Docteur

Matteo Rossi

Dr Thèse Dirigée

Dr Nicholas Brookes

Dr Marco Moretti

Michael Krisch

Thèse Soutenue

M Olivier

Isnard Professeur

M Sergio

D I Matteo

Rapporteur M Christian

M Matthieu Le Tacon

Virginie Mme

Simonet

THÈSE Pour obtenir le grade de

Keywords: symmetry, Ir-F distance, trigonal angle

La diffusion inélastique résonante de rayons X sur systèmes

Abstract

Transition-metal compounds of the 5d series have raised notable attention in the last decade following the unexpected identification of properties characteristic of strongly-correlated-electron systems. Electron correlation is supposed to be weak in heavy-transition-metal materials due to the large spatial extension of the electronic wave function. On the contrary, spin-orbit coupling (SOC) is strong and it contributes to shape the electronic structure of materials. This is especially the case of iridium oxides (iridates), where the joint action of crystal field, SOC, and electron correlation realizes the so-called j eff = 1/2 Mott insulating state. Such state has been intensively investigated and novel quantum phases and excitations have been theoretically predicted and experimentally found. Undoubtedly, one of the most employed techniques to elucidate the low-energy physics of iridates is resonant inelastic X-ray scattering (RIXS). At the iridium L 3 edge, it benefits from a favorable inelastic cross-section and a particularly good energy resolution, which matches the energy scales of the relevant excitations. The aim of the present thesis is twofold: conceive challenging instrumental upgrades that contribute to the advancement of the technique itself, and apply RIXS to inspect the magnetic and electronic properties of selected iridium compounds.

The main instrumental development concerns the design of a new RIXS spectrometer with polarization resolution. Polarization analysis of the scattered X-rays is extremely complicated when coupled to stringent requirements in terms of energy resolution and efficiency. As a matter of fact, polarization analysis is rarely exploited in current RIXS spectrometers, although it would be particularly useful to assess the nature of the spectral features. While the designed RIXS polarimeter is still under construction, the optical scheme is validated by preliminary tests. Target energy resolution with full polarization analysis is expected without efficiency reduction iii Abstract compared to current state-of-the-art RIXS spectrometers.

Additional technical developments include sample-environment equipment to perform RIXS experiments in low-temperature and highpressure conditions. The application of physical pressure allows to change the interatomic distances and thereby modify the correlation effects. By probing the magnetic and electronic properties of the perturbed system, a complete picture of the material under study can be achieved. In this respect, the magnetic excitations of the bilayer-perovskite Sr 3 Ir 2 O 7 are explored below the Néel temperature and above 10 GPa. The microscopic model Hamiltonian that is best suited to describe the magnetic structure and dynamics of Sr 3 Ir 2 O 7 is currently debated, and two alternative, opposite scenarios are proposed. The evolution of the magnetic gap as a function of pressure provides further experimental constraints against which theoretical models may be tested. Beside the specific case of Sr 3 Ir 2 O 7 , we demonstrate for the first time that RIXS experiments of the magnetic dynamics can be extended to the high-pressure regime.

In the present thesis, additional experimental work is carried out, in particular on the magnetic and electronic excitations of the post-perovskite CaIrO 3 . The lattice of CaIrO 3 contains both corner-and edge-sharing octahedra; hence this material features very different magnetic couplings in the two directions. By using RIXS, I find signatures of quasi-one-dimensional spinon-like magnetic excitations that, however, cannot be modeled with available simple model Hamiltonians. Electronic excitations are also found to have peculiar properties, too.

Finally, I focus on the structural and electronic properties of a novel class of iridium compounds, namely iridium fluorides, largely unexplored before. While their lattice features disconnected IrF 6 units, the local electronic structure resembles that of iridium oxides and thus stimulated the search for signatures of correlated-electron physics. The analysis of the RIXS spectra clearly identifies the presence of a j eff = 1/2 Mott insulating state in these compounds. The substitution of the alkali and halogen elements and the application of physical pressure reveals the robustness of the j eff = 1/2 state in these materials, thus extending the search for novel quantum phenomena in non-oxide iridium compounds.

The present work extends the potential of the RIXS technique to domains previously unexplored. The polarization analysis of the scattered X-ray beam will be especially beneficial in cases where the origin of the spectral features cannot be unambiguously determined. The high-pressure equipment will allow to perform RIXS measurements to gain information on electronic and magnetic properties of phases of matter that are unaccessible at ambient pressure. I hope that the improvements discussed in this thesis will further establish RIXS as a powerful and reliable tool to characterize elementary excitations in correlated-electron systems. iv Résumé Les oxydes de métaux de transition possèdent plusieurs propriétés physiques remarquables, dont les plus célèbres sont la supraconductivité à haute température critique dans les oxydes de cuivre (cuprates), la magnétorésistance colossale dans les oxydes de manganèse, et le multiferroïsme dans les aimants frustrés. Traditionnellement, ces propriétés sont imputées à l'effet d'une corrélation électronique forte. Par conséquent, l'étude a été concentrée sur les oxydes de métaux de transition 3d, où l'on suppose que la corrélation électronique est la plus forte. Cependant, au cours de la dernière décennie des propriétés typiques des systèmes corrélés ont été découvertes dans les oxydes de métaux de transition 5d. Ces effets sont dus au fort couplage spin-orbite qui caractérise ces composés. Depuis, plusieurs études théoriques et expérimentales ont été effectuées pour étudier les effets de l'action conjointe du couplage spin-orbite et de la corrélation électronique. La classe la plus sondée est celle des oxydes d'iridium (iridates), où les interactions entre le champ cristallin, le couplage spin-orbite et la corrélation électronique entrainent l'apparition de l'état j eff = 1/2 isolant de Mott. L'une des particularités des systèmes j eff = 1/2 est que des liaisons topologiques différentes entre les ions iridium possèdent des interactions magnétiques opposées. Plus particulièrement, les composés où les ions d'iridium sont liés par des octaèdres partageant un sommet ont un comportement magnétique similaire au cuprates avec une interaction dominante de type Heisenberg et sont possiblement caractérisés par un état supraconducteur non conventionnel. Par contre, les composés où les iridiums sont liés par des octaèdres qui partagent une arête possèdent des interactions anisotropes de type quantum compass, qui peuvent mener à l'apparition du magnétisme de Kitaev. Le Chapitre 1 de cette thèse résume la physique à basse énergie des iridates et présente l'état j eff = 1/2 en détail. Les principaux résultats qui concerv Résumé nent les propriétés électroniques et magnétiques des iridates sont passés en revue.

La diffusion inélastique résonante de rayons X (resonant inelastic Xray scattering, RIXS) est l'une des techniques les plus adoptées pour explorer les excitations magnétiques et électroniques ainsi que leur quantité de mouvement. En effet, l'utilisation de techniques plus conventionnelles de spectroscopie neutronique n'était pas réalisable dû au petit volume des échantillons qui sont actuellement synthétisés et à la large absorption des neutrons par l'iridium. Par contre, la section efficace relativement large de la spectroscopie RIXS au seuil L 3 de l'iridium et la bonne résolution en énergie ont encouragé l'emploi de cette technique. Le Chapitre 2 de cette thèse présente le processus de diffusion RIXS en détail et définit la section efficace de cette technique. La beamline ID20 de l'Installation Européenne de Rayonnement Synchrotron (European Synchrotron Radiation Facility, ESRF), où les expériences scientifiques décrites dans cette thèse ont étés effectuées, est introduite dans le Chapitre 3. La beamline a été conçue pour la diffusion inélastique résonante et non-résonante de rayons X et est équipée d'un spectromètre RIXS.

Les Chapitres du 4 au 7 sont le coeur de cette thèse. Ceux-ci comprennent la description des développements instrumentaux ayant pour but d'améliorer les possibilités offertes par la spectroscopie RIXS, et l'application du RIXS afin d'étudier la physique à basse énergie de certains iridates.

Le principal projet de développement instrumental est un analyseur de polarisation des rayons X diffusés. Celui-ci est exposé dans le Chapitre 4. La polarisation est la seule caractéristique des rayons X diffusés qui ne soit pas mesurée actuellement dû aux difficultés techniques, mais elle est essentielle pour étudier la nature de l'excitation sondée. Le nouveau spectromètre RIXS projeté avec résolution en polarisation est basé sur un nouveau schéma optique qui permet de ne compromettre ni la résolution en énergie ni l'efficience par rapport aux spectromètres RIXS courants. Même si le polarimètre n'est pas encore réalisé, le schéma optique a été testé sur la beamline ID20 de l'ESRF et les mesures effectuées ont confirmé les simulations théoriques.

Le second développement instrumental concerne l'équipement permettant la réalisation des mesures RIXS à basse température et haute pression. L'application de la pression perturbe les propriétés magnétiques et électroniques du système exploré et permet d'accéder à de nouvelles phases de la matière. La mesure des excitations à basse énergie par la spectroscopie RIXS en conditions thermodynamiques extrêmes est très compliquée et n'a jamais été réalisée. Dans le Chapitre 5, le nouvel équipement décrit est utilisé afin de mesurer la dynamique magnétique de l'isolant de Mott j eff = 1/2 Sr 3 Ir 2 O 7 à 100 K et jusqu'au 12 GPa. En effet, même si la structure magnétique de ce matériel est connue, un débat a lieu à propos les vi Résumé interactions magnétiques principales. Cette étude a pour but d'analyser l'évolution des excitations magnétiques en fonction de la pression et la comparer aux prévisions des modèles théoriques. Ces mesures démontrent que les excitations magnétiques peuvent être acquises en haute pression par la spectroscopie RIXS, démontrant ainsi la possible utilisation de cette technique dans ce nouveau domaine.

Au-delà des innovations techniques, ce projet comprend aussi l'étude des excitations élémentaires de certains composés de l'iridium. Le Chapitre 6 présente l'analyse de la dynamique magnétique et électronique de l'isolant de Mott j eff = 1/2 CaIrO 3 . La structure cristalline de ce système est particulièrement intéressante car elle est constituée d'octaèdres partageant une arête dans la direction parallèle à l'axe a et un sommet dans la direction parallèle à l'axe c. Du coup, les interactions magnétiques sont très différentes selon la direction cristallographique. La suppression du couplage de type Heisenberg pour les octaèdres partageant une arête rend ce système comparable à un aimant quasi-unidimensionnel. La dynamique magnétique de ces aimants unidimensionnels est constituée d'un continuum d'excitations avec une dépendance en énergie et quantité de mouvement typique, qui est révélé par la spectroscopie RIXS. La dynamique électronique est également mesurée et qualitativement analysée en considérant la forme des orbitales associées à l'état fondamental j eff = 1/2 ainsi qu'aux états excités.

Enfin, dans le Chapitre 7, la caractérisation de la structure cristalline et électronique des fluorures d'iridium est proposée. Ceux-ci ont étés proposés comme des isolants de Mott j eff = 1/2, une propriété ayant été jusqu'à aujourd'hui réservée aux oxydes (à l'exception de RuCl 3 ). La structure cristalline de ces composés, étudiée par diffraction de poudre, est constituée par des unités isolées d'IrF 6 . Les mesures RIXS sur Rb 2 [IrF 6 ] reportent la présence de deux excitations qui sont attribuées aux transitions entre l'état fondamental j eff = 1/2 et les états excités. L'absence d'un signal magnétique et de la dispersion de ces excitations reflètent l'absence d'interactions entre les unités IrF 6 . La substitution du métal alcalin et alcalino-terreux n'est pas efficace dans la modification de la structure électronique de ces matériels. Aussi, l'application de la pression jusqu'au 26 GPa induit de petits changements. Par contre, la substitution de l'halogène a un grand impact. En effet, la force du couplage spin-orbite est proportionnelle à la covalence de la liaison entre l'iridium et l'halogène car dans une liaison plus covalente les orbitales sont plus étendues. Elle est majeure dans les fluorures d'iridium à cause de la large électronégativité du fluor. Ces résultats suggèrent que la recherche de nouvelles phases générées par l'interaction entre le couplage spin-orbite et la corrélation électronique peut être élargie aux halogénures d'iridium.

Cette thèse accroit l'importance de l'utilisation de la spectroscopie RIXS dans des domaines qui étaient précédemment inexplorés. L'analyse de vii 1

Generic phase diagram of transition metal oxides generated by the simultaneous action of electron correlation and SOC.

1.1 Dependence of the SOC constant λ on the atomic number Z.

Introduction

Transition-metal compounds possess many fascinating physical properties, among which the most prominent examples are high-temperature superconductivity in copper oxides (cuprates) [START_REF] Keimer | From quantum matter to high-temperature superconductivity in copper oxides[END_REF] , colossal magnetoresistance in manganese oxides 2 , and multiferroicity [START_REF] Cheong | Multiferroics: a magnetic twist for ferroelectricity[END_REF] . Conventional wisdom assigns the above properties to the effect of strong electron correlation. Therefore, the scientific community has mainly focused on transition metal compounds of the 3d series, for which electron correlation is expected to be the strongest. However, in the last decade the seminal work of Kim et al. [START_REF] Kim | Novel J eff = 1/2 Mott State Induced by Relativistic Spin-Orbit Coupling in Sr 2 IrO 4[END_REF] demonstrated that correlation effects are effective also in transition metal oxides of the 5d series when supported by strong spin-orbit coupling (SOC). Since then, a number of theoretical and experimental works have been performed to explore the phase space generated by the joint action of electron correlation and SOC [START_REF] Pesin | Mott physics and band topology in materials with strong spin-orbit interaction[END_REF][START_REF] Wan | Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates[END_REF][START_REF] Burkov | Weyl semimetal in a topological insulator multilayer[END_REF][START_REF] Wan | Computational Design of Axion Insulators Based on 5d Spinel Compounds[END_REF][START_REF] Go | Correlation effects on 3d topological phases: From bulk to boundary[END_REF][START_REF] Witczak-Krempa | Correlated quantum phenomena in the strong spin-orbit regime[END_REF][START_REF] Rau | Spin-orbit physics giving rise to novel phases in correlated systems: Iridates and related materials[END_REF] . A sketchy phase diagram is displayed in Figure 1. On the left-hand side of the diagram are the conventional 3d transition metal compounds with negligible SOC. These broadly fall into two groups: simple metals or band insulators, and Mott insulators. Their properties are mainly governed by the interaction of the electrons with the periodic lattice potential and by the strong electron correlation, respectively [START_REF] Imada | Metal-insulator transitions[END_REF] . At increasing values of the SOC, a rich variety of novel quantum phases may be expected. Weakly correlated materials may turn into topological insulators, i.e. bulk insulators with surface metallic states, or semi-metals, i.e. three-dimensional magnetic systems with linearly-dispersing electronic bands, similar to graphene [START_REF] Pesin | Mott physics and band topology in materials with strong spin-orbit interaction[END_REF][START_REF] Wan | Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates[END_REF][START_REF] Go | Correlation effects on 3d topological phases: From bulk to boundary[END_REF][START_REF] Witczak-Krempa | Topological and magnetic phases of interacting electrons in the pyrochlore iridates[END_REF] . When electron correlation becomes stronger, other exotic phases are expected to become accessible, e.g. Weyl semi-metals [START_REF] Wan | Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates[END_REF][START_REF] Burkov | Weyl semimetal in a topological insulator multilayer[END_REF][START_REF] Go | Correlation effects on 3d topological phases: From bulk to boundary[END_REF] , axion insulators [START_REF] Wan | Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates[END_REF][START_REF] Wan | Computational Design of Axion Insulators Based on 5d Spinel Compounds[END_REF][START_REF] Go | Correlation effects on 3d topological phases: From bulk to boundary[END_REF] , and even-

Introduction

Spin-orbit coupling

Electron correlation tually spin liquid states [START_REF] Okamoto | Spin-Liquid State in the S = 1/2 Hyperkagome Antiferromagnet Na 4 Ir 3 O 8[END_REF][START_REF] Zhou | Quantum spin liquid states[END_REF] . To date, most attention has been drawn on iridium oxides (iridates), where the combined action of crystal field, SOC and correlation effects generates the so-called j eff = 1/2 Mott insulating ground state [START_REF] Kim | Novel J eff = 1/2 Mott State Induced by Relativistic Spin-Orbit Coupling in Sr 2 IrO 4[END_REF] . Such state has initially been introduced to explain the unexpected insulating behavior of Sr 2 IrO 4 and then extended to many other iridates. Depending on the bond topology, the j eff = 1/2 ground state may lead to very different magnetic interactions [START_REF] Jackeli | Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models[END_REF] , with relevant consequences for possible unconventional superconductivity [START_REF] Wang | Twisted Hubbard Model for Sr 2 IrO 4 : Magnetism and Possible High Temperature Superconductivity[END_REF][START_REF] Kim | Magnetic Excitation Spectra of Sr 2 IrO 4 Probed by Resonant Inelastic X-Ray Scattering: Establishing Links to Cuprate Superconductors[END_REF] and frustrated magnetism [START_REF] Chun | Direct evidence for dominant bond-directional interactions in a honeycomb lattice iridate Na 2 IrO 3[END_REF] in iridates. Chapter 1 briefly reviews the low-energy physics of these compounds. The j eff = 1/2 ground state is introduced in detail, and the main achievements in the investigation of the magnetic and electronic properties of iridates are reviewed.

Resonant inelastic X-ray scattering (RIXS) is one of the main techniques employed to study the momentum-resolved magnetic and electronic dynamics of iridates. Indeed, the small-volume samples that are currently synthesized and the large neutron absorption of iridium discourage the use of conventional neutron-based techniques. Instead, the favorable RIXS cross-section at the iridium L 3 edge and the currently-available energy resolution promoted the employment of this spectroscopy. Indeed, the energy resolution of current RIXS spectrometers matches the relevant energy scales of the magnetic and electronic excitations in iridates. Chapter 2 reviews the RIXS technique and includes the formal derivation of the scatter-Introduction ing cross-section, while Chapter 3 describes beamline ID20 of the European Synchrotron Radiation Facility (ESRF, Grenoble, France), where most of the data presented in this thesis have been collected. The beamline has been conceived for resonant and non-resonant inelastic X-ray scattering techniques and hosts a state-of-the-art end-station fully dedicated to RIXS.

Chapters 4 to 7 are the core of the present thesis. They comprise the description of instrumental developments to improve the capabilities of the RIXS technique, and the application of RIXS to study the magnetic and electronic excitations of selected iridium compounds.

The main technical development concerns the design of a RIXS spectrometer with polarization analysis capabilities. Indeed, polarization is the only characteristic of the scattered X-rays that is not analyzed today, but is crucial in some cases to assess the nature of the spectral features. So far, only few pioneering works have been done in this direction, particularly in the soft-X-ray energy range [START_REF] Braicovich | The simultaneous measurement of energy and linear polarization of the scattered radiation in resonant inelastic soft x-ray scattering[END_REF][START_REF] Minola | Collective nature of spin excitations in superconducting cuprates probed by resonant inelastic x-ray scattering[END_REF] , owing to severe technical challenges [START_REF] Ishii | Instrumental upgrades of the RIXS spectrometer at BL11XU at SPring-8[END_REF][START_REF] Gao | Development of a graphite polarization analyzer for resonant inelastic x-ray scattering[END_REF][START_REF] Gao | Toroidal silicon polarization analyzer for resonant inelastic x-ray scattering[END_REF] . In Chapter 4, I introduce a novel optical scheme for a RIXS spectrometer that includes polarization analysis without any compromise in terms of energy resolution and efficiency compared to current RIXS spectrometers.

A second technical development relates to the design of sampleenvironment equipment to perform RIXS experiments in low-temperature and high-pressure conditions. This type of measurements is very challenging and has never been performed before. In Chapter 5, I demonstrate the feasibility of performing RIXS in such conditions by investigating the magnetic dynamics of the bilayer-perovskite iridate Sr 3 Ir 2 O 7 . This compound has been selected because the best model suited to describe its magnetic properties is still debated [START_REF] Kim | Large Spin-Wave Energy Gap in the Bilayer Iridate Sr 3 Ir 2 O 7 : Evidence for Enhanced Dipolar Interactions Near the Mott Metal-Insulator Transition[END_REF][START_REF] Sala | Evidence of quantum dimer excitations in Sr 3 Ir 2 O 7[END_REF] . I propose to probe the magnetic dynamics of this compound under extreme thermodynamic conditions to shed light onto the physics of this system.

Chapters 6 and 7 deal with scientific applications of RIXS to investigate the magnetic and electronic structure of iridium compounds. In particular, in Chapter 6 I discuss the low-energy physics of CaIrO 3 . The postperovskite crystal structure of CaIrO 3 hosts both edge-and corner-sharing IrO 6 octahedra [START_REF] Ohgushi | Metal-insulator transition in Ca 1-x Na x IrO 3 with post-perovskite structure[END_REF] and therefore is expected to support fundamentallydifferent interactions along the two directions [START_REF] Jackeli | Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models[END_REF] . By analyzing the dispersion of the magnetic excitations by means of simple model Hamiltonians, I extract the main magnetic couplings. Similarly, I identify possible hopping paths for orbital excitations.

Finally, in Chapter 7, I discuss the case of non-oxide iridium compounds whose electronic structure is supposedly similar to the one of iridates [START_REF] Birol | J eff = 1/2 Mott-insulating state in Rh and Ir fluorides[END_REF] . Indeed, according to recent calculations, iridium fluorides should fall in this category [START_REF] Birol | J eff = 1/2 Mott-insulating state in Rh and Ir fluorides[END_REF] . I test this theoretical prediction and further study the effect of chemical substitution and of application of physical pressure on the crystal and electronic structure of iridium fluorides.

CHAPTER 1

The low-energy physics of iridium compounds Iridium oxides (iridates) have been the subject of a rich variety of investigations due to the theoretical prediction and experimental finding of novel quantum phases and excitations induced by relativistic spin-orbit coupling. This Chapter briefly reviews the main results of this fertile field of research.

Transition-metal oxides in the strong spin-orbitcoupling regime

Transition-metal oxides host a large class of phenomena in solid state physics, ranging from high-temperature superconductivity in copper oxides (cuprates) [START_REF] Keimer | From quantum matter to high-temperature superconductivity in copper oxides[END_REF] , to colossal magnetoresistance in manganese oxides [START_REF] Ramirez | Colossal magnetoresistance[END_REF] , to multiferroicity in frustrated magnets [START_REF] Cheong | Multiferroics: a magnetic twist for ferroelectricity[END_REF] . For many decades, research has mostly been concentrated on materials including transition metals of the 3d series. Indeed, conventional wisdom ascribes the above properties to strong electron correlation from localized narrow 3d bands with large values of the on-site Coulomb repulsion [START_REF] Keimer | From quantum matter to high-temperature superconductivity in copper oxides[END_REF][START_REF] Ramirez | Colossal magnetoresistance[END_REF][START_REF] Cheong | Multiferroics: a magnetic twist for ferroelectricity[END_REF] . The larger spatial extension of the d orbitals when moving towards heavier transition metals is naively expected to reduce both electron correlation and Coulomb repulsion. Such reduction leads to a more itinerant behavior and eventually to a metallic state, which is indeed found in many 4d and 5d compounds [START_REF] Cox | Transition Metal Oxides: An Introduction to Their Electronic Structure and Properties[END_REF] . Therefore, the observation of an insulating state in some 5d oxides (among which, for ) has been puzzling.

The explanation of the insulating behavior came from the pivotal work of Kim et al. [START_REF] Kim | Novel J eff = 1/2 Mott State Induced by Relativistic Spin-Orbit Coupling in Sr 2 IrO 4[END_REF] , who observed that the large spatial extent of the 5d orbitals is accompanied by a strong spin-orbit coupling (SOC). The SOC is a relativistic effect due to the interaction between the spin of the electron and its motion [START_REF] Griffiths | Introduction to Quantum Mechanics[END_REF] . In the electron rest frame, the circular motion of the positivelycharged nucleus generates an apparent magnetic field proportional to the orbital angular momentum l of the electron. By calculating the interaction energy between such field and the magnetic moment of the electron (proportional to the spin angular momentum s), the following spin-orbit Hamiltonian is derived [START_REF] Griffiths | Introduction to Quantum Mechanics[END_REF] :

H SOC = e 2 8π 0 m 2 c 2 Z r 3 l • s, (1.1)
where the pre-factor includes fundamental constants, Z is the atomic number, and r is the orbital radius. To obtain the value of the SOC constant, the expectation value 1/r 3 must be calculated first. It can be shown [START_REF] Griffiths | Introduction to Quantum Mechanics[END_REF] that this is proportional to (Z/n) 3 , from which one obtains a scaling factor of Z 4 /n 3 . It must be noted, however, that the origin of the proportionality of the SOC strength to Z 4 resides in the consideration of hydrogen-like wave functions [START_REF] Griffiths | Introduction to Quantum Mechanics[END_REF] . While the Z 4 scaling factor reproduces fairly well the trend within a specific series of elements, the proportionality becomes Z 2 when moving from transition metals of the 3d series to the ones of the 4d and 5d series [START_REF] Shanavas | Theoretical model for Rashba spin-orbit interaction in d electrons[END_REF] . The calculation of the SOC constant for several families of elements together with the proposed trends are shown in Figure 1.1. As a matter of fact, the SOC strength is estimated to be in the order of ∼ 50 meV for the 3d series while it amounts to ∼ 0.5 eV in iridates [START_REF] Kim | Excitonic quasiparticles in a spin-orbit Mott insulator[END_REF] .

The realization of the j eff = 1/2 state in iridates

In iridates, the cubic crystal field and the strong SOC of iridium modify the effective bandwidth of the 5d states and cooperate with the electron correlation to induce an insulating state [START_REF] Kim | Novel J eff = 1/2 Mott State Induced by Relativistic Spin-Orbit Coupling in Sr 2 IrO 4[END_REF] , as explained below.

Let us consider for simplicity a local model, where only a single iridium ion is considered within its local environment [START_REF] Kim | Novel J eff = 1/2 Mott State Induced by Relativistic Spin-Orbit Coupling in Sr 2 IrO 4[END_REF][START_REF] Jackeli | Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models[END_REF][START_REF] Ament | Theory of resonant inelastic x-ray scattering in iridium oxide compounds: Probing spin-orbit-entangled ground states and excitations[END_REF] . In most iridates, the iridium ion is tetravalent (electronic configuration: [Xe].4f 14 .5d 5 ) and surrounded by oxygen anions at the vertex of an octahedron. The role of the cubic crystal field is considered first, since it is estimated to be the dominant effect [START_REF] Kim | Novel J eff = 1/2 Mott State Induced by Relativistic Spin-Orbit Coupling in Sr 2 IrO 4[END_REF][START_REF] Sala | Crystal field splitting in Sr n+1 Ir n O 3n+1 (n = 1, 2) iridates probed by x-ray Raman spectroscopy[END_REF] . Indeed, the cubic arrangement of the anions lifts the degeneracy of the outermost 5d orbitals of iridium. This is pictorially explained in are directed towards the anions. Therefore, the latter orbitals will experience a stronger Coulomb repulsion and will be lifted to higher energies. In iridates, the splitting between t 2g and e g states due to the cubic crystal field is relatively large (∼ 3 eV [START_REF] Sala | Crystal field splitting in Sr n+1 Ir n O 3n+1 (n = 1, 2) iridates probed by x-ray Raman spectroscopy[END_REF][START_REF] Ishii | Momentum-resolved electronic excitations in the Mott insulator Sr 2 IrO 4 studied by resonant inelastic x-ray scattering[END_REF][START_REF] Sala | CaIrO 3 : A Spin-Orbit Mott Insulator Beyond the J eff = 1/2 Ground State[END_REF] ), so that the 5 electrons occupying the 5d manifold are placed in a low-spin configuration in the t 2g states. For this reason, the empty e g orbitals will be neglected in the following.

It is often convenient to map the t 2g orbitals onto the triplet of states with an effective orbital angular momentum l eff = -1 [START_REF] Jackeli | Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models[END_REF] : The effect of the SOC and of possible additional components to the crystal field due to octahedral irregularities, such as tetragonal distortion, is accounted for by considering the following Hamiltonian [START_REF] Jackeli | Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models[END_REF] :

|t 0 2g = |xy , |t 1 2g = - 1 √ 2 (i|xz + |yz ) , |t -1 2g = - 1 √ 2 (i|xz -|yz ) .
H = λl • s + ∆l 2 z , (1.3)
where λ is the SOC constant and ∆ parametrizes the strength of the tetragonal elongation (∆ > 0) or compression (∆ < 0) of the octahedron. The details of the diagonalization of the above Hamiltonian (Equation 1.3) are reported in Appendix A. The eigenvectors corresponding to the groundstate doublet are:

1 2 , - 1 2 = sin θ|t 0 2g , ↓ -cos θ|t -1 2g , ↑ , 1 2 , 1 2 = -sin θ|t 0 2g , ↑ + cos θ|t 1 2g , ↓ , (1.4)
where tan(2θ) = (2 √ 2λ)/(λ-2∆) accounts for the relative strength of λ and ∆. A sketch of the resulting energy diagram is reported in Figure 1.3 for the special case of a regular octahedron (∆ = 0). As can be seen, the t 2g manifold is split into a lower-energy filled j eff = 3/2 quartet and a higherenergy half-filled j eff = 1/2 doublet. The electron and spin densities of the j eff = 1/2 and j eff = 3/2 wave functions are also reported in Figure 1.3 for the special case of ∆ = 0. Their dependence on the relative strength of the tetragonal crystal field and the SOC is illustrated in Figure 1.4. As can be seen, in the limit of strong positive ∆ the j eff = 1/2 state is dominated by the contribution of the |t 0 2g orbital, while the j eff = 3/2 states are mainly formed by the |t ±1 2g orbitals. In the opposite scenario, i.e. strong compression of the octahedron, the ground state possesses mainly |t ±1 2g character, whereas the |t 0 2g orbital is the dominant contribution to the | 3 2 , ± 1 2 state. We note that the | 3 2 , ± 3 2 state is independent on λ and ∆, being it exactly equal to the |t ±1 2g orbital. More details can be found in Appendix A. When extending the above discussion to ions in a crystal, electron bands should be considered instead of atomic energy levels. This leads to the situ- ation illustrated in Figure 1.5. The partial filling of the t 2g band is expected to lead to a metallic state (Figure 1.5(a)). A large value of the Coulomb repulsion could in principle explain the insulating behavior of these compounds. Indeed, it would split the t 2g manifold into a lower-energy filled band and a higher-energy empty band, giving rise to a conventional S = 1/2 Mott insulator (Figure 1.5(b)). However, the value of the Coulomb repulsion would be unrealistically large [START_REF] Kim | Novel J eff = 1/2 Mott State Induced by Relativistic Spin-Orbit Coupling in Sr 2 IrO 4[END_REF] . The presence of the SOC, instead, splits the t 2g manifold into a lower-energy filled j eff = 3/2 band and a higher-energy half-filled j eff = 1/2 band (Figure 1.5(c)). The large SOC, thus, effectively reduces the bandwidth of the t 2g states into a narrow halffilled j eff = 1/2 band, so that even a small Coulomb repulsion can open a gap, leading to the realization of a j eff = 1/2 Mott insulating state (Figure 1.5(d)).

The first experimental evidences for the realization of the j eff = 1/2 ground state have been achieved on Sr 2 IrO 4 4 . Figure 1.6(a) reports angle-resolved photoemission spectroscopy (ARPES) measurements along high-symmetry directions of the Brillouin zone showing the presence of dispersing bands, none of which crosses the Fermi level, as expected for an insulator. Figure 1.6(b) displays optical conductivity measurements, which exhibit the presence of a finite gap. The maxima A and B of the conductivity correspond to transitions from the occupied bands to the unoccupied j eff = 1/2 level (Figure 1.5(d)). Figure 1.6(c) presents X-ray absorption spectra measured with two orthogonal polarizations of the incident photons. The data points (circles) are compared to theoretical expectations (solid lines) for an equal-weight superposition of |xy , |yz , and |xz orbitals in the ground state, as expected for the j eff = 1/2 state [START_REF] Kim | Novel J eff = 1/2 Mott State Induced by Relativistic Spin-Orbit Coupling in Sr 2 IrO 4[END_REF][START_REF] Sala | Orbital occupancies and the putative j eff = 1 2 ground state in Ba 2 IrO 4 : A combined oxygen K-edge XAS and RIXS study[END_REF] .

Chapter 1 The low-energy physics of iridium compounds

Bond-geometry-dependent magnetic interactions

The iridates studied in this work belong to the class of spin-orbit-induced Mott insulators, situated in the middle of the phase diagram reported in Figure 1. In this region of the phase space the SOC cooperates with the electron correlation decreasing the effective bandwidth of the system and opening an insulating gap 4 , as described above. A distinction can be made between two main families of iridates, i.e. between compounds featuring either corner-sharing or edge-sharing octahedra. Indeed, the seminal work of Jackeli and Khaliullin [START_REF] Jackeli | Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models[END_REF] showed that the magnetic interactions in these two cases are fundamentally different. Such anomalous behavior arises from the peculiar superposition of different orbitals and spin states in the ground-state wave function of iridates (see Equation 1.4).

Conventional Heisenberg interactions for corner-sharing octahedra

For corner-sharing octahedra, the Ir-O-Ir angle is 180 • and two t 2g orbitals participate to the superexchange for a given bond, e.g. the |xy and |xz orbitals for a bond along the x direction [START_REF] Jackeli | Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models[END_REF] as illustrated in Figure 1.7(a). In this case, the dominant magnetic interactions are isotropic and can be mapped onto a conventional Heisenberg Hamiltonian, where the spin moments are replaced by the j eff = 1/2 magnetic moments, usually referred to as pseudospins [START_REF] Jackeli | Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models[END_REF] . The corner-sharing geometry is found, for example, in an analogous electronic structure, where the single hole occupies a double-degenerate state (the j eff = 1/2 orbital for iridates and the |x 2 -y 2 orbital for cuprates). Similarities are found also in the magnetic structure and dynamics: the (pseudo)spins are arranged in a two-dimensional AFM square lattice [START_REF] Boseggia | Locking of iridium magnetic moments to the correlated rotation of oxygen octahedra in Sr 2 IrO 4 revealed by x-ray resonant scattering[END_REF][START_REF] Cao | Weak ferromagnetism, metal-to-nonmetal transition, and negative differential resistivity in single-crystal Sr 2 IrO 4[END_REF] and the spin-wave dispersion of Sr 2 IrO 4 is reminiscent of the one of La 2 CuO 4 [START_REF] Kim | Magnetic Excitation Spectra of Sr 2 IrO 4 Probed by Resonant Inelastic X-Ray Scattering: Establishing Links to Cuprate Superconductors[END_REF][START_REF] Coldea | Spin Waves and Electronic Interactions in La 2 CuO 4[END_REF] . This is shown in Figure 1.8, where the spin-wave dispersion of La 2 CuO 4 measured by inelastic neutron scattering (left panel) [START_REF] Coldea | Spin Waves and Electronic Interactions in La 2 CuO 4[END_REF] is compared to the magnetic dynamics revealed in Sr 2 IrO 4 by RIXS (right panel) [START_REF] Kim | Magnetic Excitation Spectra of Sr 2 IrO 4 Probed by Resonant Inelastic X-Ray Scattering: Establishing Links to Cuprate Superconductors[END_REF] .

