chosen in the intermediate range between too few points and less stable/reliable results on one hand and too many points and results impacted by the profiles' asymmetry. No error bars were assigned to the data as I estimated the noise-to-signal ratio on the PEEM image to be below 1%.

Not to mention this paragraph's title. . .

Individual snapshots appear very similar to the PEEM image in Fig.3.5 albeit with a poorer signal-to-noise ratio, hence the "blur" is not due to slow averaging on the time scale of the full image acquisition. It must be on the scale of a single snapshot at the slowest.

One may think of small, a priori negligible fluctuations/noise in the microscope's high voltage. There is also the possibility of fluctuations in the field emission itself, not to mention slow mechanical motions imparted to the sample during navigation. . .

The latter relies on procedures developed by ANDREA LOCATELLI and ONUR MENTE Ş from the Nanospectroscopy beamline (http: //www.elettra.eu/elettra-beamlines/nanospectroscopy.html) and adapted by RACHID BELKHOU and coworkers.

It is noteworthy that the post-edge tail then also match.

Strictly speaking, that would be the case too if the surface had been pure clean silicon. However, this material's resonances are far from the Co L3 and its linear absorption coefficient µSi varies in this range with approximately[41] µSi ∝ E -2.45 , where E is the photon energy. Thus, the relative variation in photoemission from Si over a 10 eV window around the Co L3 edge is about 3%.

Non-zero contrast outside the field of view has of course nothing to do with the sample.

No images with perpendicular linear polarizations were acquired, so that it cannot be inferred from this work whether part of the halo displays antiferromagnetic behaviour. This might have been possible to evidence thanks to X-Ray Magnetic Linear Dichroism (XMLD) since the FEBID halo and dots are nanocrystalline materials without texture[65].

Potential beam harmonics effects, noise in general, absorption from the platinum shell and other such disturbances are left out for simplicity. Furthermore, the average fraction of time during which the beam does not obscure the beam at location (x, y) does depend on these coordinates.

Here I remind that on the basis of the orientation of the shadow with respect to other alignment marks, the fact that the wire bends were facing the beam stands beyond doubt (see p.158).

The halo has been neglected for simplicity; yet experiments on samples absorbing resonantly the X-Ray beam were shown to experience only small temperature increase[56], below 10 K.

The core radius was chosen as r0 =37.5 nm, the outer radius rT =47.5 nm.

Since the output is only a rough estimate to be varied afterwards, I did a "manual" fit for simplicity and lack of experimental errors on this data. . .

A more precise comparison would be allowed if error bars on the reported coercive fields and remanence values had been provided.

As a reminder: a ca. 10 nm-thick layer is deposited with ALD during the pore etching; as a result, SiO2 covers both sides of the AAO membrane.

The number corresponds to a storage ring current 2/3 of that measured during the 2015 beamtime, as well as the best energy resolution (thus under the most stringent conditions on the flux in this regard), and linear polarization.

It must be remembered that with SEM, getting a precise sense of depth in the vertical direction is quite difficult.

One further interest would be the comparison between the edge shapes, yet this is not so straightforward, as will be shown later on.

In this image, one pixel corresponds to 14 nm.

Strictly speaking, from the shape of the Fe L3 peak of FeO[29], it is rather the sum of metallic iron and iron monoxide over the other oxides of iron which is monitored. However, under the hypothesis of an oxide surface layer, FeO should also vanish during sputtering: therefore, the measurement should be in principle an underestimation of the improved metallicity.

This corresponds to µd → +∞, where µ is the relevant linear X-rays absorption coefficient of the material of interest at the working photon energy, and d is the wire diameter.

This holds if one leaves the dispersive silicon photoemission cross-section aside.

Strictly speaking, it also encompasses several microscope parameters such as its detection efficiency, which are gathered in the same coefficient for the sake of simplicity.

Schematically, if this electronic mean free path increases, then the photoemission process extracts electrons in silicon from larger depths at 1410 eV than at 705 eV. This would tend to increase the PEEM intensity, contrary to the decrease in X-ray linear absorption coefficient. I do not have more detailed insight/knowledge on the matter.