The observation of strong similarities between the low-energy physics of iridates and the one of cuprates encouraged a number of works addressed to discover unconventional superconductivity in iridates. From the theoretical side, many works predicted the presence of a superconducting dome in doped Sr 2 IrO 4 17, [START_REF] Yang | Superconductivity in doped Sr 2 IrO 4 : A functional renormalization group study[END_REF][START_REF] Watanabe | Monte Carlo Study of an Unconventional Superconducting Phase in Iridium Oxide J eff =1/2 Mott Insulators Induced by Carrier Doping[END_REF][START_REF] Meng | Odd-Parity Triplet Superconducting Phase in Multiorbital Materials with a Strong Spin-Orbit Coupling: Application to Doped Sr 2 IrO 4[END_REF] . From the experimental side, surface-doped Sr 2 IrO 4 undergoes an insulator-to-metal transition through a phase reminiscent of the "pseudogap" metallic phase of cuprates, as evinced from ARPES measurements [START_REF] Kim | Fermi arcs in a doped pseudospin-1/2 Heisenberg antiferromagnet[END_REF] . The bottom panels of Figure 1.9 show that by increasing the concentration of potassium at the surface (electron doping), the Fermi surface is not fully gapped, but some intensity appears along a segment (called Fermi arc) close to the (π/2, π/2) nodal point of the twodimensional Brillouin zone [START_REF] Kim | Fermi arcs in a doped pseudospin-1/2 Heisenberg antiferromagnet[END_REF] . By increasing the doping concentration, the Fermi arcs spread through the Brillouin zone until the full Fermi surface is closed, indicative of a normal metallic phase [START_REF] Kim | Fermi arcs in a doped pseudospin-1/2 Heisenberg antiferromagnet[END_REF] . When the temperature is decreased, the Fermi arcs collapse into four point nodes with gapless excitations, while the rest of the Brillouin zone is gapped with a d-wave symmetry [START_REF] Kim | Observation of a d-wave gap in electron-doped Sr 2 IrO 4[END_REF] . Similar results have been reproduced in bulk, chemicallydoped Sr 2-x La x IrO 4 [START_REF] Kim | Observation of a d-wave gap in electron-doped Sr 2 IrO 4[END_REF][START_REF] Yan | Electron-Doped Sr 2 IrO 4 : An Analogue of Hole-Doped Cuprate Superconductors Demonstrated by Scanning Tunneling Microscopy[END_REF][START_REF] De La Torre | Collapse of the Mott Gap and Emergence of a Nodal Liquid in Lightly Doped Sr 2 IrO 4[END_REF][START_REF] Terashima | Evolution of the remnant Fermi-surface state in the lightly doped correlated spin-orbit insulator Sr 2-x La x IrO 4[END_REF] . The same phenomenology is encountered in doped cuprates (top panels of Figure 1.9) [START_REF] Yoshida | Pseudogap, Superconducting Gap, and Fermi Arc in High-Tc Cuprates Revealed by Angle-Resolved Photoemission Spectroscopy[END_REF][START_REF] Marshall | Unconventional Electronic Structure Evolution with Hole Doping in Bi 2 Sr 2 CaCu 2 O 8+δ : Angle-Resolved Photoemission Results[END_REF][START_REF] Ding | Spectroscopic evidence for a pseudogap in the normal state of underdoped high-Tc superconductors[END_REF][START_REF] Loeser | Excitation Gap in the Normal State of Underdoped Bi 2 Sr 2 Ca 2 Cu 2 O 8+δ[END_REF][START_REF] Ronning | Photoemission Evidence for a Remnant Fermi Surface and a d-Wave-Like Dispersion in Insulating Ca 2 CuO 2 Cl 2[END_REF][START_REF] Norman | Destruction of the Fermi surface in underdoped high-Tc superconductors[END_REF][START_REF] Damascelli | Angle-resolved photoemission studies of the cuprate superconductors[END_REF] .

As regards the doping dependence of the magnetic structure and excitations, when Sr 2 IrO 4 is electron doped the long-range AFM order is rapidly lost, and the system enters a paramagnetic state where short-range order is preserved [START_REF] Gretarsson | Persistent Paramagnons Deep in the Metallic Phase of Sr 2-x La x IrO 4[END_REF][START_REF] Chen | Influence of electron doping on the ground state of (Sr 1-x La x ) 2 IrO 4[END_REF][START_REF] Liu | Anisotropic softening of magnetic excitations in lightly electron-doped Sr 2 IrO 4[END_REF][START_REF] Pincini | Anisotropic exchange and spin-wave damping in pure and electrondoped Sr 2 IrO 4[END_REF] . This behavior is schematically illustrated in the phase diagram of Figure 1.10. Indeed, resonant X-ray magnetic scattering measurements reveal that for doping concentrations up to 1.5% the AFM Bragg peaks are sharp in the HK reciprocal lattice plane, whereas for larger doping concentrations they become broad and weak (see Fig- Gretarsson et al. [START_REF] Gretarsson | Persistent Paramagnons Deep in the Metallic Phase of Sr 2-x La x IrO 4[END_REF] .] ure 1.10(b)), while no correlation is observed along the L direction [START_REF] Gretarsson | Persistent Paramagnons Deep in the Metallic Phase of Sr 2-x La x IrO 4[END_REF][START_REF] Pincini | Anisotropic exchange and spin-wave damping in pure and electrondoped Sr 2 IrO 4[END_REF] . This is indicative of a two-dimensional short-range AFM order. Panels (c) and (d) of Figure 1.10 report two RIXS maps showing the evolution of the spectral features with doping along high-symmetry directions of the Brillouin zone. In particular, the maps reveal that the magnons persist in the metallic phase of doped Sr 2 IrO 4 , even though strongly damped. Moreover, the (para)magnons are strongly anisotropically softened: the dispersion is almost unchanged along the antinodal direction ((0, 0) → (0, π)), and softened along the nodal direction ((0, 0) → (π, π)) by 20-30%, in analogy to the persistence of damped anisotropically-softened paramagnons in doped cuprates [START_REF] Le Tacon | Intense paramagnon excitations in a large family of hightemperature superconductors[END_REF][START_REF] Le Tacon | Dispersive spin excitations in highly overdoped cuprates revealed by resonant inelastic x-ray scattering[END_REF][START_REF] Dean | Persistence of magnetic excitations in La 2-x Sr x CuO 4 from the undoped insulator to the heavily overdoped non-superconducting metal[END_REF] .

Bond-geometry-dependent magnetic interactions

Albeit an intensive research and strong correspondences in the crystal,

Bond-geometry-dependent magnetic interactions

electronic, and magnetic structure and dynamics of iridates and cuprates, no superconductivity has been reported in the former compounds so far. Nevertheless, the pursue of such study to discover new similarities and discrepancies between the two systems is welcome and may lead to a better understanding of the mechanisms inducing unconventional superconductivity in cuprates.

Quantum compass and Kitaev physics for edge-sharing octahedra

For edge-sharing octahedra, the Ir-O-Ir angle is 90 • , and the two orbitals responsible for the superexchange process (e.g. the yz and xz orbitals for a bond in the xy plane, see Figure 1.7(b)) destructively interfere so that the isotropic Heisenberg coupling is totally suppressed. The exchange Hamiltonian becomes anisotropic and dependent on the spatial orientation of the bond. It can be demonstrated that for a given bond perpendicular to γ = (x, y, z), the Hamiltonian takes the form [START_REF] Jackeli | Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models[END_REF] :

H γ ij = -KS γ i S γ j , (1.5)
where S γ i is the component along γ of the S = 1/2 operator for pseudospins at the lattice site i. This Hamiltonian is analogous to the compass model [START_REF] Nussinov | Compass models: Theory and physical motivations[END_REF] , which has been the subject of numerous studies, since it is the building block of the Kitaev model [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF] . The Kitaev model deals with spin-1/2 magnetic moments that interact in an Ising-like fashion on a honeycomb lattice. In view of the spatially-anisotropic nature of the magnetic Hamiltonian (Equation 1.5), each spin interacts with its neighbors through different components, thus leading to a highly-frustrated ground state. The Kitaev model is intensely sought after since it hosts a spin-liquid ground state and anyon quasiparticles obeying non-Abelian statistics [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF] , i.e. their statistical behavior belongs neither to the one of fermions nor the one of bosons, but rather ranges continuously between the two. It is believed that non-Abelian particles might play a fundamental role in quantum computation [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF][START_REF] Nayak | Non-Abelian anyons and topological quantum computation[END_REF] .

The observation that the bond-directional nature of the d orbitals can be translated into the anisotropy of the magnetic interaction when strong SOC is present [START_REF] Jackeli | Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models[END_REF] has stimulated many works on honeycomb-lattice 4d and 5d transition metal oxides, such as α-RuCl 3 , Na 2 IrO 3 , and Li 2 IrO 3 [START_REF] Rau | Spin-orbit physics giving rise to novel phases in correlated systems: Iridates and related materials[END_REF][START_REF] Chun | Direct evidence for dominant bond-directional interactions in a honeycomb lattice iridate Na 2 IrO 3[END_REF][START_REF] Chaloupka | Kitaev-Heisenberg Model on a Honeycomb Lattice: Possible Exotic Phases in Iridium Oxides A 2 IrO 3[END_REF][START_REF] Singh | Relevance of the Heisenberg-Kitaev Model for the Honeycomb Lattice Iridates A 2 IrO 3[END_REF][START_REF] Choi | Spin Waves and Revised Crystal Structure of Honeycomb Iridate Na 2 IrO 3[END_REF][START_REF] Chaloupka | Zigzag Magnetic Order in the Iridium Oxide Na 2 IrO 3[END_REF][START_REF] Katukuri | Kitaev interactions between j = 1/2 moments in honeycomb Na 2 IrO 3 are large and ferromagnetic: insights from ab initio quantum chemistry calculations[END_REF][START_REF] Sears | Magnetic order in α -RuCl 3 : A honeycomb-lattice quantum magnet with strong spin-orbit coupling[END_REF][START_REF] Chaloupka | Magnetic anisotropy in the Kitaev model systems Na 2 IrO 3 and RuCl 3[END_REF][START_REF] Banerjee | Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet[END_REF][START_REF] Ran | Spin-Wave Excitations Evidencing the Kitaev Interaction in Single Crystalline α-RuCl 3[END_REF][START_REF] Banerjee | Neutron scattering in the proximate quantum spin liquid alpha-RuCl 3[END_REF] . A direct experimental evidence for Kitaev-type magnetic interactions in Na 2 IrO 3 has been found by Chun et al. [START_REF] Chun | Direct evidence for dominant bond-directional interactions in a honeycomb lattice iridate Na 2 IrO 3[END_REF] . Figure 1.11(a) shows diffuse magnetic X-ray scattering measurements. The intensity maps are a good approximation of the spin correlation function S γγ (γ = x, y, z), and clearly reveal the presence of anisotropic interactions, as can be evinced from the spin-component-dependent intensity profile. Pure Heisenberg interactions would lead to the intensity profile of [START_REF] Chun | Direct evidence for dominant bond-directional interactions in a honeycomb lattice iridate Na 2 IrO 3[END_REF] .] component independent. Despite clear evidences of Kitaev-like interactions, these systems do not have a spin-liquid ground state, instead they exhibit magnetic order at low temperature, as revealed by magnetic susceptibility measurements [START_REF] Singh | Relevance of the Heisenberg-Kitaev Model for the Honeycomb Lattice Iridates A 2 IrO 3[END_REF][START_REF] Sears | Magnetic order in α -RuCl 3 : A honeycomb-lattice quantum magnet with strong spin-orbit coupling[END_REF] . The relatively low temperature (∼ 15 K 75 ) of the ordering transition suggests strong frustration, however they can only be considered as approximate realizations of Kitaev magnets. The presence of additional magnetic couplings beyond the Kitaev term is pointed out by inelastic neutron scattering measurements [START_REF] Choi | Spin Waves and Revised Crystal Structure of Honeycomb Iridate Na 2 IrO 3[END_REF][START_REF] Banerjee | Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet[END_REF][START_REF] Ran | Spin-Wave Excitations Evidencing the Kitaev Interaction in Single Crystalline α-RuCl 3[END_REF][START_REF] Banerjee | Neutron scattering in the proximate quantum spin liquid alpha-RuCl 3[END_REF] . The additional interactions are mainly Heisenberg-like, and may be due to direct exchange between iridium ions and/or to a departure from the ideal 90 • Ir-O-Ir bond angle caused by lattice distortions. The experimental works are supported by a large campaign of theoretical studies aiming at exploring the phase space generated by the competition of Kitaev and Heisenberg interactions [START_REF] Chaloupka | Kitaev-Heisenberg Model on a Honeycomb Lattice: Possible Exotic Phases in Iridium Oxides A 2 IrO 3[END_REF][START_REF] Chaloupka | Zigzag Magnetic Order in the Iridium Oxide Na 2 IrO 3[END_REF][START_REF] Katukuri | Kitaev interactions between j = 1/2 moments in honeycomb Na 2 IrO 3 are large and ferromagnetic: insights from ab initio quantum chemistry calculations[END_REF][START_REF] Chaloupka | Magnetic anisotropy in the Kitaev model systems Na 2 IrO 3 and RuCl 3[END_REF][START_REF] Winter | Challenges in design of Kitaev materials: Magnetic interactions from competing energy scales[END_REF] . It has been shown that a rich variety of magnetic states are possible, ranging from AFM order to a spin-liquid state [START_REF] Chaloupka | Kitaev-Heisenberg Model on a Honeycomb Lattice: Possible Exotic Phases in Iridium Oxides A 2 IrO 3[END_REF] .

The search for the accomplishment of the Kitaev model in real mate-

The search for spin-orbit-induced correlated-electron physics beyond iridium oxides

rials is a fertile field of research. Experimentally, many new compounds have been discovered and characterized, while the first studies on doped systems are currently on-going [START_REF] Rau | Spin-orbit physics giving rise to novel phases in correlated systems: Iridates and related materials[END_REF] . Theoretically, the careful identification of the additional magnetic terms and their dependence on material parameters may point the way to new model systems to explore [START_REF] Rau | Spin-orbit physics giving rise to novel phases in correlated systems: Iridates and related materials[END_REF] . The two classes of iridates presented above, namely edge-and cornersharing octahedral systems, do not span the entire field of research related to iridium compounds. An intensively-investigated family of iridates are the pyrochlores, where the frustrated pseudospins are arranged on a lattice of corner-sharing tetrahedra [START_REF] Wan | Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates[END_REF] . Iridium-based pyrochlores are expected to exhibit novel quantum phases generated by the interplay of weak electron correlation and strong SOC [START_REF] Wan | Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates[END_REF][START_REF] Witczak-Krempa | Correlated quantum phenomena in the strong spin-orbit regime[END_REF] . Pyrochlores are found in the lower part of the phase diagram shown in Figure 1. By tuning the electron correlation by substitution of the rare earth [START_REF] Yanagishima | Metal-nonmetal changeover in pyrochlore iridates[END_REF] , novel exotic phases could be realized, e.g. Weyl semi-metals [START_REF] Wan | Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates[END_REF][START_REF] Burkov | Weyl semimetal in a topological insulator multilayer[END_REF][START_REF] Go | Correlation effects on 3d topological phases: From bulk to boundary[END_REF] and axion insulators [START_REF] Wan | Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates[END_REF][START_REF] Wan | Computational Design of Axion Insulators Based on 5d Spinel Compounds[END_REF][START_REF] Go | Correlation effects on 3d topological phases: From bulk to boundary[END_REF] .

Emerging interest is also devoted to systems where the iridium ions are connected by face-sharing octahedra [START_REF] Kugel | Spin-orbital interaction for face-sharing octahedra: Realization of a highly symmetric SU(4) model[END_REF][START_REF] Khomskii | Role of local geometry in the spin and orbital structure of transition metal compounds[END_REF] . The nearest-neighbor Ir-Ir distance in face-sharing compounds may be even shorter than the one of iridium metal. The energy scale of the corresponding hopping integral is thus comparable to the one of SOC. Therefore, a proper analysis of the electronic structure should go beyond the single-ion picture and should consider molecular orbitals instead.

The search for spin-orbit-induced correlatedelectron physics beyond iridium oxides

The search for novel phases induced by the simultaneous action of SOC and electron correlation has been mainly concentrated on the family of iridium oxides in the last decade. In a recent work, Birol and Haule [START_REF] Birol | J eff = 1/2 Mott-insulating state in Rh and Ir fluorides[END_REF] proposed to extend the search to the class of iridium fluorides. Driven by the observation that a small bandwidth favors a Mott insulating state, and pushing the concept to its extreme, they analyzed the properties of Rb and related iridium fluorides are therefore predicted to be j eff = 1/2 Mott insulators.

A first indication of the realization of the j eff = 1/2 state in iridium fluorides is found by Pedersen et al. [START_REF] Pedersen | Iridates from the molecular side[END_REF] , who synthesized compounds where the IrF 6 units are isolated by the presence of molecular complexes bonded to the fluorine ligands. The X-ray absorption spectra and the magnetic circular dichroism are displayed in Figure 1.13, together with the field dependence of the magnetization. From the absorption spectra and the magnetization curve, Pedersen et al. [START_REF] Pedersen | Iridates from the molecular side[END_REF] determined the orbital and spin contributions to the total magnetic moment, finding values close to the ones expected for a j eff = 1/2 system. They therefore concluded that iridium fluorides can host spin-orbit-induced correlated physics.

Chapter 1 The low-energy physics of iridium compounds

If definitively validated, Birol and Haule's prediction extends the search for materials which display an interplay of SOC and electron correlation beyond oxides to fluoride compounds. If synthesized in structures with connected IrF 6 octahedra, these systems could exhibit magnetic ordering and give rise to novel exotic properties. Resonant inelastic X-ray scattering

The search for spin-orbit-induced correlated-electron physics beyond iridium oxides

Resonant Inelastic X-ray Scattering (RIXS) is a photon-in photon-out technique successfully employed for the study of correlated electron systems. Indeed, RIXS probes low-energy excitations in complex materials by measuring the energy, momentum and polarization dependence of the scattered photon. The information gained on the excited states of the material is directly linked to the ground-state electronic structure. In this Chapter, the scattering process of RIXS and its main features are briefly reviewed, paying particular attention to the application of RIXS in the study of 5d correlated electron systems. The last section of this Chapter is dedicated to the mathematical derivation of the generalized Kramers-Heisenberg cross-section which describes the RIXS process.

More detailed and comprehensive reviews of RIXS can be found in Schülke [START_REF] Schülke | Electron Dynamics by Inelastic X-Ray Scattering[END_REF] and Ament et al. [START_REF] Ament | Resonant inelastic x-ray scattering studies of elementary excitations[END_REF]

Features of RIXS

RIXS is a spectroscopic technique which probes different low-energy excitations, from plasmons and charge-transfer excitations at an energy of a few eV, to magnons and phonons at the meV energy scale. In the study of correlated electron systems, the most interesting excitations have orbital and/or magnetic origin. The former are electronic transitions between non-Chapter 2 Resonant inelastic X-ray scattering degenerate valence orbitals. Since in transition metal compounds the valence shell is the d band, these excitations are often called dd excitations. dd excitations are usually well-localized, but if the hopping integral between neighboring ions is large the dd excitations can propagate through the lattice, and thus show a detectable dispersion over the Brillouin zone. In this case, non-localized dd excitations are often called orbitons. Instead, magnetic excitations are perturbations of the electron's spin in the magnetic lattice. These are typically collective and disperse throughout the Brillouin zone.

It must be said that the most widely used spectroscopic technique for the study of crystal field and magnetic excitations is inelastic neutron scattering. Indeed, the typical energy resolution of neutron experiments is as good as a fraction of meV, and the relation between the scattering crosssection and the dynamical structure factor -which describes the electronic and magnetic interactions of the system under study and their evolutionis well understood. However, neutron-based techniques require large sample volumes (several mm 3 ) and cannot be used in some unfavorable cases where the material under investigation contains strong neutron-absorbing elements (such as iridium). In such cases, RIXS overcomes these issues and can be used as an alternative to inelastic neutron scattering.

RIXS features some peculiar characteristics:

• Bulk sensitivity: the penetration depth of the X-rays depends on the photon energy and on the scattering geometry, but for hard X-rays with an energy of ∼ 10 keV, the penetration depth is in the order of ∼ 50 µm. RIXS is therefore a bulk-sensitive probe where artifacts due to surface contamination or surface quality can be safely neglected. However, it must be noted that by exploiting a proper scattering geometry together with element selectivity (see below) RIXS can be used to study buried interfaces, thin films and surface states.

• Small sample volumes: the penetration depth of the X-rays is large enough to guarantee bulk sensitivity, however it is much shorter than that of neutrons. This is due to the strong interaction radiationmatter compared to neutron-matter (typical penetration depths for neutrons are in the order of cm). Hence, a smaller sample volume is required. State-of-the-art RIXS beamlines allow to focus the incident X-ray beam to ∼ 100 µm 2 and samples with this dimension can be probed. This is extremely useful in the case of iridates, where only small crystals can be synthesized, whose dimensions do not allow their investigation with neutrons.

• Element and orbital selectivity: in RIXS the incident photon energy is tuned to a specific electron binding energy (absorption edge) of one of the elements of the compound under study. Each binding energy is

Features of RIXS

characteristic of the specific element and the specific transition from one orbital to another. Thus, the resonance condition implies element and orbital selectivity. Furthermore, since the chemical environment influences the binding energy, one can disentangle the different contributions of ions which reside in non-equivalent chemical sites if the separation among the absorption edges is larger than the experimental resolution.

• Momentum resolution: photons carry a momentum k, where |k| = 2π/λ is the wave vector. During the scattering process, a fraction of the photon momentum is lost and given to the sample. The absolute value of the momentum transfer is:

|q| ≈ 4π λ sin 2θ 2 , (2.1)
where 2θ is the scattering angle. Since the wavelength of X-rays is in the same order of magnitude than the interatomic lattice spacing (i.e. λ ∼ 1 Å), the momentum transfer can span several Brillouin zones. Therefore, RIXS can probe the full dispersion of low-energy excitations in solids.

• Polarization analysis of the scattered photon: photons carry energy, momentum and polarization. The polarization change of the photon is related to the change in angular momentum. Since the total angular momentum is conserved during the RIXS process, the momentum lost by the photon is given to an excitation of the system under study. The analysis of both incident and outgoing photon polarization provides information about the symmetry and the nature of the excitation. Moreover, excitations which overlap in energy and have different polarization dependence can be disentangled in the RIXS spectrum. So far, very few RIXS beamlines allow to analyze the outgoing photon polarization, at the expense of throughput and energy resolution, but many efforts are currently being made to offer this possibility. Chapter 4 deals with the design of a new RIXS spectrometer with polarization analysis capabilities to be installed at beamline ID20 of the ESRF.

• Charge neutrality: RIXS is a photon-in photon-out technique and no charge is added to or removed from the sample. Hence, RIXS excitations preserve the neutrality of the system. This is particularly important when studying insulating samples, such as several correlated electron systems, in order to avoid charging issues.

The numerous advantages offered by the RIXS technique are, however, accompanied by two main drawbacks: the efficiency of the RIXS process, The incoming (outgoing) photons are described by their wave vector k (k ) and polarization ( ). The polarization is usually written as linear combination of two perpendicular components ( π , parallel to the scattering plane, and σ , perpendicular to it). The sample is drawn as a gray plaquette.

Chapter 2 Resonant inelastic X-ray scattering

and the energy resolution. The former is due to the small cross-section of the scattering process. In order to acquire a spectrum in a reasonable amount of time, the use of high-brilliance X-ray sources is necessary: RIXS can only be performed at third generation synchrotrons and free electron lasers. The energy resolution of a RIXS spectrometer is far from the sub-meV energy resolution of a typical neutron experiment, or to the resolution of optical spectroscopies, which is in the order of ∼ 1 meV. However, during the last years, significant improvements have been made to enhance the energy resolution of RIXS spectrometers, which can be as good as 25 meV for the favorable case of the iridium L 3 edge (∼ 11.2 keV). This allows the study of low-energy excitations with unprecedented resolving power, and allows RIXS to complement inelastic neutron scattering for the study of magnetic excitations in some materials.

The RIXS process

During the RIXS scattering process, the photon hitting the sample is scattered from the state with wave vector k and polarization to the state with wave vector k and polarization . A typical scattering geometry is sketched in Figure 2.1. The sample is drawn as a gray plaquette, while the incoming and outgoing photons are represented by their wave vectors (yellow arrows). The two wave vectors generate the scattering plane, shown as a yellow rectangle. The generic polarization of the photon is usually de- composed into a component parallel ( π , black arrows) and perpendicular ( σ , red arrows) to the scattering plane. The scattering angle, i.e. the angle between k and k , is 2θ.

The RIXS process

The microscopic picture of the RIXS process is shown in Figure 2.2. The scattering can be viewed as a two-step process. Considering the system in the ground state at the beginning, the absorption of an X-ray promotes an electron from a core level to an empty valence level. Since the energy of the incoming photon is very high (∼ 10 keV), the system is left with a highly unstable core hole. Within a very short interval of time (∼ 1 fs), an electron from one valence level radiatively decays back into the core level. The emitted X-ray may have a slightly different energy from the incoming one. The energy and momentum of the whole system (photons and electrons) are conserved before and after the scattering:

ω = ω + E exc , k = k + q, (2.2)
where ω ( ω ) and k ( k ) are the energy and momentum of the incoming (outgoing) photon, respectively, E exc is the energy loss, and q is the momentum transfer. Hence, the net result is a final excited state, where one electron has been promoted into an empty valence state. The excitation has energy E exc and carries momentum q.

An additional conservation law concerns the total angular momentum: the change of the angular momentum of the scattered photon (i.e. the change of its polarization) requires the transfer of angular momentum to Chapter 2 Resonant inelastic X-ray scattering the excitation that has been created in the material. In the particular case of magnetic excitations, the change of the electron's spin angular momentum induces the rotation of the polarization of the scattered photon. This is relevant for the study of spin-waves, although the experimental and theoretical demonstration of how RIXS can probe the dispersion of spin waves is very recent [START_REF] Ament | Theoretical Demonstration of How the Dispersion of Magnetic Excitations in Cuprate Compounds can be Determined Using Resonant Inelastic X-Ray Scattering[END_REF][START_REF] Braicovich | Magnetic Excitations and Phase Separation in the Underdoped La 2-x Sr x CuO 4 Superconductor Measured by Resonant Inelastic X-Ray Scattering[END_REF] , and is one of the key features that contributed to the great success of this technique. Indeed, when the RIXS process involves a spin-orbit-coupled intermediate state (e.g. the 2p orbitals in L-edge RIXS) the total, and not the spin, angular momentum of the system is conserved, hence spin-flip transitions may be allowed under certain circumstances.

RIXS cross-section

In this section, the cross-section that describes the RIXS process is derived. I will follow the derivation reported in the reviews by Schülke [START_REF] Schülke | Electron Dynamics by Inelastic X-Ray Scattering[END_REF] , Ament et al. [START_REF] Ament | Resonant inelastic x-ray scattering studies of elementary excitations[END_REF] and Blume [START_REF] Blume | Magnetic scattering of x rays[END_REF] , where a more complete and detailed treatment can be found.

The first step is to consider the Hamiltonian of the whole system constituted by the electrons of the sample and the radiation field. The N electrons which contribute to the scattering have momentum p, position r and spin vector σ (whose components are the Pauli matrices). The radiation field is described by its vector potential A(r), which can be expanded as a linear combination of plane waves [START_REF] Ament | Resonant inelastic x-ray scattering studies of elementary excitations[END_REF] :

A(r) = k, 2V 0 ω k a k, e ik•r + * a † k, e -ik•r , (2.3)
where the sum runs over all the modes with wave vector k and polarization , a † k, (a k, ) creates (annihilates) a photon in the mode k, , and V is the volume of the system.

The Hamiltonian describing the system contains the terms related to the

RIXS cross-section

electrons and photons alone, and to their interaction [START_REF] Ament | Resonant inelastic x-ray scattering studies of elementary excitations[END_REF][START_REF] Sakurai | Advanced Quantum Mechanics, A-W series in advanced physics[END_REF] :

H = N i=1 [p i + eA(r i )] 2 2m + e 2m N i=1 σ i • B(r i )+ + 1 2 e (2mc) 2 N i=1 σ i • E(r i ) × [p i + eA(r i )] -[p i + eA(r i )] × E(r i ) + + N i=1 e 2 ρ(r i ) 8(mc) 2 0 + 1 2 N -1 i=1 N j>i e 2 |r i -r j | - N i=1 M α=1 Z α e 2 |r i -R α | + + k, ω k a † k, a k, + 1 2 , (2.4)
where e is the elementary charge, m is the electron rest mass, c is the speed of light, and E(r) = -∇φ -∂A/∂t and B(r) = ∇ × A are the electric and magnetic fields, respectively. One can choose the scalar potential φ such that

∇ • A = 0, hence A • p = p • A.
Before proceeding with the derivation of the cross-section, the terms in the Hamiltonian 2.4 are briefly explained for clarity [START_REF] Schülke | Electron Dynamics by Inelastic X-Ray Scattering[END_REF] . The first term is the kinetic energy of the electron system in the presence of the photon field. The second term is the Zeeman energy, i.e. the energy of the magnetic moment (e /2m)σ inside the magnetic field B. The third term is the energy of the magnetic moment (e /2m)σ in the apparent magnetic field v × E, which arises from the motion of the electron with velocity v = (p + eA)/m. The pre-factor 1/2 is the Thomas correction, due to a relativistic effect related to the non-commutativity of perpendicular Lorentz transformations. It can be shown that this term is the spin-orbit interaction in an atom under the assumption of a central potential [START_REF] Sakurai | Advanced Quantum Mechanics, A-W series in advanced physics[END_REF] . The fourth term in the Hamiltonian 2.4 is the energy related to the electric charge density ρ. The fifth term is the electron-electron Coulomb repulsion, while the second to last term is the electron-nucleus attraction (Z α and R α are the atomic number and position of the αth nucleus, respectively). The last term is the energy of the photon field, i.e. the sum of harmonic oscillators with frequencies ω k .

It is possible to write the Hamiltonian 2.4 as a sum of two terms: the first one associated to the electrons and the radiation field only, and the second one describing their interaction. The second term can then be treated as a perturbation to the first one. Since RIXS is a two-step process, a second order perturbation theory is needed. During the process, a photon is scattered from the initial state |k, to the final state |k , . This induces a change in the electron system from the ground state |g to the final state |f . Fermi's golden rule to the second order gives the transition probability of Chapter 2 Resonant inelastic X-ray scattering this process:

w = 2π f f |H |g + n f |H |n n|H |g E g -E n 2 δ(E f -E g ), (2.5)
where |g = |g; k, is the ground state of the whole system with energy

E g = E g + ω k , |f = |f; k , is the final state with energy E f = E f + ω k , and |n is the intermediate state whose energy is E n .
The perturbation Hamiltonian, where only the relevant terms are kept from Equation 2.4, is:

H = N i=1 e m A(r i ) • p i + N i=1 e 2 2m A 2 (r i ) + N i=1 e 2m σ i • ∇ × A(r i ) - N i=1 e 2 (2mc) 2 σ i • ∂A(r i ) ∂t × A(r i ).
(2.6)

Quadratic terms in A are the only ones contributing to the first order term in Equation 2.5, for they are proportional to products of creation and annihilation operators (e.g. a k, a † k , ). Only the linear terms in A of H are contributing in first order to the resonant term in Fermi's golden rule. Moreover, the spin-dependent term (third term of Equation 2.6), which gives rise to resonant magnetic scattering, can be estimated to be much smaller than the other term proportional to A • p [START_REF] Schülke | Electron Dynamics by Inelastic X-Ray Scattering[END_REF][START_REF] Ament | Resonant inelastic x-ray scattering studies of elementary excitations[END_REF] . Thus, by substituting this term into Equation 2.5, one obtains:

w = 2π f e 2 2V 0 m 2 √ ω k ω k n N i,j=1 f|e -ik •r i * • p i |n n|e ik•r j • p j |g E g + ω k -E n + iΓ n 2 × δ(E f -E g + (ω k -ω k )), (2.7)
where the lifetime broadening Γ n of the intermediate states accounts for the non-radiative decay processes which are not considered in H . During a typical RIXS experiment, a monochromatic X-ray beam hits the sample, and the energy of the radiation scattered in the solid angle dΩ is analyzed with resolution d ω k . Therefore, the measured quantity is the double differential scattering cross-section, defined as the intensity of the photons scattered in the solid angle (Ω, Ω + dΩ) with energy ( ω k , ω k +d ω k ) normalized to the intensity of the incoming beam. The double differential scattering cross-section is obtained multiplying the transition rate by the density of states in the solid angle dΩ (Vk 2 dk dΩ/(2π) 3 ) and by normalizing to the incident flux c/V. By making use of the relation k 2 dk =

RIXS cross-section

(2π/c) 3 ω 2
k dω k , one obtains the expression for the double-differential scattering cross-section, also known as the generalized Kramers-Heisenberg equation:

d 2 σ dΩd ω k = V 2 ω 2 k c 4 w = πe 4 2 2 0 m 4 c 4 ω k ω k f n N i,j=1 f|e -ik •r i * • p i |n n|e ik•r j • p j |g E g + ω k -E n + iΓ n 2 × δ(E f -E g + (ω k -ω k )). (2.8)
The transition operators of the generalized Kramers-Heisenberg crosssection for RIXS are very generic and include all possible transitions from an initial to a final state through an intermediate state. However, in most cases these operators can be simplified considering that the wavelength of the photon is usually much larger than the radii of the orbitals involved in the transitions, so that the exponential in Equation 2.8 is assumed to be constant, i.e. e ik•r i = 1 + ik • r i + . . . ≈ 1. By making use of the commutator relation [p 2 i /2m, r i ] = ( /im)p i , one obtains for the absorption step:

n|e ik•r i • p i |g ≈ n| • p i |g = im • n| p 2 i 2m , r i |g ≈ im (E n -E g ) n| • r i |g ≈ imω k n| • r i |g , (2.9)
where ω k ≈ E n -E g . The operator is reduced to the well-known dipole operator. A similar relation holds for the emission step.

In the dipole approximation, the double-differential scattering crosssection becomes:

dσ 2 dΩd ω k ≈ πe 4 2 2 0 c 4 ω k ω 3 k f n f|D † |n n|D|g E g + ω k -E n + iΓ n 2 × δ(E f -E g + (ω k -ω k )), (2.10)
where D is the dipole operator: D = N i=1 • r i . The RIXS cross-section 2.10 has some important key features. Firstly, the resonance condition enhances the cross-section by several orders of magnitude. The transition matrix elements are indeed very small, and without the resonant enhancement of the cross-section the acquisition of RIXS spectra in a reasonable amount of time would not be possible even with the use of third-generation synchrotron-radiation sources. The resonance does not imply that the denominator of Equation 2.10 goes to zero, thanks to the introduction of the imaginary term related to the finite lifetime of the intermediate states. Secondly, the inspection of the numerator of Equation 2.10 Chapter 2 Resonant inelastic X-ray scattering reveals that RIXS is indeed a coherent two-step process, where an electron is excited from the ground to the intermediate state through the absorption of a photon, and then it decays from the intermediate to the final state through the emission of another photon. Thirdly, the Dirac δ function ensures the conservation of energy during the whole process: the energy lost by the scattered photon ( ω k -ω k ) is transferred to an excitation of the sample, whose energy is E exc = E f -E g . Thus, the spectral feature of a given excitation is found at constant energy transfer. Such observation provides a simple criterion to distinguish RIXS excitations from other radiative decays: for example, fluorescence peaks are found at constant outgoing photon energy, since they are transitions between two atomic levels of the system. Lastly, we note that the presence of the intermediate states is crucial in RIXS. In fact, they are responsible for the possibility to excite an electron within the same shell (e.g. dd excitations), the direct transition being dipole forbidden.

CHAPTER 3 Experimental instrumentation

Performing RIXS experiments is challenging since RIXS is a photon-hungry technique: if we define the efficiency of the RIXS process as the ratio between the number of photons reaching the detector and the ones hitting the sample, this can be as low as ∼ 10 -11 . In addition, one has to discriminate photon-energy changes extremely small compared to the photon energy itself, i.e. a resolving power as high as ∼ 10 6 is necessary. The advent of third-generation synchrotron sources and the progress in crystal production for monochromators and analyzers is such that these requirements are nowadays met.

In this Chapter, I will briefly introduce synchrotron radiation and concisely illustrate the optical components of the beamline ID20 of the European Synchrotron Radiation Facility (ESRF, Grenoble, France) and its RIXSdedicated spectrometer.

Synchrotron radiation

X-rays are electromagnetic waves whose energy is in the range between approximately 100 eV and 100 keV. Since their discovery by the German physicist Wilhelm Röntgen in 1895, the importance of X-rays was clear to the scientific community. In fact, X-rays opened a new prolific field of research that allowed to assess the properties of electrons and atoms in materials.

Nowadays, high-intensity X-ray beams are produced in synchrotron Chapter 3 Experimental instrumentation light sources, where highly-energetic charged particles are accelerated to produce electromagnetic radiation. A detailed treatment of synchrotron radiation falls outside the purpose of this thesis. The interested reader is referred to, for example, Als-Nielsen and McMorrow [START_REF] Als-Nielsen | Elements of Modern X-ray Physics[END_REF] . I will briefly explain how X-rays are produced in a modern synchrotron source, focusing particular attention to the ESRF, where the experiments discussed in this thesis were performed.

A charged particle can be accelerated by an electric or magnetic field. For a given field, the largest acceleration is obtained for particles with the smallest mass, e.g. electrons or positrons. At the ESRF, electrons are used. They are freed from atoms by exploiting the photoelectric effect or the thermionic emission, and are accelerated in a linear accelerator (Linac) by a high-voltage alternating electric field to an energy of 200 MeV. They are then transferred to the Booster synchrotron -a 300-m-long ring -where they are accelerated to 6.04 GeV by a dynamical field in a radio-frequency cavity. In order to keep the electrons on the right circular path inside the Booster, magnets are used whose magnetic-field strength is synchronized with the field in the radio-frequency cavity. Once electrons have reached their target energy, they are injected into the storage ring -a 844-m-long circular tube -where they pass through several magnets and are accelerated producing radiation.

At the ESRF, most beamlines employ undulators as a source of X-rays. A typical undulator is schematically shown in Figure 3.1. It consists of a periodic array of dipole magnets with alternating magnetic field in the direction perpendicular to the electron motion (see the inset of the Figure ). The electrons that pass through an undulator wiggle and emit radiation with a fundamental wavelength λ 96 :

λ(θ) ≈ λ u 2γ 2 1 + K 2 2 + γ 2 θ 2 , (3.1)
where λ u is the undulator periodicity, γ = E/(mc 2 ) (≈ 11.8 • 10 3 at the ESRF) is the relativistic Lorentz factor, K is the field-dependent undulator deflection parameter, and θ is the observation angle. The γ factor in Equation 3.1 takes into account that at relativistic speed the spatial period in the observer frame and in the electron frame are related by the Lorentz transform [START_REF] Als-Nielsen | Elements of Modern X-ray Physics[END_REF] , i.e. λ obs = λ u → λ u /γ = λ el . The angular dependence of Equation 3.1 is due to the Doppler effect by which the relative motion of the observer with respect to the radiation source causes a change in the radiation wavelength. The radiated energy is concentrated in a narrow cone of angle ≈ 1/γ. The deflection parameter is defined as: where B is the strength of the magnetic field inside the undulator. Therefore, the energy of the radiation produced by the undulator depends on the magnetic field. For experiments at the iridium L 3 edge (λ = 1.105 Å), one obtains K ≈ 0.61 and thus B ≈ 0.25 T at the optimized distance between the two periodic arrays of magnets which amounts to ≈ 17 mm. For K < 1, the amplitude of the sinusoidal motion of the electrons is small (within the radiation cone), and the emitted photons interfere giving high flux in a narrow energy band. The radiation produced by the undulators is linearly polarized in the electron orbital plane.

K = eBλ u 2πmc , (3.2)

Beamline ID20 at the ESRF: optical layout

Beamline ID20 at the ESRF: optical layout

Beamline ID20 at the ESRF is dedicated to hard-X-ray inelastic scattering techniques. ID20 has been recently designed and built as part of the new beamline portfolio of the ESRF in the frame of the Phase I upgrade. It is now fully operational and a flagship beamline for inelastic hard-X-ray scattering. ID20 has been conceived to guarantee some essential requirements, such as to maximize the photon flux on the sample over a wide energy range, to optimize the energy resolution, and to produce a small and stable beam. The beamline hosts two end stations equipped with two state-of-theart spectrometers for resonant and non-resonant inelastic X-ray scattering experiments [START_REF] Huotari | A large-solid-angle X-ray Raman scattering spectrometer at ID20 of the European Synchrotron Radiation Facility[END_REF][START_REF] Sala | A high-energy-resolution resonant inelastic X-ray scattering spectrometer at ID20 of the European Synchrotron Radiation Facility[END_REF] . The layout of the beamline is illustrated in Figure 3.2. The beam is produced by four undulators with a periodicity of 26 mm. The covered energy range is between 4 and 20 keV. The total-reflection collimating mirror (CM1) is the first optical component. Its importance is twofold: reject high-energy harmonics from the undulators, and collimate the beam in the vertical plane (i.e. the reflection plane of the monochromators). Harmonic

Beamline ID20 at the ESRF: optical layout

rejection is realized by coating the Si substrate with two stripes of SiC and Rh, whose cut-off energies are 12 and 23 keV, respectively. Collimation of the beam is achieved by bending the mirror into a cylindrical shape. Benders are provided to reach the optimal radius of curvature that may vary with photon energy. Due to the substantial absorbed power generated by the incident beam, the cylindrical mirror is water cooled to ensure thermal stability.

The beam divergence downstream of the collimating mirror matches the angular acceptance of the double-crystal monochromator (DCM). This is made of two silicon crystals whose parallel surfaces are cut along the (111) lattice planes. By exploiting Bragg's law, only a narrow bandwidth ∆E is reflected by the crystals:

∆E E = ω D tan θ B , (3.3)
where E is the working energy, ω D is the Darwin width of the reflection, and θ B is the Bragg angle. The quantity ∆E/E for a given reflection is angle and energy independent over a wide range. For the Si( 111) reflection, ∆E/E ≈ 1.3 × 10 -4 . The finite width of the reflection can be accounted for by the dynamical theory of diffraction [START_REF] Batterman | Dynamical Diffraction of X-Rays by Perfect Crystals[END_REF][START_REF] Authier | Dynamical theory of X-ray diffraction[END_REF] ; in particular, it is due to the interference between the phase-shifted beam that is reflected or refracted by the stack of crystal planes. The second crystal of the monochromator can translate to ensure that the reflected beam exits at a fixed height for any Bragg angle, an essential condition for the downstream optics. The DCM is liquid-nitrogen cooled to provide thermal stability, since it is exposed to a significant fraction of the beam power and a change in the crystal temperature is directly followed by an undesired change of reflected wavelength.

The energy resolution provided by the DCM is usually not enough for RIXS experiments, and a set of different post-monochromators are available to reach the required resolution at a given energy. For iridium L 3 -edge (∼ 11.2 keV) RIXS experiments, a Si(844) backscattering crystal is used, for which θ B = 85.7 • and ∆E/E ≈ 1.3×10 -6 , thus giving an energy resolution of 15 meV. The backscattering post-monochromator is a channel-cut crystal, i.e. it is machined out of a single piece of silicon. There is no need of cooling the post-monochromator since most of the beam power is absorbed by the collimating mirror and the DCM.

After the post-monochromator, a total-reflection toroidal mirror (FM2) vertically deflects downward the beam to compensate for the upward reflection of the collimating mirror. The toroidal shape allows to focus the beam in both directions to a minimal spot size of 40 × 250 µm 2 (vertical × horizontal, FWHM). The mirror is equipped with a bender that can change its tangential curvature radius to optimize the vertical focal size. The focal point of the toroidal mirror acts as a secondary source (SSA) for the downstream optical elements, and is the place where several diagnostic tools are Chapter 3 Experimental instrumentation installed to monitor the beam intensity, position and shape.

The last optical elements before the sample are the focusing Kirkpatrick-Baez (KB) mirrors [START_REF] Kirkpatrick | Formation of Optical Images by X-Rays[END_REF] . The KB system features two elliptical mirrors (FM3A and FM4A) aligned perpendicular to each other, in order to decouple vertical and horizontal focusing. The optimum spot size at the sample position is better than 10 × 20 µm 2 (vertical × horizontal, FWHM). The KB mirrors can translate by 300 mm along the beam and its focal length correspondingly adapted with benders to host bulky sample environments, e.g. for high-pressure and low-temperature experiments.

All the optical elements of the beamline are kept in ultra-high vacuum (∼ 10 -8 mbar) in order to minimize photon-flux losses.

The RIXS spectrometer

The beamline optical elements provide a focused monochromatic beam whose polarization lies in the horizontal plane. To perform a RIXS experiment, the energy and wave vector of the radiation scattered by the sample must be measured. Therefore, ID20 hosts a high-energy-resolution RIXS spectrometer that allows to measure the energy of the photons scattered into a specific solid angle. A sketch of the spectrometer is shown in To analyze their energy, the scattered X-rays are reflected by five Si(844) crystal analyzers exploiting Bragg's law and working close to backscattering to improve the energy resolution. The covered solid angle is given by the area of the five crystals. On ID20, the Rowland circle geometry is adopted for each analyzer (Figure 3.4): the source of radiation (i.e. the sample), the crystal and the detector lie on the same circumference, and the crystal is bent to a curvature radius equal to the circle diameter, which allows to focus the X-rays on the detector position. Within this geometry, the radiation is reflected from the whole crystal surface with the same Bragg angle. Therefore, the Rowland circle geometry is a way to combine energy analysis and focus of the X-rays in a single optical element.

A spherically-bent crystal analyzer grants a larger covered solid angle and better focal properties with respect to a flat crystal. However, bending a crystal introduces lattice strain that deteriorates the overall energy resolution according to the following relation [START_REF] Huotari | Improving the performance of high-resolution X-ray spectrometers with position-sensitive pixel detectors[END_REF] :

∆E E = l R 1 tan θ B 2 -ν , (3.4)
where l is the absorption length of the crystal, R is the curvature radius and ν is the Poisson ratio. For typical values of l = 0.2 µm 103 , R = 2 m and ν = 0.22, one obtains a resolution of ∆E ∼ 0.5 eV at the iridium L 3

Chapter 3 Experimental instrumentation

edge. To overcome this issue, diced crystal analyzers are used. They are produced from a large flat single crystal which is cut into a matrix of ∼ 10 4 small cubes and eventually glued onto a spherical substrate. Each single dice is strain-free and the intrinsic energy resolution of the reflection is retrieved. The improved energy resolution is accompanied by a decrease in the analyzer reflectivity due to the empty space between the cubes. Moreover, diced crystal analyzers work in a dispersive manner because of the flat surface of each single dice, i.e. a relation is introduced between photon energy and position on the detector [START_REF] Huotari | Improving the performance of high-resolution X-ray spectrometers with position-sensitive pixel detectors[END_REF][START_REF] Huotari | Resonant inelastic hard x-ray scattering with diced analyzer crystals and position-sensitive detectors[END_REF] :

dE dx = E 2R 1 tan θ B , (3.5)
where x is the direction parallel to the detector surface (black arrow in Figure 3.4). The energy range covered by the setup depends on the cube size c as [START_REF] Huotari | Improving the performance of high-resolution X-ray spectrometers with position-sensitive pixel detectors[END_REF] :

∆E c E = c R 1 tan θ B , (3.6)
and represents the energy range that can be simultaneously detected within a single acquisition, i.e. without moving the spectrometer [START_REF] Huotari | Resonant inelastic hard x-ray scattering with diced analyzer crystals and position-sensitive detectors[END_REF] . For a typical dice size c = 1 mm, one obtains ∆E c ∼ 0.4 eV. By using a position-sensitive detector -which can measure both the intensity and the position of the photons -one can take advantage of the energy-position relation to improve the energy resolution. Beamline ID20 features a position-sensitive two-dimensional detector of the Maxipix family [START_REF] Ponchut | MAXIPIX, a fast readout photon-counting X-ray area detector for synchrotron applications[END_REF] with five chips where the X-rays from the five analyzers are detected separately. Each chip is made of a matrix of 256 × 256 square pixels of size p = 55 µm. Of course, the pixel size is a limiting factor for the energy resolution, since one cannot disentangle the different energies within the same pixel. The pixel size contribution can be calculated from Equation 3.5:

∆E p E = p 2R 1 tan θ B . (3.7)
∆E p ≈ 12 meV if the pixel size of the Maxipix detector is considered. An additional contribution to the energy resolution is the Johann aberration [START_REF] Johann | Die Erzeugung lichtstarker Röntgenspektren mit Hilfe von Konkavkristallen[END_REF] , which becomes important when the Bragg angle is far from 90 • . In perfect backscattering condition, the source is at a distance R from the analyzer, and each ray coming from the source impinges at 90 • on the analyzer surface. This is not the case when the backscattering condition is not fulfilled, since the source-to-analyzer distance is R sin θ B . Thus, there is a finite dispersion of Bragg angles over the analyzer surface. The Johann aberration contributes to the energy resolution according to the following equation:

∆E J E = A 2 8R 2 sin 2 θ B 1 tan 2 θ B , (3.8)
where A is the analyzer aperture. The Johann contribution is highly asymmetric and ∆E J overestimates it because it includes the entire extension of the energy distribution. For A = 40 mm, one obtains: ∆E J ≈ 3 meV. Finally, the finite source size s also contributes to the energy resolution, since this is seen as a residual beam divergence of s/(R sin θ B ) at the analyzer position, thus:

∆E s E = s R sin θ B 1 tan θ B . (3.9)
By considering a source size of s = 10 µm, given by the focal spot of the X-ray beam at the sample position, one obtains ∆E s ≈ 4 meV. By convolving all contributions listed above, the overall energy resolution of the beamline and the spectrometer is as good as 25 meV at the iridium L 3 edge.

The crystal analyzers are installed inside a helium-filled chamber to reduce air scattering. The position of each analyzer fixes the outgoing wave vector, and therefore the momentum transfer. This can be adjusted to the desired value by rotating the analyzer chamber in the horizontal and vertical planes, although one often prefers to work in the horizontal scattering plane in the proximity of 90 • to suppress the elastic Thomson scattering signal from the sample. The five crystals allow to measure up to five momentum transfers simultaneously. The momentum resolution of the spectrometer is given by the surface area of the analyzers and can be calculated differentiating Equation 2.1:

∆|q| = 4π λ cos (2θ V ) sin (2θ H ) 2 -2 cos (2θ V ) cos (2θ H ) ∆θ H , (3.10)
where 2θ H (2θ V ) is the scattering angle in the horizontal (vertical) plane and ∆θ H = A/(R sin θ B ) is the solid angle of collection, which depends on the analyzer aperture A. A similar equation is obtained when differentiating with respect to θ V . The momentum resolution amounts to approximately 0.1 Å -1 for the iridium L 3 -edge RIXS case when considering 2θ H ≈ 90 • and 2θ V = 0. The sample is mounted on a goniometer whose motorized translations and rotations around three orthogonal directions allow to align the sample lattice in any arbitrary orientation with respect to the incoming beam. By rotating both the sample and the spectrometer chamber one can reach any desired momentum transfer.

CHAPTER 4 Polarization analysis of the scattered radiation

The RIXS spectrometers built in the past years have been designed to improve the energy-and momentum-resolution and the throughput, while the polarization of the scattered beam is usually not measured due to significant technical challenges. In this Chapter, the project of a new spectrometer for RIXS experiments with polarization resolution to be installed on beamline ID20 is described in detail.

Polarization-resolved RIXS

The analysis of the polarization of the scattered beam gives rise to new benefits for RIXS experiments. First, an accurate polarization analysis of the spectral features allows to gain information on the symmetry of the excitations and therefore on their nature. Second, features which overlap in energy but have different polarization dependence can be distinguished only with the use of a polarization filter. Third, the selection of a specific polarization channel allows to reduce the undesired non-resonant contributions to the RIXS signal, e.g. the elastic line from the sample and the spurious signal from the sample environment. This is particularly beneficial (i) in high-pressure RIXS experiments, where the strong scattering from the sample environment often covers the magnetic signal from the sample; (ii) in those cases where the single magnon lies very close to the elastic line and cannot be distinguished within the energy resolution.

An example of the use of polarization analysis to decouple two overlapping features with different symmetry, but similar energies is found in the pioneering work by Braicovich et al. [START_REF] Braicovich | The simultaneous measurement of energy and linear polarization of the scattered radiation in resonant inelastic soft x-ray scattering[END_REF] in the soft X-rays. The copper L 3edge (∼ 931 eV) RIXS spectrum of YBa 2 Cu 3 O 6.6 is reported in Figure 4.1 (black squares). As can be seen, in the spectrum without polarization analysis (black squares) the separation between elastic and magnetic scattering is not possible. However, it follows from the conservation of the angular momentum during the RIXS process that single spin-flip excitations yield a 90 • rotation of the outgoing polarization with respect to the incident one. Therefore, magnons will appear in the crossed polarization channel only (i.e. the outgoing photon polarization is orthogonal to the incoming one), while elastic scattering will appear in the non-crossed polarization channel (i.e. the outgoing and incoming photon polarizations are both parallel or perpendicular to the scattering plane). The signals from the two channels are shown as red and blue circles in Figure 4.1, respectively. It is evident that the use of a polarimeter allows to safely filter single-spin-flip excitations from elastic and multiple magnon scattering [START_REF] Braicovich | The simultaneous measurement of energy and linear polarization of the scattered radiation in resonant inelastic soft x-ray scattering[END_REF][START_REF] Minola | Collective nature of spin excitations in superconducting cuprates probed by resonant inelastic x-ray scattering[END_REF] .

For the specific case of iridium L 3 -edge RIXS, relevant scientific cases for the development of a spectrometer with polarization resolution concern, for example, the recent debate about the gapped spin waves of Sr 2 IrO 4 18,64,66,67 , and the unresolved magnetic excitations of the honeycomb iridates [START_REF] Choi | Spin Waves and Revised Crystal Structure of Honeycomb Iridate Na 2 IrO 3[END_REF] . The former scientific case has recently attracted some attention since Sr 2 IrO 4 is the prototypical j eff = 1/2 Mott insulator and has many similarities to the high-temperature cuprate superconductor parent compound La 2 CuO 4 , as thoroughly discussed in Chapter 1. In particular, the magnetic dynamics in the two systems have many common points and can be described considering an isotropic Heisenberg-like Hamiltonian [START_REF] Kim | Magnetic Excitation Spectra of Sr 2 IrO 4 Probed by Resonant Inelastic X-Ray Scattering: Establishing Links to Cuprate Superconductors[END_REF][START_REF] Coldea | Spin Waves and Electronic Interactions in La 2 CuO 4[END_REF][START_REF] Gretarsson | Persistent Paramagnons Deep in the Metallic Phase of Sr 2-x La x IrO 4[END_REF][START_REF] Liu | Anisotropic softening of magnetic excitations in lightly electron-doped Sr 2 IrO 4[END_REF] . Though, a recent RIXS study by Pincini et al. [START_REF] Pincini | Anisotropic exchange and spin-wave damping in pure and electrondoped Sr 2 IrO 4[END_REF] undermined this common behavior, pointing out the presence of a relatively large magnetic gap of ∼ 15-20 meV, as shown in Figure 4.2. The observed magnetic gap is independent on the electron doping up to a level of 10%. This finding is a signature of the presence of strong anisotropies in the exchange interactions between pseudospins, and thus breaks the parallelism between iridates and cuprates, with implications for superconductivity in the former compounds. The presence of anisotropic couplings in the basal plane is consistent with a magnetic critical scattering study by Vale et al. [START_REF] Vale | Importance of XY anisotropy in Sr 2 IrO 4 revealed by magnetic critical scattering experiments[END_REF] , and with electron spin resonance [START_REF] Bahr | Low-energy magnetic excitations in the spin-orbital Mott insulator Sr 2 IrO 4[END_REF] and Raman spectroscopy [START_REF] Gim | Isotropic and anisotropic regimes of the field-dependent spin dynamics in Sr 2 IrO 4 : Raman scattering studies[END_REF] experiments which, however, revealed the presence of a finite but much smaller gap (∼ 1 meV). A recent theoretical study [START_REF] Igarashi | Magnetic excitation in resonant inelastic x-ray scattering of Sr 2 IrO 4 : A localized spin picture[END_REF] that considers the role of anisotropic exchange couplings in undoped Sr 2 IrO 4 reveals the presence of two magnetic modes, one of which with a gap of ∼ 20 meV, in agreement with the experimental results by Pincini et al. [START_REF] Pincini | Anisotropic exchange and spin-wave damping in pure and electrondoped Sr 2 IrO 4[END_REF] . Nevertheless, the RIXS measurements by Pincini et al. [START_REF] Pincini | Anisotropic exchange and spin-wave damping in pure and electrondoped Sr 2 IrO 4[END_REF] are in contrast to previous RIXS experiments [START_REF] Kim | Magnetic Excitation Spectra of Sr 2 IrO 4 Probed by Resonant Inelastic X-Ray Scattering: Establishing Links to Cuprate Superconductors[END_REF][START_REF] Gretarsson | Persistent Paramagnons Deep in the Metallic Phase of Sr 2-x La x IrO 4[END_REF][START_REF] Liu | Anisotropic softening of magnetic excitations in lightly electron-doped Sr 2 IrO 4[END_REF] where a magnetic gap was not observed. Polarization-resolved RIXS spectra may confirm the magnetic origin of the shaded feature in Figure 4.2 and give a much more accurate value of the magnetic gap by suppressing the elastic contribution.

Polarization-resolved RIXS

The study of the magnetic dynamics of (Na,Li) 2 IrO 3 and other honey- comb compounds has been boosted in the last decade since the prediction of potential realization of the Kitaev model in these systems [START_REF] Jackeli | Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models[END_REF] . Measuring the magnetic excitations of these compounds, however, is a real challenge. On one hand large-volume single crystals are not available for inelastic neutron scattering. On the other hand, single magnons are almost unresolved in the RIXS spectra [START_REF] Gretarsson | Magnetic excitation spectrum of Na 2 IrO 3 probed with resonant inelastic x-ray scattering[END_REF] due to the relatively small exchange couplings in the order of ∼ 10 meV. Indeed, As can be seen, besides the elastic line (dashed curve), a resolution-limited feature (A 0 ) is present in the spectrum. This excitation has been ascribed to magnetic excitations on top of lattice vibrations [START_REF] Gretarsson | Crystal-Field Splitting and Correlation Effect on the Electronic Structure of A 2 IrO 3[END_REF] . The precise determination of its energy position, however, is not straightforward. Some benefits could be achieved with the use of a polarimeter that filters the magnetic signal from the elastic background.

Early experimental attempts of polarizationresolved hard-X-ray RIXS

In the hard-X-ray energy range, only three prototype RIXS spectrometers with polarization analysis have been built at beamline BL11XU of the Super Photon Ring (SPring-8, Japan) and beamline 30-ID of the Advanced Photon Source (APS, United States) [START_REF] Ishii | Instrumental upgrades of the RIXS spectrometer at BL11XU at SPring-8[END_REF][START_REF] Gao | Development of a graphite polarization analyzer for resonant inelastic x-ray scattering[END_REF][START_REF] Gao | Toroidal silicon polarization analyzer for resonant inelastic x-ray scattering[END_REF] . They are all based on the addition to the main energy analyzer of a second crystal analyzer with a Bragg angle θ as close as possible to 45 • . Indeed, this ensures the suppression of one polarization component since the Thomson scattering factor is proportional to • 90 . In particular, the polarization component parallel to the plane defined by the incoming and reflected beam is completely sup-

Early experimental attempts of polarization-resolved hard-X-ray RIXS

pressed. The polarization extinction ratio ρ = cos(2θ) 2 measures the degree of suppression of the polarization component lying in this plane. At 45 • , ρ = 0 resulting in a complete elimination of one polarization component, while 0 < ρ ≤ 1 for other Bragg angles. The other polarization component can be suppressed by rotating the analyzer to reflect the beam in a direction perpendicular to the previous one. The first prototype of polarization analyzer for RIXS experiments has been developed at beamline BL11XU of the SPring-8 and works in the proximity of the copper K-edge (∼ 8.99 keV) [START_REF] Ishii | Instrumental upgrades of the RIXS spectrometer at BL11XU at SPring-8[END_REF] . Energy and polarization analysis are decoupled by making the scattered beam undergo two reflections: a first one by a spherically-bent Ge(800) energy analyzer, and a second one by a flat pyrolytic graphite crystal exploiting the (600) reflection to obtain polarization resolution (θ ≈ 38 • , hence ρ ≈ 6%). Due to the high convergence of the X-rays reflected by the Ge(800) crystal (in the order of few degrees), only a small fraction of the beam impinges on the surface of the polarization analyzer at the right Bragg angle. Therefore, the graphite crystal has a mosaic spread that matches the convergence of the X-ray beam. Nevertheless, the efficiency of the system is ∼ 1%, and it is limited by the low reflectivity of graphite. The poor throughput results in an excessively long accumulation time. The overall energy resolution is 0.6 eV and is mainly due to the use of a spherically-bent crystal analyzer. The resolution is quite poor for most of the scientific cases studied by RIXS, even though an accurate analysis of the symmetry of the RIXS spectral features allowed to distinguish excitations to the t 2g or e g orbitals in KCuF 3 [START_REF] Ishii | Polarizationanalyzed resonant inelastic x-ray scattering of the orbital excitations in KCuF 3[END_REF] . The polarization analyzer can be rotated by 90 • in order to suppress the other linear component of the polarization.

An improvement to the above prototype has been made at beamline 30-ID of the APS, whose polarization-resolved RIXS spectrometer for copper K-edge experiments is sketched in Figure 4.4 [START_REF] Gao | Development of a graphite polarization analyzer for resonant inelastic x-ray scattering[END_REF] . The main improvements consist in the use of a diced Ge(337) energy analyzer and a concave graphite polarization analyzer to obtain better energy resolution. The diced crystal analyzer focuses the scattered X-rays into the primary focus with a well defined energy gradient. The idea is to use a properly shaped graphite crystal to image the primary focus at the detector position by preserving the energy-position relationship. Here, the graphite crystal has been shaped into a spherical surface as a compromise between two cases, the logarithmic spiral and the elliptical shape. The former would give maximum reflectivity by satisfying the Bragg condition at each point of the surface, but poor energy resolution due to bad imaging properties. The latter would have excellent imaging properties, but poor efficiency due to large deviations from the Bragg condition over the crystal surface. A mosaic graphite crystal was used to match the deviation from the Bragg angle, therefore increasing the throughput. Nevertheless, the efficiency of the spectrometer including the polarization analyzer is in the order of 1%, due to the low reflectivity of graphite. The overall energy resolution is deteriorated from the original 100 meV to 175 meV with polarization analyzer. This comes mostly from the high penetration of the X-ray beam inside the graphite crystal, which blurs the focal spot on the detector, and from the imperfect imaging properties due to the mosaicity of the analyzer.

The third prototype of RIXS spectrometer with polarization resolution has been designed for iridium L 3 -edge (∼ 11.2 keV) RIXS experiments at beamline 30-ID of the APS [START_REF] Gao | Toroidal silicon polarization analyzer for resonant inelastic x-ray scattering[END_REF] . The overall design resembles the one adopted for the copper K-edge case (Figure 4.4), with the use of a diced Si(844) energy analyzer and a concave Si(444) polarization analyzer. The main advantage with respect to the previous setup is the use of a silicon crystal, which guarantees a more accurate surface shaping and smaller X-ray penetration depth. The surface has been bent into a toroidal shape, which is again a compromise between the logarithmic spiral and the ellipse. The Bragg angle of the Si(444) reflection is 44.8 • , hence the extinction ratio is nearly ideal. Nevertheless, the overall energy resolution of 250 meV is 40% worse than the one without the polarization analyzer, and the efficiency is estimated to be ∼ 1%. Both contribute to make this setup challenging to use.

Angular-dispersive analyzers for high energy resolution applications

In the previous paragraph, I discussed early attempts to turn a RIXS spectrometer into a device with additional polarization analysis capabilities. However, the limitations of this approach appear in the system- atic degradation of energy resolution and in the unacceptable loss of efficiency of the spectrometer. Based on previous experience, we conclude that polarization-resolved RIXS should be performed with a dedicated instrument where energy and polarization resolution are intrinsically coupled.

To this aim, I discuss new concepts concerning the design of inelastic X-ray scattering spectrometers aiming at high energy-resolution applications [START_REF] Shvyd'ko | X-Ray Bragg Diffraction in Asymmetric Backscattering Geometry[END_REF][START_REF] Shvyd'ko | Using angular dispersion and anomalous transmission to shape ultramonochromatic x rays[END_REF][START_REF] Huang | An alternative scheme of angular-dispersion analyzers for high-resolution medium-energy inelastic X-ray scattering[END_REF][START_REF] Cai | The Ultrahigh Resolution IXS Beamline of NSLS-II: Recent Advances and Scientific Opportunities[END_REF][START_REF] Shvyd'ko | High-contrast sub-millivolt inelastic X-ray scattering for nano-and mesoscale science[END_REF][START_REF] Kim | Collimating Montel mirror as part of a multi-crystal analyzer system for resonant inelastic X-ray scattering[END_REF] . Their common factor is the exploitation of asymmetric Bragg reflections, i.e. reflections where the reflecting planes are not parallel to the crystal surface, as illustrated in Figure 4.5. The unusual properties of this type of reflections are well explained by the dynamical theory of diffraction which complements the kinematic theory of diffraction by including the effects of refraction at the crystal surface, interference and extinction of X-rays inside the crystal. Without going into the details of this theory, which falls outside the purpose of this work, I here report the main aspects which will be useful for the design of the polarimeter. A more accurate and detailed treatment can be found in Authier [START_REF] Authier | Dynamical theory of X-ray diffraction[END_REF] and Shvyd'ko [START_REF] Shvyd | X-Ray Optics: High-Energy-Resolution Applications[END_REF] .

X-rays impinging on the crystal surface at an angle θ B can be coherently reflected by sets of parallel lattice planes separated by a distance d H if their energy E satisfies Bragg's law: E = (hc/2d H )/ sin θ B . For a given set of lattice planes, this condition is satisfied only for certain pairs of (E, θ B ). Therefore, in the phase space defined by energy and angle, the crystal acts as a slit from which only X-rays satisfying Bragg's law are transmitted, i.e. reflected by the set of lattice planes. predicts that X-rays are reflected with R ∼ 1 within a finite narrow region in the proximity of the nominal Bragg energy and angle. The inset in Figure 4.6 zooms into a small region of the phase space and shows the finite reflection window. This window has spectral width ∆E = H E, where H is a universal parameter (energy and angle independent) of a given Bragg reflection with typical values ≤ 10 -4 for Si. The angular width of the reflection region is ∆θ = ω D , where ω D = H tan θ B is the Darwin width. The angular width of the reflection window can be seen as the angular acceptance of the particular reflection: if the divergence of the incoming X-ray beam is within the angular acceptance of the reflection, the whole beam is reflected, otherwise a fraction of the beam is lost. A similar argument applies to the spectral width of the reflection region.

When asymmetric reflections are considered, the spectral and angular widths are modified with respect to the symmetric case. In particular, the spectral width becomes:

∆E = H |b| E, (4.1)
where b = -sin(θ B -η)/ sin(θ B + η) is the asymmetry factor, defined as the ratio of the sines of the exit and incidence angles. Indeed, from inspection of Figure 4.5, it can be seen that the incidence angle is the sum of the Bragg angle θ B (angle between the wave vector of the incoming beam and the lattice planes) and the asymmetry angle η (angle between the lattice planes and the crystal surface), while the exit angle is the difference between θ B and η. The angular width of the reflection region in the asymmetric case becomes:

Angular-dispersive analyzers for high energy resolution applications

∆θ = ω D |b| . (4.2)
An important consequence of asymmetric reflections is that the phase space of the incident beam and the one of the reflected beam do not coincide any more. Indeed, the width of the reflection regions in the two phase spaces is modified in a different way in the asymmetric case. The spectral width of the reflection region in the phase space of the reflected beam is:

∆E = H |b|E, (4.3)
while the angular width is:

∆θ = ω D |b|. (4.4)
It follows that the beam properties such as bandwidth, divergence and shape are modified after an asymmetric reflection. DuMond diagrams [START_REF] Dumond | Theory of the Use of More Than Two Successive X-Ray Crystal Reflections to Obtain Increased Resolving Power[END_REF] are useful tools to quantitatively calculate the beam properties before and after a given reflection. The DuMond diagram is a simplified representation of the phase space around a pair of (E, θ) of interest. An example is shown in . This is reflected into a beam with the same bandwidth and angular spread α. We note that each energy of the polychromatic beam is reflected at a different angle, thus giving rise to an angular dispersion of reflected energies as illustrated in Figure 4.5. The angular spread α and the dispersion rate D can be calculated from the inspection of Figure 4.7:

       ∆E α = 1 D ∆E -∆E α = E tan θ B =⇒        α = 1 -|b| |b| ω D D = (1 -|b|) tan θ B E (4.5)
The physical origin of the angular dispersion resides in the refraction of X-rays at the crystal surface. Let us consider an X-ray beam inside the crystal. This can be described as a set of plane waves with wave vector k. A Bragg reflection into another set of plane waves with wave vector k is achieved if the following condition applies: k = k + H, where H is a diffraction vector connecting two points of the reciprocal lattice. This condition is equivalent to Bragg's law. After being refracted from the vacuumcrystal interface, the wave vector of the beam in vacuum will have a slightly different magnitude than the one of the wave vector of the beam inside the crystal. Their difference is related to the electric susceptibility χ of the crystal 120 : |k| = (1 + χ/2)|K|, and similarly for the diffracted beam. Since the tangential component of the wave vectors must be conserved during the refraction from the surface, it follows that the vacuum wave vectors (K, K ) differ from the ones inside the crystal by a component parallel to the surface normal z: K = k + κz, K = k + κ z. Hence, the incoming and reflected vacuum wave vectors are not related by the vector H connecting two reciprocal lattice points, but the following relations apply [START_REF] Shvyd | X-Ray Optics: High-Energy-Resolution Applications[END_REF] :

K = K + H, H = H + h, (4.6)
where h = (κ -κ)z is the additional momentum transfer, which turns out to be material and wavelength dependent. Thus, it follows from Equation 4.6 that also K is wavelength dependent, and each energy of the incident beam is reflected at a different angle. The plane defined by H and z is the dispersion plane where the angular dispersion takes place.

Angular-dispersive analyzers for high energy resolution applications

One additional consequence of asymmetric reflections is the modification of the cross-section of the beam in the dispersion plane: σ = σ/|b|, where σ and σ are the cross-section of the incident and diffracted beam, respectively.

It follows from the above Equations that two families of asymmetric reflections can be considered: -1 < b < 0 and b < -1. In the former case, the incident beam is closer to the surface than the reflected beam, i.e. the incidence angle is smaller than the reflection angle. The angular acceptance is increased with respect to the symmetric case, while the divergence of the reflected beam is decreased. The opposite occurs for b < -1.

Based on the above considerations, Shvyd'ko et al. [START_REF] Shvyd'ko | X-Ray Bragg Diffraction in Asymmetric Backscattering Geometry[END_REF] developed a new scheme to monochromate X-rays. This scheme can be used as a spectrometer in inelastic X-ray scattering experiments. The layout envisages a first collimation of the scattered beam by an optical element (e.g. a mirror), and the reflection of the collimated beam by three asymmetric crystals to achieve very high energy resolution and efficiency. A RIXS spectrometer based on Shvyd'ko's original concept and working at the iridium L 3 edge has been proposed [START_REF] Huang | Development of a Novel Resonant Inelastic X-ray Scattering Spectrometer with Resolution Better than 10 meV[END_REF][START_REF] Kim | Quartz-based flat-crystal resonant inelastic x-ray scattering spectrometer with sub-10 meV energy resolution[END_REF] and is under commissioning at beamline 30-ID of the APS, where a sub-10-meV energy resolution is targeted. A schematic representation of the spectrometer is reported in Figure 4.8. First, the X-rays scattered by the sample undergo a total reflection from a mirror, which has a twofold aim: it collects radiation over a relatively large solid angle, in the order of ∼ 10 × 10 mrad 2 ; and it collimates the beam down to a divergence smaller than 0.1 × 0.1 mrad 2 . Afterwards, the beam enters a spectrome-Chapter 4 Polarization analysis of the scattered radiation ter made of three consecutive asymmetric Bragg reflections which further shape the beam in terms of size, divergence, and energy resolution. Each reflection has a specific role. The first Si (111) crystal is asymmetrically cut in order to increase its angular acceptance to ≈ 80 µrad and match it to the beam divergence downstream of the collimating stage. The asymmetry angle is 9 • , and the asymmetry factor is b C = -0.06. The residual divergence of the beam is decreased by the factor |b C | to ≈ 5 µrad. Therefore, the beam is further collimated by the first crystal, which is usually referred to as the collimator (C). The X-rays are then reflected by the Si(844) planes of the second crystal. The asymmetry angle is 77.5 • and the asymmetry factor is b D = -0.5. After the reflection, the angular dispersion of the beam is increased to D D ≈ 0.6 µrad/meV, so that each energy of the polychromatic incident beam is reflected into a different angle. The angular dispersion is drawn as a rainbow in Figure 4.8. The second crystal is called the disperser (D). The third Si(440) crystal is asymmetrically cut in order to reduce its angular acceptance to ≈ 3 µrad. Indeed, the idea is to use the third crystal as an angular slit, which selects only a small fraction of the dispersing beam. In this way, the energy resolution is highly enhanced to approximately 7 meV. The third crystal is usually called the wavelength selector (S). The scheme is referred to as CDS analyzer from the name of the three crystals used. It offers a very good energy resolution, but no polarization analysis of the photons.

Polarization-resolved RIXS spectrometer for beamline ID20

Exploiting the concepts of dynamical diffraction theory on asymmetric reflection from crystals summarized in the previous Section, I developed a novel spectrometer to include polarization analysis of the beam and preserve high efficiency and good energy resolution. The polarizationresolved RIXS spectrometer is designed to work at the iridium L 3 edge.

The polarimeter is shown in Figure 4.9 and it features a collimating optical element and only two asymmetrically-cut crystals. The first optical element is essential to collect a sufficiently large solid angle and to collimate the X-rays. Two crystals are then used to perform energy and polarization analysis of the beam. The reduced number of elements compared to the conventional CDS scheme aims at simplifying the alignment and utilization of the polarimeter, while preserving its performance in terms of energy resolution and efficiency. The collimating element and the crystal optics are described in more detail in the following.