In practice, if it is still inside the wire's thin section; no XMCD-PEEM experiments of mine have shown cases of domain wall pinning in a wire's thick section.

Rare cases in which e.g. no reliable information concerning the applied field was available constitute not interpretable observations.

This applied field is the maximum field reached during the pulse sent using MÁRLIO BONFIM's pulse generator.

In such structures where pinning might be present, the switching field is a lower bound on the nucleation field.

The error given here is evaluated from a calibration of the used electromagnet by the Circe beamline's staff.

The latter simulations deal with vortex-like domain walls, which are the analogues of BPWs in soft tubes but without a singular point thanks to the empty space inside the tube.

After all, such an object can be viewed as a small metallic spring. . .

Due to the way of probing, the composition is laterally averaged along the X-ray beam propagation direction.

Remerciements

The contrast is laterally averaged over a distance barely larger than the shadow's width. The abscissa increases from the bottom left to the top right of the XMCD-PEEM image in Fig. 3.14), the extremities corresponding to the blue crosses on that same image. Despite the travelled distance through the cobalt core, on the order of 70 nm i.e. more than four times the nominal X-Ray mean free path 2/(µ R + µ L ) 16 nm [34], the contrast levels are very low.

Although 768. 65 eV is closer to the L 3 edge's maximum of absorption (see shadow XAS in Fig. 3.13), the signal amplitude is definitely higher at 768.20 eV. This I attribute mainly to the fact that photoemission from the halo is about doubled from 768.20 eV to 768.65 eV (see Fig. 3.13) while the shadow's mean absorption increases. At first, such a reasoning may seem plainly wrong because more absorption from the sample should result in larger dichroism: the latter is expected to saturate at infinite sample length for instance. However, such a saturation can only happen in the absence of background in the normalization factor defining the contrast C(x, y). In the present case, instead of normalizing with a denominator that contains only intensity affected by transmission through the wire, we are including a very large parasitic intensity to this denominator. The proper limit to take from Eq.(3.2) is exp (-´wire µ R/L dl) 2(1u)/u; thus, the contrast reads:

Both exponentials in the numerator from Eq.(3.3) become smaller if there is an increase in the linear absorption coefficients, which is what happens between 768.20 eV and 768.65 eV. Qualitatively, this is why the contrast is lower at the latter energy. Quantitatively, discussing the evolution of the dichroic signal's amplitude requires considering that at pre-edge energies, there is no difference between µ L and µ R and hence C = 0. Then, as the photon energy increases, this difference builds up while the background is not yet too overwhelming; in other words, the absorption through the wire is not yet so strong that both exponentials in Eq.(3.2) are very much smaller than unity. Then, as the difference µ Lµ R increases, so does the average absorption (µ L + µ R )/2: the latter brings the contrast to the limit in Eq. (3.3) where strong absorption means that dichroism is a smaller and smaller difference while the background is unchanged. Therefore, the signal-to-noise ratio must decrease at some point.

Based on the above, there must be an optimal photon energy that cannot be the L 3 edge's peak. During the beamtime, XMCD-PEEM imaging was performed at several target photon energies: 767. 40

IV Helicity investigations IV.1 Nanowire Cicero

In the above-mentioned wire Cicero (see section II.4 e.g. Fig. 4.22, p.203), no magnetic contrast could be retrieved from the bulk of the thick wire sections' shadow. This is a consequence of the aforementioned (see section II.4 p.202) background intensity [30] that might be partially caused by the beam's harmonics. Some changes on the rim are visible, whereas the shadow's bulk (in the thick section) displays a contrast independent of the thin section's configuration or of the thick section's rim. However, the thin section's shadow provided XMCD contrast sufficient for monitoring its magnetic state. A significant number (28) of nucleation events followed by pinning could be triggered in this nanostructure. Several of them are reproduced in Fig. 4.34: the frames a1) through e2) are grouped in pairs, the top [a1), b1) etc.] and bottom [a2), b2) etc.] views are XMCD-PEEM images of the wire before and after application of field, cap is "flown" i.e. it is maintained at fixed height above the sample substrate without mechanical contact. This is illustrated with a generic PCB on the photograph to the right of Fig. 4.51. Measurements performed by MICHAEL FOERSTER et al. [70] at Alba indicated a loss of amplitude of about 40% for pulses of width 10 ns (and amplitude 1 V) after a full round-trip through the microscope cable. This would mean about 20% amplitude loss upon reaching the sample. Therefore, based on the discussions from Chapter 2, section IV.2.d p.123, the application of fields with amplitudes sufficient for influencing/triggering domain wall motion should be possible.