The collimating element

As in the case of the CDS setup described above, a Montel mirror seems to be a good choice for the collimating optics for its compactness, ease of alignment and mechanical stability [START_REF] Kim | Collimating Montel mirror as part of a multi-crystal analyzer system for resonant inelastic X-ray scattering[END_REF] . A Montel mirror is essentially made of two perpendicular optical surfaces mounted side by side. By shaping the surfaces to a parabolic shape, the X-rays from a point source are collimated in two dimensions after being reflected twice by the mirror surfaces. The surface of the Montel mirror is coated with a multilayer structure. The multilayer mirror works in a relatively narrow (1%) bandwidth around the iridium L 3 edge. A sketch of the multilayer Montel mirror is reported in Figure 4.10.

In the design of the Montel mirror, two main aspects should be considered: the maximization of the collected solid angle Ω, and the minimization of the residual beam divergence ∆ θ in the dispersion plane. The former is given by:

Ω = L f sin φ 2 , (4.7)
where L is the length of the mirror, f is the focal distance (sample to mir-

Chapter 4 Polarization analysis of the scattered radiation

ror center), and φ is the incidence angle. The residual beam divergence, instead, can be approximated by:

∆ θ = s f 2 + (2Σ) 2 , (4.8)
where s is the source size, and Σ is the slope error of the mirror surface. To find a good set of parameters, one should keep in mind that φ is related to the multilayer periodicity d by Bragg's law: sin φ = λ/(2d), and d is technologically limited to 2 nm or above. Hence, φ is in the order of 1-2 • . Furthermore, to guarantee enough space for the sample environment, such as low-temperature and/or high-pressure equipment, the mirror edge should be at least 20 cm away from the sample position. This gives additional constraints on L and f . Finally, both the collected solid angle and the residual divergence share the same dependence on the focal distance, thus a trade-off value must be chosen. The final adopted parameters of the Montel mirror are reported in Table 4.1. In particular, the collected solid angle amounts to 16 × 16 mrad 2 , i.e. comparable to the one of a conventional RIXS spectrometer with an analyzer aperture of 35 mm at approximately 2 m. The residual divergence of ≈ 40 µrad in the dispersion plane of the asymmetric reflections is estimated considering that the beam size at the sample position is 10 µm. This contribution dominates the residual beam divergence, but a potential modification of the beamline focusing optics during the ESRF-EBS upgrade might reduce the spot size down to 4 µm, in which case the source size contribution (∼13 µrad) will be reduced to below the slope error contribution (∼24 µrad). The beam divergence in the non-dispersive plane is dominated by the source size (∼30 µm) in this direction, but has little influence on the energy resolution because it couples to the Bragg angle only in second order. We note that the source size in the non-dispersive plane becomes significant when the polarization analyzer is rotated by 90 • around the sample-to-mirror axis (dashed blue line in Figure 4.9) in order to suppress the other polarization component. Nevertheless, this issue will be settled after the ESRF-EBS upgrade since a sub-10-µm source size is expected in both dispersive and non-dispersive planes.

The two-crystal setup

In order to quantitatively calculate the energy resolution of the spectrometer and the degrading effects of the residual beam divergence, I make use of the DuMond diagrams [START_REF] Dumond | Theory of the Use of More Than Two Successive X-Ray Crystal Reflections to Obtain Increased Resolving Power[END_REF] . The phase space of the incident beam for the asymmetric Si(311) reflection is shown in Figure 4.11(a). From Equation 4.2, one can estimate the angular acceptance of the reflection (angular width of the green band in the Figure ): where b 1 = -0.06 has been chosen in order to match the angular acceptance to the residual beam divergence downstream of the Montel mirror. This corresponds to the asymmetry angle η 1 = 17.5 • . The spectral width of the reflection region is given by Equation 4.1:

∆θ 1 = ω D,1 |b 1 | ≈ 40 µrad, (4.9)
θ 1 * Δθ´1 slope = 1/D 1 Δθ 2 Δθ´2 slope = 1/D 2 (a) (b) (c) θ 1 θ´1 θ 2 θ´2 E ΔE 1 ΔE´1 ΔE 2 ΔE´2 ΔE α 1 α 2
∆E 1 = ω D,1 |b 1 | E tan θ B,1 ≈ 1.2 eV, (4.10)
Let us first neglect the effects of the beam divergence and consider perfectly collimated X-rays delivered by the Montel mirror (vertical blue line in ). The green band is steeper than the orange one since θ B,2 > θ B,1 . A narrow intersection region between the two reflection bands is found, in particular between the orange band and the blue line. The bandwidth ∆E is given by:

∆E = ω D,2 |b 2 | E tan θ B,2 + D 1 E ≈ 14.4 meV. (4.11)
The X-rays transmitted to the phase space of the reflected photons of the second crystal are plotted as a blue line in panel (c), whose slope is:

D 2 = |b 2 |D 1 + (|b 2 | -1) tan θ B,2 E ≈ 0.9 µrad/meV, (4.12)
and whose angular spread is α 2 = D 2 ∆E ≈ 13 µrad. The treatment above is valid for the ideal case of perfectly collimated X-rays. I now consider the real case of the Montel mirror delivering a beam with residual divergence ∆ θ. The DuMond diagrams representing the reflections from the Si(311) and Si(444) crystals in the case of a divergent beam are shown in Figure 4.12. The effects of the beam divergence are taken into account considering all photons inside the blue region of width ∆ θ, which are then propagated through the system. Figure 4.12(a) represents the phase space of the incident beam for the Si(311) crystal. It is evident from the DuMond diagram that there is an additional contribution to the energy resolution due to the divergence, which is:

Polarization-resolved RIXS spectrometer for beamline ID20

(a) (b) (c) θ 1 θ´1 θ 2 θ´2 E α 2 ΔE α 1 Δθ Δθ 1 ΔE Δθ 2 ΔE 1 ΔE max ΔE 1
∆ Ē1 = E∆ θ tan θ B,1 ≈ 1.2 eV, (4.13)
where ∆ θ ≈ 40 µrad. Figure 4.12(b) shows the phase space of the reflected photons of the first crystal and the overlapping phase space of the incident photons of the second crystal. The blue area has slope equal to 1/D 1 and width ∆ θ -D 1 ∆ Ē1 . The intersection between the orange and blue regions represents the reflected beam from the second crystal. The additional contribution to the energy resolution due to the divergence is then:

∆ Ē = ∆ Ē1 tan θ B,1 -D 1 E tan θ B,2 + D 1 E . (4.14)
However, because of the dispersive nature of the setup, the energy resolution of the spectrometer has as upper limit the energy ∆E max :

∆E max = E tan θ B,1 + tan θ B,2 (∆θ 1 + ∆θ 2 ) ≈ 25 meV, (4.15)
defined by the intersection between the green and orange reflectivity bands. A residual beam divergence that would make ∆E exceed ∆E max will result in a loss of the spectrometer efficiency, but not in a deterioration of the energy resolution. We can therefore define the maximum value ∆ θmax below which the residual beam divergence does not cause an intensity loss. It can be estimated with the help of Equations 4.13 and 4.14 by imposing ∆ Ē1 = (∆E max ) 2 -(∆E) 2 . It turns out that: which roughly coincides with the expected residual divergence of the X-ray beam. The energy resolution of the spectrometer is finally given by:

∆ θmax = tan θ B,1 (∆E max ) 2 -(∆E) 2 E tan θ B,2 + D 1 E tan θ B,1 -D 1 E ≈ 39 µrad, (4.16)
∆E spec = min (∆E) 2 + ∆ Ē 2 , ∆E max . (4.17)
The overall energy resolution is obtained convolving the spectrometer contribution to the 15-meV-bandwidth of the Si(844) post-monochromator. A resolution of approximately 30 meV is achieved. Therefore, the proposed spectrometer provides the same energy resolution currently available at beamline ID20, and has the great advantage of introducing polarization resolution. The main parameters of the two asymmetric crystals are summarized in Table 4

Ray-tracing simulations

The DuMond diagrams offer a simple way to visualize the effects of the asymmetric reflections, and to calculate approximate values of the energy resolution. For more accurate results, I also performed ray-tracing simulations using the software Shadow [START_REF] Sanchez Del Rio | SHADOW3: a new version of the synchrotron X-ray optics modelling package[END_REF] . The package Shadow implements the equations of the dynamical theory of diffraction to give very accurate results of the optical system. I limit the simulation of the optical layout to the two asymmetric crystals for simplicity. The X-rays delivered by the Montel mirror are modeled by a point source of polychromatic radiation with a divergence of 40 µrad. The beam then undergoes two reflections from the asymmetric Si(311) and Si(444) crystals, respectively. Figure 4.13 plots the energy distribution of the photons at a plane placed after the two crystals and orthogonal to the beam direction. The histogram can be fitted to a Gaussian function with FWHM = 21 meV. The efficiency of the system, defined as the number of photons within the FWHM normalized to the initial number of photons within the same bandwidth, is 45%.

The overall efficiency, including also the reflectivity of the Montel mirror, is 27%. This is estimated to be approximately a factor 2 less than a conventional RIXS spectrometer covering the same solid angle [START_REF] Sala | High energy-resolution set-up for Ir L3 edge RIXS experiments[END_REF] .

The analysis based on DuMond diagrams shows that the beam divergence is a crucial parameter for the polarimeter, as it greatly affects its efficiency and energy resolution. Hence, I investigated the effects of the beam divergence more accurately with ray-tracing simulations. The results of the ray-tracing simulation and of the DuMond diagram analysis are shown in Figure 4.14. The quantitative agreement is fairly good: as expected, the en- ergy resolution increases with the beam divergence up to ∆ θmax ≈ 40 µrad, and then reaches a plateau at approximately 25 meV. The efficiency of the two crystals is maximum (≈ 73%) for a perfectly collimated beam and decreases monotonically with the beam divergence.

While day-1 operations will be carried out with a residual beam divergence of 40 µrad, it may improve in the near future. As already mentioned, the future refurbishment of the beamline focusing optics in the framework of the ESRF-EBS upgrade project will lead to a smaller beam size at the sample position, resulting in a residual beam divergence of ∼ 30 µrad downstream of the Montel mirror. Figure 4.14 suggests that the energy resolution will improve to ≈ 18 meV and the efficiency to ≈ 55%.

Feasibility test of the polarimeter

In order to validate the design of the polarimeter, we carried out preliminary tests on the two-crystal setup. The Si(311) and Si(444) crystals have been asymmetrically cut with high precision by the X-ray optics laboratory of the ESRF. The crystals have been installed in the beamline in the dis- persive geometry of the polarimeter, and their energy resolution has been determined by rocking the Si(844) post-monochromator and by recording the intensity of the reflected beam after the two asymmetric crystals. The results are reported in Figure 4.15 for three different values of the beam divergence: 18, 31 and 96 µrad. The corresponding energy widths are 20, 24 and 26 meV, respectively. By subtracting the contribution of the incident bandwidth, the energy resolution of the two-crystal setup reduces to 14, 19 and 22 meV. Polarization resolution was tested by rotating the polarization of the incident beam by 90 • and recording a similar energy scan. We observed an intensity drop of the transmitted beam by more than 5 orders of magnitude, in line with an extinction ratio ρ of 3 × 10 -5 for the Si(444) reflection. The performance in terms of energy resolution and polarization analysis is perfectly consistent with the simulated ones, as can be seen in Figure 4.14, making us confident that the polarimeter will work as expected.

Conclusions and future work

Polarization analysis of the scattered radiation during RIXS experiments has been proven to be beneficial in many cases [START_REF] Braicovich | The simultaneous measurement of energy and linear polarization of the scattered radiation in resonant inelastic soft x-ray scattering[END_REF][START_REF] Minola | Collective nature of spin excitations in superconducting cuprates probed by resonant inelastic x-ray scattering[END_REF][START_REF] Ishii | Polarizationanalyzed resonant inelastic x-ray scattering of the orbital excitations in KCuF 3[END_REF] . For example, a full polarization analysis of a spectral feature brings information on its symmetry and specific nature; moreover, two unresolved features in the RIXS spectrum can be disentangled if their polarization response is different; fi-Chapter 4 Polarization analysis of the scattered radiation nally, polarization analysis could help to improve the RIXS signal-to-noise ratio by filtering out the spurious non-resonant scattering from the sample itself or from the sample environment.

Recent upgrades in the RIXS beamlines of the SPring-8 and the APS allowed the measurement of the polarization of the scattered beam [START_REF] Ishii | Instrumental upgrades of the RIXS spectrometer at BL11XU at SPring-8[END_REF][START_REF] Gao | Development of a graphite polarization analyzer for resonant inelastic x-ray scattering[END_REF][START_REF] Gao | Toroidal silicon polarization analyzer for resonant inelastic x-ray scattering[END_REF] . However, the simple design consisting in the decoupling of the energy and polarization analysis by the use of two crystal analyzers resulted in extremely long counting times and a deterioration of the energy resolution to few hundreds of meV.

I reported in this Chapter the concept of a new RIXS spectrometer with polarization analysis capabilities based on the CDS analyzer by Shvyd'ko et al. [START_REF] Shvyd'ko | X-Ray Bragg Diffraction in Asymmetric Backscattering Geometry[END_REF] . This scheme has been recently adapted for high-resolution inelastic X-ray scattering applications [START_REF] Shvyd'ko | Using angular dispersion and anomalous transmission to shape ultramonochromatic x rays[END_REF][START_REF] Huang | An alternative scheme of angular-dispersion analyzers for high-resolution medium-energy inelastic X-ray scattering[END_REF][START_REF] Cai | The Ultrahigh Resolution IXS Beamline of NSLS-II: Recent Advances and Scientific Opportunities[END_REF][START_REF] Shvyd'ko | High-contrast sub-millivolt inelastic X-ray scattering for nano-and mesoscale science[END_REF][START_REF] Kim | Collimating Montel mirror as part of a multi-crystal analyzer system for resonant inelastic X-ray scattering[END_REF][START_REF] Huang | Development of a Novel Resonant Inelastic X-ray Scattering Spectrometer with Resolution Better than 10 meV[END_REF][START_REF] Kim | Quartz-based flat-crystal resonant inelastic x-ray scattering spectrometer with sub-10 meV energy resolution[END_REF] . I proposed a simplified CDS design in order to simultaneously analyze the energy and polarization. The scattered photons are first collimated by a Montel multilayer mirror, which covers a solid angle of 16 × 16 mrad 2 and has an efficiency of ≈ 60%. The residual beam divergence is estimated to be ∼ 40 µrad. Then, the photons undergo two asymmetric reflections through a first Si(311) crystal and a second Si(444) crystal. As the Bragg angle of the Si(444) crystal is very close to 45 • at the iridium L 3 edge, only one component of the polarization is reflected. The energy resolution of the polarimeter, simulated with the ray-tracing software Shadow, is 21 meV and its efficiency ∼ 27%, i.e. only a factor 2 less than the one of the current RIXS spectrometer installed at beamline ID20. The energy resolution of the two-crystal ensemble was tested on the beamline and a value perfectly consistent with the simulated one was measured. A future upgrade of the beamline during the ESRF-EBS upgrade project will allow to decrease the residual beam divergence after the Montel mirror to ∼ 30 µrad, thus improving the performances of the polarimeter in terms of energy resolution and efficiency.

The polarimeter will be installed in the Experiment Hutch 2 of beamline ID20. The conceptual design is shown in Figure 4.16. The Montel mirror and the two asymmetric crystals will be mounted on a metallic frame that can be easily installed in place of the current RIXS spectrometer. The spectrometer features a 90 • rotation about the scattered X-ray beam propagation, thus allowing to disentangle the polarization of the beam in the two perpendicular directions.

The multilayer Montel mirror was recently purchased and its delivery is expected in 2018. The asymmetric crystals were realized by the X-ray optics laboratory of the ESRF and are already available. When the Montel mirror will be delivered and all the equipment mounted on the beamline, a commissioning phase is foreseen, where the characteristics of the new spectrometer will be tested. After this initial phase, the polarimeter will be available to the user community of the beamline. In this Chapter, I present recent instrument developments made on beamline ID20 at the ESRF that allow extending the study of magnetic excitations with RIXS to the high-pressure regime. This field is rather unexplored as the technique usually exploited to study the magnetic dynamics, inelastic neutron scattering, is limited to the ∼ 1 GPa pressure range by the experimental complexity involving both low-temperature equipments and tiny single-crystal samples 126 .

Introduction

RIXS is nowadays an established and reliable spectroscopic technique to study low-energy excitations of materials. The main advantages and drawbacks of the technique have been briefly reviewed in Chapter 2. Although RIXS is used since many decades to study elementary excitations in different compounds, the theoretical and experimental demonstration of the possibility to measure single-spin-flip excitations [START_REF] Ament | Theoretical Demonstration of How the Dispersion of Magnetic Excitations in Cuprate Compounds can be Determined Using Resonant Inelastic X-Ray Scattering[END_REF][START_REF] Braicovich | Magnetic Excitations and Phase Separation in the Underdoped La 2-x Sr x CuO 4 Superconductor Measured by Resonant Inelastic X-Ray Scattering[END_REF] contributed enormously to the ascent of this technique. Indeed, the discovery has opened a new field of research, where spin waves are studied with X-rays in a way complementary to inelastic neutron scattering.

Chapter 5 Performing RIXS at high pressure and low temperature: the case of Sr 3 Ir 2 O 7

RIXS in the hard-X-ray energy range can be coupled to complex sample environments to study the dynamics of matter under extreme conditions, e.g. high pressure [START_REF] Rueff | Inelastic x-ray scattering by electronic excitations under high pressure[END_REF] . High-pressure studies of condensed matter are extremely valuable, since pressure can be effectively used to alter the electron density and, for example, induce structural, electronic, and magnetic phase transitions.

I here present recent instrument developments made on beamline ID20 at the ESRF that allow performing RIXS measurements of electronic and magnetic excitations under high pressure and low temperature conditions. This work is motivated by the widespread interest in the study of magnetic dynamics in correlated-electron systems, which we aim to extend to the high pressure regime.

The developed setup is employed to probe the pressure evolution of the magnetic excitations of Sr 3 Ir 2 O 7 . This compound has raised some attention in the last years since general consensus on the microscopic Hamiltonian that describes its magnetic interactions has not been reached yet. While the data that I present do not provide a definitive answer to the question, they enrich the available database. In addition, to the best of my knowledge, the first observation of single-magnon excitations at pressures above 10 GPa is produced 126 .

Crystal and magnetic structure of Sr 3 Ir 2 O 7

The crystal structure of Sr 3 Ir 2 O 7 is shown in Figure 5.1 together with the one of Sr 2 IrO 4 for comparison. In both compounds, iridium ions (gray spheres) are surrounded by octahedra of oxygen ions (red spheres). The octahedra share corners along both a and b crystallographic axes, thus generating perovskite-like layers [START_REF] Crawford | Structural and magnetic studies of Sr 2 IrO 4[END_REF][START_REF] Cao | Anomalous magnetic and transport behavior in the magnetic insulator Sr 3 Ir 2 O 7[END_REF][START_REF] Cao | Weak ferromagnetism, metal-to-nonmetal transition, and negative differential resistivity in single-crystal Sr 2 IrO 4[END_REF][START_REF] Subramanian | Single crystal structure determination of double layered strontium iridium oxide Sr 3 Ir 2 O 7[END_REF] . In Sr 2 IrO 4 , which is the first member of the Ruddlesden-Popper series Sr n+1 Ir n O 3n+1 (n = 1), the perovskite-like layers are isolated [START_REF] Crawford | Structural and magnetic studies of Sr 2 IrO 4[END_REF][START_REF] Cao | Weak ferromagnetism, metal-to-nonmetal transition, and negative differential resistivity in single-crystal Sr 2 IrO 4[END_REF] . Instead, in Sr 3 Ir 2 O 7 , the following member of the Ruddlesden-Popper series (n = 2), two layers are connected by bridging oxygens along the c axis [START_REF] Cao | Anomalous magnetic and transport behavior in the magnetic insulator Sr 3 Ir 2 O 7[END_REF][START_REF] Subramanian | Single crystal structure determination of double layered strontium iridium oxide Sr 3 Ir 2 O 7[END_REF] .

The presence of the bilayers influences the physics of Sr 3 Ir 2 O 7 with respect to the one of the single-layer Sr 2 IrO 4 . Though both compounds show an AFM long-range order on a square lattice below T N ≈ 280 K [START_REF] Cao | Anomalous magnetic and transport behavior in the magnetic insulator Sr 3 Ir 2 O 7[END_REF][START_REF] Cao | Weak ferromagnetism, metal-to-nonmetal transition, and negative differential resistivity in single-crystal Sr 2 IrO 4[END_REF] , the j eff = 1/2 magnetic moments (orange arrows in Figure 5.1) in Sr 2 IrO 4 align in the ab plane [START_REF] Boseggia | Locking of iridium magnetic moments to the correlated rotation of oxygen octahedra in Sr 2 IrO 4 revealed by x-ray resonant scattering[END_REF] , while in Sr 3 Ir 2 O 7 they are forced to align parallel to the c axis by the finite interlayer coupling [START_REF] Boseggia | On the magnetic structure of Sr 3 Ir 2 O 7 : an x-ray resonant scattering study[END_REF][START_REF] Kim | Dimensionality driven spin-flop transition in layered iridates[END_REF] . Both systems are insulators [START_REF] Cao | Anomalous magnetic and transport behavior in the magnetic insulator Sr 3 Ir 2 O 7[END_REF][START_REF] Cao | Weak ferromagnetism, metal-to-nonmetal transition, and negative differential resistivity in single-crystal Sr 2 IrO 4[END_REF] , although the bilayer compound is on the verge of a metal-to-insulator transition as can be evinced by the softening of the charge gap with respect to the single-layer system [START_REF] Moon | Dimensionality-Controlled Insulator-Metal Transition and Correlated Metallic State in 5d Transition Metal Oxides Sr n+1 Ir n O 3n+1 (n = 1, 2, and ∞)[END_REF] . Therefore, Sr 3 Ir 2 O 7 is an intriguing case for investigating the role of the j eff = 1/2 wave function on a bilayer square lattice, where interlayer exchange potentially coexists with intralayer cou- pling. Indeed, RIXS measurements of the magnetic excitations of Sr 3 Ir 2 O 7 strongly differ from the ones of Sr 2 IrO 4 18,25 . Figure 5.2 reports the momentum dependence of the low-energy excitations of Sr 3 Ir 2 O 7 . Three features are present in the spectra: a quasielastic peak (A), an intense band with gap of ≈ 90 meV and width of ≈ 70 meV (B), and a weak broad feature (C) ascribed to bimagnon excitations [START_REF] Kim | Large Spin-Wave Energy Gap in the Bilayer Iridate Sr 3 Ir 2 O 7 : Evidence for Enhanced Dipolar Interactions Near the Mott Metal-Insulator Transition[END_REF] . Feature B is ascribed to a single magnon excitation [START_REF] Kim | Large Spin-Wave Energy Gap in the Bilayer Iridate Sr 3 Ir 2 O 7 : Evidence for Enhanced Dipolar Interactions Near the Mott Metal-Insulator Transition[END_REF] . To explain the magnetic dynamics, a first model has been proposed by Kim et al. [START_REF] Kim | Large Spin-Wave Energy Gap in the Bilayer Iridate Sr 3 Ir 2 O 7 : Evidence for Enhanced Dipolar Interactions Near the Mott Metal-Insulator Transition[END_REF] , who considered a linear spin-wave approach with dominant intralayer (J) over interlayer (J c ) magnetic interaction (J J c ). The two exchange paths are shown in Figure 5.1. Magnetic interactions that arise from this model are conventional spin waves, and for a bilayer acoustic and optical branches are in principle allowed [START_REF] Reznik | Direct observation of optical magnons in YBa 2 Cu 3 O 6.2[END_REF] . Since only one feature is observed in the RIXS spectra of Sr 3 Ir 2 O 7 , Kim et al. [START_REF] Kim | Large Spin-Wave Energy Gap in the Bilayer Iridate Sr 3 Ir 2 O 7 : Evidence for Enhanced Dipolar Interactions Near the Mott Metal-Insulator Transition[END_REF] conclude that the two magnetic branches must be almost degenerate. ear spin-wave model and bond-operator mean-field approach succeed in describing the evolution of the magnetic dynamics of this material [START_REF] Lu | Doping Evolution of Magnetic Order and Magnetic Excitations in (Sr 1-x La x ) 3 Ir 2 O 7[END_REF][START_REF] Hogan | Disordered dimer state in electrondoped Sr 3 Ir 2 O 7[END_REF] .

I here propose an experiment to discriminate between the two magnetic models by investigating the magnetic excitations of Sr 3 Ir 2 O 7 at high pressure. Indeed, the pressure evolution of the crystal structure has already been measured [START_REF] Donnerer | Pressure dependence of the structure and electronic properties of Sr 3 Ir 2 O 7[END_REF] . From this study, one can infer the pressure dependence of the intralayer and interlayer superexchange couplings, and therefore the evolution of some magnetic properties, such as the spin-wave gap.

High-pressure low-temperature equipment

Diamond anvil cells (DACs) are standard tools to perform experiments at pressures up to several hundreds of GPa [START_REF] Jayaraman | Diamond anvil cell and high-pressure physical investigations[END_REF][START_REF] Bassett | Diamond anvil cell, 50th birthday[END_REF] . The working principle is simple: a sample placed between two opposed diamond anvils is exposed to pressure when a force pushes the two diamonds together. Between the two anvils, a metal foil (the gasket) with a drilled hole that serves as sample chamber is placed to confine the sample inside a pressure-transmitting medium that ensures hydrostatic conditions. On the left-hand side of Figure 5.5 the basic constituents of a generic DAC are sketched. To probe the magnetic dynamics at high pressure, the DAC must be placed inside a cryostat and cooled below the Néel temperature, where the magnetic order sets in.

Before describing in detail the DAC and the cryostat utilized for highpressure low-temperature RIXS experiments at beamline ID20, I list a few technical challenges that must be overcome when attempting such experiments. First of all, RIXS measurements require a flexible scattering geometry, especially when carried out on single crystals. This is necessary in order to orient the sample with respect to its crystallographic axes on the goniometer stage and to scatter along a precise direction in reciprocal space. For this purpose, spectrometer and sample stage are typically equipped with the necessary degrees of freedom. This flexibility should be preserved as much as possible also in the case of complex sample environments. Therefore, both DAC and cryostat must be equipped with large opening windows to minimize the constraints on the scattering geometry.

Another important aspect is the relatively low efficiency of RIXS experiments. It is therefore imperative to maximize the solid angle of collection, minimize the X-ray absorption through the sample environment, and avoid any loss of intensity due to X-rays not impinging on the sample, also taking into account the X-ray penetration depth of a given material. Roughly speaking, the latter implies that the smallest measurable sample has the dimensions of the beam focal spot size times the penetration depth (for example, ∼ 10×20×50 µm 3 in the case of Sr 3 Ir 2 O 7 for photons of ∼ 11.2 keV). Given that the sample size imposes limitations on the maximum reachable pressure, a compromise must be made.

Finally, scattering of X-rays from the sample environment generates background noise. This is typically due to non-resonant elastic and inelastic X-ray scattering from the gasket and/or the diamonds of the DAC. The spurious signal might be as strong as or even stronger than the signal from the sample. This is reduced by adopting a scattering geometry as close as possible to 90 • to minimize the Thomson elastic scattering and by introducing shielding elements that prevent the X-rays from reaching the crystal analyzer and thus the detector. Note that elastically-scattered X-rays from a source slightly off the sample position will impinge on the crystal analyzer at a wrong Bragg angle and could be erroneously interpreted as inelastic features, thus giving rise to artifacts in the RIXS spectrum. In addition, X-ray scattering from crystalline diamonds will feature sharp elastic and phonon peaks, whose energy loss (up to ≈ 170 meV 138 ) depends on the momentum transfer relative to the diamond orientation. Similarly, the gasket is also a source of scattering. In case amorphous beryllium is used, the phonon signal is a broad distribution, resembling the phonon density of states, which extends up to ≈ 80 meV [START_REF] Schmunk | Lattice Dynamics of Beryllium[END_REF][START_REF] Stedman | Phonon spectrum of beryllium at 80k[END_REF] .

Taking into account all the above considerations, a scattering geometry where the X-rays go through the gasket of the DAC, both in their way to and from the sample, has been adopted. In addition, the high-pressure cryostat has been equipped with a system of slits to screen the X-rays scattered by the environment surrounding the sample. The extra-high 16-sided (100)-oriented diamond anvils are in the middle of the opening window. They are based on the type Ia Boehler-Almax design [START_REF] Boehler | New anvil designs in diamondcells[END_REF] (diameter of 3.1 mm, height of 2.72 mm, angle of the diamond faces of 25 • ). Diamonds of culet size ranging from 250 µm (maximum pressure ∼ 80 GPa) to 500 µm (maximum pressure ∼ 30 GPa) are used to cover different pressure ranges.

Diamond anvil cell and gasket material

A cylinder-piston system (modified LeToullec design [START_REF] Letoullec | The membrane diamond anvil cell: A new device for generating continuous pressure and temperature variations[END_REF] ) allows the movement of the diamond anvils along the compression axis. The anvils are pushed together by the application of force on one diamond through a gas-driven membrane that is connected to a pressure controller.

For the scattering geometry adopted in our experiments, the choice of the gasket material is restricted to beryllium, because it is a light element with low absorption of hard X-rays. The main disadvantages are the low shear strength and brittleness, which limit the maximum reachable pressure. The gasket with an outer diameter of 5 mm and a thickness of 0.2 mm is pre-indented to approximately 1/10 of the culet size (i.e. ∼ 30-50 µm) before the experiment, and a sample chamber with diameter approximately equal to 50-60% of the culet size is laser-drilled in the center to accommodate the sample.

A pressure-transmitting medium is used to ensure hydrostatic compression of the sample. We choose neon as pressure medium during our experiments, since it is an inert gas and it is almost transparent to X-rays. In principle, helium can be used as well. However, shrinkage of the sample chamber with pressure is less severe when neon is used: the diameter of the sample chamber shrinks by approximately a factor two from ambient pressure to 20 GPa (10 GPa) when Ne (He) is used. The Panoramic DAC has been gas loaded in the dedicated Gas Loading System available in the ESRF High-Pressure Lab.

The pressure is determined by measuring the energy position of the laser-induced R 1 fluorescence line of a ruby (Al 2 O 3 :Cr 3+ ) sphere placed next to the sample inside the sample chamber [START_REF] Mao | High-pressure physics: the 1-megabar mark on the ruby r1 static pressure scale[END_REF] . The pressure-by-rubyluminescence (PRL) equipment includes a laser, a spectrometer and a detector.

An image of a typical sample loading is shown in the sample and the ruby sphere after the gasket has been put between the diamond anvils. The laser-drilled sample chamber with diameter of 150 µm is visible.

Cryogenic equipment

The Panoramic DAC is accommodated inside a cryostat, in such a way that the compression axis is vertical and the wide opening windows are horizontal in the laboratory reference frame. An overview of the cryostat is shown in Figure 5.7.

The outer part of the cryostat is the vacuum chamber. It is made of stainless steel to withstand ambient pressure, and it is fixed on the RIXS sample goniometer. Two large windows have been opened in the walls of the chamber and covered by an X-ray-transparent 0.1-mm-thick Kapton foil. The vacuum is as good as ≤ 10 -7 mbar during the measurements. The vacuum chamber hosts the entries for the siphon, the vacuum pumping port, the electric wiring connectors, and a feedthrough for the capillary that connects the pressure controller to the cell membrane.

The Panoramic DAC is placed inside a copper main heat exchanger, which is cooled down by a continuous flow of helium. Its temperature is measured by a Cernox R sensor and regulated by resistive Kapton foil heaters. Helium then flows through stainless steel capillaries into a copper secondary heat exchanger before it is evacuated through the input siphon. The temperature of the secondary exchanger is measured and regulated in a similar fashion as the main heat exchanger. It is designed such that the contact surface between the secondary exchanger and the vacuum chamber is minimized and the thermal paths are maximized for thermal decoupling. Furthermore, thermally-insulating washers are utilized to fix the two elements. Because of the cylindrically-symmetric design of the cryostat, the position of the DAC moves by less than ≈ 40 µm in the horizontal plane and by ≈ 160 µm along the vertical axis when cooled from room temperature to 100 K.

The cryostat chamber furthermore hosts a system of motorized slits. These screen the X-rays scattered by the sample environment and thus reduce the background signal. Ideally, they should be placed in close contact with the sample, but this is obviously not possible. In our case, they ro-

The magnetism of Sr 3 Ir 2 O 7 at high pressure

tate around the DAC compression axis at a distance of 16.5 mm from the sample with an angular resolution of 0.001 • . The apertures are cut in a 0.1mm-thick tantalum foil and have widths of 0.32 mm, 0.16 mm, and 0.08 mm. Due to the finite analyzer aperture A in the scattering plane and finite slit-to-sample distance l, X-ray scattering from the sample surrounding will still leak through into the spectrometer. Indeed, the "field of view" f of the spectrometer is given by

f = sL + Al L -l , (5.1)
where L is the sample-to-analyzer distance and s is the slit aperture. In our case, L ≈ 2000 mm and l = 16.5 mm; for a slit aperture s = 0.08 mm, the analyzer aperture has been set to A = 15 mm in order to collect the Xrays scattered by a region of only f ≈ 0.2 mm around the sample position, roughly equal to the size of the sample chamber.

The magnetism of Sr 3 Ir 2 O 7 at high pressure

The experiment was carried out with two panoramic DACs with diamond culet sizes of 500 µm and 250 µm. The sample was cleaved to a dimension of approximately 50×50×15 µm 3 and placed at the center of the sample chamber, as shown in Figure 5.6. The incident energy of 11.216 keV was chosen in order to maximize the signal from the magnetic excitations. The first feature is ascribed to the single magnon excitation [START_REF] Kim | Large Spin-Wave Energy Gap in the Bilayer Iridate Sr 3 Ir 2 O 7 : Evidence for Enhanced Dipolar Interactions Near the Mott Metal-Insulator Transition[END_REF][START_REF] Sala | Evidence of quantum dimer excitations in Sr 3 Ir 2 O 7[END_REF] , while the second feature is the excitation of a hole from the j eff = 1/2 ground state to the j eff = 3/2 band, in accordance to works on the sister compound Sr 2 IrO 4 4, [START_REF] Kim | Magnetic Excitation Spectra of Sr 2 IrO 4 Probed by Resonant Inelastic X-Ray Scattering: Establishing Links to Cuprate Superconductors[END_REF][START_REF] Kim | Excitonic quasiparticles in a spin-orbit Mott insulator[END_REF] . The pressure dependence at room temperature of the latter excitation has been already investigated by Ding et al. [START_REF] Ding | Pressure-Induced Confined Metal from the Mott Insulator Sr 3 Ir 2 O 7[END_REF] , who have found that the peak position changes by at most ∼ 15% from ambient pressure to approximately 65 GPa, while the width does not undergo a significant change. I here focus on the pressure evolution of the single magnon. Note that in the work of Ding et al. [START_REF] Ding | Pressure-Induced Confined Metal from the Mott Insulator Sr 3 Ir 2 O 7[END_REF] this feature is not visible because the experiment was carried out at room temperature and the resolution of ∼ 110 mev was not sufficient [START_REF] Kim | Advances in high-resolution RIXS for the study of excitation spectra under high pressure[END_REF] . going from ambient pressure to 12 GPa, as highlighted by the vertical solid lines below the RIXS spectra representing the fitted position of the single magnon.