Unfortunately, the electrical contact between CPW tracks and termination impedance out of NiCr failed during the beamtime. This connection, which had been repaired by CHRISTOPHE THIRION using silver paint (see section I.5), could not be restored despite intense endeavours; as a result, the DC termination impedance as measured from the microscope rack was ca. 25 kΩ. Still, I have tried to use the pulsed field setup in a case of pinned BPW in nanowire D12 (see section IV.2). Starting from the pinning site associated to domain wall nucleation in this wire, I have applied a DC field with the sample holder's electromagnet of 11 ± 1 mT, resulting in motion above the instrumental resolution but below the domain wall width, without depinning. Then, the application of the same field in combination with a pulse sent through the CPW with the Avtech R pulse generator led to clear depinning. This sequence of field and subsequent shadow XMCD-PEEM imaging is reported in Fig. 4.52.

Further imaging (not shown here) using either only low field from the electromagnet or only the Avtech R pulse generator allowed to conclude that the latter's effect was below 3 mT, based on domain wall pinning behaviour. As a result, considering the very adverse configuration in terms of termination impedance and the experimental error on the applied field, not to mention the plausible interplay of temperature in depinning, it is clearly not possible to decisively conclude on the effect of the nanosecond-scale field pulse.

On the positive side, it must be mentioned that the fracture corresponding to the absence of metal connecting CPW and NiCr termination impedance was very narrow, below 10 µm. This was sufficient to yield very high DC impedance values, but far into the AC regime, the impact of this unfortunate feature is not as clear. Furthermore, the likely current pulse echo induced by the strong impedance mismatch at the CPW's termination must have been weaker than the first pulse passing underneath the micromanipulated nanowire. This follows from the non-zero linear attenuation through the CPW.

Taking all this into account, it is only possible to state that the sequence from Fig. 4.52 does not rule out a positive, significant influence of the applied nanosecond field pulse, but on the other hand that too many unknowns come into play to make this influence likely. The thermally activated depinning coupled to the unknown precise behaviour of this particular pinning site, as well as the unfavourable impedance mismatch (leading to a reflection coefficient Γ ∼ 1, see Chapter 2, section IV.1.c and the DC applied field value plus its associated experimental error are strong arguments speaking against any measurable effect of the pulsed fields. the local composition 24 through n Fe (x) = 1n Ni (x) and the bulk linear absorption coefficients µ 0 Fe and µ 0 Ni . From the publications by Nakajima [32] and Henke [18], they read (after proper averaging of the µ + and µ -): µ 0

Fe (E 1 ) = 0.071/nm, µ 0 Fe (E 2 ) = 0.0098/nm and µ 0 Ni (E 1 ) = 0.0025/nm, µ 0 Ni (E 2 ) = 0.046/nm. Finally, L is the length travelled through the wire, it should be very close if not equal to the wire diameter, d 180 ± 20 nm. Since dark field correction and normalization have been done, one must now adjust the intensities on the substrate similarly to XMCD: this balances the different products of flux and photoemission cross-sections at the two working energies. In other words, this makes I 1 and I 2 equal. Now, if we compute the contrast from these corrected intensities, in the absence of parasitic background (as mentioned in JAMET et al. [30], perhaps caused by the beam harmonics), we may derive:

Thus, the fraction of nickel reads:

At present, we only need to extract the contrast values at different positions. Let x 1 , x 2 and x 3 be the respective positions along the wire towards the abrupt diameter modulations, x 1 in the Ni-poorest area, x 2 in the Ni-richest one, x 2 > x 1 (x-axis going to the right in the PEEM images), and x 3 in the intermediate region beyond the pinning site.