Before discussing the implications of the magnetic mode softening, I would like to briefly address the pressure dependence of the magnetic structure of Sr 3 Ir 2 O 7 . Figure 5.9(a) and (b) display the elastic signal acquired while scanning across the magnetic Bragg reflections in the reciprocal lattice at fixed L (a) and at fixed H, K (b). The magnetic reflections are broadened and damped with pressure, but overall the long-range AFM order survives at least until 12 GPa. Above this pressure value, we noticed a degradation of the sample quality, as testified by a drastic broadening and suppression of the diffraction peaks. Therefore, results can be safely discussed only up to 12 GPa. From an experimental point of view, how- departure of the in-plane Ir-O-Ir bond angle from the straight (180 • ) bond geometry implies a stronger inhibition of the intralayer superexchange coupling [START_REF] Jackeli | Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models[END_REF] (J in Figure 5.1), which is therefore naively expected to decrease with pressure. On the other hand, the interlayer superexchange coupling (J c in Figure 5.1) should be less affected and, even if so, it should increase because of the reduction of the bond length. Figure 5.11 displays the dependence of the magnetic gap on the parameters J and J c , which are scaled as expected, i.e. J decreases while J c increases with respect to the values at ambient pressure. The other magnetic couplings are assumed to be unchanged by pressure for simplicity. As can be seen, the linear spin-wave model by Kim et al. [START_REF] Kim | Large Spin-Wave Energy Gap in the Bilayer Iridate Sr 3 Ir 2 O 7 : Evidence for Enhanced Dipolar Interactions Near the Mott Metal-Insulator Transition[END_REF] predicts a softening of the magnetic gap, while on the contrary the gap hardens according to the quantum-dimer approach by Moretti Sala et al. [START_REF] Sala | Evidence of quantum dimer excitations in Sr 3 Ir 2 O 7[END_REF] . The softening of the magnetic mode revealed by our experiment is consistent with the model by Kim et al. [START_REF] Kim | Large Spin-Wave Energy Gap in the Bilayer Iridate Sr 3 Ir 2 O 7 : Evidence for Enhanced Dipolar Interactions Near the Mott Metal-Insulator Transition[END_REF] , which considers a dominant intralayer interaction whose reduction naturally explains the decrease of the magnetic gap [START_REF] Kim | Large Spin-Wave Energy Gap in the Bilayer Iridate Sr 3 Ir 2 O 7 : Evidence for Enhanced Dipolar Interactions Near the Mott Metal-Insulator Transition[END_REF] . While the qualitative considerations above seem to favor the linear spin-wave model, accurate first principle quantum chemical calculations that account for the precise evolution of the lattice under pressure are needed to definitively discriminate between the Chapter 5 Performing RIXS at high pressure and low temperature: the case of Sr 3 Ir 2 O 7 two theoretical approaches.

Conclusions and outlook

In this Chapter, I presented the equipment to perform RIXS experiments at high pressure and low temperature. The recently-developed apparatus, particularly useful to probe the magnetic dynamics at high pressure, was successfully commissioned on beamline ID20 at the ESRF. High-pressure low-temperature RIXS experiments are particularly challenging due to severe requirements such as flexible scattering geometry, minimization of the absorption through the complex setup, and reduction of the background noise from the sample environment. The instrumental upgrades include a Panoramic DAC with wide opening windows suitable for RIXS experiments, and a customized cryostat, which incorporates a system of rotating slits that filters out the non-resonant spurious signal from the environment.

The feasibility of high-pressure low-temperature RIXS experiments was demonstrated through the measurement of the magnetic excitations of Sr 3 Ir 2 O 7 up to a maximum pressure of 12 GPa, above which the sample quality degraded significantly. To the best of my knowledge, this is the highest pressure at which single magnons have ever been recorded.

The bilayer perovskite Sr 3 Ir 2 O 7 was chosen for this study since it shares some of the interesting physical properties of the single-layer-analog Sr 2 IrO 4 [START_REF] Cao | Anomalous magnetic and transport behavior in the magnetic insulator Sr 3 Ir 2 O 7[END_REF][START_REF] Boseggia | On the magnetic structure of Sr 3 Ir 2 O 7 : an x-ray resonant scattering study[END_REF][START_REF] Kim | Dimensionality driven spin-flop transition in layered iridates[END_REF][START_REF] Moon | Dimensionality-Controlled Insulator-Metal Transition and Correlated Metallic State in 5d Transition Metal Oxides Sr n+1 Ir n O 3n+1 (n = 1, 2, and ∞)[END_REF][START_REF] Hogan | First-order melting of a weak spin-orbit mott insulator into a correlated metal[END_REF][START_REF] Lu | Doping Evolution of Magnetic Order and Magnetic Excitations in (Sr 1-x La x ) 3 Ir 2 O 7[END_REF][START_REF] Hogan | Disordered dimer state in electrondoped Sr 3 Ir 2 O 7[END_REF] . Two fundamentally-different models have been proposed to interpret its magnetic dynamics, the strong difference consisting in the proposed dominant magnetic interactions, being of either intralayer [START_REF] Kim | Large Spin-Wave Energy Gap in the Bilayer Iridate Sr 3 Ir 2 O 7 : Evidence for Enhanced Dipolar Interactions Near the Mott Metal-Insulator Transition[END_REF] or interlayer [START_REF] Sala | Evidence of quantum dimer excitations in Sr 3 Ir 2 O 7[END_REF] character. Our results showed that the long-range AFM order of Sr 3 Ir 2 O 7 persists to at least 12 GPa. Moreover, the single magnon softens with pressure at a rate of ≈ 1.5 meV/GPa. By extrapolating the pressure trend of the magnetic mode, it was found that the magnetic gap collapses at a pressure of ∼ 55-60 GPa, a value very similar to the pressure associated to the structural phase transition in this compound [START_REF] Donnerer | Pressure dependence of the structure and electronic properties of Sr 3 Ir 2 O 7[END_REF] . The softening of the magnetic gap seems to be consistent with the spin-wave model by Kim et al. [START_REF] Kim | Large Spin-Wave Energy Gap in the Bilayer Iridate Sr 3 Ir 2 O 7 : Evidence for Enhanced Dipolar Interactions Near the Mott Metal-Insulator Transition[END_REF] , who consider a leading intralayer over interlayer superexchange coupling. Indeed, the pressure evolution of the crystal structure suggests a decrease of the intralayer interaction that naturally explains the softening of the magnetic gap predicted by the model by Kim et al. [START_REF] Kim | Large Spin-Wave Energy Gap in the Bilayer Iridate Sr 3 Ir 2 O 7 : Evidence for Enhanced Dipolar Interactions Near the Mott Metal-Insulator Transition[END_REF] .

I hope that the detailed description of the sample environment necessary for high-pressure low-temperature RIXS experiments as well as the quality of the collected data will open the way for further studies of magnetic excitations in the unexplored high-pressure domain.

The magnetic and electronic dynamics of CaIrO 3

In this Chapter, I investigate the magnetic and electronic dynamics of CaIrO 3 by means of iridium L 2,3 -edge RIXS in order to determine the principal interactions governing the magnetic and electronic properties of this system.

Introduction

The j eff = 1/2 ground state is generated from the superposition of the iridium 5d-t 2g active orbitals with different spin components [START_REF] Kim | Novel J eff = 1/2 Mott State Induced by Relativistic Spin-Orbit Coupling in Sr 2 IrO 4[END_REF][START_REF] Jackeli | Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models[END_REF] , as discussed in Chapter 1. Intersite interactions are determined by the spin densities within the ground-state wave function and the symmetry of the bond geometry [START_REF] Jackeli | Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models[END_REF] . In particular, Jackeli and Khaliullin [START_REF] Jackeli | Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models[END_REF] demonstrated that the superexchange interaction between two pseudospins connected by one oxygen (corner-sharing octahedra) turns out to be mainly Heisenberglike, whereas isotropic coupling is strongly suppressed for bond geometries where the pseudospins are bridged by two oxygens (edge-sharing octahedra) [START_REF] Jackeli | Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models[END_REF] . The fundamentally-different nature of the interactions for the two bond geometries gives rise to very distinct magnetic behaviors. Specifically, corner-sharing octahedra display a cuprate-like magnetism, with long-range AFM order and dispersing spin waves [START_REF] Kim | Magnetic Excitation Spectra of Sr 2 IrO 4 Probed by Resonant Inelastic X-Ray Scattering: Establishing Links to Cuprate Superconductors[END_REF][START_REF] Crawford | Structural and magnetic studies of Sr 2 IrO 4[END_REF] . Instead, edgesharing honeycomb iridates are expected to realize the Kitaev model, with

Chapter 6 The magnetic and electronic dynamics of CaIrO 3 a spin-liquid ground state and non-Abelian anyonic excitations [START_REF] Kitaev | Anyons in an exactly solved model and beyond[END_REF] .

In this Chapter, I address the case of CaIrO 3 , whose peculiar crystal structure among iridates features a two-dimensional network of cornerand edge-sharing octahedra [START_REF] Ohgushi | Metal-insulator transition in Ca 1-x Na x IrO 3 with post-perovskite structure[END_REF][START_REF] David | High-pressure structure and bonding in CaIrO3: The structure model of MgSiO3 postperovskite investigated with time-of-flight neutron powder diffraction[END_REF] . Hence, the two types of interactions described above simultaneously contribute to the realization of the magnetic and electronic ground state of CaIrO 3 . Specifically, the inhibition of Heisenberg superexchange along the edge-sharing direction is expected to effectively reduce the system to a one-dimensional magnet. The aim of this study is to investigate the magnetic and electronic dynamics of CaIrO 3 by iridium L 2,3 -edge RIXS in order to identify the principal interactions governing superexchange couplings and intersite hoppings. [START_REF] Ohgushi | Metal-insulator transition in Ca 1-x Na x IrO 3 with post-perovskite structure[END_REF][START_REF] David | High-pressure structure and bonding in CaIrO3: The structure model of MgSiO3 postperovskite investigated with time-of-flight neutron powder diffraction[END_REF] . The crystal structure, often referred to as post-perovskite, is reported in Figure 6.1(a). It comprises IrO 6 octahedra which are connected to each other by sharing edges and corners along the a and c directions, respectively (Figures 6.1(b, c)), and are separated by calcium layers along the b axis. The octahedra are compressed along the local z axis with a bond-length ratio of 0.97 and are rotated by 23 • around the a axis [START_REF] Ohgushi | Metal-insulator transition in Ca 1-x Na x IrO 3 with post-perovskite structure[END_REF] .

Crystal and magnetic structure

The system undergoes an AFM transition at the Néel temperature of T N ∼ 110 K [START_REF] Ohgushi | Metal-insulator transition in Ca 1-x Na x IrO 3 with post-perovskite structure[END_REF][START_REF] Gunasekera | Mott insulator-to-metal transition in yttrium-doped CaIrO 3[END_REF][START_REF] Gunasekera | Magnetic fluctuations driven insulator-to-metal transition in Ca(Ir 1-x Ru x )O 3[END_REF] . The magnetic structure has been refined by means of resonant x-ray diffraction [START_REF] Ohgushi | Resonant X-ray Diffraction Study of the Strongly Spin-Orbit-Coupled Mott Insulator CaIrO 3[END_REF] , and the arrangement of the magnetic moments (pseudospins) is reported in Figure 6.1(a). As can be seen, the pseudospins are ferromagnetically aligned along the edge-sharing direction (Figure 6.1(b)), while they are antiferromagnetically aligned along the corner-sharing direction (Figure 6.1(c)). The moments are canted along the b axis, as can be argued from the parasitic ferromagnetism along this direction [START_REF] Ohgushi | Resonant X-ray Diffraction Study of the Strongly Spin-Orbit-Coupled Mott Insulator CaIrO 3[END_REF] .

Magnetic dynamics

Iridium L 3 -edge RIXS spectra of CaIrO 3 measured at momentum transfer Q = (0.5, 2, 9) r.l.u., T = 295 K (red circles) and T = 20 K (blue circles) are reported in Figure 6.2(a). The temperature-independent features above 0.4 eV will be described in the next Section. I will here discuss the excitations below 0.4 eV. When the sample is cooled below the Néel temperature, an additional spectral weight centered at 34 ± 3 meV appears on top of an asymmetric temperature-independent continuum. The new feature, shaded in blue in the Figure, can be clearly distinguished in the RIXS spectra starting from ∼ 20 K above the Néel temperature, it reaches its maximum intensity at T N and then decreases, but remains finite, as the temperature is further lowered, as displayed in Figure 6.2(b). The FWHM has a monotonic trend instead, with an abrupt change around the magnetic ordering temperature and an energy-resolution-limited minimum at the lowest temperature (Figure 6.2(c)). The temperature dependence of the spectral weight and FWHM of the narrow feature suggests that this excitation has magnetic origin and is related to the development of long-range AFM order occurring at T N ∼ 110 K [START_REF] Ohgushi | Metal-insulator transition in Ca 1-x Na x IrO 3 with post-perovskite structure[END_REF][START_REF] Gunasekera | Mott insulator-to-metal transition in yttrium-doped CaIrO 3[END_REF][START_REF] Gunasekera | Magnetic fluctuations driven insulator-to-metal transition in Ca(Ir 1-x Ru x )O 3[END_REF] .

Having discussed the temperature dependence of the narrow magnetic feature, I now turn to the study of its momentum dependence along the principal crystallographic directions. The measurements were taken at 40

Magnetic dynamics

K. A global inspection of Figure 6.3(a, c, e) suggests that the narrow feature sets the low energy boundary of a continuum of excitations with characteristic dispersion in reciprocal space. Along the edge-sharing H direction (Figure 6.3(a)), a tiny dispersion of 23 ± 1.7 meV with periodicity of 1 r.l.u. is detected. No momentum dependence is observed along the K direction (Figure 6.3(c)), which evidences the total lack of magnetic interactions along the crystallographic b direction, where the superexchange path is suppressed by the presence of calcium layers. Finally, along the cornersharing L direction (Figure 6.3(e)), the RIXS map shows that the low energy boundary has a dispersion of about 120 ± 4 meV with periodicity of 1 r.l.u., while the upper higher energy dome has double periodicity. Features in the magnetic region of the RIXS spectra were fitted using a Lorentzian function for the narrow low-energy feature and an asymmetric Lorentzian distribution for the continuum. The peak position of the narrow excitation is reported in panels (b), (d), and (f) of Figure 6.3 as white squares. The gray bands around them represent the corresponding FWHM. In order to quantitatively determine the nature and strength of the magnetic interactions in CaIrO 3 , the experimental dispersions were fitted to numerical solutions of model Hamiltonians computed with the SpinW code [START_REF] Toth | Linear spin wave theory for single-Q incommensurate magnetic structures[END_REF] , as explained below.

For a pair of nearest-neighbor pseudospins Si and Sj , the most general bilinear Hamiltonian contains both isotropic and anisotropic terms [START_REF] Moriya | Anisotropic Superexchange Interaction and Weak Ferromagnetism[END_REF] :

H = i J Si • Si+1 + D • ( Si × Si+1 ) + Si Γ Si+1 , (6.1)
where the scalar J is the isotropic Heisenberg superexchange, the vector D is the antisymmetric Dzyaloshinskii-Moriya (DM) anisotropic interaction, and the traceless tensor Γ is the symmetric part of the superexchange anisotropy. Following symmetry considerations, some of the nine independent couplings of Equation 6.1 may not be allowed. In the specific case of CaIrO 3 , the bond along the a axis has C 2h symmetry (Figure 6.1(b)). Due to the presence of an inversion center, the DM interaction vanishes [START_REF] Moriya | Anisotropic Superexchange Interaction and Weak Ferromagnetism[END_REF] . The application of the symmetry operations for the C 2h point group, i.e. the C 2 rotation axis parallel to the a axis and the mirror plane σ h perpendicular to the C 2 axis, reveals that the symmetric tensor Γ a has the form:

Γ a =   A 0 0 0 B C 0 C -A -B   . (6.2)
Further constraints can be introduced by considering the explicit form of the wave function and the geometry of the magnetic bond. Following Jackeli and Khaliullin [START_REF] Jackeli | Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models[END_REF] , for pseudospins connected by edge-sharing octahedra a quantum compass interaction is expected. In the specific case of CaIrO

Magnetic dynamics

this interaction takes the form of an Ising-like term K Sz 1

Sz

2 , where z is the axis orthogonal to the plane defined by the two iridium ions and the two bridging oxygens [START_REF] Jackeli | Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models[END_REF] . Note that the rule by Jackeli and Khaliullin [START_REF] Jackeli | Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models[END_REF] applies for j eff = 1/2 systems and results from the relative phase of the |xz and |yz orbitals, which is preserved for tetragonal octahedral distortions, such as the case of CaIrO 3 . The z axis is rotated by θ = 23 • with respect to the crystallographic c axis [START_REF] Ohgushi | Metal-insulator transition in Ca 1-x Na x IrO 3 with post-perovskite structure[END_REF] and in the lattice reference frame the magnetic Hamiltonian can be casted in the following form:

H a = J a S1 • S2 + S1   -K 0 0 0 K(sin θ) 2 -K cos θ sin θ 0 -K cos θ sin θ K(cos θ) 2   S2 . (6.3)
The isotropic Heisenberg coupling J a has been included to account for possible direct exchange mechanisms and/or incomplete suppression due to octahedral distortions [START_REF] Chaloupka | Kitaev-Heisenberg Model on a Honeycomb Lattice: Possible Exotic Phases in Iridium Oxides A 2 IrO 3[END_REF][START_REF] Chaloupka | Zigzag Magnetic Order in the Iridium Oxide Na 2 IrO 3[END_REF] . The minimal Hamiltonian, containing only two free parameters, has been numerically solved with the SpinW code [START_REF] Toth | Linear spin wave theory for single-Q incommensurate magnetic structures[END_REF] and the spin-wave dispersion has been fitted to the experimental one. The result is plotted as a red solid line in Figure 6.3(b). The best fit is obtained considering a dominant FM Ising interaction K = -7 meV and an AFM Heisenberg coupling J a = 4 meV. These values are consistent with the magnetic structure of CaIrO 3 [START_REF] Ohgushi | Resonant X-ray Diffraction Study of the Strongly Spin-Orbit-Coupled Mott Insulator CaIrO 3[END_REF] . By inspection of the bond along the c axis (Figure 6.1(c)), a C 2 rotation axis parallel to the b axis, and two mirror planes parallel to the ab and bc planes are found. Thus, the bond has C 2v symmetry. In this case, only the component of the DM interaction pointing along the a axis is allowed: D = (D, 0, 0). The application of the symmetry operations of this point group reveals that the symmetric tensor is diagonal:

Γ c =   A 0 0 0 B 0 0 0 -A -B   . (6.4)
Following the recipe of Jackeli and Khaliullin [START_REF] Jackeli | Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models[END_REF] for pseudospins connected by corner-sharing octahedra, a pseudodipolar interaction of the form J z Sz 1 Sz 2 is theoretically derived. The minimal Hamiltonian for the bond along the c axis therefore reads:

H c = J c S1 • S2 + D • ( S1 × S2 ) + S1       - J z 3 0 0 0 - J z 3 0 0 0 2J z 3       S2 . (6.5)
The best fit to the experimental dispersion is obtained by J c = 170 meV, J z = 4 meV and D = 3 meV. The calculated dispersion is plotted as a red solid line in Figure 6.3(f). The DM interaction gives rise to a canting angle of ∼ 1 • , in agreement with the value of 2-4 • reported in the literature [START_REF] Sala | CaIrO 3 : A Spin-Orbit Mott Insulator Beyond the J eff = 1/2 Ground State[END_REF][START_REF] Ohgushi | Resonant X-ray Diffraction Study of the Strongly Spin-Orbit-Coupled Mott Insulator CaIrO 3[END_REF] . Overall, the magnetic structure generated by the superexchange couplings listed above is consistent with the measured one by Ohgushi et al. [START_REF] Ohgushi | Resonant X-ray Diffraction Study of the Strongly Spin-Orbit-Coupled Mott Insulator CaIrO 3[END_REF] . Indeed, the magnetic moments are AFM aligned along the c direction and FM aligned along the a axis. Ab initio quantum chemical calculations on CaIrO 3 by Bogdanov et al. [START_REF] Bogdanov | Post-perovskite CaIrO 3 : A j = 1/2 quasi-one-dimensional antiferromagnet[END_REF] provide values of the Heisenberg superexchange J a = -7.3 meV and J c = 121 meV, hence they agree on the strong suppression of the isotropic coupling for edge-sharing octahedra. We note, however, that the computed value of J c = 121 meV disagrees with the value obtained from the fit of the experimental dispersions. Moreover, linear spin-wave theory fails in predicting the magnetic continuum and its momentum dependence. These issues reveal the inadequacy of linear spin-wave theory to quantitatively interpret the complex dynamics of CaIrO 3 . Indeed, the dominant Heisenberg interaction for corner-sharing octahedra and its suppression for edge-sharing bonds effectively reduces the dimensionality of CaIrO 3 , which can be thought of as weakly-coupled quasi-one-dimensional AFM chains extending along the c axis. For a one-dimensional antiferromagnet, the characteristic excitations are quasi-particles called spinons, as explained below.

Spinons are quasi-particles that can be thought of as domain walls of a one-dimensional AFM chain [START_REF] Giamarchi | Quantum Physics in One Dimension[END_REF] . They are created from the fractionaliza-

Magnetic dynamics

tion of magnons, a phenomenon in which a magnon carrying spin quantum number of 1 breaks into two spinons carrying fractional spin quantum number of 1/2 each [START_REF] Giamarchi | Quantum Physics in One Dimension[END_REF] . Spinons are deconfined along the chain since they can independently propagate with no energy cost. Their independent motion gives rise to the peculiar energy-momentum continuum, which arises from the sum of the dispersion relations of the single spinons [START_REF] Giamarchi | Quantum Physics in One Dimension[END_REF] .

An evidence for quasi-one-dimensional magnetic behavior is the observation of the excitation continuum with characteristic periodicity of the lower and upper boundaries and intensity profile [START_REF] Karbach | Two-spinon dynamic structure factor of the onedimensional s = 1/2 Heisenberg antiferromagnet[END_REF][START_REF] Caux | The four-spinon dynamical structure factor of the Heisenberg chain[END_REF][START_REF] Caux | The two-spinon transverse structure factor of the gapped Heisenberg antiferromagnetic chain[END_REF][START_REF] Lake | Quantum criticality and universal scaling of a quantum antiferromagnet[END_REF][START_REF] Lake | Confinement of fractional quantum number particles in a condensed-matter system[END_REF][START_REF] Mourigal | Fractional spinon excitations in the quantum Heisenberg antiferromagnetic chain[END_REF] . We try to model the RIXS data with the two-spinon dynamical structure factor of the onedimensional AFM XXZ chain. The XXZ chain is made of spin-1/2 magnetic moments interacting according to the following magnetic Hamiltonian 157 :

H XXZ = J i S x i S x i+1 + S y i S y i+1 + δS z i S z i+1 (6.6)
where the sum runs over the chain length, J is the AFM Heisenberg superexchange coupling, and δ is the anisotropy. The presence of the anisotropy is necessary since a pure AFM Heisenberg chain would exhibit gapless excitations [START_REF] Karbach | Two-spinon dynamic structure factor of the onedimensional s = 1/2 Heisenberg antiferromagnet[END_REF][START_REF] Caux | The four-spinon dynamical structure factor of the Heisenberg chain[END_REF] . In our case, δ is an effective symmetric anisotropy that induces the opening of the magnetic gap. The exact two-spinon dynamical structure factor S(Q, ω) of the XXZ chain has been calculated following the work of Caux et al. [START_REF] Caux | The two-spinon transverse structure factor of the gapped Heisenberg antiferromagnetic chain[END_REF] and has been convoluted with the resolution function for comparison to the data. Figure 6.4 compares the observed magnetic dynamics along the chain direction (left) with the calculated spin dynamical structure factor of the AFM XXZ model (right panel). The best fit to the boundaries of the magnetic continuum (white solid lines) is obtained for values of J = 70 meV and δ ≈ 2. As can be seen in Figure 6.4, the two-spinon dynamical structure factor of the AFM XXZ chain reproduces fairly well the measured spectra. This allows us to ascribe the continuum to (pseudo)spinon excitations. Figure 6.5 compares RIXS spectra of CaIrO 3 (black circles) measured at three different momentum transfers with the corresponding two-spinon dynamical structure factor of the AFM XXZ chain (red solid line). S(Q, ω) has been calculated considering J = 70 meV and δ ≈ 2, and convolved with the resolution function. The spectra have been normalized according to the sum rules [START_REF] Caux | The two-spinon transverse structure factor of the gapped Heisenberg antiferromagnetic chain[END_REF] . We note that S(Q, ω) fits well the spectrum at Q = 0, but the fit worsens when going away from the Brillouin zone center. The disagreement between RIXS data and the two-spinon dynamical structure factor can be imputed to two main reasons. First, the XXZ chain oversimplifies the magnetic Hamiltonian, since it does not account for either the DM interaction or the coupling between the chains. The latter would confine the spinons, thus giving the system a two-dimensional character which would make the high-energy continuum less pronounced [START_REF] Lake | Confinement of fractional quantum number particles in a condensed-matter system[END_REF] . Second, besides the dynamical structure factor, the RIXS magnetic response 0.2 0.0 0.2 0.4 0.6 0.8 1.0 Energy loss (eV)

Intensity (arb. u.) Q = 0 Q = /4 Q = /2 S(Q, )
Data points contains the additional contribution of the spin-exchange structure factor T (Q, ω) [START_REF] Klauser | Spin-Exchange Dynamical Structure Factor of the S = 1/2 Heisenberg Chain[END_REF] , therefore it cannot be compared directly to S(Q, ω). Indeed, the RIXS intensity is:

I(Q, ω) = |W (Q)| 2 S(Q, ω) + |Y (Q)| 2 T (Q, ω), (6.7)
where |W (Q)| 2 and |Y (Q)| 2 are the scattering amplitudes for singlemagnon and elastic scattering, respectively. The calculation of the scattering amplitudes within a single-ion approach is reported in Appendix B. In particular, |Y (Q)| 2 can be computed from Equations B.10 and B.12, while |W (Q)| 2 from Equation B.11. By noticing that the single-magnon scattering amplitude vanishes at the iridium L 2 edge 162 , one could isolate the spin-exchange structure factor from iridium L 2 -edge RIXS data and remove this contribution from the iridium L 3 -edge RIXS spectra, eventually obtaining the mere two-spinon dynamical structure factor S(Q, ω). An attempt is shown in Figure 6.6. The spectra acquired at the L 3 and L 2 edges are first normalized to the integrated intensity of the features above 0.4 eV and then multiplied by the edge-dependent scattering amplitudes for these features. The elastic line has been removed from the L 2 -edge spectra, which are eventually multiplied by

|Y L 3 (Q)| 2 /|Y L 2 (Q)| 2 .
Note that the spectra, plotted as red squares, are now proportional to the spin-exchange structure factor. The two-spinon dynamical structure factor is obtained by subtracting this contribution from the normalized L 3 -edge RIXS data (black circles). The result is plotted as blue diamonds in Figure 6.6. The analysis has been made for the spectra measured at Q = 0 and Q = π/2. It seems that the high-energy continuum is entirely accounted for by the spin-exchange structure factor, while the two-spinon dynamical structure factor only contains a symmetric peak centered at ∼ 30 meV and ∼ 170 meV at Q = 0 and Q = π/2, respectively. This result, however, seems unrealistic because for Q = 0 the magnetic continuum should extend above the peak position for Q = π/2. Limitations to this analysis may be due to the crude approximation used to calculate |W (Q)| 2 and |Y (Q)| 2 , here estimated using a single-ion model, and self absorption, which may be different for L 2 and L 3 edges. A more reliable analysis could be obtained by directly comparing the iridium L 2 -edge RIXS spectra to the spin-exchange structure factor. The latter can be computed from the work by Klauser et al. [START_REF] Klauser | Spin-Exchange Dynamical Structure Factor of the S = 1/2 Heisenberg Chain[END_REF] . However, such Chapter 6 The magnetic and electronic dynamics of CaIrO 3 computation is very involved and falls outside the purpose of this work.

Electronic dynamics

I now turn to the discussion of the RIXS spectral features above 0.4 eV. Figure 6.7(a, c, e) displays the RIXS spectra of CaIrO 3 measured at T = 40 K along the orthogonal high-symmetry directions H, K, and L of the reciprocal lattice. From the inspection of the RIXS maps, three features can be distinguished and are named A, B, and C following the nomenclature of Refs. 112 and 39. These excitations show characteristic dispersions over the Brillouin zone. Since features A and B are energetically close, the fit is unsuccessful in precisely determining their energy position. To enhance the sensitivity to the peak position, the second derivative of each spectrum has been numerically calculated. It is reported in Figure 6.7(b, d, f). The dispersion of the excitations is now more evident, and is highlighted by the red dashed lines, which are guides to the eye. Let us consider first features B and C. Along the H direction, feature B disperses with minimum at ≈ 0.6 eV and maximum at ≈ 0.74 eV. Feature C, centered at ≈ 1.25 eV, is dispersionless instead. Along the K direction, excitations B and C are centered at approximately 0.64 eV and 1.25 eV, respectively, and do not display a detectable momentum dependence. Finally, along the L direction, excitation B exhibits maxima at the one-dimensional Brillouin zone center (Q = (1, 2, 9 ± {0, 1}) r.l.u. = Y) and minima at the boundaries (Q = (1, 2, 9 ± 0.5) r.l.u. = T) (refer to the Brillouin zone reported in Figure 6.8 for clarity on the notation used). Its dispersion amounts to ≈ 0.11 eV. Dispersionless excitation C is centered at approximately 1.25 eV. Excitations B and C have been studied in a recent work by Moretti Sala et al. [START_REF] Sala | CaIrO 3 : A Spin-Orbit Mott Insulator Beyond the J eff = 1/2 Ground State[END_REF] , who have ascribed them to excitations of a hole from the j eff = 1/2 ground state to the j eff = 3/2 excited states, in analogy to the electronic excitations of other iridates [START_REF] Kim | Magnetic Excitation Spectra of Sr 2 IrO 4 Probed by Resonant Inelastic X-Ray Scattering: Establishing Links to Cuprate Superconductors[END_REF][START_REF] Kim | Excitonic quasiparticles in a spin-orbit Mott insulator[END_REF][START_REF] Gretarsson | Crystal-Field Splitting and Correlation Effect on the Electronic Structure of A 2 IrO 3[END_REF] . These excitations are usually referred to as spin-orbit excitons [START_REF] Kim | Magnetic Excitation Spectra of Sr 2 IrO 4 Probed by Resonant Inelastic X-Ray Scattering: Establishing Links to Cuprate Superconductors[END_REF] . The non-degeneracy of the j eff = 3/2 energy levels is due to the tetragonal distortion of the IrO 6 octahedra. By utilizing a single-ion model, Moretti Sala et al. [START_REF] Sala | CaIrO 3 : A Spin-Orbit Mott Insulator Beyond the J eff = 1/2 Ground State[END_REF] have found the main energies at play in the ground state of CaIrO 3 , i.e. the SOC λ and the tetragonal crystal field ∆. Neglecting the 5d-e g orbitals, estimated to be ∼ 3 eV higher due to the cubic crystal field [START_REF] Sala | CaIrO 3 : A Spin-Orbit Mott Insulator Beyond the J eff = 1/2 Ground State[END_REF] , the energy of the 5dt 2g orbitals occupied by the single hole are functions of the SOC λ and the tetragonal distortion ∆. By considering a single iridium site, non-local effects are disregarded and any possible renormalization is included in the effective values of λ and ∆. The mathematical derivation of the energy levels within a single-ion model is reported in Appendix A, see in particular Equation A. 13. By imposing the calculated energy levels to coincide with the experimentally-determined excitation energies, the effective values of λ = 0.52 eV and ∆ = -0.71 eV have been determined [START_REF] Sala | CaIrO 3 : A Spin-Orbit Mott Insulator Beyond the J eff = 1/2 Ground State[END_REF] . These values are in agreement with quantum chemical calculations [START_REF] Bogdanov | Post-perovskite CaIrO 3 : A j = 1/2 quasi-one-dimensional antiferromagnet[END_REF][START_REF] Kim | Nature of the Insulating Ground State of the 5d Postperovskite CaIrO 3[END_REF][START_REF] Singh | Post-perovskite CaIrO 3 : a conventional Slater type antiferromagnetic insulator[END_REF] and suggest a stabilization of the |t ±1 2g against the |t 0 2g orbital in the ground-state wave function (Equation 1.4). The occupancy of the former is estimated to be ≈ 90% [START_REF] Sala | CaIrO 3 : A Spin-Orbit Mott Insulator Beyond the J eff = 1/2 Ground State[END_REF] . Such scenario is consistent with the tetragonal compression of the IrO 6 octahedra along the local z axis [START_REF] David | High-pressure structure and bonding in CaIrO3: The structure model of MgSiO3 postperovskite investigated with time-of-flight neutron powder diffraction[END_REF] 2g , ± on the excited site and the ground-state orbital on the neighboring site via the bridging oxygens in both corner-and edge-sharing directions. Therefore, a large hopping integral is expected to produce a dispersive, non-localized excitation. An evidence for the strong coupling between |t ±1 2g , ± orbitals on neighboring sites is also given by the magnitude of J c determined above from the magnetic dynamics. Instead, the orbitals |xy, ± on the excited site do not overlap with the ligands neither along the corner-nor along the edge-sharing direction. Hence, the hopping integral vanishes in first approximation. Direct exchange may occur along the edge-sharing direction for an ideal j eff = 1/2 ground state, where the |xy, ± contributes as much as the |t ±1 2g , ± to the wave function. However, the |xy, ± contribution is suppressed in CaIrO 3 , and so is the di-6.4 Electronic dynamics rect exchange. An indication in this direction comes from the small value of J a . The absence of effective couplings between the |xy, ± excited state on one site and the j eff = 1/2 ground state on the neighboring site eventually leads to a non-dispersive, localized excitation.

Electronic dynamics

In passing, I here envisage some consequences of the above discussion for Kitaev materials. The physical realization of the Kitaev model in real materials is often spoiled by the presence of additional magnetic couplings that compete and sometimes exceed the Kitaev term. This is for example the case of Na 2 IrO 3 , where direct-exchange-like interactions lead to longrange magnetic order [START_REF] Chun | Direct evidence for dominant bond-directional interactions in a honeycomb lattice iridate Na 2 IrO 3[END_REF][START_REF] Chaloupka | Kitaev-Heisenberg Model on a Honeycomb Lattice: Possible Exotic Phases in Iridium Oxides A 2 IrO 3[END_REF][START_REF] Choi | Spin Waves and Revised Crystal Structure of Honeycomb Iridate Na 2 IrO 3[END_REF][START_REF] Chaloupka | Zigzag Magnetic Order in the Iridium Oxide Na 2 IrO 3[END_REF][START_REF] Banerjee | Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet[END_REF][START_REF] Ran | Spin-Wave Excitations Evidencing the Kitaev Interaction in Single Crystalline α-RuCl 3[END_REF][START_REF] Banerjee | Neutron scattering in the proximate quantum spin liquid alpha-RuCl 3[END_REF] . Direct exchange for edge-sharing octahedra arises from the overlap of the |xy, ± orbitals on neighboring ions and can be effectively suppressed by playing with the distortion of the IrO 6 octahedra. Indeed, as it is the case for CaIrO 3 , a compressive distortion leads to a suppression of the |xy, ± orbital contribution to the ground state and eventually suppresses direct exchange mechanisms. This may be of interest when designing materials relevant for Kitaev physics.

Let us now consider the resolution-limited excitation A. Along the edge-sharing H direction, it is located at ≈ 0.45 eV at the Brillouin zone center (Q = (0, 2, 9) r.l.u. = Γ) as well as at the boundary (Q = (1, 2, 9) r.l.u. = Y), and disperses upwards by ≈ 0.14 eV until it reaches its maximum at Q = (0.5, 2, 9) r.l.u. Along the K direction, excitation A does not display a detectable momentum dependence. Along the corner-sharing L direction, it manifests a dispersion of ≈ 0.05 eV out of phase with respect to the one of feature B. Excitation A is found in the RIXS spectra of several iridates, including Sr 2 IrO 4 [START_REF] Kim | Excitonic quasiparticles in a spin-orbit Mott insulator[END_REF] and Na 2 IrO 3 112 . Being feature A much sharper than B and C, it is reasonable to think that it has a different origin. Its nature is debated: Gretarsson et al. [START_REF] Gretarsson | Crystal-Field Splitting and Correlation Effect on the Electronic Structure of A 2 IrO 3[END_REF] suggest that it may be due to the excitation of a particle and hole pair across the charge gap, which amounts approximately to 0.3-0.4 eV in Sr 2 IrO 4 [START_REF] Kim | Novel J eff = 1/2 Mott State Induced by Relativistic Spin-Orbit Coupling in Sr 2 IrO 4[END_REF] and Na 2 IrO 3 [START_REF] Comin | Na 2 IrO 3 as a novel relativistic mott insulator with a 340-mev gap[END_REF] . We note, however, that the charge gap of CaIrO 3 is smaller (≈ 0.34 eV [START_REF] Ohgushi | Metal-insulator transition in Ca 1-x Na x IrO 3 with post-perovskite structure[END_REF][START_REF] Jang | The electronic structure of epitaxially stabilized 5d perovskite Ca 1-x Sr x IrO 3 (x = 0, 0.5, and 1) thin films: the role of strong spin-orbit coupling[END_REF] ) than the energy of A, therefore casting doubts on the particle-hole-excitation option. Plotnikova et al. [START_REF] Plotnikova | Jahn-Teller Effect in Systems with Strong On-Site Spin-Orbit Coupling[END_REF] propose that the three electronic excitations visible in the RIXS spectra arise from the interplay of superexchange and Jahn-Teller interactions. Indeed, even though the Jahn-Teller mechanism is not allowed for a hole in a single j eff = 1/2 state due to the absence of orbital degrees of freedom, the mechanism is allowed for the j eff = 3/2 excited states [START_REF] Plotnikova | Jahn-Teller Effect in Systems with Strong On-Site Spin-Orbit Coupling[END_REF] . By considering the Jahn-Teller effect, a new channel for delocalization is established alongside the superexchange process [START_REF] Plotnikova | Jahn-Teller Effect in Systems with Strong On-Site Spin-Orbit Coupling[END_REF] . The calculated momentum dependence of the RIXS spectral features is shown in Figure 6.9. Although the calculation was made considering Sr 2 IrO 4 , it qualitatively agrees with our observations. At the Brillouin zone center feature A is minimum while B is maximum, and vice versa at the zone boundary. Feature C does not exhibit a significant dispersion instead. The three observed excitations show analogous properties and could thus be ascribed to the superexchange and the Jahn-Teller-mediated motion of the excited j eff = 3/2 holes.

Conclusions

In this Chapter, I investigated the magnetic and electronic excitations of CaIrO 3 by iridium L 2 -and L 3 -edge RIXS. The crystal structure of CaIrO 3 is peculiar among iridates since it features both edge-and corner-sharing IrO 6 octahedra [START_REF] Ohgushi | Metal-insulator transition in Ca 1-x Na x IrO 3 with post-perovskite structure[END_REF][START_REF] David | High-pressure structure and bonding in CaIrO3: The structure model of MgSiO3 postperovskite investigated with time-of-flight neutron powder diffraction[END_REF] . The presence of the two types of bond geometries is of particular interest, owing to the prediction of fundamentally-different intersite interactions in the two cases [START_REF] Jackeli | Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models[END_REF] .

By probing the magnetic dynamics of this compound along the highsymmetry directions of the Brillouin zone, I found that the dominant superexchange coupling along the corner-sharing direction has an isotropic nature. Instead, for nearest-neighbor pseudospins sharing an edge the Heisenberg superexchange is strongly suppressed and the dominant magnetic interaction is anisotropic, Ising-like. Such observations agree with the predictions by Jackeli and Khaliullin [START_REF] Jackeli | Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models[END_REF] .

I tried to interpret the magnetic dynamics in two ways, by using linear spin-wave theory on a three-dimensional magnet, and using the AFM XXZ model for a one-dimensional chain. In the former case, the dispersions computed considering minimal model Hamiltonians for the two directions agree well with the experimental ones with dominant couplings K a = -7 meV and J c = 170 meV. However, this analysis does not take

Conclusions

into account the high-energy tails of the magnetic excitations and likely leads to an overestimation of the Heisenberg coupling. The inhibition of isotropic superexchange for edge-sharing octahedra leads to the formation of quasi-one-dimensional AFM chains along the corner-sharing direction. The hallmark of such systems is the presence of spinon-like excitations bounded within a characteristic energy-and momentum-continuum [START_REF] Karbach | Two-spinon dynamic structure factor of the onedimensional s = 1/2 Heisenberg antiferromagnet[END_REF][START_REF] Caux | The four-spinon dynamical structure factor of the Heisenberg chain[END_REF][START_REF] Caux | The two-spinon transverse structure factor of the gapped Heisenberg antiferromagnetic chain[END_REF] , which motivates the use of the XXZ model for one-dimensional chains. The reasonably-good agreement between data and model validates the effective dimensionality reduction of CaIrO 3 . The limitation of this analysis resides in the impossibility to quantitatively reproduce the continuum, probably because interchain interactions are neglected.

The different nature of the magnetic interactions for the two bond geometries implies different electronic dynamics along the two directions. Indeed, the electronic excitations exhibited distinct momentum dependences along the edge-and corner-sharing directions. In particular, three electronic excitations were observed in the RIXS spectra, centered at approximately 0.45, 0.65, and 1.25 eV, and named A, B, and C, respectively. Feature B was found to exhibit a dispersion of ∼ 0.14 eV along the H direction with a minimum at the Brillouin zone center, while along L it disperses by ∼ 0.11 eV. Feature C is dispersionless instead. A simple explanation of the momentum dependence of the two excitations is achieved considering the overlap of the wave functions associated to the j eff = 3/2 excited states with the ligands. Feature A is ubiquitous in several iridates [START_REF] Kim | Excitonic quasiparticles in a spin-orbit Mott insulator[END_REF][START_REF] Gretarsson | Crystal-Field Splitting and Correlation Effect on the Electronic Structure of A 2 IrO 3[END_REF] and its origin is still debated [START_REF] Gretarsson | Crystal-Field Splitting and Correlation Effect on the Electronic Structure of A 2 IrO 3[END_REF][START_REF] Plotnikova | Jahn-Teller Effect in Systems with Strong On-Site Spin-Orbit Coupling[END_REF] . The model proposed by Plotnikova et al. [START_REF] Plotnikova | Jahn-Teller Effect in Systems with Strong On-Site Spin-Orbit Coupling[END_REF] that includes the hopping channels due to the superexchange process and the Jahn-Teller mechanism qualitatively explains the presence of three excitations in the RIXS spectra and their momentum dependence. However, a microscopic model Hamiltonian that accurately reproduces the dispersions of the electronic excitations in CaIrO 3 has not yet been proposed.

Possibility to realize spin-orbit-induced correlated physics beyond iridium oxides

Following the theoretical prediction of realization of spin-orbit-induced correlated-electron physics in a class of iridium and rhodium fluorides by Birol and Haule [START_REF] Birol | J eff = 1/2 Mott-insulating state in Rh and Ir fluorides[END_REF] , this Chapter deals with the experimental check of the theoretical study by means of RIXS. The crystal structures of the investigated iridium fluorides have also been refined by X-ray diffraction on powder and single-crystal samples. I also analyzed the evolution of the electronic structure with respect to chemical substitution and physical pressure.

Part of the results presented in this Chapter is published in Physical Review B 95, 235161 (2017) as "Possibility to realize spin-orbit-induced correlated physics in iridium fluorides" [START_REF] Rossi | Possibility to realize spin-orbit-induced correlated physics in iridium fluorides[END_REF] .

Introduction

The recent work by Birol and Haule 28 predicts the realization of a j eff = 1/2 Mott insulating state in Rb 2 [IrF 6 ] and related iridium fluorides. This prediction would extend the search for novel phases induced by the interplay of SOC and electron correlation to the family of iridium fluorides. Indeed, dynamical mean-field theory calculations by Birol and Haule ). The trigonal distortion can be quantified as (β -β 0 )/β 0 , where β is the angle between the Ir-F bond and the trigonal axis, and β 0 = arccos(1/ √ 3) ≈ 54.7 • is the corresponding angle in perfect cubic symmetry [START_REF] Avram | Optical properties of 3d-ions in crystals: Spectroscopy and crystal field analysis[END_REF] . The calculated distortion is 3.9% for Rb 2 [IrF 6 ].

XRPD patterns of Rb 2 [IrF 6 ] were collected at several temperatures in the range between 100 and 400 K. An analysis of the peak positions and shapes reveals that the crystal structure is stable in the investigated temperature range. Figure 7.1(b) shows the temperature dependence of the diffraction peaks associated to the (100) and the (001) reflections (first two peaks of the diffractogram). Their continuous variation can be directly associated to a smooth change of the a and c lattice parameters, as reported in Figure 7.1(c), and suggests that no structural phase transition occurs between 100 K and 400 K.

A similar analysis has been carried out for all the compounds at room temperature. The results of the crystal structure refinement are summarized in Table 7.1. In particular, Na 2 [IrF 6 ] is isostructural to Na 2 [PtF 6 ] 171 and belongs to the space group P 321, whereas (K,Rb,Cs) 2 [RbF 6 ] belong to the space group P 3m1, in agreement with existing literature [START_REF] Smolentsev | Rubidium hexafluoridoiridate(IV)[END_REF][START_REF] Smolentsev | Caesium hexafluoridoiridate(IV)[END_REF][START_REF] Fitz | Einkristalluntersuchungen an LiMF 6 (M = Rh, Ir), Li 2 RhF 6 und K 2 IrF 6[END_REF] , and Ba[IrF 6 ] belongs to the R 3 space group [START_REF] Smolentsev | Three hexafluoridoiridates(IV)[END_REF] . The common feature to all systems is the presence of isolated IrF 6 units with comparable trigonal distortions of the octahedral cage.

The electronic structure of iridium fluorides

After characterizing the samples from a structural point of view, I now turn to the investigation of their electronic structure with RIXS. A representative iridium L 3 -edge RIXS spectrum of Rb 2 [IrF 6 ] single crystal is

The electronic structure of iridium fluorides

shown in Figure 7.2(a). The spectrum was collected at momentum transfer Q = (1.5, 0, 6) r.l.u. and T = 20 K. The incident photon energy was fixed at 11.2165 keV, where excitations within the t 2g shell were found to resonate. This energy is ∼ 3 eV below the main absorption line, in agreement with works on iridates [START_REF] Sala | Crystal field splitting in Sr n+1 Ir n O 3n+1 (n = 1, 2) iridates probed by x-ray Raman spectroscopy[END_REF][START_REF] Ishii | Momentum-resolved electronic excitations in the Mott insulator Sr 2 IrO 4 studied by resonant inelastic x-ray scattering[END_REF][START_REF] Sala | CaIrO 3 : A Spin-Orbit Mott Insulator Beyond the J eff = 1/2 Ground State[END_REF][START_REF] Lefrançois | Anisotropic interactions opposing magnetocrystalline anisotropy in Sr 3 NiIrO 6[END_REF] . The black dots in Figure 7.2(a) correspond to the background-subtracted data points, while the red solid line is the fit to the data. As can be seen, no spectral excitations are found below 0.7 eV. This is indicative of the absence of magnetic interactions among iridium ions. At higher energy losses, two features (named A and B in the Figure) are clearly distinguished. They are fitted to two Lorentzian functions (blue dashed lines) and their energy positions are 805 ± 0.4 meV and 915 ± 1.3 meV for feature A and B, respectively. Considering the resonance behavior of the two features, we ascribe them to transitions from the j eff = 1/2 to the j eff = 3/2 states, in line with previous RIXS studies of iridium oxides [START_REF] Kim | Magnetic Excitation Spectra of Sr 2 IrO 4 Probed by Resonant Inelastic X-Ray Scattering: Establishing Links to Cuprate Superconductors[END_REF][START_REF] Kim | Excitonic quasiparticles in a spin-orbit Mott insulator[END_REF][START_REF] Sala | CaIrO 3 : A Spin-Orbit Mott Insulator Beyond the J eff = 1/2 Ground State[END_REF][START_REF] Gretarsson | Crystal-Field Splitting and Correlation Effect on the Electronic Structure of A 2 IrO 3[END_REF][START_REF] Lefrançois | Anisotropic interactions opposing magnetocrystalline anisotropy in Sr 3 NiIrO 6[END_REF] .

The momentum and temperature dependence of features A and B has also been analyzed. Figure 7.2(b) displays the RIXS spectra of Rb 2 [IrF 6 ] zoomed on the energy range that includes the j eff = 3/2 excitations and measured at different high-symmetry points of the Brillouin zone. No dispersion can be detected within the experimental uncertainties (see the vertical gray dashed lines corresponding to the peaks position), suggesting that the IrF 6 octahedra behave as isolated units. Moreover, features A and B do not show any temperature dependence, as illustrated in Figure 7.2(c). Our findings support the scenario of a strong insulating character of iridium fluorides, with narrow bands and no tendency to develop long-range magnetic order, in line with theoretical predictions [START_REF] Birol | J eff = 1/2 Mott-insulating state in Rh and Ir fluorides[END_REF] .

Similar measurements were carried out for all samples in powder form. Figure 7.3 shows a stack of the corresponding RIXS spectra. Interestingly, the overall shape is very similar and closely resembles the RIXS spectrum of Figure 7.2(a). All spectra clearly show a splitting of the j eff = 3/2 states.

To get a better insight into the electronic structure of iridium fluorides, we collaborated with Dr. M. Retegan (ESRF) to perform quantum chemical calculations using the ORCA software [START_REF] Neese | The ORCA program system[END_REF] . State-averaged complete active space self-consistent field (CASSCF) and N -electron valence perturbation theory (NEVPT2) [START_REF] Angeli | Introduction of n-electron valence states for multireference perturbation theory[END_REF] calculations were performed to determine the energy of the excited states and the SOC constant [START_REF] Atanasov | Detailed ab initio first-principles study of the magnetic anisotropy in a family of trigonal pyramidal iron(ii) pyrrolide complexes[END_REF] . All computations were done using the embedded cluster approach in order to account for the environment [START_REF] Staemmler | The cluster approach for the adsorption of small molecules on oxide surfaces[END_REF] . Models constructed starting from the crystal structure as determined by XRPD consisted of a central IrF 6 octahedron (the quantum cluster) surrounded by point charges. The active space included five electrons distributed over the five 5d orbitals. A detailed description of the computational method can be found in Rossi et al. [START_REF] Rossi | Possibility to realize spin-orbit-induced correlated physics in iridium fluorides[END_REF] . Table 7.2 compares the calculated splittings of the j eff = 3/2 excited states to the experimental values found by fitting the RIXS spectra. Although the calculated splittings are Compound

E exp A E exp B E exp B -E exp A E calc B -E calc A λ ∆ Na 2 [

The electronic structure of iridium fluorides

The estimated values of λ and ∆ for all samples are reported in Table 7.2. As can be seen, estimates for |∆| vary between ≈ 150 and ≈ 190 meV. The relatively large trigonal contribution to the crystal field contrasts with the prediction of an isotropic electronic state close to the SU(2) limit for iridium fluorides [START_REF] Birol | J eff = 1/2 Mott-insulating state in Rh and Ir fluorides[END_REF] . Indeed, SU(2) symmetry would require the degeneracy of the 5d-t 2g states. The estimated values of the SOC constant λ are very similar (≈ 570 meV) for all the iridium fluorides studied here. We note that the value of λ is 10-40% larger than in iridium oxides, where it ranges between 380 [START_REF] Kim | Excitonic quasiparticles in a spin-orbit Mott insulator[END_REF] and 520 meV [START_REF] Sala | CaIrO 3 : A Spin-Orbit Mott Insulator Beyond the J eff = 1/2 Ground State[END_REF] . Ligand-field parameters, SOC constant, and interelectronic repulsion terms have also been calculated by fitting the full configuration interaction matrix elements obtained from the CASSCF/NEVPT2 calculation to the matrix elements of a model Hamiltonian containing those interactions [START_REF] Atanasov | Detailed ab initio first-principles study of the magnetic anisotropy in a family of trigonal pyramidal iron(ii) pyrrolide complexes[END_REF] . We compute λ ≈ 550 meV for all iridium fluorides, in agreement with the experimentally-fitted values. Using the same theoretical approach, we calculate λ ≈ 520 meV for Sr 2 IrO 4 . In order to rule out the possibility that the reduction of SOC in iridium oxides compared to fluorides arises from differences in their crystal structure, we computed the SOC constant for an IrF 6 octahedron where the F ions have been placed at the positions of the O ions in Sr 2 IrO 4 . We obtain λ ≈ 550 meV, i.e. a larger value than the one obtained considering the IrO 6 octahedron. This confirms that the nature of the coordinating ion, rather than the crystal structure, determines the differences in the SOC constant. The larger value of the SOC in iridium fluorides than in oxides is indicative of a smaller degree of covalency in the former compounds [START_REF] Jørgensen | Spectroscopy of transition-group complexes[END_REF][START_REF] Al-Mobarak | The effect of covalency on the spin-orbit coupling constant[END_REF] . Indeed, since the SOC constant is inversely proportional to the radial extension of the orbitals [START_REF] Griffiths | Introduction to Quantum Mechanics[END_REF] (see Equation 1.1), the larger ionic character of the Ir-F bond compared to the Ir-O bond induces a stronger localization of the electron cloud in iridium fluorides and therefore an increased value of the SOC. This is of particular interest because covalency in Sr 2 IrO 4 is thought to be responsible for strong orbital anisotropies, in view of the increased spatial extent of the 5d-t 2g orbitals reaching the nearest-neighbor iridium atoms and beyond [START_REF] Agrestini | Long-range interactions in the effective low-energy Hamiltonian of Sr 2 IrO 4 : A core-to-core resonant inelastic x-ray scattering study[END_REF] . Iridium fluorides might therefore be the ideal playground for studying spin-orbit-induced correlated physics because correlation effects might be enhanced by the more localized nature of the electronic states, whereas long-range anisotropies, which contribute to deviate the j eff = 1/2 from the SU(2) symmetric limit, are strongly suppressed.

As a final remark, oxides and fluorides are discussed in relation to the similarities between cuprates and iridates. Let us first consider the copperbased compounds La 2 CuO 4 and K 2 CuF 4 . Although they share the same K 2 [NiF 4 ]-type crystal structure and are both insulating, their magnetic properties are very different: La 2 CuO 4 is an AFM insulator [START_REF] Vaknin | Antiferromagnetism in La 2 CuO 4-y[END_REF][START_REF] Chen | Electronic states in La 2-x Sr x CuO 4+δ probed by soft-x-ray absorption[END_REF] , while K 2 CuF 4 has a FM ground state [START_REF] Kugel | The Jahn-Teller effect and magnetism: transition metal compounds[END_REF] . Indeed, in La 2 CuO 4 a strong tetragonal crystal field splits the 3d-e g states and stabilizes the x 2 -y 2 orbital, which gives rise to ferro-orbital ordering and ultimately to AFM coupling Chapter 7 Possibility to realize spin-orbit-induced correlated physics beyond iridium oxides on a straight (180 • ) bond. On the contrary, in K 2 CuF 4 the CuF 6 octahedra are regular and the degeneracy of the 3d-e g states is essentially preserved. Therefore, the Cu 2+ ions are Jahn-Teller active. The so-called cooperative Jahn-Teller effect sets in and x 2 -z 2 /y 2 -z 2 alternating orbital ordering is stabilized leading to FM long-range order in the ground state [START_REF] Kugel | The Jahn-Teller effect and magnetism: transition metal compounds[END_REF] . When moving to Sr 2 IrO 4 , much of the physics of La 2 CuO 4 is retained, as explained in Chapter 1. In Sr 2 IrO 4 , the only active orbital is the j eff = 1/2, which is branched off from the 5d-t 2g by virtue of the strong SOC. One important consequence of such strong SOC is that, no matter how undistorted the system is, the Jahn-Teller mechanism is not supported in the Ir 4+ compounds [START_REF] Plotnikova | Jahn-Teller Effect in Systems with Strong On-Site Spin-Orbit Coupling[END_REF] . We therefore speculate that while the lack of magnetism in the studied iridium fluorides can basically be attributed to the isolation of the IrF 6 units, the ground state of a hypothetical iridium fluoride with an undistorted K 2 [NiF 4 ]-type crystal structure would probably never support FM order (unlike copper fluorides). Instead, it might even be closer to the Heisenberg AFM state found in copper oxides than iridium oxides. In this respect, the parallelism between copper oxides/fluorides and iridium oxides/fluorides is broken.

Crystal and electronic structure of Rb 2 [IrF 6 ] at high pressure

As discussed in the previous section, the substitution of the alkali or alkaline earth metal is not effective in modifying the low-energy electronic structure of iridium fluorides. However, the application of physical pressure may be. We therefore characterized the crystal and electronic structure of Rb 2 [IrF 6 ] at pressures up to 55 GPa and 26 GPa, respectively.

Crystal structure refinement

XRD measurements at high pressure were performed on beamline ID27 at the ESRF. The incident X-ray beam was monochromated to λ = 0.3738 Å, and a KB mirror was used to focus the beam at the sample position to a spot size of 3×3 µm 2 . The diffracted beam was collected by a MAR165 twodimensional CCD detector. The diffractometer was calibrated using standard samples, i.e. powder CeO 2 for powder rings and single-crystalline enstatite (MgSiO 3 ) for angle-resolved diffraction peaks. A standard-design DAC was used to apply pressure. The diamond culet was 300 µm in diameter, which allowed us to reach pressures above 50 GPa. A rhenium gasket was chosen for its mechanical hardness. The gasket, with initial thickness of 200 µm, was pre-indented to ∼ 40 µm and a sample chamber was created by laser-drilling a hole of ∼ 180 µm in the middle. Both powder and single-crystalline Rb 2 [IrF 6 ] were loaded in the sample chamber. Helium was used as pressure-transmitting medium. The pressure was determined by measuring the laser-induced fluorescence line of a ruby sphere placed inside the sample chamber [START_REF] Mao | High-pressure physics: the 1-megabar mark on the ruby r1 static pressure scale[END_REF] . A picture of the loading with sample and chamber dimensions is shown in Figure 7.5.

Powder diffraction data of Rb 2 [IrF 6 ] were used to refine the lattice parameters at each pressure point. Powder rings were collected by a continuous rotation of the DAC by ±30 • . Typical acquisition times were ∼ 1 min per pressure point. The diffraction pattern was obtained by a radial integration of the intensity along the detector surface. The diffraction peaks were indexed using the Rietveld refinement implemented in the GSAS and EXPGUI [START_REF] Toby | EXPGUI, a graphical user interface for GSAS[END_REF] packages. Single-crystal diffraction data of Rb 2 [IrF 6 ] were used to determine the atomic positions within the unit cell. The angle-resolved diffraction images were collected by scanning the DAC within a range of ±30 • with discrete steps of 0.5 • . Typical acquisition times were ∼ 30 s per pressure point. Indexing of the diffraction peaks and their intensity integration were performed with the automatic peak hunting routine implemented in the software CrysAlisPro. The atomic coordinates were calculated using the software Shelx 187 . Figure 7.6 shows the reconstructed Ewald sphere of Rb 2 [IrF 6 ] projected onto the (H, K, 0) reciprocal lattice plane at ambient pressure (a) and at 55 GPa (b). The crystal is very well preserved, as evinced by the presence of narrow diffraction peaks and the absence of powder rings.

The pressure dependence of the a and c lattice parameters is reported in Figure 7.7(a). Their smooth variation with pressure is indicative of the absence of a structural phase transition in the investigated pressure range. We note that the c lattice parameter, which corresponds to the nearest-neighbor Ir-Ir bond distance, assumes a value of ≈ 4 Å at the highest pressure point, and is thus comparable to that of Sr 2 IrO 4 (≈ 3.9 Å) at ambient pressure [START_REF] Crawford | Structural and magnetic studies of Sr 2 IrO 4[END_REF] . Therefore, a stronger magnetic interaction between nearest-neighbor iridium ions might be expected, although the establishment of long-range magnetic order is still very unlikely since the IrF 6 units remain isolated. Figure 7.7(b) reports the compression curve, i.e. the pressure dependence of the unit-cell volume normalized to the volume at ambient pressure. It is remarkable that despite the strong reduction of the unit-cell volume by ≈ 40%, the crystal does not undergo a phase transition at least until 55 GPa. Figure 7.7(b) also reports the pressure dependence of the axial ratio c/a. Our data display a monotonic decrease of the c/a ratio until ∼ 20 GPa, implying that the crystal is compacting more along the c axis rather than in the ab plane. Above ∼ 20 GPa, the axial ratio exhibits an inversion in its pressure evolution until it saturates at ∼ 45 GPa.

From the angle-resolved diffraction images collected on the singlecrystal sample, we were able to refine the atomic coordinates at each pres-beyond iridium oxides sure point. In particular, we reconstructed the pressure evolution of the IrF 6 octahedra. The dependence of the Ir-F bond distance on pressure is reported in Figure 7.8(a). As can be seen, the fluorine anions tend to approach the central iridium ion. The data (green circles) are fitted to a linear trend (dashed line) with slope of ≈ -1.87 × 10 -3 Å/GPa. Figure 7.8(b) reports the pressure evolution of the F-Ir-F bond angle (green circles). The data are fitted to a first-degree polynomial. As can be seen, the pressure dependence of the F-Ir-F bond angle cannot be detected within the experimental uncertainties. The bond angles are within the range 86.3 • ± 1.6 • , giving rise to a trigonal distortion ranging from 2.5% to 7%.

The pressure evolution of the electronic structure

The pressure evolution of the electronic structure of Rb 2 [IrF 6 ] was investigated with RIXS. The high-pressure experimental setup is similar to the one described in Chapter 5. A panoramic DAC with culet diameter of 250 µm was used to apply pressure. A beryllium gasket with initial thickness of 200 µm was pre-indented to ∼ 30 µm and a sample chamber of ∼ 150 µm in diameter was laser drilled in the middle. The sample chamber was completely filled with powder Rb 2 [IrF 6 ] and no pressure-transmitting medium was used. The non-hydrostatic conditions resulted in a pressure gradient inside the sample chamber of at most ±10%, as measured from the fluorescence of multiple ruby spheres placed inside the sample chamber.

The RIXS spectra of powder Rb 2 [IrF 6 ] measured at high pressure are reported in Figure 7.9 (filled circles). The spectra were collected at 5 pressure points until a maximum pressure of 26 ± 2 GPa. Two features at approximately 0.8 and 0.9 eV can be clearly distinguished at all pressure points. In analogy to the spectrum measured at ambient pressure (Figure 7.2), the two features are named A and B and are ascribed to transitions from the j eff = 1/2 ground state to the non-degenerate j eff = 3/2 excited states. The fit of the RIXS spectra are reported as solid lines in Figure 7.9. The red triangles are the fitted positions of features A and B.

To study the pressure evolution of the electronic structure of Rb 2 [IrF 6 ] we adopt a single-ion model. This allows us to find quantitative values of the SOC constant λ and of the effective trigonal distortion ∆ to the cubic crystal field, as explained in the previous section. We note that λ is related to the center of mass of features A and B, while ∆ is related to their energy splitting. The pressure dependence of the energy splitting of features A and B and of their center of mass is shown in Figure 7.10(a) and (b), respectively. Green circles are the results of the fit to the data, while the dashed black lines are linear fits and should be used as guides to the eye. As can be seen, the splitting of features A and B increases by ≈ 24 meV (i.e. in the order of the energy resolution) from ambient pressure to 26 GPa. The center of mass seems to have a positive pressure trend as well, however it . This variation can be ascribed to the compression of the unit cell together with the IrF 6 octahedra when pressure is applied, as reported above. The increase of the effective trigonal crystal field can be attributed to the stronger interaction between the iridium ion and the surrounding anions, thus enhancing the lift of the orbital degeneracy.

The effect of halogen substitution on the electronic structure

The application of chemical pressure through the substitution of the alkali and alkaline earth metal as well as the application of physical pressure have little effect on the electronic structure of the iridium fluorides. Indeed, little differences are measured in the RIXS signal in both cases. Therefore, we decided to take another route and substitute the fluorine ion with different members of the halogen group, namely chlorine and bromine. The replacement of the fluorine ligands with other elements is expected to affect the electronic structure more severely, since the nature of the iridium-ligand bond strongly affects the crystal field and the SOC, as discussed in Section 7. RIXS. This calls for a deeper investigation of the crystal structure by means of high-resolution X-ray diffraction. The study of the electronic structure with other techniques is also desirable to confirm our observations. Nevertheless, the smaller splitting of features A and B in K 2 [IrCl 6 ] and K 2 [IrBr 6 ] than in K 2 [IrF 6 ] indicates that the ground state in the former is undoubtedly more isotropic than in the latter.

The unavailability of single crystals precluded momentum-dependent RIXS studies to unveil the presence of correlated-electron physics in these systems. However, we demonstrated that the local electronic structure of iridium compounds can be more effectively tuned by chemical substitution of the halogen.

Conclusions

Motivated by the theoretical prediction of the realization of spin-orbitinduced correlated physics in a novel class of iridium compounds [START_REF] Birol | J eff = 1/2 Mott-insulating state in Rh and Ir fluorides[END_REF] , I investigated the crystal and electronic structure of Na ] and the other iridium fluorides studied here are characterized by a j eff = 1/2 electronic ground state [START_REF] Birol | J eff = 1/2 Mott-insulating state in Rh and Ir fluorides[END_REF] . The absence of low-energy features, as well as momentum and temperature dependence in the RIXS spectra of Rb 2 [IrF 6 ] single crystal suggests that interactions beyond the first coordination shell of iridium ions are negligible, thus precluding long-

Conclusions

range magnetic order in the investigated temperature and pressure ranges. At the same time, I found a substantial splitting of the j eff = 3/2 excited states, which is indicative of a deviation from the isotropic limit. Consistently, quantum chemical calculations on a single IrF 6 cluster reproduced the experimental trend observed among the various compounds at ambient pressure and elucidate that the low-energy electronic structure of iridium fluorides is ascribed to characteristic local distortions of the IrF 6 cage without significant influence from neighboring ions. Rb 2 [IrF 6 ] preserves its crystal structure up to at least 55 GPa. The j eff = 1/2 electronic state is robust against the application of pressure up to at least 26 GPa. I found a pressure-induced increase of the splitting of the RIXS spectral features, which was ascribed to an increase of the effective trigonal crystal field strength. This is due to a stronger interaction between the central iridium ion and the surrounding fluorine anions, whose relative distance is reduced when pressure is applied.

I also reported an increase of the spin-orbit coupling in iridium fluorides as compared to iridium oxides. This finding is corroborated by quantum chemical calculations and suggests that the larger electronegativity of fluorine compared to oxygen reduces the degree of covalency in the system. This has important consequences: i) the spatial extension of the 5dt 2g orbitals is reduced and correlation effects might indeed be enhanced; ii) long-range anisotropies are mitigated and an isotropic j eff = 1/2 ground state is more likely to stabilize.

The study of Na ] pointed out that the substitution of the alkali and alkaline earth has little effect on the electronic structure of these compounds. Likewise, the application of physical pressure is poorly effective in modifying the electronic states, and the j eff = 1/2 state departs even more from the isotropic limit. Another pursued direction was the substitution of fluorine with another member of the halogen group. I observed a reduction of the SOC strength when moving from K 2 [IrF 6 ] to K 2 [IrCl 6 ] and K 2 [IrBr 6 ], indicative of a larger degree of covalency in the Ir-Cl and Ir-Br bonds than in the more ionic Ir-F bond. Finally, a smaller splitting of the j eff = 3/2 states was observed, indicative of a more isotropic ground state.

This comprehensive study of selected iridium fluorides, chlorides, and bromides demonstrated that the j eff = 1/2 state is robust in these materials upon chemical substitution and application of physical pressure. We suggest that the search for novel quantum phases induced by the interplay of SOC and electron correlation may be extended to non-oxide materials. If synthesized in crystal structures with connected units, such hypothetical iridium compounds might indeed support features characteristic of spinorbit-induced correlated physics and might even more closely resemble the low-energy physics found in copper oxides than it is the case of iridium oxides.

Conclusions and future perspectives

My work focused on the study of the magnetic and electronic structure and dynamics of iridium-based compounds by means of RIXS. These systems have raised particular attention in the last decade due to the identification of correlation effects induced by the strong SOC of iridium. RIXS has been one of the leading techniques in this field. Indeed, the synthesis of smallvolume samples and the strong neutron absorption of iridium made the use of conventional neutron-based techniques particularly unfavorable to study elementary excitations in iridates with momentum resolution. Instead, the sufficiently-large RIXS cross-section at the iridium L 3 edge and the energy resolution of currently-available RIXS spectrometers favored the use of this technique.

In particular, I carried out RIXS experiments to probe the low-energy physics of selected iridium compounds. I investigated the magnetic and electronic excitations of CaIrO 3 . This system is especially interesting since its post-perovskite crystal structure features both edge-and corner-sharing IrO 6 octahedra along the a and c crystallographic axes, respectively. The magnetic interactions strongly depend on the bond topology: for cornersharing octahedra, an isotropic Heisenberg-like exchange coupling was found to be the dominant interaction, while for edge-sharing octahedra, the Heisenberg exchange is strongly suppressed and a weak anisotropic Isinglike coupling was detected. The strong anisotropy of the magnetic couplings makes this material a quasi-one-dimensional magnet with character-Conclusions and future perspectives istic spinon-like magnetic excitations. However, the details of the magnetic dispersion could not be entirely reproduced by a simple one-dimensional XXZ model. Specifically, the dispersion of magnetic excitations along the direction perpendicular to the chains can only be explained if an interchain coupling is introduced. The behavior of orbital excitations in CaIrO 3 was also probed. A qualitative explanation of the momentum dependence of these excitations was provided, based on the electronic density associated to the ground-and excited-states wave functions, and consistent with the anisotropy of the magnetic couplings. A more quantitative analysis of the magnetic and electronic excitations of CaIrO 3 based on quantum chemical calculations is on-going.

I also addressed the study of non-oxide iridium compounds. In particular, I investigated the crystal and electronic structure of Na ] by means of X-ray diffraction and RIXS, respectively. Iridium fluorides were proposed to exhibit spin-orbit-induced correlated-electron physics. Moreover, they were predicted to display unprecedented proximity to the isotropic j eff = 1/2 ground state. I found that the electronic excitations of these compounds can indeed be modeled within the j eff = 1/2 scenario, but a sizable crystal field due to the trigonal distortion of the IrF 6 octahedra should be included. Moreover, only dispersionless spectral features were detected in single-crystal compounds, suggesting the absence of iridium-iridium interactions, likely due to the isolation of the IrF 6 octahedra. Little, but significant differences between the various compounds could be ascribed to slight modifications of the IrF 6 cages, as confirmed by quantum chemical calculations. In the spirit of tuning the electronic structure of iridium fluorides, I also studied the dependence of their excitations on pressure and chemical substitution. Physical pressures up to 26 GPa were found to modify, but not in an effective manner, the electronic structure of iridium fluorides. On the other hand, chemical substitution of the halogen element had a larger impact. In particular, I found that the SOC scales with the degree of covalency of the iridium-ligand bond, and it is the strongest for iridium fluorides owing to the larger electronegativity of fluorine. I also found that the j eff = 1/2 state is closer to the isotropic limit in K 2 [IrCl 6 ] and K 2 [IrBr 6 ] than in K 2 [IrF 6 ]. My studies on iridium fluorides, and on the evolution of their electronic structure with pressure and chemical substitution, suggest that the search for quantum phenomena relying on the j eff = 1/2 model can be extended to non-oxide materials. If crystal structures with connected units could be stabilized, possibly with the application of pressure, such hypothetical iridium compounds might indeed support features characteristic of spinorbit-induced correlated physics.

Part of my PhD activity was devoted to instrumental developments. I proposed the design of a RIXS spectrometer with polarization resolution to be used at the iridium L 3 -edge. The polarimeter is based on Bragg re-Conclusions and future perspectives flections from asymmetrically-cut crystals and features a Montel multilayer mirror to collect the photons scattered by the sample on a sufficiently large solid angle, and collimate the scattered beam. The expected energy resolution and efficiency of the polarimeter were estimated to be approximately 21 meV and 27% by ray-tracing simulations, respectively. These values are comparable to the ones of current state-of-the-art RIXS spectrometers. Preliminary measurements on the two-crystal setup were carried out on beamline ID20 at the ESRF with encouraging results. The device is currently under construction and will soon be delivered and commissioned. When operational, the polarimeter will allow RIXS measurements with full characterization of the scattered X-rays, providing useful information about the symmetry and thus the nature of the excitations. For example, the polarimeter will open new perspectives in the study of magnetic excitations: magnetic features are associated to a rotation of the photon polarization and can be easily uncovered if polarization analysis of the scattered X-rays is performed.

Further developments of the polarimeter can be easily envisaged to cover the relevant absorption edges of other elements, namely the K and L edges of 3d and 5d transition metals, respectively. In addition, the optical scheme based on the use of flat asymmetric crystals leaves room for straightforward improvements of the energy resolution at the expenses of efficiency. I expect that RIXS experiments with sub-10-meV energy resolution will be routinely performed in the near future [START_REF] Kim | Quartz-based flat-crystal resonant inelastic x-ray scattering spectrometer with sub-10 meV energy resolution[END_REF] .

Finally, sample-environment equipment to perform RIXS experiments under extreme conditions of pressure and temperature was designed and commissioned. High-resolution RIXS of low-energy features at high pressure and low temperature are extremely challenging and, to the best of my knowledge, have never been performed before. The equipment was successfully employed to probe the magnetic excitations of the bilayerperovskite Sr 3 Ir 2 O 7 at 100 K and up to 12 GPa. These measurements may help to define the model that is best suited to describe the magnetic structure and dynamics of Sr 3 Ir 2 O 7 , for which consensus has not been reached yet. In general, performing RIXS measurements in extreme thermodynamic conditions will eventually lead to a better understanding of the lowenergy physics of materials. Besides hydrostatic pressure, uniaxial [START_REF] Hicks | Piezoelectric-based apparatus for strain tuning[END_REF] and epitaxial strain can also be used to effectively modify the interatomic distances and thereby correlation effects. In this context, iridates grown as thin films or multilayers will be certainly explored in the future.