The contrasts are: C(x 1 ) -0.47 ± 0.02, C(x 2 ) -0.07 ± 0.02, C(x 3 ) -0.27 ± 0.02. The error on the composition can be estimated by estimated the lowest and upper bound with the proper bounds on the contrast and lengths 25 . For position 1, the factor with the logarithm is always positive, whereas it is always negative for positions 2 and 3 (and more so for the maximum value of contrast, not in absolute value). Hence, for the latter, the lower bound is taken for the maximum contrast and smaller diameter estimate, and the upper bound for the minimum contrast and larger diameter estimate. On the contrary, for position 1, the lower bound is taken for the maximum contrast and larger diameter estimate, while the upper bound corresponds to the minimum contrast and smaller diameter estimate.

With this, the estimated compositions read n Ni (x 1 ) = 0.53 ± 0.01, n Ni (x 2 ) = 0.577 ± 0.003, and n Ni (x 3 ) = 0.553 ± 0.004. It must be remembered that the calculation completely omits any parasitic beam intensity, which would reduce the contrast just like in shadow XMCD-PEEM. In other words, the estimated composition variations are likely to be underestimated in this approach.

Summary

The thesis is concerned with the observation of ferromagnetic domain walls in cylindrical nanowires, and their dynamics under applied magnetic fields. These nanostructures were electrodeposited by colleagues of mine into nanoporous alumina templates with a tailored pore geometry. The materials are soft FeNi or CoNi alloys; the diameters range from 150 nm to 250-300 nm, with a typical length of 30 µm.

My work first comprised experimental developments of sample holders and high-frequency electronics towards field-induced domain wall motion. The latter I investigated with X-ray Magnetic Circular Dichroism coupled to transmission PhotoEmission Electron Microscopy (XMCD-PEEM). This synchrotron-based technique allows to monitor the internal domain wall configuration before and after displacement; due to the stringent requirements of time-resolved XMCD-PEEM experiments in terms of reproducibility, the real-time dynamics is out of reach as of yet.

The response of ferromagnetic domain walls to applied magnetic fields is notably characterized by their mobility, i.e. the ratio of attained velocity to field strength. In cylindrical nanowires, a novel ingredient emerges in the case of one domain wall type that is absent in flat strips: the Bloch point domain wall. Not only does this domain wall host a micromagnetic singularity, that is to say a point where magnetization vanishes (the Bloch point), but it also possesses a discrete degree of freedom representing the sense of magnetization winding around the nanowire axis. It has been predicted that Bloch point wall motion under sufficiently high fields leads to this degree of freedom selecting one of its only two possible values. In other words, one winding becomes unstable. I report in this thesis experimental evidence of such a selection in a majority of Bloch point wall motion events.

Although mobility measurements could not be carried out, my experiments have furthermore evidenced transformations between domain wall types that had not been predicted in simulations. Since the Bloch point wall contains a topological defect (the Bloch point itself), this unexpected behaviour questions the sometimes argued protection attributed to topologically non-trivial textures. While reminiscent of the well-known conversion between transverse and vortex walls in strips, these transformations in cylindrical nanowires involve topologically non-equivalent micromagnetic configurations, in contrast with the aforementioned transverse and vortex walls. Moreover, the observed only relative stability of domain wall types suggests caution in the interpretation of future mobility measurements in such systems, if the internal wall configuration cannot be resolved.

Aside from such electrodeposited samples, I have also studied an upright core-shell nanowire grown by colleagues with Focused-Electron-Beam-Induced Deposition. This nanostructure featured a nanocrystalline cobalt core and a platinum shell. Its magnetic configuration was investigated with transmission XMCD-PEEM as well. Contrary to the aforementioned horizontally-lying wires, the core-shell sample was vertical with no diameter modulations. On the other hand, the geometry featured bends engineered to favour domain wall pinning. In this novel imaging configuration, the challenge was to recover as much of the nanowire's magnetic state as possible. I was able to demonstrate the presence of at least one domain wall.