One of the main limiting factors in high-pressure RIXS experiments is the sample size, which ultimately depends on the focal spot size of the Xray beam. In this respect, I here note that the upcoming upgrade project of the ESRF storage ring (Extreme Brilliant Source, EBS) will improve the source characteristics of the beamlines. In the specific case of ID20, the ESRF-EBS upgrade will provide a smaller source size, which eventually Conclusions and future perspectives leads to a narrower focal spot size at the sample position. With an estimated beam size of 4 × 4 µm 2 , high-pressure RIXS experiments should become more accessible and pressures as high as 100 GPa may be achieved. In this pressure range, many of the materials studied in this thesis undergo or are expected to undergo structural phase transitions leading to dramatic changes of their low-energy physics.

Of course, the study of 5d transition metal compounds is mainly focused on -but not limited to -iridium-based systems. The interplay between SOC and electron correlation has been investigated also in materials containing osmium and tantalum [START_REF] Calder | Enhanced spin-phonon-electronic coupling in a 5d oxide[END_REF][START_REF] Jeong | Direct experimental observation of the molecular J eff = 3/2 ground state in the lacunar spinel GaTa 4 Se 8[END_REF] , and in the future it will likely be extended to other 5d transition metal compounds. I expect that RIXS will play a major role in the characterization of their magnetic and electronic structure, especially if the instrumental developments proposed in this thesis will be routinely implemented.

Chapter A Single-ion model

i.e. linear combinations of spherical harmonics Y m l of the form [START_REF] Griffiths | Introduction to Quantum Mechanics[END_REF] :

|xy = i √ 2 |Y -2 2 -|Y 2 2 , |yz = i √ 2 |Y -1 2 + |Y 1 2 , |xz = 1 √ 2 |Y -1 2 -|Y 1 2 , |x 2 -y 2 = 1 √ 2 |Y -2 2 + |Y 2 2 , |3z 2 -r 2 = |Y 0 2 . (A.1)
As explained in Chapter 1, the octahedral (O h ) symmetry of the oxygen anions surrounding the central iridium ion lifts the degeneracy of the 5d orbitals, leading to lower-energy t 2g = {|xy , |yz , |xz } states and higherenergy e g = {|x 2 -y 2 , |3z 2 -r 2 } orbitals. In iridates, the splitting between energy levels due to the cubic crystal field is relatively large (10Dq ∼ 3 eV [START_REF] Sala | Crystal field splitting in Sr n+1 Ir n O 3n+1 (n = 1, 2) iridates probed by x-ray Raman spectroscopy[END_REF][START_REF] Ishii | Momentum-resolved electronic excitations in the Mott insulator Sr 2 IrO 4 studied by resonant inelastic x-ray scattering[END_REF][START_REF] Sala | CaIrO 3 : A Spin-Orbit Mott Insulator Beyond the J eff = 1/2 Ground State[END_REF] ), so that the 5 electrons in the 5d shell are placed in a low-spin configuration in the t 2g states 4 , whereas the e g states are empty and will be neglected in the following.

One can notice from the definition of cubic harmonics (Equation A.1) that the orbital angular momentum of an electron occupying the e g states is quenched. Indeed, the expectation values of the projection of the orbital angular momentum along the quantization axis is x 2 -y 2 |l z |x 2 -y 2 = 3z 2 -r 2 |l z |3z 2 -r 2 = 0. The expectation values of l x and l y also vanish since they can be written as linear combinations of ladder operators l ± = l x ± il y , which would create states with l z = ±1 while the e g states only have l z = 0, ±2.

It is often convenient to map the t 2g orbitals onto the triplet of states with an effective orbital angular momentum l eff = -1 [START_REF] Jackeli | Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models[END_REF] , and with complex wave functions written as linear combinations of |xy , |yz and |xz states:

|t 0 2g = |xy , |t 1 2g = - 1 √ 2 (i|xz + |yz ) , |t -1 2g = - 1 √ 2 (i|xz -|yz ) . (A.2)
The corresponding electron density is displayed in Chapter A Single-ion model transformation matrix must be used:

T =                        i √ 2 0 0 0 0 0 0 i √ 2
0 0 0 0 0 0 -i 0 0 0 0 0 0 -i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 i 0 0 0 0 0 0 i

- i √ 2 0 0 0 0 0 0 - i √ 2 0 0 0 0                        . (A.7)
In the B ef f basis set, the matrix representation of the SOC Hamiltonian is:

H SOC =                   0 0 0 0 0 - λ √ 2 0 0 - λ √ 2 0 0 0 0 - λ √ 2 λ 2 0 0 0 0 0 0 - λ 2 0 0 0 0 0 0 - λ 2 0 - λ √ 2 0 0 0 0 λ 2                   , (A.8)
where the matrix element ij is the expectation value of the SOC Hamiltonian A.4 on the i-th and j-th element of B ef f , e.g. the second element in the third row is t 1 2g , ↓ |H SOC |t 0 2g , ↑ = -λ/ √ 2.

The matrix H SOC can be diagonalized as Λ = U -1 H SOC U, where Λ is the matrix of the eigenvalues: where each column corresponds to the eigenvectors

Λ =        
J = {| 1 2 , -1 2 , | 1 2 , 1 2 , | 3 2 , -3 2 , | 3 2 , -1 2 , | 3 2 , 1 2 , | 3 2 , 3 2
}. Note that the matrix elements of U correspond to the Clebsch-Gordan coefficients for the addition of angular momenta l = 1 and s = 1/2 [START_REF] Griffiths | Introduction to Quantum Mechanics[END_REF] .

The SOC Hamiltonian applied to the t 2g states gives two nondegenerate energy levels, i.e. the t 2g states are split into a lower-energy filled j eff = 3/2 quartet and a higher-energy half-filled j eff = 1/2 doublet 4 . The splitting of the j eff states is in agreement with Hund's third rule for more-than-half-filled t 2g orbitals. The rule states that the highest value of the total angular momentum (l s) lies lowest in energy. The wave functions of the j eff = 1/2 doublet are:

1 2 , - 1 2

A.3 The effect of octahedral distortion

In real compounds, the octahedral cage surrounding the iridium ion is rarely regular, and diffraction measurements reveal that it always appears to be distorted. The distortion is usually in the order of few percent, still its effects on the ground and excited states can be detected during a spectroscopy measurement. The compounds studied in this work display either a tetragonal or trigonal distortion of the octahedron. I will therefore consider these two cases only.

A.3.1 Tetragonal distortion

In this case, the octahedron is elongated or compressed along the local z axis and the distances between the iridium ion and the 6 ligands are no The matrix representation of the tetragonal crystal field is diagonal in the B ef f basis set and reads:

H ∆ =        
∆ 0 0 0 0 0 0 ∆ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

        , (A.12)
where ∆ quantifies the strength of the tetragonal crystal field and it is positive for an elongation of the octahedron. By diagonalizing the total Hamiltonian H = H SOC + H ∆ , one obtains the following eigenvalues: The distances between the iridium ion and the 6 surrounding anions are the same, but the angles are different from 90 • . It is useful to write the t 2g orbitals considering the trigonal axis as the quantization axis [START_REF] Ballhausen | Introduction to ligand field theory, McGraw-Hill series in advanced chemistry[END_REF] :

E | 1 2 ,± 1
|t 0 2g = |Y 0 2 , |t 1 2g = 2 3 |Y -2 2 + 1 3 |Y 1 2 , |t -1 2g = - 1 3 |Y -1 2 + 2 3 |Y 2 2 .
(A.17)

From the inspection of the electron densities associated to these orbitals ( with effective orbital angular momentum l eff = -1, and t 0 2g |l z |t 0 2g = 0, t ±1 2g |l z |t ±1 2g = ∓1. The matrix representation of the trigonal crystal field Hamiltonian in the B trig ef f basis set is analogous to Equation A.12, and the total Hamiltonian including also the SOC reads:

H trig =                   ∆ 0 0 0 0 - λ √ 2 0 ∆ λ √ 2 0 0 0 0 λ √ 2 λ 2 0 0 0 0 0 0 - λ 2 0 0 0 0 0 0 - λ 2 0 - λ √ 2 0 0 0 0 λ 2                   , (A.18)

B.1 Calculation of the matrix elements

In order to calculate the transition amplitude from a specific d orbital to a specific p orbital, a change of basis must be made. The following matrix allows to obtain the spin-orbit-split p states { 1 2 , + 1 2 , 1 2 , -1 2 , 3 2 , + 3 2 , 3 2 , + 1 2 , 3 2 , -1 2 , 3 2 , -3 2 } as linear combinations of l p = 1 spherical harmonics:

P =                 0 0 0 0 0 1 0 - 2 3
f |D|i = P † f ΣT V i . (B.8)
The calculation of the transition amplitudes from the intermediate p states to the final d states is straightforward if one considers the Hermitian of matrix B.6. The two transition amplitudes are then multiplied to obtain the RIXS scattering amplitude. It is convenient to express the scattering amplitude in a way such that the incident-and outgoing-photon polarizations are explicit in the equation. It can be shown that:

A f ≈ A f , (B.9)
where A f is the scattering tensor.

We will consider as ground state of the system the wave function corresponding to the j eff = 1/2 state: |g = 1 2 , 1 2 . The expression of this state in the B ef f basis set is reported in Equation A. 16.

       . (B.11)
We note that the elements on the diagonal are all zero, implying that the excitation of a magnon requires the rotation of the photon polarization. The scattering tensors for transitions involving the j eff = 3/2 orbitals are omitted since they are never considered in this work.

B.3 L 2 -edge RIXS scattering amplitude

We now consider the scattering tensor for L 2 -edge RIXS. For elastic scattering ( 12 , 1 2 → 1 2 , 1 2 ) we obtain: The tensor for the single spin-flip transition ( 1 2 , 1 2 → 1 2 , -1 2 ) is identically zero, meaning that this transition is forbidden at the L 2 edge.

A | 1 2 , 1 2 =      1

1. 2 2 63 5. 1 3 79 6. 1 90 xv LIST OF FIGURES 6 . 6

 2263137919066 Pictorial view of the electron density of the 5d orbitals inside the octahedral cage. . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Splitting of the t 2g orbitals into a lower-energy j eff = 3/2 quartet and a higher-energy j eff = 1/2 doublet due to the effect of the SOC λ. . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Dependence of the electron density of the j eff = 1/2 and j eff = 3/2 states on the relative strength of the tetragonal crystal field and the SOC. . . . . . . . . . . . . . . . . . . . . 1.5 Modification of the outermost t 2g energy band due to the action of SOC and Coulomb repulsion. . . . . . . . . . . . . . . 1.6 Experimental evidences of the realization of the spin-orbitinduced j eff = 1/2 ground state in Sr 2 IrO 4 . . . . . . . . . . .1.7 Corner-sharing and edge-sharing IrO 6 octahedra and active t 2g orbitals in the superexchange process. . . . . . . . . . . . 1.8 Spin-wave dispersion in La 2 CuO 4 and Sr 2 IrO 4 . . . . . . . . . 1.9 Evolution of the Fermi surface of La 2 CuO 4 and Sr 2 IrO 4 upon hole and electron doping, respectively. . . . . . . . . . . . . . xiii LIST OF FIGURES 1.10 (a) Schematic phase diagram of La-doped Sr 2 IrO 4 . (b) Doping dependence of the magnetic Bragg peak (0.5, 0.5, 32) r.l.u. (c,d) Magnetic and electronic excitations of La-doped Sr IrO 4 for doping concentrations of x = 0 (c) and x = 0.1 (d). 1.11 Diffuse magnetic X-ray scattering of Na 2 IrO 3 . . . . . . . . . . 1.12 DMFT calculations on Rb 2 IrF 6 . . . . . . . . . . . . . . . . . . 1.13 Iridium L 2,3 -edge X-ray absorption spectra and X-ray magnetic circular dichroism of molecular iridium fluorides. . . . 2.1 Scattering geometry in RIXS. . . . . . . . . . . . . . . . . . . 2.2 Schematic representation of the RIXS process. . . . . . . . . . 3.1 Schematic representation of an undulator. . . . . . . . . . . . 3.2 Layout of beamline ID20 at the ESRF. . . . . . . . . . . . . . . 3.3 Sketch of the RIXS spectrometer installed at beamline ID20. . 3.4 Rowland circle geometry adopted for the RIXS spectrometer of beamline ID20. . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 RIXS spectrum of YBa 2 Cu 3 O 6.6 , decomposed into the crossed polarization component and non-crossed polarization component. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 RIXS spectra of (Sr 1-x La x) 2 IrO 4 as a function of doping x. . 4.3 RIXS spectrum of Na 2 IrO 3 measured at T = 10 K. . . . . . . 4.4 Configuration of the spectrometer with polarization resolution used at beamline 30-ID of the APS. . . . . . . . . . . . . 4.5 Asymmetric Bragg reflection. . . . . . . . . . . . . . . . . . . 4.6 X-ray reflectivity R of a Si(311) crystal in the phase space defined by energy and angle. . . . . . . . . . . . . . . . . . . 4.7 Example of DuMond diagram. . . . . . . . . . . . . . . . . . 4.8 CDS spectrometer for sub-10-meV RIXS experiments at the iridium L 3 edge. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Sketch of the polarization analyzer for RIXS experiments designed for beamline ID20. . . . . . . . . . . . . . . . . . . . . 4.10 Sketch of the multilayer Montel mirror. . . . . . . . . . . . . 4.11 DuMond diagrams representing the reflection of a collimated beam from two asymmetrically-cut crystals. . . . . . . 4.12 DuMond diagrams representing the reflection of a divergent beam from two asymmetrically-cut crystals. . . . . . . . . . . 4.13 Ray-tracing simulation of the energy distribution of the photons after the two asymmetric reflections. . . . . . . . . . . . 4.14 Dependence of the energy resolution and efficiency of the two-crystal setup on the beam divergence downstream of the Montel mirror. . . . . . . . . . . . . . . . . . . . . . . . . . xiv LIST OF FIGURES 4.15 Rocking curve of the Si(844) post-monochromator for different values of the beam divergence recorded by a diode placed downstream of the two asymmetric reflections. . . . . 61 4.16 Mechanical model of the polarization-resolved RIXS spectrometer installed at beamline ID20. . . . . . . . . . . . . . . Crystal and magnetic structure of Sr 2 IrO 4 and Sr 3 Ir 2 O 7 . . . . 67 5.2 RIXS spectra of Sr 3 Ir 2 O 7 collected at T = 30 K along highsymmetry directions of the two-dimensional Brillouin zone. 68 5.3 (a) RIXS spectra of Sr 3 Ir 2 O 7 collected at T = 20 K along highsymmetry directions of the two-dimensional Brillouin zone. (b) Closeup on limited region of the Brillouin zone. . . . . . 68 5.4 Schematic phase diagram of La-doped Sr 3 Ir 2 O 7 . . . . . . . . 69 5.5 Panoramic DAC utilized for high-pressure RIXS experiments at beamline ID20. . . . . . . . . . . . . . . . . . . . . . 71 5.6 Sr 3 Ir 2 O 7 and ruby sphere loaded inside the DAC. . . . . . . 73 5.7 Helium-flow cryostat for high-pressure low-temperature RIXS experiments. . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.8 Iridium L 3 -edge RIXS spectra of Sr 3 Ir 2 O 7 measured at low temperature and high pressure. . . . . . . . . . . . . . . . . . 76 5.9 Pressure dependence of the magnetic Bragg reflections of Sr Ir 2 O 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.10 Linear extrapolation of the pressure evolution of the magnon softening. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.11 Evolution of the magnetic gap when the intralayer superexchange J decreases and the interlayer coupling J c increases. Crystal and magnetic structure of CaIrO 3 . . . . . . . . . . . . 83 6.2 Iridium L 3 -edge RIXS spectra of CaIrO 3 measured at momentum transfer Q = (0.5, 2, 9) r.l.u. and at temperature T = 295 K and T = 20 K; temperature dependence of the integrated intensity and of the FWHM of the narrow feature. 84 6.3 Momentum transfer dependence of the magnetic excitations of CaIrO 3 measured along the H, K, and L high-symmetry directions of the reciprocal lattice. . . . . . . . . . . . . . . . . 86 6.4 Momentum dependence of the RIXS spectra of CaIrO 3 compared to the exact calculation of the dynamical structure factor of the AFM XXZ chain. . . . . . . . . . . . . . . . . . . . . 88 6.5 Comparison between the experimental data collected at Q = 0, π/4 and π/2, and the two-spinon dynamical structure factor of the one-dimensional AFM XXZ chain. . . . . . . . . . . Comparison between iridium L 3 and L 2 edge RIXS spectra measured at Q = 0 and Q = π/2. The subtraction between spectra measured at the L 3 and L 2 edge is proportional to the dynamical structure factor. . . . . . . . . . . . . . . . . . 6.7 Momentum transfer dependence of the electronic excitations of CaIrO 3 measured along the H, K, and L high-symmetry directions of the reciprocal lattice. . . . . . . . . . . . . . . . . 6.8 The Brillouin zone of CaIrO 3 . . . . . . . . . . . . . . . . . . . 6.9 Momentum-and energy-dependence of the spin-orbit exciton computed considering both superexchange and Jahn-Teller interactions. . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 XRPD pattern of Rb 2 IrF 6 and temperature dependence of the lattice constants. . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Iridium L 3 -edge RIXS spectrum of Rb 2 IrF 6 . . . . . . . . . . . 7.3 Iridium L 3 -edge RIXS spectra of A 2 IrF 6 (A = Na, K, Rb, Cs) and BaIrF 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Comparison between the energy splitting of the j eff = 3/2 excitations obtained from the experimental data and from the quantum chemical calculations. . . . . . . . . . . . 7.5 Powder and single-crystal Rb 2 IrF 6 and ruby sphere loaded at the center of the diamond culet. . . . . . . . . . . . . . . . 7.6 Projection of the Ewald sphere onto the (H, K, 0) reciprocal lattice plane at ambient pressure and at 55 GPa. . . . . . . . 7.7 (a) Pressure dependence of the lattice parameters of Rb 2 IrF 6 . (b) Isothermal compression curve at ambient temperature and pressure dependence of the axial ratio c/a.

Figure 1 :

 1 Figure 1: Generic phase diagram of transition metal oxides generated by the simultaneous action of electron correlation and SOC. [Figure from Witczak-Krempa et al. 10 ]

Fig- ure 1 . 2 . 2 1. 2 Figure 1 . 1 : 2 Figure 1 . 2 :

 122211212 Figure 1.1: Dependence of the SOC constant λ on the atomic number Z. The Z 4 dependence reproduces the trend within a specific series of elements, whereas the systematic trend among different series is proportional to Z 2 . [Figure from Shanavas et al. 34 ]t 2g e g

Figure 1 . 3 :

 13 Figure 1.3: Splitting of the t 2g orbitals into a lower-energy j eff = 3/2 quartet and a higher-energy j eff = 1/2 doublet due to the effect of the SOC λ. The electron and spin densities of the j eff = 1/2 and j eff = 3/2 wave functions are also shown for the special case ∆ = 0.

1. 2 Figure 1 . 4 :

 214 Figure 1.4: Dependence of the electron density of the j eff = 1/2 and j eff = 3/2 states on the relative strength of the tetragonal crystal field ∆ and the SOC λ.

Figure 1 . 5 :Figure 1 . 6 :Figure 1 . 7 :

 151617 Figure 1.5: Modification of the outermost t 2g energy band due to the action of SOC ζ SO and Coulomb repulsion U . (a) SOC and Coulomb repulsion are not included; (b) without SOC and with an unrealistically-large Coulomb repulsion U ; (c) with SOC and without Coulomb repulsion; (d) with both SOC and Coulomb repulsion. [Figure from Kim et al. 4 ]

Figure 1 . 8 :

 18 Figure 1.8: Spin-wave dispersion in La 2 CuO 4 (left panel) and Sr 2 IrO 4 (right panel), showing the striking similarities in the magnetic dynamics of the two materials. [Figure adapted from Coldea et al. 48 and Kim et al. 18 ]

Figure 1 . 9 :Figure 1 . 10 :

 19110 Figure 1.9: Evolution of the Fermi surface of La 2 CuO 4 (top panels) and Sr 2 IrO 4 (bottom panels) upon hole and electron doping, respectively. [Figure adapted from Yoshida et al. 49 and Kim et al. 50 .]

Figure 1 .

 1 11(b), which is spin-Chapter 1 The low-energy physics of iridium compounds

Figure 1 . 11 :

 111 Figure 1.11: Diffuse magnetic X-ray scattering of Na 2 IrO 3 integrated over the energy range between 0 and 100 meV covering the entire magnetic excitations and representing a good approximation of the spin correlation function. (a) Spin-component-resolved correlation functions showing evidences for bond-directional magnetic interactions. (b) Spin-componentintegrated correlation functions. [Figure from Chun et al. 19 .]

Figure 1 . 12 :

 112 Figure 1.12: DMFT calculations on Rb 2 [IrF 6 ] show that this compound has a wide gap and narrow j eff = 1/2 bands. [Figure from Birol and Haule 28 ]

Figure 1 . 13 :

 113 Figure 1.13: Iridium L 2,3 -edge X-ray absorption spectra (positive values) and X-ray magnetic circular dichroism (negative values) measured in a magnetic field of 17 T. Samples 1 (green) and 2 (purple) contain IrF 6 units connected to molecular complexes, while sample 3 (red) contains IrCl 6 units. The inset shows the magnetic field dependence of the magnetization for sample 2. [Figure from Pedersen et al.89]

Figure 2 . 1 :

 21 Figure 2.1: Scattering geometry in RIXS.The incoming (outgoing) photons are described by their wave vector k (k ) and polarization ( ). The polarization is usually written as linear combination of two perpendicular components ( π , parallel to the scattering plane, and σ , perpendicular to it). The sample is drawn as a gray plaquette.

Figure 3 . 1 :

 31 Figure 3.1: Schematic representation of an undulator used at the ESRF as a high-flux X-ray source. [Figure from the website of the ESRF.]

Figure 3 . 2 :

 32 Figure 3.2: Layout of beamline ID20 at the ESRF. Following the X-ray beam direction, from bottom to top: primary slits (ps), collimating mirror (CM1), secondary slits (ss) and double-crystal monochromator (DCM) are installed in the first optics hutch (OH1); the post-monochromators (post-monos) and the focusing toroidal mirror (FM2) are located inside the second optics hutch (OH2); the secondary sources for the RIXS and non-resonant IXS (NIXS) spectrometers (SSA and SSB, respectively) are placed in the first experimental hutch (EH1); the second experimental hutch (EH2) hosts the focusing KB mirrors (FM3A + FM4A) and the RIXS spectrometer, while in the last experimental hutch (EH3) the focusing KB mirrors (FM3B + FM4B) and the NIXS spectrometer are installed.

  Figure 3.3.

Figure 3 . 3 :Figure 3 . 4 :

 3334 Figure 3.3: Sketch of the RIXS spectrometer installed at beamline ID20. [Courtesy of C. Henriquet.]

Figure 4 . 1 :

 41 Figure 4.1: RIXS spectrum of YBa 2 Cu 3 O 6.6 (black squares), decomposed into the crossed polarization component (closed red circles) and non-crossed polarization component (open blue circles). [Figure from Braicovich et al. 20 ]

Figure 4 . 2 :

 42 Figure 4.2: RIXS spectra of (Sr 1-x La x ) 2 IrO 4 as a function of doping x at momentum transfers corresponding to the crystallographic center (0, 0) and the antiferromagnetic zone center (π, π). The shaded lineshapes correspond to the single magnon, as results from the fit of the data points. [Figure from Pincini et al.67]

Figure 4 . 3 :

 43 Figure 4.3: (a) RIXS spectrum of Na 2 IrO 3 measured at T = 10 K. (b) Detailed view of the low-lying RIXS excitations. [Figure from Gretarsson et al. 112 ]

  Figure 4.3(a) reports the RIXS spectrum of Na 2 IrO 3 measured at T = 10 K, a zoom into the low-energy region of the spectrum is shown in Figure 4.3(b).

Figure 4 . 4 :

 44 Figure 4.4: Configuration of the spectrometer with polarization resolution used at beamline 30-ID of the APS. The beam scattered by the sample is energy analyzed by the main diced analyzer and focused at the primary focus. The polarization analyzer images the beam spot from the primary focus onto the strip detector. [Figure from Gao et al. 23 ]

Figure 4 . 5 :

 45 Figure 4.5: An asymmetric Bragg reflection. The incident beam is reflected by a family of lattice planes forming an angle η with respect to the crystal surface. The angular dispersion of the reflected beam is also shown.

Figure 4 .

 4 6 shows the reflectivity R of the Si(311) family of lattice planes. The black solid line represents the pairs of (E, θ B ) satisfying Bragg's law. The dynamical theory of diffraction Chapter 4 Polarization analysis of the scattered radiation

Figure 4 . 6 :

 46 Figure 4.6: X-ray reflectivity R of a Si(311) crystal in the phase space defined by energy and angle. The solid line represents the position of the peak reflectivity as given by Bragg's law. The inset zooms into a narrow region of the phase space and shows the finite reflection window with spectral width ∆E and angular width ∆θ predicted by the dynamical theory of diffraction. The curves close to the inset represent cuts along the white paths across the reflection region.

Figure 4 . 7 :

 47 Figure 4.7: Example of DuMond diagram. Panel (a) shows the phase space of the incident beam, and panel (b) the one of the reflected beam. The green region represents the total reflection region with spectral and angular widths as results from the dynamical theory of diffraction. Please note that the x axis of panel (b) is reversed, as indicated by the arrow pointing to the left.

Figure 4 . 7 :

 47 panel (a) represents the phase space of the incident beam, and panel (b) the one of the reflected beam in the case |b| < 1. The green region represents the total reflection window with spectral and angular widths as predicted by the dynamical theory of diffraction. It is assumed that all X-rays that fall inside the green region of panel (a) are transmitted (i.e. totally reflected by the lattice planes) into the green region of panel (b), while all other X-rays are lost. The slope of the green region can be calculated by differentiating Bragg's law in the proximity of the Bragg angle: dE/dθ = E/ tan θ B . Let us consider first an incident monochromatic beam with divergence ∆θ (horizontal blue line in Figure 4.7(a)). The photons are transmitted (black dashed line) to the phase space of the reflected beam giving rise to a monochromatic beam with the same energy and divergence equal to ∆θ (horizontal blue line in Figure 4.7(b)). It follows from Equation 4.2 and 4.4 that the beam divergence has changed by a factor |b|. Let us now consider the case of a collimated beam with bandwidth ∆E (vertical blue line in Figure 4.7(a))

Figure 4 . 8 :

 48 Figure 4.8: CDS spectrometer for sub-10-meV RIXS experiments at the iridium L 3 edge. The beam is first collimated by a mirror and then reflected by three asymmetrically-cut crystals. [Figure from the website of the crystal optics group of the APS.]

4. 4 Figure 4 . 9 :

 449 Figure 4.9: Sketch of the polarization analyzer for RIXS experiments designed for beamline ID20. The polarimeter is made of two units: a collimating Montel mirror, and a set of two asymmetrically-cut crystals, one of which with a Bragg angle close to 45 • .

Figure 4 . 10 :

 410 Figure 4.10: The multilayer Montel mirror has a parabolic surface to collimate the beam coming from a point source (the sample). It delivers a beam whose size is ∼ 5 mm and divergence is ∼ 40 µrad in the dispersion plane.

Figure 4 . 11 :

 411 Figure 4.11: DuMond diagrams representing the reflection of a collimated beam from two asymmetrically-cut crystals. (a) Phase space of the incident beam for the Si(311) reflection with asymmetry factor b 1 = -0.06. (b) Phase space of the reflected beam for the Si(311) crystal and phase space of the incident beam for the Si(444) reflection with asymmetry factor b 2 = -8.33. (c) Phase space of the reflected beam for the Si(444) crystal. The solid blue line represents the photons within the reflection region that can be transmitted from the phase space of the incident beam to the one of the reflected beam.

Figure 4 .

 4 11(a)). The X-rays are transmitted in the phase space of reflected photons (Figure4.11(b)) giving rise to an angular-dispersive beam with the same bandwidth. The angular spread α 1 of the reflected beam and the dispersion rate D 1 can be calculated from Equation4.5, and the following values are obtained: α 1 ≈ 37 µrad and D 1 ≈ 0.03 µrad/meV. Let us now consider the effects of the second asymmetric crystal (Figure 4.11(b)). The phase space of the incident beam for the Si(444) reflection overlaps with the phase space of the reflected beam for the Si(311) crystal, hence they are plotted in the same diagram. The orange region represents the reflection window of the Si(444) asymmetric crystal, with spectral width ∆E 2 ≈ 19 meV and angular width ∆θ 2 ≈ 1.7 µrad corresponding to an asymmetry angle η 2 = -38 • (b 2 = -8.33

Figure 4 . 12 :

 412 Figure 4.12: DuMond diagrams representing the reflection of a divergent beam from two asymmetrically-cut crystals.

  .

2 .I

 2 conclude by noticing that the footprint of the beam on the first crystal surface is l 1 = σ/ sin(θ B,1 -η 1 ) ≈ 130 mm, where σ ≈ 5 mm is the beam size after the Montel mirror. The footprint on the second crystal surface is l 2 = l 1 / sin(θ B,2 -|η 2 |) ≈ 80 mm, while the beam height after the two reflections is σ 2 = σ/|b 1 b 2 | ≈ 10 mm. The beam size and crystal lengths are adequate for silicon-based X-ray optics and technological issues related to the operation of the polarimeter are not expected.

58 4Figure 4 . 13 :

 58413 Figure 4.13: Ray-tracing simulation of the energy distribution of the photons after the two asymmetric reflections. E = 11.215 keV.

Figure 4 . 14 :

 414 Figure 4.14: Dependence of the energy resolution and efficiency of the two-crystal setup on the beam divergence downstream of the Montel mirror. Blue circles and red squares represent the energy resolution estimated with the DuMond diagrams and the ray-tracing simulations, respectively. The black stars represent the resolution measured during the feasibility test. The efficiency of the system is plotted as black open diamonds.

Figure 4 . 15 :

 415 Figure 4.15: Rocking curve of the Si(844) post-monochromator for different values of the beam divergence recorded by a diode placed downstream of the two asymmetric reflections. Solid lines represent the fit of the data points to a Gaussian profile. E = 11.215 keV.

Figure 4 . 16 :

 416 Figure 4.16: Mechanical model of the polarization-resolved RIXS spectrometer installed at beamline ID20. The horizontal (a) or vertical (b) polarization of the scattered beam is measured. The inset of panel (a) shows the mounting of the Montel mirror and the asymmetric crystals inside the chamber. [Courtesy of C. Cosculluela and C. Henriquet.]

5. 2 Figure 5 . 1 :

 251 Figure 5.1: Crystal and magnetic structure of Sr 2 IrO 4 (left) and Sr 3 Ir 2 O 7 (right). Iridium ions (gray spheres) are surrounded by octahedra of oxygen ions (red spheres) giving rise to layered-perovskite structures. The pseudospins (orange arrows) are antiferromagnetically aligned parallel to the ab plane in Sr 2 IrO 4 and parallel to the c axis in Sr 3 Ir 2 O 7 . The main superexchange paths J and J c mentioned throughout the Chapter are also shown.

Figure 5 . 2 :Figure 5 . 3 :

 5253 Figure 5.2: RIXS spectra of Sr 3 Ir 2 O 7 collected at T = 30 K along highsymmetry directions of the two-dimensional Brillouin zone. Feature B is ascribed to the single magnon. [Figure from Kim et al. 25 ]

5 :

 5 Panoramic DAC utilized for high-pressure RIXS experiments at beamline ID20. On the left-hand side of the picture, the opposed diamond anvils, the gasket and the sample are sketched.

Figure 5 .

 5 Figure 5.5 shows a drawing of the Panoramic DAC 141 utilized in our experiments. It has been designed by the ESRF High-Pressure Sample-

Figure 5 .

 5 6. Panel (a) shows the culet of one anvil, in the middle of which a single crystal of Sr 3 Ir 2 O 7 is loaded. The sample has been polished roughly to the desired size. Sr 3 Ir 2 O 7 naturally cleaves into layers orthogonal to the c axis, with sharp edges parallel either to the (100) or to the (110) axes, as illustrated in the Figure. A ruby sphere is placed next to the sample.

Figure 5 . 6 Figure 5 . 6 :

 5656 Figure 5.6: (a) Sr 3 Ir 2 O 7 and ruby sphere loaded at the center of the diamond culet with diameter of 250 µm. The sample has a triangular shape with edges of ≈ 65 µm parallel to high-symmetry directions of the lattice. (b) Sr 3 Ir 2 O 7 and ruby sphere after the pre-indented laser-drilled gasket has been placed between the diamond anvils.

Chapter 5 Figure 5 . 7 :

 557 Figure 5.7: Helium-flow cryostat for high-pressure low-temperature RIXS experiments. The panoramic DAC is placed inside the main heat exchanger.

Figure 5 .

 5 8(a) displays iridium L 3 -edge RIXS spectra of Sr 3 Ir 2 O 7 measured at T = 150 K and at momentum transfer Q = (3.5, 3.5, 0) r.l.u. corresponding to the twodimensional Brillouin zone boundary (π, π) where the magnetic excitations are the strongest. The two spectra measured at ambient pressure (black circles) and at P = 3.1 GPa (red diamonds) are qualitatively similar: besides the elastic line at 0 eV, both spectra feature a sharp (FWHM ≈ 0.04 eV) peak at ∼ 0.1 eV and a broad (FWHM ≈ 0.45 eV) excitation centered at ∼ 0.75 eV.

Figure 5 .

 5 8(b) shows the magnetic region of the RIXS spectra measured at T = 100 K and at momentum transfer Q = (3.475, 3.475, 0) r.l.u. = (0.95π, 0.95π, 0). The single magnon excitation is clearly softened when

Figure 5 . 8 :

 58 Figure 5.8: (a) Iridium L 3 -edge RIXS spectra of Sr 3 Ir 2 O 7 measured at ambient pressure (black circles) and at P = 3.1 GPa (red diamonds). The spectra were collected at T = 150 K and Q = (3.5, 3.5, 0) r.l.u. corresponding to the two-dimensional Brillouin zone boundary (π, π). (b) Magnetic region of the RIXS spectra measured at T = 100 K and Q = (3.475, 3.475, 0) ≈ (π, π, 0) showing the softening of the magnetic mode with pressure.

Figure 5 . 9 :

 59 Figure 5.9: Pressure dependence of the magnetic Bragg reflections of Sr 3 Ir 2 O 7 measured by scanning along high-symmetry directions of the reciprocal lattice at fixed L (a) and at fixed H, K (b). The peak at H = K = 3.58 r.l.u. in panel (a) is an artifact due to scattering from beryllium. The scans were collected at T = 100 K.

Figure 5 . 11 :

 511 Figure 5.11: Evolution of the magnetic gap when the intralayer superexchange J decreases and the interlayer coupling J c increases with respect to their values at ambient pressure. Predictions are reported according to the linear spin-wave model (black line) and the quantum-dimer model (red line). The other magnetic interactions are assumed to be constant with pressure.

CaIrO 3

 3 single crystals were flux grown by Dr. K. Ohgushi and collaborators in the Institute for Solid State Physics of the University of Tokyo, as explained in detail in Ref. 148. CaIrO 3 crystallizes in the orthorhombic space group Cmcm (63) with lattice constants a = 3.147 Å, b = 9.863 Å, and c = 7.299 Å

Figure 6 . 1 :

 61 Figure 6.1: (a) Crystal structure of CaIrO 3 : the iridium ions (gray spheres) are surrounded by octahedra of oxygen ions (red spheres). The IrO 2 layers are separated by calcium planes (green spheres) along the b axis. The pseudospins (orange arrows) are ferromagnetically aligned along the a axis and antiferromagnetically aligned along the c axis. (b) The IrO 6 octahedra share one edge along the a direction, the point group symmetry is C 2h . (c) The IrO 6 octahedra share one corner along the c direction, the point group symmetry is C 2v .

Figure 6 . 2 :

 62 Figure 6.2: (a) Iridium L 3 -edge RIXS spectra of CaIrO 3 measured at momentum transfer Q = (0.5, 2, 9) r.l.u. and at temperature T = 295 K (red circles) and T = 20 K (blue circles). (b) Temperature dependence of the integrated intensity of the shaded feature. (c) Temperature dependence of the FWHM of the shaded feature. Black solid lines are guides to the eye.

Figure 6 . 3 :

 63 Figure 6.3: Momentum transfer dependence of the magnetic excitations of CaIrO 3 measured along the H (a, b), K (c, d) and L (e, f) highsymmetry directions of the reciprocal lattice. Panels (a), (c), and (e) report the raw data, while panels (b), (d), and (f) display the fitted energy position of the single magnon (white squares) and the calculated dispersion using the minimal two-spin Hamiltonian (red solid lines). The gray region around the fitted position represents the FWHM of the magnon as results from the fit. The insets in panels (b) and (d) show a zoomed energy region around the dispersion.

Chapter 6 The magnetic and electronic dynamics of CaIrO 3 -Figure 6 . 4 :

 6364 Figure 6.4: Momentum dependence of the RIXS spectra of CaIrO 3 (lefthand side) compared to the exact calculation of the dynamical structure factor of the AFM XXZ chain, convoluted with the resolution function (right-hand side). White solid lines are the best fit to the boundaries of the magnetic continuum, obtained by setting J = 70 meV and δ ≈ 2.

Figure 6 . 5 :

 65 Figure 6.5: Comparison between the experimental data (black dots) collected at at Q = 0, π/4 and π/2, and the two-spinon dynamical structure factor of the one-dimensional AFM XXZ chain (red solid line).

2 Figure 6 . 6 :

 266 Figure 6.6: Comparison between iridium L 3 (black circles) and L 2 (red squares) edge RIXS spectra measured at Q = 0 (a) and Q = π/2 (b). The spectra have been normalized as explained in the text. The subtraction between spectra measured at the L 3 and L 2 edges, proportional to the dynamical structure factor, is plotted as blue diamonds.

Figure 6 . 7 :

 67 Figure 6.7: Momentum transfer dependence of the electronic excitations of CaIrO 3 measured along the H (a, b), K (c, d) and L (e, f) highsymmetry directions of the reciprocal lattice. Panels (a), (c), and (e) report the raw data, while panels (b), (d), and (f) display the second derivative of the data. Red dashed lines are guides to the eye highlighting the dispersion of features A and B. The electron density of the excited states are also shown.

Chapter 6 Figure 6 . 8 :

 668 Figure 6.8: The Brillouin zone of CaIrO 3 . We note that the onedimensional Brillouin zone consists of the segment centered at Y and extending to T.

  which favors the hole occupation of the |t ±1 2g orbitals. A qualitative interpretation of the different momentum dependence of features B and C can be given by looking at the shapes of the wave functions of the ground and excited states on neighboring sites. By naively considering the single-ion model and the values of the parameters determined by Moretti Sala et al. 39 , we approximate the ground-state wave function as | 1 2 , ± 1 2 ≈ ∓|t ±1 2g , ∓ , and the excited-state wave functions as | 3 2 , ± 1 2 ≈ |t 0 2g , ± = |xy, ± , and | 3 2 , ± 3 2 = |t ±1 2g , ± . The corresponding electron densities are shown in Figure 6.7. A strong overlap is expected between the orbitals |t ±1

Figure 6 . 9 :

 69 Figure 6.9: Momentum-and energy-dependence of the spin-orbit exciton computed considering both superexchange and Jahn-Teller interactions. [Figure from Plotnikova et al.167]

28

 28

7. 2 Figure 7 . 1 :

 271 Figure 7.1: (a) XRPD pattern of Rb 2 [IrF 6 ] collected at T = 295 K with an incident photon wavelength of λ = 0.3999 Å. Red dots are the data; the black solid line displays the Rietveld refinement. The refined crystal structure is shown in the inset. (b) The smooth angular variation of the (100) and (001) reflections with temperature reveals that the a and c lattice constants possess different temperature rates and that no phase transition is observed in the range between 100 K and 400 K. (c) Temperature dependence of the lattice constants and of their ratio. [Figure from Rossi et al.168]

168 ] 7

 1687 Figure 7.1: (a) XRPD pattern of Rb 2 [IrF 6 ] collected at T = 295 K with an incident photon wavelength of λ = 0.3999 Å. Red dots are the data; the black solid line displays the Rietveld refinement. The refined crystal structure is shown in the inset. (b) The smooth angular variation of the (100) and (001) reflections with temperature reveals that the a and c lattice constants possess different temperature rates and that no phase transition is observed in the range between 100 K and 400 K. (c) Temperature dependence of the lattice constants and of their ratio. [Figure from Rossi et al.168]

Figure 7 . 2 :Chapter 7 Table 7 . 2 :

 72772 Figure 7.2: (a) Iridium L 3 -edge RIXS spectrum of Rb 2 [IrF 6 ] measured at T = 20 K. Black dots are the background-subtracted experimental points, the red solid line is the total fit, and the dotted blue curves are the fit to the j eff = 3/2 excitations. (b) Momentum transfer dependence of the j eff = 3/2 excitations at the high-symmetry points of the Brillouin zone. (c) Temperature dependence of the j eff = 3/2 excitations between 20 K and 300 K, measured on powder Rb 2 [IrF 6 ]. [Figure from Rossi et al. 168 ]

Figure 7 . 4 :

 74 Figure 7.4: Comparison between the energy splitting of the j eff = 3/2 excitations obtained from the experimental data and from the quantum chemical calculations. [Figure from Rossi et al. 168 ]

7. 4 Figure 7 . 5 :

 475 Figure 7.5: Powder and single-crystal Rb 2 [IrF 6 ] loaded at the center of the diamond culet, together with a ruby sphere to determine the pressure.

Figure 7 . 6 :Figure 7 . 7 :

 7677 Figure 7.6: Projection of the Ewald sphere onto the (H, K, 0) reciprocal lattice plane at ambient pressure (a) and at 55 GPa (b). The bright spots arranged in a hexagonal lattice are the Bragg diffraction peaks from the sample.

7. 4 Figure 7 . 8 :

 478 Figure 7.8: Pressure dependence of the Ir-F bond distance (a) and of the F-Ir-F bond angle (b). The dashed lines are obtained by fitting the data to linear polynomials.

7. 4 Figure 7 . 9 :

 479 Figure 7.9: Iridium L 3 -edge RIXS spectra of powder Rb 2 [IrF 6 ] measured at different pressures up to 26 GPa. Experimental data points are shown as filled circles, whereas solid lines are the fit to the data.

Figure 7 . 10 :

 710 Figure 7.10: Pressure evolution of the energy splitting (a) and of the center of mass (b) of features A and B of Rb 2 [IrF 6 ]. The dashed black lines are obtained by fitting the data to linear polynomials.

Figure 7 . 12 :

 712 Figure 7.12: SOC strength as a function of the electronegativity of the halogen. The dashed line is a linear fit to the data points.

Figure A. 1 .

 1 The |t 0 2g orbital corresponds to the |xy orbital, whereas the other two are obtained by a rotation around the z axis of the |yz or |xz orbitals, the direction of rotation corresponding to the value of the effective orbital angular momentum l z = ±1. Note that t 0 2g |l z |t 0 2g = 0, whereas t ±1 2g |l z |t ±1 2g = ∓1 (in units of ).

A. 3

 3 The effect of octahedral distortion and U is the matrix of the eigenvectors:

  can occupy either states.

Figure A. 2 :

 2 Figure A.2: Schematic representation of the tetragonal (a) and trigonal (b) distortion of the octahedron.

2 -Figure A. 3 :Figure A. 4 :

 234 Figure A.3: (a) Tetragonal crystal field dependence of the eigenvalues of the total Hamiltonian H = H SOC + H ∆ . The solid line is the ground state in the hole representation, whereas the dashed lines are the excited states. (b) Tetragonal crystal field dependence of the orbital occupancies of the ground state wave function (Equation A.16), and corresponding electron density (top). A typical value of the SOC constant λ = 0.5 eV has been chosen.

Figure A. 4 )

 4 , it is straightforward to realize that a trigonal elongation or compression splits the |t 0 2g orbital (whose lobes point along the (111) direction) from the |t ±1 2g orbitals (whose lobes point far from the trigonal axis and thus experience a different Coulomb repulsion due to the anions). As for the tetragonal case, the |t 0 2g and |t ±1 2g states represent a basis subset B trig ef f

7 )

 7 The matrices T (Equation A.7) and V (Equation A.14) allow to change basis set from the l d = 2 spherical harmonics (B sph ) to the t 2g states with effective l eff = 1 (B ef f ), and from the latter to the Kramers doublets (K), respectively. Therefore, the transition amplitudes from the Kramers doublets to the intermediate p states are contained in the tensor whose elements are:

4 √ 2 cos θ -2 sin θ 2 - i 4 √ 2 cos θ -2 sin θ 2 0 i 4 √ 2 cos θ -2 sin θ 2 1 4 √ 2

 42424242

  Iridium L 3 -edge RIXS spectra of powder Rb 2 IrF 6 measured at different pressures up to 26 GPa. . . . . . . . . . . . . . . . 7.10 Pressure evolution of the energy splitting and of the center of mass of features A and B of Rb 2 IrF 6 . . . . . . . . . . . . . . 7.11 Iridium L 3 -edge RIXS spectra of K 2 IrF 6 , K 2 IrCl 6 , and K 2 IrBr 6 . 7.12 SOC strength as a function of the electronegativity of the halogen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tetragonal crystal field dependence of the eigenvalues of the total Hamiltonian H = H SOC +H ∆ . (b) Tetragonal crystal field dependence of the orbital occupancies of the ground state wave function, and corresponding electron density. . . 132 A.4 Electron density of the trigonal-symmetric t 2g orbitals with orbital angular momentum equal to 0, -1, and 1. . . . . . . . 133 xvii

	A.1 Electron density of the t 2g orbitals with orbital angular mo-
	mentum equal to 0, -1, and 1. . . . . . . . . . . . . . . . . . .
	A.2 Schematic representation of the tetragonal and trigonal dis-

. . . . . . . . 7.8 Pressure dependence of the Ir-F bond distance and of the F-Ir-F bond angle. . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 tortion of the octahedron. . . . . . . . . . . . . . . . . . . . . xvi LIST OF FIGURES A.3 (a)

Chapter 1 The low-energy physics of iridium compounds example

  , Sr 2 IrO 4 30 , Sr 3 Ir 2 O 7 31 , Na 2 IrO 3 32 , and CaIrO 3 27

Chapter 1 The low-energy physics of iridium compounds layered

  -perovskite crystal structures where the iridium ions are arranged in a two-dimensional square lattice and the magnetic moments are antiferromagnetically (AFM) aligned. Among other iridates, this is the case of Sr 2 IrO 4 30,41,42 , Sr 3 Ir 2 O 7 31,43,44 , and Ba 2 IrO 4 45,46 . Particular attention has been drawn on Sr 2 IrO 4 owing to striking similarities with the isostructural cuprate parent compound La 2 CuO 4 . The analogies are present in the Mott insulating phase as well as in the doped metallic regime. Undoped Sr 2 IrO 4 shares with La 2 CuO 4

  2 [IrF 6 ] and related compounds, whose crystal structure is made of isolated IrF 6 units 88 .

	By means of dynamical mean-field theory (DMFT) calculations, they com-
	puted the realization of a spin-orbit-induced j eff = 1/2 Mott insulating state
	in Rb 2 [IrF 6 ]. Their calculations, reported in Figure 1.12, display the pres-
	ence of a wide Mott-like gap of ∼ 2 eV. Moreover, the first occupied and
	unoccupied bands are very narrow and have j eff = 1/2 character. Rb 2 [IrF 6 ]

Table 4 . 1 :

 41 Construction parameters of the multilayer Montel mirror in the dispersion plane of the asymmetric crystals and in the orthogonal plane.= estimate from the manufacturer.

	Parameter	Dispersion plane Perpendicular plane
	Working energy	11.2 ± 0.1 keV	11.2 ± 0.1 keV
	Substrate	Si(100)	Si(100)
	Coating	Ru/C	Ru/C
	Source size s	10 µm	30 µm
	Mirror length L	230 mm	230 mm
	Focal distance f	315 mm	315 mm
	Incidence angle φ	1.27 •	1.27 •
	Periodicity d	2.5 nm	2.5 nm
	Collected solid angle Ω	16 mrad	16 mrad
	Source size contribution s/f	32 µrad	95 µrad
	Slope error Σ	12 µrad	12 µrad
	Residual divergence ∆ θ	40 µrad	100 µrad
	Exit beam size σ	5 mm	5 mm
	Reflectivity	78%	78%
	Δθ 1

Table 4 . 2 :

 42 Summary of the main parameters of the two asymmetric crystals. A beam divergence of 40 µrad is assumed.

	Parameter	First crystal	Second crystal
	Working energy E	11.215 keV	11.215 keV
	Reflection	Si(311)	Si(444)
	Bragg angle θ B	19.73 •	44.84 •
	Darwin width ω D	10.04 µrad	4.93 µrad
	Asymmetry angle η	17.5 •	-38 •
	Asymmetry factor b	-0.06	-8.33
	Incidence angle	2.23 •	82.84 •
	Exit angle	37.23 •	6.84 •
	Intrinsic energy resolution	1.2 eV	14.4 meV
	Divergence contrib. to resolution	1.2 eV	20.93 meV
	Total energy resolution	1.8 eV	25.4 meV
	Dispersion rate D	0.03 µrad/meV 0.9 µrad/meV
	Footprint l	130 mm	80 mm
	Beam size σ	80 mm	10 mm

Table 7 . 1 :

 71 Space group, site symmetry, Ir-F distance, trigonal angle (β), and octahedral distortion at T = 295 K for A 2 [IrF 6 ] (A = Na, K, Rb, Cs) and Ba[IrF 6 ]. 87 • and 93 • . This is indicative of a slight compression of the IrF 6 octahedral cage along the local trigonal axis (corresponding to the crystallographic c axis for Rb 2 [IrF 6 ]

	Compound	Space group symmetry Site	Ir-F (Å)	β ( • )	β-β 0 β 0 (%)
	Na 2 [IrF 6 ]	P 321	32./3..	1.939/1.948 56.48/55.70 3.2/1.75
	K 2 [IrF 6 ]	P 3m1	3m.	1.940	57.19	4.5
	Rb 2 [IrF 6 ]	P 3m1	3m.	1.975	56.89	3.9
	Cs 2 [IrF 6 ]	P 3m1	3m.	1.941	56.52	3.3
	Ba[IrF 6 ]	R 3	3.	1.937	51.89	-5.2

Table 7 . 3 :

 73 Center of mass and energy splitting of features A and B of Rb 2 [IrF 6 ], and calculated values of the SOC constant λ and of the effective trigonal crystal field ∆ as a function of pressure. All values are in meV. a fraction of the energy resolution and higher pressures are required for an unambiguous conclusion. The values of the center of mass and of the peak splitting, together with the calculated values of λ and ∆ for each pressure point are reported in Table7.3. The calculation reveals that the SOC constant is not affected much by the application of pressure, since its value only changes by approximately 1%. The effective trigonal crystal field, instead, changes by ∼ 20%

	P (GPa) (E A + E B )/2 E B -E A	λ	∆
	0	843 ± 13	98 ± 26 557 -140
	5 ± 0.3	855 ± 6	104 ± 12 564 -148
	10.3 ± 0.4	858 ± 8	113 ± 17 566 -159
	17 ± 2	852 ± 13	110 ± 26 562 -156
	26 ± 2	855 ± 19	121 ± 39 563 -171
	increases by only

  3. The measurements were carried out at room temperature and ambient pressure on K 2 [IrF 6 ], K 2 [IrCl 6 ], and K 2 [IrBr 6 ]. The crystal structure evolves from trigonal (space group P 3m1) for K 2 [IrF 6 ] to cubic (space group F m 3m) for K 2 [IrCl 6 ] 188 and K 2 [IrBr 6 ]. Diffraction measurements

		0.60
	(eV)	0.45 0.50 0.55	K 2 [IrCl 6 ]	K 2 [IrF 6 ]
		0.40	K 2 [IrBr 6 ]
		0.35	3.0	3.5 Electronegativity (Pauling units)	4.0

  2 [IrF 6 ], K 2 [IrF 6 ], Rb 2 [IrF 6 ], Cs 2 [IrF 6 ], and Ba[IrF 6 ] by means of XRPD, iridium L 3 -edge RIXS, and quantum chemical calculations. Rb 2 [IrF 6 ] was also investigated at high pressure by XRD and RIXS. The results support the theoretical prediction that Rb 2 [IrF 6

  2 [IrF 6 ], K 2 [IrF 6 ], Rb 2 [IrF 6 ], Cs 2 [IrF 6 ], and Ba[IrF 6

Acknowledgments

I would like to thank my supervisor Dr. Marco Moretti, whom I am deeply indebted to for his guidance, confidence, support, and above all friendship. His passion and devotion are a source of inspiration where to draw in the dark hours as well as a driving force to brilliant achievements. I had the privilege to be supervised also by Dr. Michael Krisch, who encouraged me throughout these years to overcome challenges with novel ideas beyond conventional schemes. I am grateful to Dr. Nicholas Brookes, whose advices and support were always appreciated.

I would like to acknowledge Christian Henriquet for assistance during the experiments and for support on the technical developments presented in this thesis. His help was beneficial and his mere presence on the beamline a relief. I am grateful to Christoph, Chiara, Anna, and Blanka for adopting me in the big family of ID20. Margherita, Roberto, Yi, and Manuel participated in some of my thesis projects. Together we worked, learned, and achieved so much. Thanks to all the friends I met in Grenoble, with whom I spent three amazing years, I shared weekends in the mountains, and I enjoyed wonderful "soirées Grenobloises". You were my travel companions along this journey, and I will always remember the lovely days spent together.

I recognize this PhD research would not have been possible without the financial support of the ESRF. The ESRF is also acknowledged for providing beamtime and financing the polarimeter project. Many are the ESRF staff members who contributed to this work. In particular, I would like to thank Jeroen, Gaston, and the ESRF high-pressure lab for precious help and assistance during high-pressure experiments; Ray and Christian for support on the polarimeter project; Carlos for taking care of the mechanical issues related to the polarimeter; Roberto for providing crystals; Pieter for valuable advices in many occasions and for organizing stimulating spec-Acknowledgments troscopy meetings; Marius for performing quantum chemical calculations; Carlotta for assistance during XRPD experiments and related data analysis; and Volodymyr for assistance during single-crystal X-ray diffraction experiments and related data analysis.

I would like to acknowledge Prof. Desmond McMorrow, Davide Pincini, and Christian Donnerer for fruitful and pleasant collaborations; Prof. Alexander Gubanov for providing samples, and for bringing additional samples that we investigated with great success; Prof. Kenya Ohgushi for supplying samples; Prof. Krzysztof Wohlfeld for useful and clarifying discussions; Dr. Sándor Toth for help with the SpinW code; and Prof. Giulio Monaco for enlightening discussions.

I would like to thank Prof. Sergio Di Matteo, Prof. Olivier Isnard, Prof. Matthieu Le Tacon, Prof. Christian Rüegg, and Dr. Virginie Simonet. They agreed to referee this work and honor me with their presence in the PhD panel.

Finally, I express my profound gratitude to my parents for their invaluable support throughout my life, and to my fiancée Giusy for her unconditional love and encouragement. I owe you all my accomplishments.

(left panel). In order to describe the magnetic excitations of Sr 3 Ir 2 O 7 , including the previously-undetected feature D, Moretti Sala et al. [START_REF] Sala | Evidence of quantum dimer excitations in Sr 3 Ir 2 O 7[END_REF] developed a bond-operator mean-field model that considers a dominant interlayer over intralayer superexchange (J c J) and dimer-like pseudospin interactions. Magnetic excitations are singlet-to-triplet transitions, which occur when a single pseudospin is excited [START_REF] Sala | Evidence of quantum dimer excitations in Sr 3 Ir 2 O 7[END_REF] . Two branches of magnetic excitations (transverse and longitudinal) are allowed within this model.

Both approaches explain well the magnetic dynamics of Sr 3 Ir 2 O 7 . The doping dependence of the magnetic excitations of this compound offers a first possibility to discriminate between the two models. When Sr 3 Ir 2 O 7 is electron doped by lanthanum substitution, the system undergoes a transition from an insulating AFM state to a metallic paramagnetic state [START_REF] Hogan | First-order melting of a weak spin-orbit mott insulator into a correlated metal[END_REF] . The transition occurs at doping concentrations of ≈ 4%, as shown in the phase diagram reported in Figure 5.4. The magnetic excitations of doped Sr 3 Ir 2 O 7 are severely damped and softened [START_REF] Lu | Doping Evolution of Magnetic Order and Magnetic Excitations in (Sr 1-x La x ) 3 Ir 2 O 7[END_REF][START_REF] Hogan | Disordered dimer state in electrondoped Sr 3 Ir 2 O 7[END_REF] . Surprisingly, both lin- ever, much higher pressures can be reached. Note that the elastic channel acquired at 12 GPa (red diamonds in Figure 5.9(a)) is contaminated by spurious signal from beryllium. The artifact is visible in the scan and centered at H = K = 3.58 r.l.u. This testifies the complexity of this experiment, which is even increased when inelastic signal is targeted.

The softening of the single magnon with pressure is reported in Figure 5.10 where the fitted positions at Q = (3.5, 3.5, 0) r.l.u. (black circles) and Q = (3.475, 3.475, 0) r.l.u. (black squares) have been merged in view of the proximity of the two momentum transfers. The magnetic mode softens by ≈ 1.5 meV/GPa, as indicated by the linear trend (red solid line). In passing, we notice that the extrapolated pressure value where the magnetic gap collapses (∼ 55-60 GPa) almost coincides with the pressure at which the structural phase transition is found in Sr 3 Ir 2 O 7 (≈ 54 GPa) [START_REF] Donnerer | Pressure dependence of the structure and electronic properties of Sr 3 Ir 2 O 7[END_REF] , therefore suggesting some interrelation. While the structural phase transition occurs also at 295 K [START_REF] Donnerer | Pressure dependence of the structure and electronic properties of Sr 3 Ir 2 O 7[END_REF] , this is only 15 K above the Néel temperature of Sr 3 Ir 2 O 7 [START_REF] Cao | Anomalous magnetic and transport behavior in the magnetic insulator Sr 3 Ir 2 O 7[END_REF] . Further studies are needed to investigate the connection, if any, between the collapse of the magnetic excitations and the structural phase transition.

The RIXS results are now discussed in relation to the debate about the best suited magnetic model that describes the magnetic dynamics of Sr 3 Ir 2 O 7 . High-pressure diffraction measurements revealed a higher compressibility of the a and b axes with respect to the c axis [START_REF] Donnerer | Pressure dependence of the structure and electronic properties of Sr 3 Ir 2 O 7[END_REF] . The compression of the unit cell is accompanied by a rotation of the IrO 6 octahedra around the c axis while the out-of-plane Ir-O-Ir bond angle is unchanged [START_REF] Donnerer | Pressure dependence of the structure and electronic properties of Sr 3 Ir 2 O 7[END_REF][START_REF] Ding | Pressure-Induced Confined Metal from the Mott Insulator Sr 3 Ir 2 O 7[END_REF] . The Chapter 7 Possibility to realize spin-orbit-induced correlated physics beyond iridium oxides display the presence of a wide Mott-like gap and electronic bands with j eff = 1/2 character (see Figure 1.12).

Early measurements by Pedersen et al. [START_REF] Pedersen | Iridates from the molecular side[END_REF] on IrF 6 molecular complexes seem to validate the theoretical prediction (Figure 1.13). However, these materials are model systems and may have little correspondence with the real compounds proposed by Birol and Haule. Therefore, I investigated the crystal and electronic structure of selected iridium fluorides (Rb

and Ba[IrF 6 ]) with the aim of understanding differences and similarities between the low-energy physics of iridium fluorides and oxides. The results are supported by quantum chemical calculations. The pressure evolution of the crystal and electronic structure of Rb 2 [IrF 6 ] is also investigated by means of X-ray diffraction and RIXS, respectively. In addition, the effects of the substitution of the halogen element on the electronic properties of iridium fluorides are analyzed. [START_REF] Rossi | Possibility to realize spin-orbit-induced correlated physics in iridium fluorides[END_REF] .

The crystal structure of iridium fluorides

In order to refine the crystal structure of the samples, we performed X-ray powder diffraction (XRPD) on beamline ID22 at the ESRF. The incoming X-rays were monochromated to λ = 0.3999 Å by a Si (111) doublecrystal monochromator. The X-rays diffracted by the sample were collimated by 9 Si (111) analyzers and collected by a Cyberstar scintillation detector.

The XRPD pattern of Rb 2 [IrF 6 ] at T = 295 K is shown in Figure 7.1(a) (red dots). The peak positions and intensities were refined with a standard Rietveld procedure using the software Topas [START_REF] Coelho | Topas: General Profile and Structure Analysis Software for Powder Diffraction Data[END_REF] . The result is shown as a black solid line that reproduces well the experimental data points. Rb 2 [IrF 6 ] crystallizes in the K 2 GeF 6 structure (space group P 3m1 (164)) with lattice parameters a = b = 5.9777(0) Å and c = 4.7986(5) Å, and angles α = β = 90 • , and γ = 120 • at 295 K, supporting a previous refinement [START_REF] Smolentsev | Rubidium hexafluoridoiridate(IV)[END_REF] . The crystal structure is reported in the inset of Figure 7.1(a). As can be seen, it is made of disconnected IrF 6 octahedra, with fluorine ions occupying symmetry-equivalent lattice positions. The Ir-F bond lengths are all the same and equal to 1.975 Å, while the F-Ir-F bond angles are smaller than the experimental ones, the calculations reproduce the trend among the different compounds. As displayed in Figure 7.4, the calculated and experimental splittings are found to correlate very nicely. Na 2 [IrF 6 ] is omitted from this analysis because there are two inequivalent iridium sites in this compound. Overall, the agreement between experiments and quantum chemical calculations suggests that there is little or no influence of the alkali (alkaline earth) metal on the low-energy electronic structure of iridium fluorides. This is mainly the consequence of the isolation of the IrF 6 units in the crystal structure. Instead, the trend reflects the trigonal distortion of the IrF 6 octahedra.

In order to compare iridium fluorides and oxides, I adopt a single-ion model where a Hamiltonian containing the SOC λ and the effective trigonal distortion ∆ to the cubic crystal field acts on the iridium 5d-t 2g orbitals. This model, often used in the literature of iridates, is thoroughly discussed in Appendix A. From the diagonalization of the Hamiltonian matrix, the dependence of the eigenvalues on λ and ∆ can be found. This is reported in Equation A. 13. By constraining the eigenvalues of the single-ion-model Hamiltonian to the experimental energy positions of the features A and B, two possible solutions are obtained, corresponding either to an elongation (∆ > 0) or to a compression (∆ < 0) of the octahedra. The sign of ∆ was chosen in accordance to the octahedral distortion determined by XRPD.

7.5

The effect of halogen substitution on the electronic structure 0.0 0.2 0.4 0.6 0.8 1.0 Energy loss (eV) reveal that the IrCl 6 and IrBr 6 octahedra are regular. The RIXS spectra of the three compounds are displayed in Figure 7.11. Besides the elastic line, two features are detected in all spectra, in analogy to the measurements on the iridium fluorides reported above. Therefore, we assume that the j eff = 1/2 scenario holds also for K 2 [IrCl 6 ] and K 2 [IrBr 6 ], and the two features are associated to electronic excitations from the j eff = 1/2 ground state to the j eff = 3/2 excited states.

The two spectral features are named A and B. A first analysis reveals that the center of mass of the features decreases when going from K 2 [IrF 6 ] to K 2 [IrCl 6 ] and to K 2 [IrBr 6 ]. We recall that the center of mass of features A and B is related to the SOC constant within a single-ion model. The progressive reduction of the SOC strength reflects the larger degree of covalency of the Ir-Cl and Ir-Br bonds with respect to the more ionic Ir-F bond, which makes the iridium orbitals more extended [START_REF] Jørgensen | Spectroscopy of transition-group complexes[END_REF][START_REF] Al-Mobarak | The effect of covalency on the spin-orbit coupling constant[END_REF] . As a matter of fact, the SOC constant scales almost linearly with the electronegativity of the halogen, as shown in Figure 7.12.

The splitting of the j eff = 3/2 states in the RIXS spectra of K 2 [IrCl 6 ] and K 2 [IrBr 6 ] is puzzling in view of the cubic symmetry of the IrX 6 (X = Cl, Br) octahedra detected by diffraction measurements. A possible explanation is the presence of small but finite distortions, invisible to diffraction, but that may cause sizable effects to the electronic structure as detected by In this Appendix, a simple local model is introduced to explain the lowenergy physics of iridates. This elementary model is a first approximation to the complicated physics of these systems, however it has been extensively used in the literature of iridates [START_REF] Kim | Novel J eff = 1/2 Mott State Induced by Relativistic Spin-Orbit Coupling in Sr 2 IrO 4[END_REF][START_REF] Jackeli | Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models[END_REF][START_REF] Ament | Theory of resonant inelastic x-ray scattering in iridium oxide compounds: Probing spin-orbit-entangled ground states and excitations[END_REF][START_REF] Sala | CaIrO 3 : A Spin-Orbit Mott Insulator Beyond the J eff = 1/2 Ground State[END_REF][START_REF] Kim | Phase-Sensitive Observation of a Spin-Orbital Mott State in Sr 2 IrO 4[END_REF][START_REF] Sala | Resonant X-Ray Scattering and the j eff =1/2 Electronic Ground State in Iridate Perovskites[END_REF][START_REF] Matteo | Magnetic ground state of Sr 2 IrO 4 and implications for second-harmonic generation[END_REF] .

As a first approximation, only the outermost 5d shell of a single iridium ion is considered. The main parameters contributing to the electronic and magnetic structure of iridates are the crystal electric field, the SOC, and the electron correlation. I will explain the nature of these interactions and their effect on the physics of iridium-based systems.

A.1 The effect of local symmetry

In the compounds considered in this work, the tetravalent iridium ion has electronic configuration [Xe].4f 14 .5d 5 . The outermost half-filled 5d orbitals are described in terms of hydrogen-like atomic wave functions ψ nlm (r) = R nl (r)Y m l (Θ, Φ), where R nl (r) is the radial part and Y m l (Θ, Φ) is the angular part. The former can be neglected in the following since its dependence on the principal (n) and azimuthal (l) quantum numbers is shared by all 5d orbitals. Instead, the angular part changes for the different orbitals, since it depends on the magnetic quantum number m. It is often useful to write the angular part of the wave function as cubic harmonics,

A.2 The effect of spin-orbit coupling

The SOC is a relativistic effect whose nature has been introduced in Chapter 1. In iridium compounds, the SOC amounts to approximately 0.5 eV [START_REF] Kim | Novel J eff = 1/2 Mott State Induced by Relativistic Spin-Orbit Coupling in Sr 2 IrO 4[END_REF][START_REF] Kim | Excitonic quasiparticles in a spin-orbit Mott insulator[END_REF][START_REF] Kim | Phase-Sensitive Observation of a Spin-Orbital Mott State in Sr 2 IrO 4[END_REF] . In order to take this interaction into account, the spin angular momentum must be added to the wave function of the electron. If we take the z axis as the quantization axis, the following spinors correspond to the spin states "up" and "down", respectively:

By multiplying the above spin states to the spherical harmonics, one obtains a spin-dependent basis set for the 5d orbitals:

2 , ↑ }. The spin-orbit Hamiltonian H SOC = λl • s can be straightforwardly applied on the basis set B sph by writing the dot product l • s in terms of ladder operators for the orbital (l ± ) and spin (s ± ) angular momenta:

and by using the following equations:

where |j, m j is the general state with angular momentum j and projection along the quantization axis m j 33 . Even though it is easy to calculate the matrix representation of the SOC Hamiltonian in the spherical harmonics basis set, it is useful to write it in terms of the t 2g basis set with effective orbital angular momentum

A.3 The effect of octahedral distortion

SOC constant λ = 0.5 eV. As can be seen, for ∆ = 0 (regular octahedron) one recovers the scenario where only the SOC was taken into account and the j eff = 3/2 energy levels were degenerate

). Instead, a finite tetragonal distortion lifts the degeneracy of the j eff = 3/2 states, giving rise to three non-degenerate levels.

The eigenvectors of the total Hamiltonian are:

where the columns are the coefficients of the linear combinations giving rise to the Kramers doublets [START_REF] Cheong | Multiferroics: a magnetic twist for ferroelectricity[END_REF] 2 }. The parameters θ and φ account for the relative strength of the SOC and the tetragonal crystal field according to the following equations:

The highest-energy doublet, occupied by the single hole, is: As can be seen, for an elongated octahedron (∆ > 0) the shape of the wave function resembles that of the |t 0 2g orbital. Indeed, the hole feels a stronger attraction to the basal anions than to the apical anions. Instead, for a compressed octahedron, the opposite situation is realized and the shape of the wave function resembles that of the |t ±1 2g orbitals.

A.3.2 Trigonal distortion

In this case, the octahedron is elongated or compressed along the local trigonal axis, i.e. one of the axes belonging to the (111) where ∆ quantifies the strength of the trigonal crystal field and it is positive for an elongation of the octahedron. The trigonal (111) axis is chosen as the quantization axis. The diagonalization of the above matrix reveals that the eigenvalues are the same as the ones reported in Equation A.13. The matrix of the eigenvectors is:

where the columns are the coefficients of the linear combinations giving rise to the Kramers doublets [START_REF] Cheong | Multiferroics: a magnetic twist for ferroelectricity[END_REF] 2 }. The parameters θ and φ account for the relative strength of the SOC and the trigonal crystal field. The relationships reported in Equation A.15 still hold.

The trigonal crystal field dependence of the eigenvalues and of the orbital occupancies of the ground-state wave function is analogous to the one reported in Figure A.3. Of course, the electron densities will be different than the ones displayed in the Figure . Indeed, for an elongated octahedron the shape of the wave function will resemble that of the |t 0 2g orbital, whereas for a compressed octahedron the shape will approach the one of the |t ±1 2g orbitals, reported in Figure A.4.

A.4 Magnetic moments

After discussing the electronic structure, I now turn to the evaluation of the magnetic moment of a hole occupying the j eff = 1/2 state. The magnetic moment can be calculated by evaluating separately the contribution of the orbital and spin moments 4 :

(A.20)

The above expressions amount to ∓2/3, ∓1/6, and ∓1, respectively, for a regular octahedron (∆ = 0). The orbital moment dominates the magnetic moment, being twice as large as the spin contribution. We notice that by replacing l z → -l z , we obtain a total magnetic moment of 1/3, which is the A.4 Magnetic moments value of a pure j = 1/2 state obtained from an antiparallel alignment of orbital moment l = 1 and spin moment s = 1/2 4 . The magnetic moment of a j eff = 1/2 particle is analogous to the one of a free electron with spin s = 1/2. Hence, the j eff = 1/2 magnetic moment can be treated in a similar fashion to the s = 1/2 moment.

The j eff = 1/2 wave functions reported in Equations A.11 and A.16 are limited to the case of magnetic moment oriented parallel to the z axis. Indeed, the wave functions are eigenvectors of the operators l z and s z with eigenvalues reported in Equation A.20. For a magnetic moment in any direction, a linear combination of the j eff = 1/2 wave functions must be considered [START_REF] Matteo | Magnetic ground state of Sr 2 IrO 4 and implications for second-harmonic generation[END_REF] :

where β = 0 and β = π/2 correspond to the magnetic moment oriented along ±z, respectively, whereas β = π/4 corresponds to the case where the magnetic moment is orthogonal to the z axis; γ is the angle with respect to the x axis. The evaluation of the expectation values of the operators l z and s z and of the total magnetic moment is straightforward considering the results reported in Equation A.20.

APPENDIX B

Calculation of L 2,3 -edge RIXS scattering amplitudes

In Chapter 2, I introduced the general concepts of the RIXS technique and derived the cross-section that mathematically describes the scattering process (Equation 2.10). In this Appendix, I will calculate the amplitude of the RIXS cross-section for specific transitions from an initial 5d state through the intermediate 2p orbitals to a final 5d state. The starting point is the double-differential scattering cross-section derived in Chapter 2 (Equation 2.10) by making use of the dipole approximation. The terms in the cross-section can be grouped as follows:

where A( , , ω k , ω k ) = A f is the scattering amplitude [START_REF] Ament | Resonant inelastic x-ray scattering studies of elementary excitations[END_REF] :

In the following, we will assume that for a given absorption edge the lifetimes of all intermediate states are equal. Hence, the above expression can be simplified as:

B.1 Calculation of the matrix elements

To calculate the scattering amplitude of a given RIXS transition, matrix elements of the form f |D|i (where i and f are the initial and final states, respectively) must be computed. Their calculation is carried out following the procedure for copper L 2,3 -edge RIXS described in Ref. where R np,lp (r)Y mp lp (Θ, Φ) is the wave function of a specific p orbital split into the radial and angular parts. n p , l p , and m p are the principal, azimuthal, and magnetic quantum numbers of the p orbital. A similar argument applies for the wave function of the d orbital. In the following, the radial part of the integral will be neglected since it is common for all L 2,3edge transitions. By computing the above integral for all possible combinations of spherical harmonics with l p = 1 and l d = 2, the following transition matrix is obtained:

B.2 L 3 -edge RIXS scattering amplitude

Let us calculate the scattering tensor for L 3 -edge RIXS. For elastic scattering ( 1 2 , 1 2 → 1 2 , 1 2 ) we obtain: