Ktao

Llg Landau-Lifshitz-Gilbert

Remanent Rh

Hysteresis

Uhv Ultra

Nicolas Bergeal

Gerbold Ménard

Guilhem Saiz

Espci

Maximilien Cazayous

Pierre Hemme

Nicolas Olivier

François Muller

Hélène Magnan

Delphine Cabaret

LEED low energy electron diffraction LH linear horizontal

Keywords: ferroelectricity, spin-orbitronics, spin-charge interconversion, oxide heterostructures AFM atomic force microscope AHE anomalous Hall effect UPS ultraviolet photoemission spectroscopy

Ferroelectric control of charge

Titre: Contrôle du transport de charge et de spin par la ferroélectricité dans des gaz d'électrons bidimensionnels de type Rashba Mots clés: ferroélectricité, spin-orbitronique, interconversion spin-charge, hétérostructures d'oxydes Résumé: La demande énergétique croissante venant du secteur des technologies de l'information et de la communication et les limites physiques liées à la miniaturisation des composants logiques nécessitent de développer de nouvelles technologies plus économes et de repenser les paradigmes de traitements de données. Partant de ce constat, le domaine de la spintronique qui exploite le spin des électrons comme nouveau support de l'information pour le stockage et le traitement de données a su proposer de nombreuses avancées technologiques au cours des dernières années permettant notamment d'augmenter la densité du stockage d'informations. Les courants de spin purs, qui n'ont pas d'effet de dissipation thermique, pourraient également amener à une révolution dans le transport et le stockage d'informations, bien que cela requière des systèmes d'interconversion spin-charge à haute efficacité pour permettre une pleine intégration à l'électronique moderne. Au-delà des métaux lourds, des systèmes possédant un couplage spin-orbit de type Rashba comme les hétérostructures d'oxydes semblent des candidats idéaux pour générer des courants de spins sans l'aide de matériaux ferromagnétiques, bien que ces derniers servent souvent de support d'information. De plus, l'intégration de systèmes Rashba dans de futurs dispositifs aurait l'avantage d'une modulabilité de leur couplage spin-orbite par des champs électriques, permettant d'envisager la réalisation de composants logiques.

Dans cette thèse, de nouvelles solutions permettant de réaliser du calcul en mémoire ont été proposées, exploitant les fortes efficacités de conversion spin-charge des hétérostructures d'oxydes et la commutabilité électrique à faible puissance des ferroélectriques. Des gaz 2D d'électrons à l'interface AlO x /SrTiO 3 ont d'abord étudiés comme plateformes modèles pour l'interconversion spin-charge. En ap-pliquant un modèle théorique développé pour les mesures de magnétotransport dans le plan d'un système Rashba, le coefficient Rashba spin-orbite a pu être extrait et on a démontré le contrôle de son amplitude et de son signe par un champ électrique. En outre, l'application de forts champs électriques a permis d'obtenir un contrôle non-volatile de la conversion spin-charge à basse température par des expériences de pompage de spin à résonance ferromagnétique, en induisant un état ferroélectrique dans le titanate de strontium (STO). Une autre approche permettant de rendre STO ferroélectrique par une substitution du Sr par des ions Ca a été exploitée pour comprendre les propriétés diélectriques et de transport à l'interface AlO x /Ca:STO, dévoilant un contrôle ferroélectrique de diverses propriétés de transport et de l'amplitude de conversion charge-spin. Par ailleurs, un comportement ferromagnétique a également été incorporé dans des gaz 2D d'électrons "ferroélectriques" dans des hétérostructures de LaAlO 3 / EuTiO 3 /Ca:SrTiO 3 . Des mesures de spectroscopie d'absorption de rayons X et de magnétotransport ont ainsi permis de confirmer le caractère multiferroïque de ces gaz 2D d'électrons offrant en outre, un couplage magnétoélectrique. Pour finir, on a exploré un autre type d'oxyde à plus fort couplage spin-orbite, à savoir le tantalate de potassium (KTO), qui peut également être rendu ferroélectrique par l'introduction de Nb à la place des ions Ta. On a ainsi montré que des systèmes AlO x /KTaNbO 3 pouvaient présenter des cycles de polarization électriques inattendus, indiquant la présence de deux sousréseaux de déplacements polaires opposés se compensant partiellement en l'absence d'un champ électrique externe. Enfin, des mesures d'effet Hall à l'interface AlO x /KTaNbO 3 permettent d'envisager de contrôler le type de porteurs de charges majoritaire (électrons ou trous) selon l'état de polarisation du matériau. of information and communication technologies and the physical limitations inherent to the miniaturization of electronics components call for new technological developments to improve the energy efficiency and rethink computational paradigms. In these regards, the expanding field of spintronics, which relies on the spin of electrons to carry and process information instead of its charge, has been able to provide numerous technological advances improving data storage density and longevity, for instance. Pure spin currents, which are dissipation-less, could also bring a new revolution in the transport and processing of information but require efficient spincharge interconverters for a full integration with modern electronics. Aside from heavy metals or topological insulators, systems hosting a Rashba-type of spin-orbit coupling such as oxide heterostructure interfaces are very promising candidates to generate spin currents without the help of ferromagnets. In addition, the integration of Rashba systems in future devices may benefit from the tunability of their inherent spin-orbit coupling by electric fields, offering ways to perform logic operations.

In this thesis, we propose and study new routes to achieve in-memory computing, harnessing the high conversion efficiency of oxide heterostructure interfaces and the low power electrical switchability of ferroelectrics. We first use the AlO x /SrTiO 3 two-dimensional electron gas (2DEG) as a model platform to study spincharge interconversion phenomena under the application of external electric fields. Applying a theoretical model to our in-plane magnetotransport measurements, we extract the Rashba spin-orbit interaction coefficient and demonstrate the modulation of both its amplitude and sign. Going further, we even achieve a non-volatile electrical control of spin-to-charge conversion at low temperature by spin pumping by ferromagnetic resonance (SP-FMR), by inducing a ferroelectric state in the bulk of the otherwise quantum paraelectric strontium titanate SrTiO 3 (STO). We then consider another approach to make strontium titanate SrTiO 3 (STO) ferroelectric by a minute isovalent ionic substitution of Sr by Ca and study the dielectric and interfacial transport properties of AlO x /Ca:STO systems, unveiling a ferroelectric control of their sheet resistance, carrier density and possibly of the charge-to-spin conversion amplitude. Building on these results, we also show how ferromagnetism can be incorporated with "ferroelectric 2DEGs" in LaAlO 3 /EuTiO 3 /Ca:SrTiO 3 epitaxial heterostructures and using X-ray absorption spectroscopy (XAS) and related techniques, we actually demonstrate the coexistence of ferromagnetism and switchable polar moments in the two-dimensional electron gas (2DEG) region, three usually contra-indicated properties. Using a ensemble of magnetotransports measurements then confirms the strong coupling of these properties as we demonstrate the presence of anomalous Hall effect (AHE), and negative out-of-plane magnetoresistance with different amplitudes depending on the polarization state and a cyclability by an applied electric field. Finally, we start exploring another oxide material with larger intrinsic spin-orbit coupling, namely potassium tantalate KTaO 3 (KTO), which can also become ferroelectric by the introduction of Nb in place of Ta ions and be used to generate 2DEGs at its surface by similar methods as for STO. We show that the resulting AlO x /KTaNbO 3 systems display unexpected hysteretic polarization cycling with the electric field, as a first hint of the presence of two subnetworks or regions of polarization partially compensating each other in absence of an external electric field. These systems reveal other surprises in their transport properties as we show from simple Hall measurements a remarkable change of dominant carrier type (electrons or holes) depending on the remanent polarization state.

Résumé de la thèse en français

Avec le développement et l'intégration industrielle des matériaux semiconducteurs ces cinquante dernières années, les technologies de l'information et de la télécommunication (TIC) ont considérablement investi les activités quotidiennes de milliards d'êtres humains, si bien que certaines prédictions annoncent que ce secteur pourrait dépasser les 8 TW h de consommation énergétique annuelle en 2030, c'est-à-dire plus 21 % de la demande totale en énergie attendue [START_REF] Jones | Data Centres Are Chewing up Vast Amounts of Energy[END_REF][START_REF] Andrae | On Global Electricity Usage of Communication Technology: Trends to 2030[END_REF].

Dans un contexte où l'empreinte écologique humaine sur les écosystèmes et les changements climatiques s'amplifient [START_REF]Climate Change 2022: Impacts, Adaptation and Vulnerability 3-33[END_REF], la recherche de sobriété énergétique est une piste à privilégier dans tous les secteurs économiques [START_REF]IPCC. Climate Change 2022: Mitigation of Climate Change isbn[END_REF][START_REF]A Task of Terawatts[END_REF]. Au delà des efforts à fournir pour repenser et réduire nos usages de ces technologies dans certains domaines, il devient clair qu'une plus grande efficacité énergétique des appareils électroniques et processeurs est également nécessaire. Étant donné les limites physiques liées à la miniaturisation des composants logiques, avec des tailles de gravure approchant les quelques nanomètres [START_REF] Moore | Cramming More Components onto Integrated Circuits[END_REF][START_REF] Frank | Approaching the Physical Limits of Computing in 35th International Symposium on Multiple-Valued Logic[END_REF][START_REF] Markov | Limits on Fundamental Limits to Computation[END_REF][START_REF] Bohr | CMOS Scaling Trends and Beyond[END_REF], il semble donc urgent de développer de nouvelles technologies plus économes quitte à repenser les paradigmes de traitements de données par exemple avec du calcul en mémoire (in-memory logic ou memory in logic).

WRITE READ

Figure 1: Dispositif magnetoelectric spin-orbit logic (MESO) proposé par Intel, comprenant un élément ferromagnétique, servant de support de l'information stockée, et qui peut être écrite via une tension en entrée grâce au couplage avec un élément magnétoélectrique (ME). La lecture en sortie se fait en faisant passer un courant à travers le ferroaimant ce qui polarise le courant en spin et ce dernier peut ensuite être re-converti en un simple courant de charge grâce à l'élément à fort couplage spin-orbite.

Partant de ce constat, les acteurs majeures de l'industrie de l'électronique et des semiconducteurs ont commencé à proposer de nouveaux dispositifs et architectures, exploitant les propriétés de matériaux émergents [START_REF] Bohr | CMOS Scaling Trends and Beyond[END_REF][START_REF] Nikonov | Benchmarking of Beyond-CMOS Exploratory Devices for Logic Integrated Circuits[END_REF]. Parmi ces solutions potentielles, le domaine de la spintronique, qui exploite le spin des électrons comme nouveau support de l'information pour le stockage et le traitement de données, pourrait être particulièrement prometteur pour l'introduction de nouveaux paradigmes. En xiii parallèle, un important travail de recherche académique et de développements industriels ont permis d'intégrer de nouveaux procédés aux lignes de fabrication pré-existantes et ainsi de voir l'apparition sur le marché de nouveux composants de mémoire basés sur la spin-orbitronique et sur des matériaux possédant différents ordres ferroïques.

Dans leur recherche de nouveaux composants, les acteurs industriels comme Intel songent donc sérieusement à combiner la commutabilité des matériaux ferroïques et leur paramètre d'ordre (ferromagnétisme, ferroélectricité, ...) avec les phénomènes liés au couplage spin-orbite, et ont par exemple proposé un nouveau type de dispositif appelé "MESO" (voir fig. 1). Dans ces dipositifs, l'information est encodée par la direction de l'aimantation d'un matériau magnétique et peut être contrôlée par l'application d'une faible tension (100 mV) aux bornes d'un matériau magnétoélectrique (ME) en entrée. Pour lire l'information binaire ainsi stockée, un courant est polarisé en spin en passant à travers la couche magnétique et un matériau avec fort couplage spin-orbite permet de générer un courant de charge dans une direction ou une autre (selon la direction d'aimantation) et qui peut se propager à d'autres dispositifs similaires placés en aval.

Tandis que les matériaux et hétérostructures magnétoélectriques à base d'oxydes comme la ferrite de bismuth (BiFeO 3 ) sont d'ores et déjà envisagés pour l'écriture dans ces dispositifs "MESO", une des difficultés reste l'intégration de ces composants et l'obtention d'un couplage magnétique suffisant entre la couche magnétoélectrique et celle ferromagnétique placée en contact. D'autres hétérostructures d'oxydes à fort couplage spin-orbite ont également démontré leur haute efficacité cette fois-ci pour l'interconversion entre courants de spin et de charge, notamment aux interfaces LaAlO 3 /SrTiO 3 [START_REF] Lesne | Highly Efficient and Tunable Spin-to-Charge Conversion through Rashba Coupling at Oxide Interfaces[END_REF], AlO x /SrTiO 3 [START_REF] Vaz | Mapping Spin-Charge Conversion to the Band Structure in a Topological Oxide Two-Dimensional Electron Gas[END_REF] ou dans les gaz bidimensionnels d'électrons basés sur KTaO 3 [START_REF] Vicente-Arche | Spin-Charge Interconversion in KTaO 3 2D Electron Gases[END_REF][START_REF] Zhang | Thermal Spin Injection and Inverse Edelstein Effect of the Two-Dimensional Electron Gas at EuO-KTaO 3 Interfaces[END_REF], ce qui renforce l'attractivité pour cette famille de matériaux. En effet, différents mécanismes (transfert de charge, réaction d'oxydo-réduction) permettent d'obtenir dans ces hétérostructures d'oxydes une interface conductrice où les électrons sont confinés sur quelques nanomètres. Ce confinement et plus généralement la présence de forts champs électriques brisant la symmétrie d'inversion impacte les propriétés de transport et de couplage spin-orbite en renforçant ce dernier par une interaction de type Rashba, ce qui permet in fine d'obtenir une interconversion spin-charge efficace à travers les effets Edelstein direct et indirect.

Sans être limité à ces phénomènes, les matériaux à base d'oxydes de métaux de transition présentent une large gamme de propriétés physiques avec des diagrammes de phase riches et complexes. Par exemple, la ferroélectricité, c'est-à-dire la présence d'une polarisation (déplacements polaires ordonnés) spontanée et renversable dans un matériau, peut créer un champ électrique particulièrement aux surfaces opposées du matériau (ou à l'interface avec d'autres matériaux) le long de la direction de polarisation. Ce champ électrique peut alors être exploité pour créer ou renforcer le couplage spinorbite d'interface et ainsi obtenir des textures de spin complexes, par exemple à l'interface entre un ferroélectrique et un matériau possédant déjà un fort couplage spin-orbite atomique [START_REF] Tao | Perspectives of Spin-Textured Ferroelectrics[END_REF][START_REF] Gosteau | Calculs ab initio des effets spin-orbite de type « Rashba » dans des oxydes de métaux de transition et leurs interfaces[END_REF] comme cela a été prédit pour le système Pt/PbZr x Ti 1-x O 3 [START_REF] Fang | Tuning the Interfacial Spin-Orbit Coupling with Ferroelectricity[END_REF].

Bien que le contrôle des textures de spin de type Rashba par la ferroélectricité ne soit pas limité aux oxydes, et que le concept ait déjà été prédit [START_REF] Di Sante | Electric Control of the Giant Rashba Effect in Bulk GeTe[END_REF] et démontré [START_REF] Rinaldi | Ferroelectric Control of the Spin Texture in GeTe[END_REF][START_REF] Varotto | Room-Temperature Ferroelectric Switching of Spin-to-Charge Conversion in Germanium Telluride[END_REF] dans le semiconducteur GeTe par exemple, les hétérostructures d'oxydes semblent tout de même présenter un intérêt du fait de leur large efficacité de conversion et de leurs propriétés ferroélectriques mieux étudiées, plus robustes et plus facilement renversables. : Schéma conceptuel du contrôle ferroélectrique du couplage spin-orbite interfacial dit "Rashba" et des textures de spin associées. Le panneau supérieur illustre le type d'hétérostructure qui peut être réalisé pour cela, avec un matériau ferroélectrique dont la polarisation est contrôlée par une tension appliquée aux bornes de ce matériau. Ici, le système "spin-orbite" correspond au gaz 2D d'électrons dans lequel l'effet Rashba est attendu avec deux contours de Fermi concentriques présentant des enroulement de spin opposés comme illustré dans le panneau central. Le retournement de la polarisation doit idéalement inverser la chiralité des deux contours. En utilisant l'effet Edelstein inverse par exemple, pour un même courant de spin injecté, celui-ci sera converti en un courant de charge avec des directions opposées, simplement par l'application de pulses de tension.

De manière plus générale, le couplage entre polarisation ferroélectrique et texture de spin liée à l'effet Rashba permettrait à terme de s'affranchir de tout élément ferromagnétique, dans des dispositifs dits "FESO" (ferroélectrique + spin-orbite). L'information serait alors encodée par la direction de la polarisation, et l'on pourrait ainsi idéalement modifier la texture de spin le long des contours de Fermi et donc l'amplitude et/ou le signe de la conversion entre courants de charge et courants de spin, simplement par l'application momentanée d'une tension électrique (voire fig. 2).

Dans cette thèse, de nouvelles solutions permettant de réaliser du calcul en mémoire ont ainsi été proposées, exploitant les fortes efficacités de conversion spin-charge des hétérostructures d'oxydes et la commutabilité électrique à faible puissance des ferroélectriques.

AlO x /SrTiO 3 : un système modèle pour étudier l'interconversion spin-charge Des gaz 2D d'électrons à l'interface AlO x /SrTiO 3 ont d'abord étudiés comme plateformes modèles pour l'interconversion spin-charge. Des dispositifs en forme de croix de Hall ont été lithographiés par xv un procédé simple, permettant de caractériser proprement les propriétés de magnétotransport des gaz 2D d'électrons ainsi obtenus, particulièrement en fonction d'une tension de grille appliquée depuis l'arrière de l'échantillon. (a) Dépendance en angle du signal harmonique de rang 2 pour diverses tensions de grille et sous un champ magnétique de 9 T dans le plan de l'échantillon. Les points colorés correspondent aux données mesurées après soustraction de contributions d'arrière-plan, tandis que les courbes ajustées par rapport au modèle théorique sont représentées par des lignes continues. Seules quelques coourbes sont montrées et décalées verticalement pour un soucis de clarté. (b) Amplitude de la BMR extraite des ajustements paramétriques des dependances en angle, en fonction de la tensionde grille. Les barres d'erreur sont obtenues lors des ajustements, correspondent à l'erreur résiduelle et sont toutes au moins deux ordres de grandeur en dessous des valeurs d'amplitude. (c) Résistance planaire extraite du signal à la seconde harmonique en fonction du champ magnétique placé dans le plan de l'échantillon et orthogonal au courant, rampé entre +9 T et -9 T. Le signal est mesuré à V g = -150 V (en orange) et +75 V (en vert). Les lignes sont uniquement à titre indicatif.

En appliquant un modèle théorique développé pour les mesures de magnétotransport de systèmes Rashba avec un champ magnétique appliqué dans le plan de l'échantillon, le coefficient Rashba spinorbite a pu être extrait et on a pu démontrer grâce à la présence d'un magnétorésistance bilinéaire (BMR) le contrôle de l'amplitude et du signe de la conversion charge-spin par un champ électrique (voir fig. 3), tout en appréciant les limites du modèle pensé pour un système idéal limité à une bande parabolique. Cependant, étant donné le changement du signe de la conversion spin-charge, des modèles de dispositifs logiques universels basés sur des conversions bidirectionnelles sucessives entre courant de spin et courant de charge ont été proposés, en exploitant la modulabilité de ces effets de conversion par une tension de grille.

xvi

En outre, l'application de forts champs électriques a permis de "réveiller" une phase ferroélectrique dans le titanate de strontium (SrTiO 3 ou STO) survivant jusqu'à 60 K environ et d'obtenir en plus un contrôle électrique non-volatile de la conversion spin-charge dans le gaz 2D d'électrons à basse température, obtenu lors d'expériences de pompage de spin à résonance ferromagnétique. Cela constitue ainsi une première étape significative vers la réalisation de dispositifs FESO. ) μ 0 (H-H res ) (mT) After V g > 0 After V g < 0 After V g > 0 After V g < 0 Figure 4: Démonstration du contrôle électrique non-volatile de la conversion spin-charge par injection de spin à la résonance ferromagnétique. (a) Dépendance du courant de charge généré par l'effet Edelstein inverse avec la tension de grille arrière appliquée, montrant un comportement hystérétique. (b) Courbes de courant de charge correspondantes obtenues (A) dans l'état initial puis successivement à +200 V (B), à rémanence positive (C), -200 V (D), rémanence négative (E) puis à nouveau à +200 V (F). Le courant de charge subit un changement de signe non seuelement lorsque la tension est appliquée mais également à rémanence entre les états de polarisation up ou down. (c) Courant de charge obtenu après des pulses successifs de ±200 V pendant 500 ms démontrant la cyclabilité du phénomène. (d) Des mesures similaires permettent de montrer la différence entre les courants à rémanence positive ou négative en fonction de la température, et ainsi d'observer un maximum du courant de charge généré autour de 30 K suivi d'un déclin corrélé avec la transition ferroélectrique vers paraélectrique de l'échantillon.

xvii Contrôle des propriétés de transport dans un gaz 2D "ferroélectrique": AlO x /Ca:SrTiO 3 Une autre approche permettant de rendre STO ferroélectrique par une substitution du Sr par des ions Ca a été exploitée pour comprendre les propriétés diélectriques et de transport à l'interface AlO x /Ca:STO. Ces investigations par des mesures électriques, de spectroscopie Raman et de magnétotransport ont dévoilé un contrôle ferroélectrique de diverses propriétés de transport simples comme la résistance et la densité de porteurs, mais indiquent aussi la possibilité de contrôler l'amplitude du couplage Rashba comme cela transparaît avec le phénomène d'antilocalisation faible. V g = 0 V after +70 V V g = 0 V after 70 V En plus de cela, des expériences préliminaires de magnétotransport avec le champ magnétique dans le plan de l'échantillon ont été réalisées de manière similaire au système AlO x /SrTiO 3 , révélant un possible contrôle rémanent de l'amplitude de la conversion charge-spin par la ferroélectricité.

xviii

Gaz 2D d'électrons magnétoélectrique

Par ailleurs, un comportement ferromagnétique a également pu être incorporé dans des gaz 2D d'électrons "ferroélectriques" dans des hétérostructures où une fine couche de titanate d'europium (EuTiO 3 ) a été insérée: LaAlO 3 /EuTiO 3 /Ca:SrTiO 3 . Des mesures de spectroscopie d'absorption de rayons X ont dans un premier temps pu confirmer le caractère ferromagnétique du gaz d'électrons ainsi que la présence de déplacements polaires à l'interface dont l'amplitude est modulable après application d'une tension positive ou négative, c'est-à-dire selon l'état de polarisation ferroélectrique de l'échantillon. Des mesures des propriétés ferroélectriques et de magnétotransport ont aussi permis de révéler une empreinte de la ferroélectricité et du ferromagnétisme sur les propriétés de transport de ces gaz 2D d'électrons, notamment via la présence d'un effet Hall extraordinaire modulable à rémanence selon l'état de polarisation. Ces systèmes métalliques bidimensionnels offrent donc en outre un couplage xix magnétoélectrique.

Exploration de matériaux ferroélectriques alternatifs et leur interface conductrice: AlO x /KTaO 3 :Nb Pour finir, on a exploré un autre type d'oxydes à plus fort couplage spin-orbite, à savoir le tantalate de potassium (KTO), qui peut également être rendu ferroélectrique par l'introduction de Nb à la place des ions Ta. On a ainsi montré que des systèmes AlO x /KTaNbO 3 pouvaient présenter des cycles de polarization électriques inattendus, indiquant la présence de deux sous-réseaux de déplacements polaires opposés se compensant partiellement en l'absence d'un champ électrique externe, typique d'un comportement "antiferroélectrique". Enfin, des mesures d'effet Hall à l'interface AlO x /KTaNbO 3 à basse température pour une concentratin de 1.5 % de niobium Nb permettent d'envisager de contrôler le type de porteurs de charges xx majoritaire (électrons ou trous) selon l'état de polarisation du matériau.

xxi

I have met a lot of warm and generous friends along these four years, starting from those I met in the SMNO Master, and those thanks to whom the few months of internship and years of PhD have been an amazing journey. Merci à Diane et Yanis d'avoir été de tels exemples de gentillesse et de générosité, si naturelles chez vous, à Pauline pour les conversations toujours agréables et les instants privilégiés d'un trek autrichien hors du temps (et à refaire !), à Laurette pour ton sourire et ta bonne humeur si communicative et à toute la team des Grognons d'avoir gardé cette cohésion et permis à un fort sentiment d'appartenance et d'inclusion d'exister, toujours.

I will never have enough words to describe how extremely grateful I am of the moments lived and shared with all members of the Oxitronics group and all the Great Palaisians and all the feelings I have for all of you. First, grazie mille Sara, for the mutual support at work and outside, the precious life moments and music shared together, for accepting and composing with who I am, but also for all the lessons taught and for helping me to grow wiser with you, to understand myself and let some of the craziness in me come out and show itself. It has indeed been quite a crazy -sometimes tough-but invaluable journey, and I hope you know how much our friendship means to me. I promise you, everything is going to be alright. To Srijani, thank you for being such a nice colleague and close friend at all times, always listening, sharing, and dedicated to help. Please excuse me for all the past and future "sorry"'s you could receive from me. Thank you Luis, for guiding me from the start and making this journey almost side-by-side; even if you don't tell much, your heart says a lot. Many thanks as well to Lucia for being who you are, a deeply caring person with a fierce character, with the will to improve things around you as well as yourself, I cannot thank you enough for the support and kindness you showed me. Thank you also Fernando for being so natural, always here to help and sorry that my departure made you change your diet so drastically tronco, I hope you will get to like vegetables! More generally, big thanks to the Spanish team for the nice mood, dinners and parties and for teaching me rudiments of Spanish for my future life, to Sougata for your insights into magnetism and help with the PPMS setup improvements and precious help with the movings. Of course, I cannot forget all the curaazy times spent with the best office mates although there were many. In particular, I will forever be nostalgic of the wonderful days, nights and weekends full of laughter spent with my dear and multi-talented friend Jin, although my heart had to go on without you since you left one year and half ago; I now wish you the best, hoping to see you again some day soon, and to hear again your music skills! Thank you also Felix for your infinite knowledge and patience teaching me technicalities, science of 2DEGs and so much more. Thank you Victor for sharing your musical tastes and simplicity with me. To Raphaël, thank you for communicating your values, political views and commitments and your passion for cinema with so much enthusiasm, but also for opening up in more difficult times and keeping the link even after you left the lab. Big thanks and good luck to Hugo, Gabriel, Gaëtan, Ruchi and Thomas in taking over as the next generation of oxide Rashba experts, I hope this thesis will serve you well.

I also want to thank all collaborators, and in particular the Naples team: Marco Salluzzo, Daniela Stornaiuolo, Yu Chen and Maria D'Antuono, for the very nice work done together, the knowledge shared and the very warm (not only due to the heatwave) and welcoming feeling I had with you xxiv

Introduction

The advent of semiconductor technologies in the last half-century has allowed information and communications technologies (ICT) to invest our every day lives beyond what could be imaginable only decades ago, so much so that the global projected electricity demand from this sector could reach 21 % (more than 8 TW h) of the total production by 2030 [START_REF] Jones | Data Centres Are Chewing up Vast Amounts of Energy[END_REF][START_REF] Andrae | On Global Electricity Usage of Communication Technology: Trends to 2030[END_REF].

In a context of increased human footprint on the environment and climate change [START_REF]Climate Change 2022: Impacts, Adaptation and Vulnerability 3-33[END_REF], the search of sobriety should be a priority for all economical sectors [START_REF]IPCC. Climate Change 2022: Mitigation of Climate Change isbn[END_REF][START_REF]A Task of Terawatts[END_REF]. In addition to making efforts on rethinking and reducing our usage of information and communications technologies (ICT), a clear need for higher energy efficiency of computing devices and processors is also needed.

While most computing paradigms nowadays are based on the von Neumann architecture, the latter is becoming highly inefficient for the treatment of large datasets as it requires a continuous transfer of data back and forth between the physically separated memory and processing units. While over the last 50 years, the shrinkage of complementary metal-oxide-semiconductor (CMOS) transistors size has helped increasing the working frequency and number of transistors per unit area while containing the power consumption of processors (see fig. 8), it seems that a physical limit has been reached recently, as sizes of 10 nm are reached [START_REF] Moore | Cramming More Components onto Integrated Circuits[END_REF][START_REF] Frank | Approaching the Physical Limits of Computing in 35th International Symposium on Multiple-Valued Logic[END_REF][START_REF] Markov | Limits on Fundamental Limits to Computation[END_REF][START_REF] Bohr | CMOS Scaling Trends and Beyond[END_REF].

To overcome these physical limitations, leaders of the semiconductor electronics industry started looking at new architectures, devices and materials to continue reducing the power consumption and bring new computing paradigms in their processors [START_REF] Bohr | CMOS Scaling Trends and Beyond[END_REF][START_REF] Nikonov | Benchmarking of Beyond-CMOS Exploratory Devices for Logic Integrated Circuits[END_REF]. In-memory computing as such allows to circumvent the problems inherent to the von Neumann architecture by bringing processing units to the same physical location as individual bits of data storage, thereby reducing the power consumption and increasing the throughput.

While several approaches may be considered for beyond CMOS technologies, the field of spintronics which relies on the storage and transfer of spin information instead of the electron's charge, may be particularly promising for the introduction of these new paradigms and the reduction of power consumption. In parallel, important research and industrial developments on new types of non-volatile memory based on spin-orbitronics and materials with various ferroic Reproduced from [START_REF] Rupp | Microprocessor Trend Data/50-Years Processor Trend[END_REF] orders have seen the day, making it compatible with the current semiconductor industry and large scale fabrication processes.

In the search of new components, industrial players like Intel have thus considered the use of collective switching of order parameter in ferroic materials (ferroelectrics, ferromagnets, ...) combined with spin-orbit phenomena as a serious option to achieve the desired goals and they proposed a new type of transistor called the magnetoelectric spin-orbit logic (MESO) device (see 9) [START_REF] Manipatruni | Spin-Orbit Logic with Magnetoelectric Nodes: A Scalable Charge Mediated Nonvolatile Spintronic Logic[END_REF][START_REF] Manipatruni | Beyond CMOS Computing with Spin and Polarization[END_REF].

This device is comprised of a magnetoelectric writing element, coupled to a ferromagnetic material which stores the desired information. A spin-orbit material or heterostructure on the output side is then used to read the state of the ferromagnet by generating a charge current in one direction or another, which can then propagate to similar cascaded devices. WRITE READ Figure 9: Magnetoelectric spin-orbit logic (MESO) device proposed by Intel, comprising a ferromagnetic information storage element that can be written via a magnetoelectric (ME) element on the input side, and read out by flowing a current through it which becomes spin-polarized and converted back into a charge current thanks to the spin-orbit (SO) element.

Magnetoelectric oxides such as BiFeO 3 are particularly promising platforms for the realization of such devices but the achievement of large spin-charge interconversion efficiencies at oxide interfaces such as LaAlO 3 /SrTiO 3 [START_REF] Lesne | Highly Efficient and Tunable Spin-to-Charge Conversion through Rashba Coupling at Oxide Interfaces[END_REF], AlO x /SrTiO 3 [START_REF] Vaz | Mapping Spin-Charge Conversion to the Band Structure in a Topological Oxide Two-Dimensional Electron Gas[END_REF] or KTaO 3 -based two-dimensional electron gases (2DEG) [START_REF] Vicente-Arche | Spin-Charge Interconversion in KTaO 3 2D Electron Gases[END_REF][START_REF] Zhang | Thermal Spin Injection and Inverse Edelstein Effect of the Two-Dimensional Electron Gas at EuO-KTaO 3 Interfaces[END_REF] increases even more the potential of this family of materials, also for the spin-orbit output part of the magnetoelectric spin-orbit logic (MESO) device [START_REF] Varignon | A New Spin for Oxide Interfaces[END_REF][START_REF] Vaz | Oxide Spin-Orbitronics: New Routes towards Low-Power Electrical Control of Magnetization in Oxide Heterostructures[END_REF]. Indeed, the electronic confinement at these interfaces and more generally due to interfa-cial electric fields entails the presence of a Rashba-type spin-orbit coupling which then allows spin-charge interconversion phenomena to be exploited in such systems.

Not limited to these effects, a wide array of physical properties can be found in the oxides family. For example, the existence of ferroelectricity, i.e. the presence of a spontaneous and electrically switchable polarization in a material, often leads to electric fields at opposite surfaces of the material and can in fact be utilized to create or enhance the spin-orbit coupling and complex spin textures at heterostructure interfaces [START_REF] Tao | Perspectives of Spin-Textured Ferroelectrics[END_REF][START_REF] Gosteau | Calculs ab initio des effets spin-orbite de type « Rashba » dans des oxydes de métaux de transition et leurs interfaces[END_REF], and has been demonstrated at the Pt/PbZr x Ti 1-x O 3 interface, for instance [START_REF] Fang | Tuning the Interfacial Spin-Orbit Coupling with Ferroelectricity[END_REF].

Although ferroelectric control of Rashba spin textures is not limited to oxide materials and the concept has been transposed theoretically [START_REF] Di Sante | Electric Control of the Giant Rashba Effect in Bulk GeTe[END_REF] and demonstrated experimentally in GeTe [START_REF] Rinaldi | Ferroelectric Control of the Spin Texture in GeTe[END_REF][START_REF] Varotto | Room-Temperature Ferroelectric Switching of Spin-to-Charge Conversion in Germanium Telluride[END_REF] and other materials, oxides do present an interest.

Building on the expertise around spin-orbitronics and oxides acquired in the Oxitronics group in Unite Mixte de Physique CNRS, Thales, Universite Paris Saclay (UMPhy), we will thus focus in this thesis on investigations of charge and spin transport properties at oxides interfaces and the possibility to control them electrically by ferroelectricity.

In chapter 1, we will first remind the reader of some basic knowledge in the field of spintronics and related to ferroelectricity and then propose a review of the current literature on few oxides used in this thesis and 2DEGs obtained at their surface and interface with other materials. Finally, we will introduce the concept of "ferroelectric" metals, a counter-intuitive notion still under debate, and implying the combination of polar ordering and electrical conduction by free carriers in the same material. This will be thus constitute one of the motivations of this thesis in which we utilize ferroelectric materials to introduce carriers at their surface.

The second chapter will then cover the main ensemble of techniques used during this thesis work, from sample fabrication to the characterization of their structural, magnetic, electrical and transport properties, in addition to spin-charge interconversion by spin pumping by ferromagnetic resonance (SP-FMR).

Then, in chapter 3 we will briefly review spin-charge interconversion characterization techniques and efficiencies associated to different materials, before showing how the charge-to-spin interconversion can be electrically tuned in AlO x /SrTiO 3 patterned samples. In a second part, we will also investigate the possiblity of making STO ferroelectric by applicaiton of large electric fields as was demonstrated in the past [START_REF] Gränicher | Induzierte Ferroelektrizitat von SrTiO3 bei sehr tiefen Temperatur und uber dieKalterzeugung durch adiabatic Entpolarisierung[END_REF][START_REF] Hemberger | Quantum Paraelectric and Induced Ferroelectric States In[END_REF] and use this functionality to obtain a non-volatile electric control of spin-to-charge conversion by spin pumping by ferromagnetic resonance (SP-FMR).

Chapter 4 will then explore another way to induce ferroelectricity in STO this time by minute Ca substitution and we will use AlO x /Ca:STO heterostructures to obtain a ferroelectric control of charge transport properties at the interface, but also insights of a ferroelectric control of charge-to-spin conversion in this system.

Going one step further, we will show in chapter 5 how we design and achieve the realization of multiferroic 2DEGs in LaAlO 3 /EuTiO 3 /Ca:SrTiO 3 heterostructures, with the coexistence and coupling of both ferroelectric and ferromagnetic orders, and imprints in both spectroscopic and magnetotransport measurements.

Finally, we will present an alternative system to obtain two-dimensional conductive system CHAPTER 1

Background

In this chapter, we provide some background knowledge and overview of the literature to introduce or remind the reader about useful concepts throughout the rest of this thesis. We start by introducing concepts relative to spin (and orbital) transport and the links that can be made with classical pure charge transport, as well as some mechanisms of interconversion between the two. We then go on with concepts of ferroic orders and in particular ferroelectricity and some neighboring special dielectric properties. We show how these properties -among other very interesting ones-can be found in the transition metal oxides studied in this thesis. In particular, we review how conductivity can be obtained and confined at the interfaces of such materials and the extraordinary richness of properties they offer.

Finally, we introduce the concept of "ferroelectric" or "polar" metals which is still a concept under debate and on which we try to bring some new light in this work.

. Spin-orbitronics and exotic transport phenomena

Devising new concepts and paradigms often requires to take a step back and look at the object or topic of interest from a broader view and then to allow oneself to change perspective and consider properties that were ignored in the first place. In this section we try to cover different aspects and properties of a well-known particle that is the electron, and transport phenomena related to its various properties.

. The electron beyond its charge

Indeed, electronics as we know it today is mostly relying on one fundamental property of the underlying particle, that is the electron's charge q e = -e (or hole charge q h = e, e being the elementary charge constant) and phenomena related to charge transport.

However, besides carrying a charge, the electron also has a mass m e which in most cases governs its kinetics. More interestingly for us, since the electron is a fermion, it has a spin of ± 1 2 and its quantized spin angular momentum is S = ℏ 2 σ where σ is the vector of Pauli matrices. When projected along an aribtrary ẑ axis, S takes values of S z = ± ℏ 2 which we usually refer to as "spin-up" or "spin-down". Due to its nature, the spin agular momentum also gives rise to an associated magnetic moment:

µ spin = -g e e 2m e S = -g s µ B ℏ S (1.1)
where g s = |g e | is the dimensionless electron g-factor approximately equal to 2, and µ B = eℏ 2me is used as a unit of magnetic moment and called the Bohr magneton. Being a fermion, the electron is also subject to the Pauli exclusion principle which states that no two electrons can co-exist with the same set of properties, i.e. same energy, same spin, etc. Some of these properties reach yet another level of complexity and significance when the electrons are not free but bound to atoms in which case the electrons gain an orbital angular momentum L and an additional magnetic moment associated to it:

µ orb = -g L e 2m e L = -g L µ B ℏ L (1.2)
where g L is the electron orbital g-factor ideally equal to 1.

The change is even more important in solid state systems and particularly in periodic crystals, on which we focus for the matter of this thesis. In this situation, because of the accumulation of many electrons and the Pauli exclusion principle, each material will show a specific electronic band structure depending on many intrinsic properties (physical properties of individual constituting atoms and impurities, types of bonding and interactions, ...) and on the environment (temperature, strain, ...).

The shape of this band structure relating the electrons' momentum to their energy will govern how electrons near the Fermi level (or chemical potential) can move, and a deviation of their effective mass m * = αm e compared to the electron's mass is usually observed, depending on the direction of propagation. The orbital character of the bands near the Fermi level, the filling of these bands as well as their spin-degeneracy or dependence will also give rise to new interesting properties as we will see along this thesis.

Finally, while most of the transport of information in modern electronics is based on the intrinsic charge of electrons, storage and readout of information have been relying heavily also on other degrees of freedom, and more specifically on magnetic properties, starting the field of spintronics about 25 years ago with the discovery of giant magnetoresistance (GMR) [START_REF] Baibich | Giant Magnetoresistance of (001)Fe/(001)Cr Magnetic Superlattices[END_REF][START_REF] Binasch | Enhanced Magnetoresistance in Layered Magnetic Structures with Antiferromagnetic Interlayer Exchange[END_REF] used later on for magnetic read-heads of "hard disks" [START_REF] Chappert | The Emergence of Spin Electronics in Data Storage[END_REF]. Nowadays, new paradigms have started to emerge, exploring the possibility to use spin and even orbital angular momentum as a medium to carry information instead or in combination with the charge [START_REF] Nikonov | Benchmarking of Beyond-CMOS Exploratory Devices for Logic Integrated Circuits[END_REF][START_REF] Manipatruni | Spin-Orbit Logic with Magnetoelectric Nodes: A Scalable Charge Mediated Nonvolatile Spintronic Logic[END_REF][START_REF] Manipatruni | Beyond CMOS Computing with Spin and Polarization[END_REF][START_REF] Datta | Electronic Analog of the Electro-optic Modulator[END_REF][START_REF] Allwood | Magnetic Domain-Wall Logic[END_REF][START_REF] Behin-Aein | Proposal for an All-Spin Logic Device with Built-in Memory[END_REF][START_REF] Nikonov | Benchmarking Spintronic Logic Devices Based on Magnetoelectric Oxides[END_REF].

. Spin and charge currents

As we start to speak about not one single electron but rather about collective behaviors of an ensemble of spin or charge carriers, we can use the concepts of charge carrier density n and spin density s. It is also useful to think of these two quantities in terms of the density of carriers for a spin-up n ↑ or spin-down n ↓ orientation along a well-defined quantization axis ûs :

n = n ↑ + n ↓ (1.3a) s = n ↑ -n ↓ , (1.3b)
from which we can also define a formula to quantify the carrier spin polarization as the ratio between the spin (carrier) density and the total carrier density:

P n = s n ûs .
(1.3c)

For ferromagnetic (FM) materials for instance, there is a net spin density in equilibrium since the density of states near the chemical potential (Fermi level) is different between spin-up and spin-down. On the contrary, for non-magnetic materials, the spin density should be zero unless one finds a way to inject a non-equilibrium spin population and we will see how in further sections.

One basic approach to define and understand the concept of spin transport is to look back at some definitions related to charge transport. [START_REF] Maekawa | Spin Current isbn[END_REF][START_REF] Maekawa | Spin Current: Experimental and Theoretical Aspects[END_REF][START_REF] Shi | Proper Definition of Spin Current in Spin-Orbit Coupled Systems[END_REF][START_REF] Zucchetti | Spin-Charge Interconversion in Ge-based Structures[END_REF][START_REF] Fabian | [END_REF] Figure 1.1: The total variation of charge inside the volume V is equal to the sum of currents penetrating or leaving the oriented closed surface Ω: an increase (resp. decrease) in charge within Ω is equivalent to a negative (resp. positive) current.

Let's consider a volume V enclosed by an oriented surface Ω with given charge density ρ = nq, where q is the charge of a single carrier (see schematics on fig. 1.1). Due to the conservation of charge, any increase or decrease in the carrier density must originate from a flow of charge entering or leaving the volume V , i.e. a charge current density j c passing across Ω in one direction or another, as seen on fig. 1

.2:

˚V ∂ρ

∂t dV = - ‹ Ω j c • dΩ.
(1.4)

Using the divergence theorem, we can rewrite this equation as:

∂ρ ∂t = -∇ • j c , (1.5)
or, focusing on the carrier density rather than the charge density, the eq. (1.5) is then equivalent to:

∂n ∂t = - ∇ • j c q .
(1.6a)

And since by definition the charge current is the time derivative of the product between charge and a position observable j c ≜ ∂ ∂t (ρr), where r is the position vector of the carriers (an electron, a hole or a charge wavepacket), and because the charge must be conserved (i.e., the time derivative of ρ is zero), we have:

j c = nqv = q (nµE + D∇n) , (1.6b) with v = ∂r ∂t
being the velocity of the carriers, µ being their mobility, E the electric field and D the diffusion coefficient. This expression also called charge drift-diffusion equation relates how a charge current can be generated either by applying an external electric field E (drift) or by a spatial imbalance ∇n of carriers (diffusion).

These continuity equations eqs. (1.6a) and (1.6b) are more broadly transposable to other quantities such as spins or more simply to a subpart of the total carrier density n σ for one spin population σ =↑ (↓) (while -σ =↓ (↑)). However, because the spin angular momentum is non-conservative contrary to the charge, there is a probability w that a carrier's spin is flipped from σ to -σ within the time τ such that the spin flip rate is w/τ which results in:

∂n σ ∂t = - ∇ • j σ q -w (n σ -n -σ ) , (1.7a)
with: j σ = q (n σ µE + D∇n σ ) .

(1.7b)

Intuitively, we can define the total charge current and the spin current as we did for the carrier density and spin density in eqs. (1.3a) and (1.3b) either by adding up or substracting contributions of one spin channel to the other:

j c = j ↑ + j ↓ = q (n ↑ v ↑ + n ↓ v ↓ ) (1.8a) j s = j ↑ -j ↓ = q (n ↑ v ↑ -n ↓ v ↓ ) .
(1.8b)

Here, we allowed for different velocities v σ for each spin channel and thus, possibly different transport properties (scattering time, mobility, conductivity, ...), between the spin-up and spin-down channels, which was first conjectured in 1936 by Mott [START_REF] Mott | The Electrical Conductivity of Transition Metals[END_REF] based on early magnetoresistance measurements by Thomson [START_REF] Thomson | On the Electro-Dynamic Qualities of Metals:-Effects of Magnetization on the Electric Conductivity of Nickel and of Iron[END_REF], and later actually demonstrated in Ni-and Fe-based ferromagnetic metallic alloys [42][START_REF] Fert | Two-Current Conduction in Nickel[END_REF][START_REF] Fert | Electrical Resistivity of Ferromagnetic Nickel and Iron Based Alloys[END_REF]. Similar to eq. (1.3c), we also define a spin polarization of the current:

P j = j s j c .
(1.9)

Indeed, with these definitions, it becomes easy to picture different scenarios where we can have either non-polarized charge current flowing (j c ̸ = 0 and j s = 0, so that P j = 0) or charge current with an imbalance of spin population and thus, a net spin polarization (j c ̸ = 0 and j s ̸ = 0, so that P j ̸ = 0), as is shown in fig. 1.2. One can even imagine a situation where the currents due to either spin populations are equal and opposite (j s ̸ = 0 and j ↑ = -j ↓ ), in which case no net charge current is flowing, j c = 0, which results in the current polarization diverging: P j → ∞. This condition is referred to as the case of a pure spin current. In fact, spin current displays one major advantage over charge current in requiring no net displacement of charge and being dissipationless, an appealing feature for low-power devices. Finally, if we apply eqs. (1.7a) and (1.7b) to each spin population σ =↑ and σ =↓ and we use these simple definitions of spin and charge currents, we can either end up with the conservation and drift-diffusion equations of charge (from eq. (1.8a)) as defined earlier in eqs. (1.5) and (1.6b), or obtain the equivalent for spin (from eq. (1.8b)) as follows:

∂s ∂t = - ∇ • j s q - s τ sf , (1.10a)
where τ sf = 1/2w has been defined as the spin relaxation time and:

j s = q (sµE + D∇s) .
(1.10b)

As it was the case for charge transport, the spin conservation and drift-diffusion equations are fundamental to the description of spin transport and resolving them with different boundary conditions allows one to understand the time evolution and spatial distribution of spins in the case of spin injection or spin diffusion, for instance. While we will let the reader refer to ref [START_REF] Fabian | [END_REF] for a more complete and thorough resolution of these equations in different situations, we note few important results here.

In the case where we neglect the drift contribution, which means no electric field is applied, i.e. E = 0, considering one dimensional transport only and assuming a spin density at t = 0 all concentrated locally at x = 0: s(x, 0) = s 0 δ(x), (1.11) the general solution of the spin diffusion equation becomes:

s(x, t) = s 0 √ 4πDt e -x 2
/4Dt e -t/τ sf .

(1.12)

We observe two exponential terms: one expressing the spatial spin diffusion and the second the time decay of the spin polarization (or spin relaxation). From this equation, we can also introduce what is called the spin diffusion length, which relates how far the spin diffuses before relaxation occurs:

λ s = Dτ sf .
(

1.13)

There are several mechanisms responsible for the loss of spin information in the form of spin relaxation and spin dephasing phenomena which we review briefly in section 1.1.4. Some materials like Al or Cu offer a large value of spin diffusion length λ s ≈ 100 nm and allow the propagation of spin information over a long distance, making them ideal as a spin carrier medium [START_REF] Laczkowski | Spin-Dependent Transport Characterization in Metallic Lateral Spin Valves Using One-Dimensional and Three-Dimensional Modeling[END_REF]. This thesis however will focus more on materials suitable for the high efficiency generation of spin currents from charge current and vice versa, a property which often comes at the cost of a shorter λ s as in the prototypical Pt and Ta [START_REF] Rojas-Sánchez | Spin Pumping and Inverse Spin Hall Effect in Platinum: The Essential Role of Spin-Memory Loss at Metallic Interfaces[END_REF][START_REF] Zahnd | Spin Diffusion Length and Polarization of Ferromagnetic Metals Measured by the Spin-Absorption Technique in Lateral Spin Valves[END_REF].

As a last note, the definition of the spin current j s used here, corresponds to a spin-polarized charge current density, with dimension of charge times a velocity. If one wants to consider the flow of spin particles or information only, dividing eq. (1.10b) by the charge q is necessary (and adding a factor ℏ 2 for the flow of spin angular momentum). Indeed, although in this section and in this manuscript we mainly consider spin currents mediated by a flow of charge carriers -in particular electrons (or holes) -many other mechanisms and associated pseudo-particles allowing transport of spin information (such as magnons, spinons, spin-triplet pair, ...) can also be found depending on the studied system [START_REF] Han | Spin Current as a Probe of Quantum Materials[END_REF].

We should also remind that in the form used until now, the spin current expression as a vector j s is valid for a well-defined quantization axis ûs of the spins. In a more general case, the spin current should be written as a second rank tensor, with components j s ij , where the first index i denotes the direction of the flow and the second index j refers to the orientation of the spins in three-dimensional space (i, j = x, y, z). Similarly, the scalar spin (carrier) density s can then be transformed into its vector form s or accompanied by the index j = x, y, z to specify the orientation. These specifications will be helpful when considering spin-charge current interconversion due to spin-orbit coupling, where the geometry matters.

. Spin-orbit coupling

So far, we have treated the spin and charge currents indendently from each other as if the spin and orbital angular momentum of electrons where decorrelated. However, spin-orbit This magnetic field interacts with the spin magnetic moment of the electron so that depending on the relative direction between the orbit around the nucleus and the spin, L and S may be (b) parallel or (c) antiparallel, leading to two energy-split states. Reused from [START_REF] Vaz | Spin-to-Charge Current Conversion in SrTiO3-based Two-Dimensional Electron Gases[END_REF].

coupling (SOC) (or spin-orbit interaction (SOI)) gives us a link between the electrons' spin and their motion in a landscape of electric potentials.

Let us use a simplistic approach to illustrate the concept with the case of atomic spin-orbit coupling, starting from a classical picture where an electron is orbiting the nucleus of an atom. Although there is apparently no magnetic field acting on the electron moving with a velocity v in the rest-frame of the nucleus, by a Lorentz transformation, it becomes as if the nucleus was orbiting around the electron and as a result the electron feels an induced effective magnetic field in its own frame of motion: .14) where c is the speed of light and E is the electric field due to the proximity with the nucleus. As a result of the cross-product between the electric field E and the velocity v, the magnetic field B so must be perpendicular to both of the former and oriented along the same axis as the orbital angular momentum L as seen on fig. 1.3. The coupling between this effective magnetic field and the spin magnetic moment via a Zeeman-like interaction introduces an additional positive or negative energy contribution depending on the relative orientation between the spin magnetic moment µ s and the induction field B so , ultimately linking the spin S and orbital L angular momentum for an electron located in the vicinity of a charged nucleus:

B so = E × v c 2 1 -v 2 c 2 , ( 1
H Zeeman = -µ spin • B so . (1.15)
Given some proper relativistic treatment and after replacing the velocity by v = p/m and µ spin and B by their previous definition, eq. (1.15) yields the following general forms of the SOC hamiltonian:

H so = eℏ 4m 2 e c 2 σ • (∇V × p) = eℏ 2 4m 2 e c 2 (σ × k) • E, (1.16
)

where E = -∇V is the electric field caused by the potential V , p = -iℏ∇ is the quantum mechanical momentum operator which eigenvalues in the case of an electron gas are p = ℏk.

ALthough the above model applies for a system with one elctron, the general idea of spin-orbit coupling is that the spins carried by electrons interact with a momentum-dependent effective magnetic field B so (k) caused by an electric field.

It becomes apparent that the spin-orbit coupling is strongly related to the strength of the local electric field felt by electrons. For atomic spin-orbit coupling, although some interactions between electrons exist, the dominant contribution arises from the nuclear electric field with a stronger effect on electrons at smaller distance from the nucleus, but also a strong dependence on the atomic number Z (B so ∝ Z 2 ) with, in general, a higher SOC for heavier compounds.

In solids, in addition to the atomic number of the materials constituents, the main electric field that is considered is the one arising from the periodic potential of the crystal lattice and to impurities and defects and might be modulated in some cases by certain structural or intrinsic electrical properties, or by external electric fields.

Finally, in a similar spirit as for eqs. (1.1) and ( 1.2), we introduce the total angular momentum J = L + S associated to a total magnetic moment µ J .

. Symmetry considerations

The interplay between SOC and the breaking of time reversal T and/or space inversion Î symmetries usually give rise to exotic transport phenomena, some of which will be covered in the next sections. Before that, we provide a brief reminder of the definition of these symmetries and their implications. Time reversal's intuitive meaning is that we halt the system's time evolution and play it backwards. It comes down to the application of the time-reversal operator: T : t → -t, which keeps the spatial coordinates of a particle preserved but reverses its velocity v and its momentum k as well as its spin polarization direction σ (see fig. 1.4b:

k T -→ -k, σ T -→ -σ (or ↑ (↓) T -→↓ (↑)).
(1.17)

In a system that is time-reversal symmetric, for particles with half-integer spin (such as electrons) and given energy eigenstate, the Kramers' theorem requires that there is another eigenstate with equal energy under the time-reversal operation, i.e. with opposite spin and momentum:

ε k,↑(↓) T = ε -k,↓(↑) .
( 1.18) This spin degeneracy is typically lifted in presence of an external magnetic field or in a ferromagnetic material as we end up with one spin population's energy dispersion shifted to lower energy while the other is lifted up due to the Zeeman effect as illustrated in fig. 1.4d. Another phenomenological consequence that results from a magnetic field is the deviation of charged particles due to the Lorentz force that changes sign under time-reversal operation, which means the trajectory is not the same and thus time-reversal T symmetry is broken. On the other hand, space-inversion symmetry implies the presence of an inversion center in the system of interest as one of its symmetry elements and applying the inversion operation noted Î : r → -r also reverses the momentum k while keeping the spin identical:

(a) (b) (d) (c)
k Î -→ -k, ↑ (↓) Î -→↑ (↓).
(1.19

)
One example is given in fig. 1.4a for a centrosymmetric crystal, where for any atoms (or actually any point) at coordinates r = (x, y, z) relative to the inversion center P 0 of the 3D space, there is an equivalent atom (or point with exact same properties) at the opposite side of P 0 in -r = (-x, -y, -z). Because of the eq. (1.19), having a system displaying spatial inversion Î symmetry means that the energy of an electron with momentum k and spin-up ↑ for instance is the same as an electron with opposite momentum -k and same spin orientation, i.e. ↑:

ε k,↑(↓) Î = ε -k,↑(↓) .
(1.20)

. Dresselhaus and Rashba effects

There are usually two ways spatial inversion symmetry can be broken in crystals and heterostructures. The first one is the bulk inversion asymmetry (BIA) which is present when the unit cell of the material itself is not centrosymmetric (as should be the case for semiconductors with zinc-blende crystal structure, for instance), while the second one arises from electric fields typicaly at surfaces and heterostructure interfaces (for instance due to a confining potential as found in 2DEGs) and is called structural inversion asymmetry (SIA). Breaking the space inversion symmetry can lift the spin degeneracy by inducing a SOI and thus lead to either the Dresselhaus effect in the case of BIA or the Bychkov-Rashba type SOI in the case of SIA (although as we will see in section 1.5, bulk Rashba can also exist in bulk polar materials [START_REF] Ishizaka | Giant Rashba-type Spin Splitting in Bulk BiTeI[END_REF]).

Dresselhaus was seemingly the first to recognize this kind of phenomenon in non-centrosymmetric bulk semiconductors of the zinc-blende type [START_REF] Dresselhaus | Spin-Orbit Coupling Effects in Zinc Blende Structures[END_REF] such as GaAs or InSb, where the effective magnetic field due to spin-orbit coupling is dependent on the momentum vector k. In fact, since the SOC alone must preserve the time-reversal symmetry, the spin-orbit field must be odd in electron momentum. As such, the additional Hamiltonian term due to the Dresselhaus effect in 3D-space, and usually derived from a k • p perturbation model, contains odd-powers of the momentum and is notably described by the k 3 -term [START_REF] Fabian | [END_REF][START_REF] Dresselhaus | Spin-Orbit Coupling Effects in Zinc Blende Structures[END_REF][START_REF] Manchon | New Perspectives for Rashba Spin-Orbit Coupling[END_REF][START_REF] Dyakonov | Spin Relaxation of Two-Dimensional Electrons in Noncentrosymmetric Semiconductors[END_REF]:

H D = ℏ 2 β D ℏ 3 2m 3 e E g κ(k) • σ, where κ(k) =   k x k 2 y -k 2 z k y k 2 z -k 2 x k z k 2 x -k 2 y   , (1.21)
where E g is the semiconductor band gap, β D is the Dresselhaus SOC strength and depends on the material, and k i are the momentum components along the main crystal axes.

The other term we are interested in comes from the Bychkov-Rashba effect (also commonly called Rashba effect, for ease of use) which typically arises due to SIA in quantum wells and two-dimensional electron gases and is expressed as [START_REF] Bychkov | Properties of a 2D Gas with Lifted Spectral Degeneracy[END_REF]:

H R = α R (k × ẑ) • σ, (1.22)
where the Rashba coefficient α R is proportional to the amplitude and sign of the interfacial electric field E = E z ẑ, and again can be estimated in semiconductors by a proper k • p perturbation model generally around the Γ point, by a tight-binding approach [START_REF] Fabian | [END_REF][START_REF] Yu | Fundamentals of Semiconductors: Physics and Materials Properties 4th ed[END_REF] or by experiments through weak anti-localization (WAL) for instance [START_REF] Knap | Weak Antilocalization and Spin Precession in Quantum Wells[END_REF][START_REF] Dresselhaus | Observation of Spin Precession in GaAs Inversion Layers Using Antilocalization[END_REF][START_REF] Maekawa | Magnetoresistance in Two-Dimensional Disordered Systems: Effects of Zeeman Splitting and Spin-Orbit Scattering[END_REF][START_REF] Hikami | Spin-Orbit Interaction and Magnetoresistance in the Two Dimensional Random System[END_REF] or Shubnikov-de Haas (SdH) oscillations. The actual electric field governing the strength of the Rashba SOC is in fact the one causing valence band offsets near the interface rather than the electric field in the lowest conduction band [START_REF] Fabian | [END_REF][START_REF] Zawadzki | Spin Splitting of Subband Energies Due to Inversion Asymmetry in Semiconductor Heterostructures[END_REF], although higher conduction bands introducing anisotropic effective mass might also affect α R [START_REF] Lassnig | p� Theory, Effective-Mass Approach, and Spin Splitting for Two-Dimensional Electrons in GaAs-GaAlAs Heterostructures[END_REF]. In summary, interband contributions seem to be determining for both Dresselhaus and Rashba amplitude coefficients.

Incorporating the Rashba term in a nearly free electrons model, i.e. with parabolic bands, gives the following total hamiltonian:

H = ℏ 2 k 2 2m * + α R (k × ẑ) • σ, (1.23)
with k-linear eigenvalues:

ε k,↑(↓) = ℏ 2 k 2 2m * ± α R |k|, (1.24)
where ↑ (↓) spin quantization axis is dependent on k.

The first consequence of the Rashba effect is that it splits the spin-up and spin-down energy dispersions in opposite directions, not along the energy axis but along the k direction (in the three dimensions), and by an amount k R = m * α R /ℏ 2 so that the energy dispersion reaches a minimum along a circle of radius k R shifted down by an energy

ε R = ε(k = k R ) = α R m *
2ℏ 2 with respect to the crossing at k = 0. At k = k R , the total energy difference between the spin-"up" ε + and spin-"down" ε -dispersions is maximal and amounts to:

∆ε max = 2α R k R = 2m * α 2 R ℏ , (1.25)
where:

α R = ε R 2k R (1.26)
Although at the band bottom we still have only one contour, if the chemical potential level is above the crossing point between ε + and ε -, an energy cut at this energy actually shows a division of the Fermi surface in two inner and outer contours. Second, similar to the Dresselhaus effect, the Rashba effect also causes spin-momentum locking with the spins σ in the plane of the 2DEG, i.e. the x, y-plane, only this time with the condition that σ ⊥ k as can be understood from eq. (1.23) and as we see from the Fermi surface in fig. 1.6. Both contours have well-defined spin-momentum locking with opposite chirality and the average spin density over the Brillouin zone (BZ) is zero as opposed to the case of Zeeman effect in ferromagnets. As an addition, this chirality can be modulated or even inverted by applying an external electric field via the Rashba coefficient α R which will be useful for spinorbitronics applications, in which gate voltage can control spin-charge interconversion in a device, as we will see in section 1.1.5 and in chapter 3. Finally, it should be noted that both Dresselhaus and Rashba effect can coexist in the same system. An extensive analysis and useful information on both effects can be found in the works of Fabian et al. [39] and Ganichev et al. [62]. The presence of both effects at the same time makes it more difficult to extract one or the other parameter of α R and β D since they both lead to spin-momentum locking but usually do not have the same k-dependence of the spin texture, so that especially when they are of similar amplitudes, the spin-splitting can vanish in some k directions and lead to a modification or a canceling out of some macroscopic phenomena [START_REF] Ganichev | Experimental Separation of Rashba and Dresselhaus Spin Splittings in Semiconductor Quantum Wells[END_REF] like the lack of Shubnikov-de Haas (SdH) beats [START_REF] Averkiev | Suppression of Spin Beats in Magneto-Oscillation Phenomena in Two-Dimensional Electron Gas[END_REF], spin relaxation anisotropy [START_REF] Averkiev | Giant Spin Relaxation Anisotropy in Zinc-Blende Heterostructures[END_REF][START_REF] Averkiev | Spin Relaxation Anisotropy in Two-Dimensional Semiconductor Systems[END_REF] or WAL disappearance [START_REF] Knap | Weak Antilocalization and Spin Precession in Quantum Wells[END_REF]. The Dresselhaus effect is often ignored when one tries to extract the (supposedly) dominating Rashba term in interfacial systems [START_REF] Maekawa | Spin Current isbn[END_REF], although some experiments also allow under certain conditions to disentangle both contributions to the spinorbit interaction.

k E k R ∆E E F E R (a) k y k x (b) k y k x (c)
The Dresselhaus effect can also be reduced to its k-linear term for instance in the presence of strain along the (001) direction or in the case of a two-dimensional electron gas, by averaging over the ẑ-direction considering small values of k and thus neglecting higher-order terms, so that the spin-orbit hamiltonian in the 2DEG becomes:

H so = α R (k x σ y -k y σ x ) + β D (k x σ x -k y σ y ), (1.27)
Similarly, a k-cubic Rashba term was observed in hole-doped semiconductor quantum wells [START_REF] Winkler | Rashba Spin Splitting in Two-Dimensional Electron and Hole Systems[END_REF][START_REF] Winkler | Spin Orientation of Holes in Quantum Wells[END_REF][START_REF] Moriya | Cubic Rashba Spin-Orbit Interaction of a Two-Dimensional Hole Gas in a Strained-Ge / SiGe Quantum Well[END_REF], in the ferroelectric bulk Rashba semiconductor GeTe [START_REF] Di Sante | Electric Control of the Giant Rashba Effect in Bulk GeTe[END_REF], but also interestingly in Sr-TiO 3 Ti-t 2g surface states through magnetoresistance measurements particularly for the higher lying, anisotropic and heavier d xz/yz bands [START_REF] Nakamura | Experimental Evidence of Cubic Rashba Effect in an Inversion-Symmetric Oxide[END_REF][START_REF] Shanavas | Theoretical Study of the Cubic Rashba Effect at the SrTiO 3 (001) Surfaces[END_REF].

. Consequences of spin-orbit coupling on transport properties

In general, scattering events correspond to a redefinition of a carrier's momentum direction only (elastic scattering, with frequency τ -1 el ) and/or of its amplitude (in which case we call it inelastic scattering, with frequency τ -1 in ) and as we have seen, loss of spin information (spin-flip events) might also occur at random times, with frequency τ -1 sf .

Spin relaxation phenomena

In absence of any magnetic field, the spin population of a non-magnetic material tends to relax back to an equilibrium state with no particular ordering and an average zero spin polarization P n , with the characteristic spin relaxation time τ s . It is of paramount importance when controlling spin currents, to understand how far a spin population can propagate in a material before it gets depolarized, as we saw in section 1.1.2, and we have expressed this distance as the spin diffusion length.

Without entering in too many details, different scenarios have been proposed to explain how this relaxation phenomena could occur for instance at a ferromagnetic/non-magnetic interface. In the general case, we distinguish four important mechanisms that can lead to spin relaxation:

(i) The Elliott-Yafet mechanism, in which there is a small probability that the spin flips upon a momentum scattering event off impurities or phonons. As a result, the spin relaxation time is simply proportional to the momentum relaxation time τ sf ∼ τ .

(ii) The D'yakonov-Perel mechanism arises in systems lacking spatial inversion symmetry, as a consequence of spin-orbit coupling. As we saw in the previous section, the spin momentum locking manifests itself as an effective k-dependent magnetic field B so (k) which forces the spin to realign after some time. In fact, the carrier's spin precesses around that field with a Larmor frequency Ω(k) = ( ege 2me )B so (k) until the carrier itself gets scattered and changes its momentum to k ′ which changes the axis and frequency of precession to the new Ω(k ′ ). Counter-intuitively, since the spin relaxes between momentum scattering events, the more often these events occur, the less time the spin will have to change its orientation along the new field every time, resulting in longer time before the spin orientation information is lost (i.e. longer spin relaxation time).

(iii) The Bir-Aronov-Pikus mechanism is mostly dominant in p-doped semiconductors due to exchange interaction between the electrons and holes which causes a spin-flip of either particles. The total spin is preserved in this exchange process, however a strong spin-orbit coupling affecting the valence band may result in faster spin relaxation of holes with the Eliott-Yafet mechanism.

(iv) When nuclear spins are present, the hyperfine interactions may also cause spin flips, which can usually be observed in semiconductors at low temperatures.

For a more complete understanding and derivation of these phenomena, we invite the reader to refer to Fabian et al. [START_REF] Fabian | [END_REF] and Dyakonov [71].

In presence of spin-orbit coupling, not only can there be different mechanims modifying the spin during or in between scattering events, but the change in momentum direction can also be affected by the direction of the incoming (charge) carrier's spin.

Anomalous Hall effect

As we will see in following chapters, the Hall effect discovered in 1879 by Hall [START_REF] Hall | On a New Action of the Magnet on Electric Currents[END_REF] has had a very important impact until today. In this effect, the Lorentz force felt by carriers of charge q subject to an out-of-plane magnetic field B = µ 0 H z , and moving at a velocity v under the electric field E, causes a deflection due to the magnetic field:

F L = q(E + v × B).
The accumulation of charges at opposite edges of a sample causes a counteracting electromotive force in response and as we will see in section 2.4.1, the resulting voltage V H proportional B z allows to retrieve the carrier density in the sample with remarkable simplicity.

Hall [START_REF] Hall | On the "Rotational Coefficient" in Nickel and Cobalt. The London, Edinburgh[END_REF] reported two years later that a similar effect could be observed in ferromagnets (Co and Ni) but with an amplitude ten times what could be measured in nonmagnetic conductors, an effect later coined anomalous Hall effect (AHE).

Actually, while in nonmagnetic metals the Hall (transverse) resistance simply exhibits a linear relationship with the applied magnetic field (see fig. 1.7b), what one can observe for the case of a magnetic material is a strong slope at low field, and once some critical field is exceeded, the Hall curve returns to the expected linear trend of the ordinary Hall effect (see fig. 1.7d). After it was evidenced that this effect could be correlated to the magnetization, an empirical relationship was suggested to link the transverse resistivity to the applied magnetic field strength B = H z and the magnetization M z [START_REF] Pugh | Hall Effect and the Magnetic Properties of Some Ferromagnetic Materials[END_REF][START_REF] Pugh | and Intensity of Magnetization[END_REF]: (1.28) in which the first term represents the well established Hall effect while the second term is the anomalous Hall effect (AHE) term proprortional to the magnetization of the sample. What this equation tells is that in a ferromagnetic material, a finite transverse resistance exists even in absence of an external magnetic field due to this imbalance in the spin population.

ρ xy = R 0 H z + R A M z ,
It took a lot of time and extensive theoretical and experimental efforts to understand and formalize the exact origins and mechanisms that could lead to the anomalous Hall effect (AHE). In 1954, [START_REF] Karplus | Effect in Ferromagnetics[END_REF] suggested that in presence of spin-orbit coupling in a ferromagnetic conductor subject to an electric field, electrons could acquire an additional contribution to their group velocity. The anomalous velocity perpendicular to the applied electric field and summed over all occupied band states could be nonvanishing, implying a contribution to the Hall conductivity In the case of the ordinary Hall effect, the transverse resistance is continuously linear in magnetic field, while for the anomalous Hall effect, a low-field regime with a strong slope appears, with saturation beyond some value of magnetic field. Reused with permission from [START_REF] Vaz | Spin-to-Charge Current Conversion in SrTiO3-based Two-Dimensional Electron Gases[END_REF][START_REF] Vicente-Arche | Quantum Oxide Heterostructures for Spin-Charge Interconversion[END_REF].

σ xy . While this theoretical concept helped significantly in the understanding of the AHE, it had the flaw of ignoring scattering-related contributions which had been observed in some systems throughout the large datasets acquired over time and a lot of controversy surrounded the microscopic mechanism of the phenomenon [START_REF] Nagaosa | Anomalous Hall Effect[END_REF].

Nowadays, the AHE (and the related spin Hall effect (SHE) explained in the next section) is usually divided into three contributions: the intrinsic contribution as described by [START_REF] Karplus | Effect in Ferromagnetics[END_REF] (see fig. 1.8c), and the extrinsic contributions from the skew scattering [START_REF] Smit | The Spontaneous Hall Effect in Ferromagnetics I[END_REF][START_REF] Smit | The Spontaneous Hall Effect in Ferromagnetics II[END_REF] and side-jump scattering [START_REF] Berger | Side-Jump Mechanism for the Hall Effect of Ferromagnets[END_REF] (see figs. 1.8a and 1.8b respectively). The latter two describe an asymmetric scattering amplitude for either spin-up or spin-down electrons and have different dependence on the concentration of scattering impurities which can help to disentangle one from the other.

The skew scattering mechanism is related to the asymmetric scattering due to SOC on the impurity, leading to a change in the wave-vector direction as the electrons depart from the impurity in different directions, depending on the incoming electron's spin. Instead, for the side-jump scattering, a lateral displacement of the electron occurs without any change in the final momentum vector. In the limit of a system clean from defects and impurities, the intrinsic contribution becomes dominant. This is the case initially described by the ideal model of Karplus & Luttinger and extended much later with the concept of Berry curvature.

In a semiclassical picture of transport, electrons in the crystal are considered as wavepackets, centered in r and with momentum ℏk. Concretely, they obey the following equations of motion:

   ṙ = 1 ℏ ∂ε n (k) ∂k -k × Ω n (k) ℏ k = -e(E -ṙ × B) (1.29)
The second line of this system of equations shows simply the effect of the Lorentz force, when the carriers are subject to an electric field E and a magnetic field B, while the first line is the group velocity. This group velocity contains an additional component (the anomalous velocity) with respect to the first kinetic term, orthogonal to the external force (here, the Lorentz force) and related to the Berry curvature Ω n (k) associated with the n-th band (in a single band approximation, i.e. neglecting interband coupling). This added contribution acts similarly to a magnetic field in the momentum space and it derives directly from the band structure, which is why it is intrinsic to the material. In a periodic crystal the Bloch electrons are characterized by the periodic part of the eigenstates |u n,k ⟩ for each band of index n, and the Berry curvature can be expressed as [START_REF] Berry | Quantal Phase Factors Accompanying Adiabatic Changes. Proceedings of the Royal Society of London[END_REF][START_REF] Xiao | Berry Phase Effects on Electronic Properties[END_REF]:

Ω n (k) = i ∂u n,k ∂k × ∂u n,k ∂k .
(1.30)

In a material with a large spin-orbit splitting like Pt or in a Rashba system, the Berry curvature is spin-dependent Ω(k) ∼ α R σ, where α R the Rashba coefficient is a measure of the SOC strength.

As a consequence, the anomalous Hall velocity takes opposite sign for the spin up or spin down electrons. In a ferromagnetic medium, because there is a spontaneous imbalance of the spin population, more majority spin carriers (electrons) at the Fermi level will be deviated in one direction than minority spin carriers, leading to a net charge accumulation at either side of the sample and thus, adding up to the ordinary Hall effect. As we shall see in section 1.1.5, in a nonmagnetic system, the two spin populations are equal so that a transverse pure spin current is formed (SHE).

Introducing the Berry phase and Berry curvature

We have briefly exposed one form of the Berry curvature, applicable for the case of the AHE. The formalism proposed by Berry [82] relies on few key concepts which we will try to decipher in this small section, although we also invite the reader to refer to other sources in the literature for a more complete and thorough derivation of each concept [START_REF] Berry | Quantal Phase Factors Accompanying Adiabatic Changes. Proceedings of the Royal Society of London[END_REF][START_REF] Xiao | Berry Phase Effects on Electronic Properties[END_REF][START_REF] Bohm | The Geometric Phase in Quantum Systems[END_REF].

Three quantities are often used within this formalism: the Berry phase, Berry connection and Berry curvature, the first and the latter being actually linked to measurable physical quantities.

We start by considering a physical system described by a Hamiltonian H which depends on a set of parameters denoted R which can evolve with time, i.e.:

H = H(R) and R = R(t).
(1.31)

We will try to observe some properties of that system as it evolves slowly (i.e. it follows an adiabatic evolution), or in other words as R(t) moves slowly along a path C in the parameter space. Solving the Schrödinger equation with the hamiltonian H will give us an orthonomal basis from the eigenstates |φ n (R)⟩ with energy ε n (R):

H(R) |φ n (R)⟩ = ε n (R) |φ n (R)⟩ . (1.32)
In the adiabatic approximation, each eigenstate |φ n (R(0))⟩ will remain an eigenstate of the hamiltonian at time t, H(R(t)) throughout the process, up to a phase factor chosen to be single-valued along the path C.

The quantum state at time t is given as:

|ψ n (t)⟩ = e iγn(t) e [-i ℏ ´t 0 dt ′ εn(R(t ′ ))] |φ n (R)⟩ , (1.33)
where we can identify the second exponential as the dynamical phase factor, while the first term can be isolated by inserting eq. (1.33) into the time-dependent Schrödinger equation and multiplying from the left by ⟨φ n (R)|:

γ n = ˆC dR • A n (R), (1.34)
where A n (R) is a vector potential and is called the Berry connection:

A n (R) = i ⟨φ n (R)| ∂ ∂R |φ n (R)⟩ . (1.35)
Now if we consider that the adiabatic process corresponds to the evolution of the parameter R around a closed path, γ n is then well-defined modulo an integer multiple of 2π and is a geometric phase actually called the Berry phase. It is now given by: (1.36) Finally, if the parameter space is three-dimensional, we can define a general form of the Berry curvature, a vector field derived from the vector potential A n (R):

γ n = ˛C dR • A n (R),
Ω n (R) = ∇ R × A n (R).
(1.37)

We note that, according to the Stokes theorem, the Berry phase eq. (1.36) can be re-written as an integral over the surface S enclosed by the path C, and where we can also incorporate the Berry curvature instead of the Berry connection:

γ n = ˆS dS • Ω n (R). (1.38)
Things get interesting for us when we try to apply these general concepts to particular cases, for instance in a periodic crystal where various Berry phase effects can be expected. In such a case, we transpose the general parameter space to the momentum space within a Brillouin zone (R becomes k), and use the basis defined by the Bloch wavefunctions (we can actually recover the egeinstates as defined earlier |u n,k ⟩, i.e. the cell-periodic part of the Bloch function). If the crystal momentum k is forced to vary in the momentum space, then the Bloch state should pick up a Berry phase:

γ n = ˛C dk • ⟨u n,k | ∂ ∂k |u n,k ⟩ , (1.39)
while the Berry curvature will recover the same form as we introduced in eq. (1.30) for the AHE. It is an intrinsic property of the band structure and provides very powerful way to describe the dynamics of Bloch electrons, especially in systems that break either or both of the spatial inversion and time-reversal symmetries. The Berry curvature is a central concept used as a basis tool to describe many different phenomena, not only to compute the AHE but also the quantum Hall effect, the valley Hall effect, orbital magnetization, etc. and importantly it has been used by Resta [START_REF] Resta | Theory of the Electric Polarization in Crystals[END_REF] and to define the modern theory of polarization, actually based on the adiabatic change of polarization over time (i.e. a displacement of ions within the unit cell), particularly representative of the case of ferroelectric switching, which we will cover in section 1.2. It is also used fundamentally for topological systems (magnetic skyrmions and other peculiar spin textures, for instance), either directly or indirectly through the related concept of Chern number.

Finally, as we mentioned before, this tool may also be useful to describe spin-charge interconversion phenomena such as the spin Hall effect (SHE) or the Rashba-Edelstein effect (REE)

. Spin-charge interconversion phenomena

In section 1.1.2, where we have derived charge and spin drift-diffusion equations eqs. (1.6b) and (1.10b), the situation we have described is valid only without considering any coupling between these two quantities. The definition of spin-charge interconversion (SCI) encompasses all phenomena that convert a charge current into a spin current or vice versa. The presence of spin-orbit coupling (SOC) is the key ingredient for spin-charge interconversion (SCI) since it generates an asymmetry in the transport of carriers depending on their spin polarization.

In fact, we need to introduce some additional terms in each of the two equations simply to account for the spin-orbit coupling related phenomena, which in the case of a three-dimensional system can result in the phenomenological equations [71]:

j c = q (µnE + D∇n + γ so µE × s + γ so D∇ × s) (1.40a) j s ij = q µs j E i + D ∂s j ∂x i + ϵ ijk γ so µnE k + γ so D ∂n ∂x k , (1.40b)
where γ so is a dimensionless parameter proportional to the spin-orbit interaction, ϵ ijk is the unit antisymmetric tensor, x k defines a generic direction in space and the other parameters are as we defined them in section 1.1.2. While these equations are written for the three-dimensional case, their principle are also transposable to the two-dimensional case if we modify the electric field, space gradients and all currents to have only components lying in the 2D plane. In 1.40a, the term γ so µE × s expresses how a charge current induced by an electric field couples to the spin density and gives rise to a transverse charge flow (some kind of Hall current). For instance, this term can explain the appearance of the anomalous Hall effect (AHE) in ferromagnets, which indeed is proportional to the magnetization but cannot be solely explained by it, and that is where the SOC term plays a role. Non-equilibrium spin-polarization can also be produced by optical means as was shown by [START_REF] Miah | Observation of the Anomalous Hall Effect in GaAs[END_REF] in GaAs illuminated with circularly polarized light, or simply by spin injection. The term γ so D∇ × s simply describes an electric current generated by an inhomogeneous spin density, demonstrated for the first time by Bakun et al. [88] in Ga x Al 1-x As and later called inverse spin Hall effect (ISHE).

Conversely, a spin current can originate from a transverse drift (γ so µnE k ) or diffusion (γ so D ∂n/∂x k ) charge currents via the SOC. This phenomenon is called the spin Hall effect (SHE) and was originally predicted by Dyakonov & Perel [89] in 1971 and observed experimentally more than 30 years later by Kato et al. [90] using Kerr rotation microscopy.

Direct and inverse spin Hall effect

The SHE is the generation of a pure spin current j s resulting from the flow of a charge current j c in a nonmagnetic material having significant SOC. This spin current is perpendicular to both j c and the direction of the spin polarization of carriers ûs , and can be expressed from the term j s ij = qϵ ijk γ so µnE k in eq. (1.40b), neglecting the spin and charge inhomogeneities in the sample as well as the second order term qµs j E i since the only spin density is proportional to the electric field itself: j s,SHE = θ SHE j c × ûs .

(1.41)

The Onsager reciprocal inverse spin Hall effect (ISHE) allows to convert a spin current into a charge current: j c,ISHE = θ SHE j s × ûs .

(1.42)

The conversion efficiency here is given by the spin Hall angle θ SHE which is proportional to the SOC constant γ so . The SHE is often described by the spin Hall conductivity tensor σ k ij (where i, j and k indices can take values x, y and z) as, for instance: which, considering that the charge current can simply be obtained by Ohm's law j c x = σ xx E x , allows us to express the spin Hall angle as a ratio between the spin Hall conductivity and the electrical (longitudinal) conductivity σ xx :

σ z xy = - j yz E x , (1.43)
θ SHE = σ z xy σ xx .
(1.44)

The value of the spin Hall angle is indeed dimensionless since we defined both charge and spin currents in the same units of a charge flow density and the efficiency of conversion will be particularly high in heavy metals (e.g. Ta, Pt, ...) and alloys thereof. Conversion between charge and spin currents via the spin Hall effect has also been observed in semiconductors, such as GaAs [START_REF] Kato | Observation of the Spin Hall Effect in Semiconductors[END_REF][START_REF] Zhao | Coherence Control of Hall Charge and Spin Currents[END_REF], ZnSe [START_REF] Stern | Current-Induced Polarization and the Spin Hall Effect at Room Temperature[END_REF], and in 2DEGs such as the AlGaAs(110) quantum wells [START_REF] Sih | Spatial Imaging of the Spin Hall Effect and Current-Induced Polarization in Two-Dimensional Electron Gases[END_REF][START_REF] Wunderlich | Experimental Observation of the Spin-Hall Effect in a Two-Dimensional Spin-Orbit Coupled Semiconductor System[END_REF].

As for the AHE, the SHE can be of extrinsic or intrinsic origin and gives rise to an accumulation of spins of opposite orientation at the opposite edges of the sample, except that the resulting Hall voltage is zero. In the intrinsic case, it is caused by the anomalous velocity of the electrons as formalized previously through the Berry curvature. In a Rashba system, the latter is a k-dependent effective magnetic field, since the spin-quantization axis is always perpendicular to k. The generation of a spin current from the Rashba magnetic field is explained in [START_REF] Sinova | Universal Intrinsic Spin Hall Effect[END_REF]. As shown in fig. 1.10, when injecting a charge current (applying an electric field -E x ) in the system, after a time interval t 0 (much shorter than the momentum relaxation time τ m ), the circular Fermi contour is shifted from the origin in momentum space by the amount |eE x t 0 |/ℏ. As they move in momentum space, electrons are subject to an effective torque and their spins acquire an additinal component along +z or -z depending if their k y momentum component is positive or negative respectively. In this way, a spin current polarized along the z axis is created along the y direction.

We will now see another spin-charge interconversion phenomenon relevant to Rashba systems, namely the (Rashba-)Edelstein effect.

Direct and inverse Edelstein effect

Systems displaying a locking of the spin of carriers with their momentum like Rashba systems allow another type of spin-to-charge conversion mechanism to occur: the so-called spin galvanic effect [START_REF] Ganichev | Spin-Galvanic Effect[END_REF].

The spin-galvanic effect was first observed in 2001 in GaAs quantum wells [START_REF] Ganichev | Spin-Galvanic Effect[END_REF][START_REF] Ganichev | Conversion of Spin into Directed Electric Current in Quantum Wells[END_REF] but the first work that "introduced and quantitatively characterized" [START_REF] Shen | Microscopic Theory of the Inverse Edelstein Effect[END_REF] the effect was published by Rojas Sánchez et al. [START_REF] Rojas Sánchez | Spin-to-Charge Conversion Using Rashba Coupling at the Interface between Non-Magnetic Materials[END_REF] only 10 years later. It is more recently named inverse Edelstein effect (IEE) in reference to its Onsager-reciprocal effect, the direct Edelstein effect (DEE), a phenomenon predicted by Edelstein [100] by which a non-equilibrium spin polarization can be generated when applying a charge current.

Let us work with an example situation where we apply an electric-field across a typical Rashba system (e.g. a 2DEG) in the xy-plane, along the -x direction E = -E x ûx to obtain a current flow j c = σE.

We recall from section 1.1.3.2 that the Fermi surface of a Rashba system actually comprises two concentric contours with opposite winding chirality of the spins along the inner vs outer contours, so that the spins σ are always orthogonal to the momentum vector k in one way or another. We note the inner and outer Rashba contours with aor + sign, each with a corresponding radius (or Fermi wavevector length) k F ± such that the difference between both contours can be related to the Rashba coefficient α R by: ∆k F = 2m * α R ℏ 2 . In a semiclassical picture of transport, due to the electric field, both contours shift in the same direction by an amount ∆k x ≈ ∆k x± = eE x τ ± /ℏ (τ ± being the respective scattering time of each contour), yielding two excess spin densities δs -and δs + for the inner and outer contours respectively, with opposite spin-polarization along y, as illustrated on fig. 1.11a. In a simplified nearly-free electron model, we observe that each of these two spin densities are not equivalent since the respective Fermi surfaces (roughly proportional to k F ± ∆k x± ) are not the same, and each of them can be expressed as [101]: 1.45) where m * is the effective mass of the electrons, j c ± is the charge current hosted by each contour such that .46) and ûy is the unit vector along y, representing the spin polarization axis orthogonal to the charge current density and the vector normal to the symmetry-breaking electric field, i.e. the out-of-plane vector ẑ. Both spin densities cannot compensate for each other and it results in a net spin density, uniform across the area of the sample (where the current flows): 1.47) where ε F is the Fermi energy. This spin density can of course diffuse over a certain distance in adjacent layers and away from the stimulated current channel, following the drift-diffusion and continuity equations as defined in this section or section 1.1.2. Conversely, when injecting vertically a uniform non-equilibrium density of spins polarized along y, the occupied density of states for the corresponding spin orientation increases with respect to the other, which can be essentially viewed as a shift of the inner and outer contours in opposite ±k x directions (see fig. 1.11b). Because the outer contour is of course larger by definition, not only are there more states "hosting" the injected spin carriers but the smaller k-vector of carriers in the inner contour cannot compensate for that of the carriers from the outer contour. In the end, a net displacement of carriers in the direction of the majority spin carriers (transverse to the spin direction) develops, i.e we create a charge current [START_REF] Rojas Sánchez | Spin-to-Charge Conversion Using Rashba Coupling at the Interface between Non-Magnetic Materials[END_REF]:

δs ± = ± m * eℏk F j c ± ûy , (
j c = j c + + j c -≈ eℏk 2 F 2πm * ∆k x , (1
⟨δs⟩ = α R m * 2πℏ ∆k x ûy = α R m * 2eε F j c ûy , (
j c ≈ - eα R ℏ ⟨δs⟩ ûx .
(1.48)

For practical analysis of experimental results and comparisons between the direct and inverse Edelstein as well as the (inverse) spin Hall effect, it is also convenient to relate the charge current density j c , with the spin density ⟨δs⟩ and the spin current j s . For instance in a spin-pumping experiments (see section 2.5), the non-equilibrium spin density ⟨δs⟩ generated in the 2D Rashba system is the result of a spin current injection j s and we can relate them by [START_REF] Rojas Sánchez | Spin-to-Charge Conversion Using Rashba Coupling at the Interface between Non-Magnetic Materials[END_REF]:

j s e = ⟨δs⟩ τ s , (1.49)
where τ s reflects the balance between spin injection and the interfacial spin density and is an effective spin-momentum relaxaton time in the strong spin-orbit coupling regime or equivalent to the momentum relaxation time in the weak SOC regime [START_REF] Shen | Microscopic Theory of the Inverse Edelstein Effect[END_REF][START_REF] Rojas Sánchez | Spin-to-Charge Conversion Using Rashba Coupling at the Interface between Non-Magnetic Materials[END_REF]. Combining eqs. (1.48) and (1.49), we can finally obtain the relation between the charge current and spin current:

j c = λ IEE j s , where λ IEE = α R τ s ℏ , (1.50)
where j s is the 3D spin current injected along -z from the ferromagnetic overlayer, in A m -2 , j c is the 2D charge current generated within the interface plane in A m -1 and thus λ IEE has the dimension of a length expressed in m as opposite to the dimensionless spin Hall angle θ SHE . Note that in the intrinsic limit, the relaxation time in this equation becomes the momentum relaxation time τ as argued by Shen et al. [98]. Finally, proposed a way to compare (I)SHE efficiencies in 3D systems with (I/D)EE in 2D systems, by calculating an inverse spin Hall effect length as:

λ ISHE = θ SHE λ sf , (1.51)
where λ sf is the spin diffusion length defined earlier.

For a more complete derivation of the equations of the charge and spin current in this section, one can refer to refs [START_REF] Zucchetti | Spin-Charge Interconversion in Ge-based Structures[END_REF][START_REF] Rojas Sánchez | Spin-to-Charge Conversion Using Rashba Coupling at the Interface between Non-Magnetic Materials[END_REF][101][102][103].

Both Rashba-Edelstein and spin Hall effects play a major role in spintronics, as they allow for an electric control of the spin-charge interconversion, with promise to redefine new computing paradigms. As we will see in chapter 3, the possibility of controlling the sign and amplitude of the spin-to-charge conversion by different methods has been demonstrated in different systems under the application of a gate voltage, either via the Edelstein effect as in LaAlO 3 /SrTiO 3 2DEGs [START_REF] Lesne | Highly Efficient and Tunable Spin-to-Charge Conversion through Rashba Coupling at Oxide Interfaces[END_REF], and in topological insulators, or via the spin Hall effect in graphene [104] or WS 2 / graphene heterostructures [105]. In this thesis work, we focus on the possibility to not only control the spin-charge interconversion with a gate voltage in a volatile way, but also to add remanence to the system utilizing ferroelectricity.

. Ferroelectricity

. Ferroic orders

Ferroic materials are systems that show phase transitions upon which a spontaneous longrange ordering of some physical property (the order parameter) emerges, potentially causing breaking of either or both of the time reversal T and space inversion Î symmetries. Typically, these phase transitions occur in the temperature dependence at a specific material-dependent temperature, named Curie temperature and denoted T C . The directional order parameter, usually accompanied by a lowering of the point group symmetry of the crystal, will have at least two low energy states, ideally degenerate. A uniform long-range ordering will be seen across regions that we also call ferroic domains while their boundaries form domain walls with different geometries of transition from one domain to the next. These domains can be expanded by progressively switching the order parameter of all domains towards the same direction under the application of an external conjugate field and the dynamic response of domain nucleation and expansion (or domain wall movement) follow different regimes depending on the amplitude and dynamics of the applied field itself.

The switching between two states is a hysteresic process, in that it requires the conjugate field to overcome a certain energy barrier for the order parameter to switch from its initial stable position. Once all domains propagate through the crystal and coalesce so that the order parameter is oriented the same way in the entire material, i.e. for a high enough field, the ferroic order is said to have reached saturation.

One property that makes ferroic materials particularly interesting and largely exploited in technological applications is to retain a finite remanent ordering even after removing the applied conjugate field, and that is dependent on the sample's history, which is ideal for information storage in memory devices.

The four classes of ferroic orders are ferroelasticity (spontaneous and cyclable strain ϵ under the application of a stress σ), ferroelectricity (sponteanous and cyclable electric polarization P responding to electric fields E), ferromagnetism (magnetic ordering M responding to magnetic fields H) and ferrotoroidicity (ferrotoroidic order T switchable by ferrotoroidic field S).

Antiferroic ordering may also exist and consists in the ordering of one ordering parameter coexisting on different intertwinned sublattices with opposite orientation, leading to a compensation of the macroscopic measurable order parameter. Imperfect compensation of one lattice by the other may also give rise to what is called ferri(-electric or -magnetic) orrdering, where finally the macroscopic value of the order parameter is non-zero.

. Phenomenology of ferroelectrics

While classical dielectrics only show a polarizability with electric field that vanishes to zero when the electric field is removed, ferroelectric materials display spontaneous ordered electrical dipole moments which macroscopically give a finite electric polarization P, even without external electric field.

They are a result of polar displacements, i.e. relative displacements of cations (positively charged) with respect to anions (negatively charged) creating a net electric dipole moment at the scale of a unit cell, which by definition breaks bulk inversion symmetry of the crystal: it is then polar and thus, non-centrosymmetric. One prototypical example in the family of ferroelectric oxide perovskite (ABO 3 ) is PbZr x Ti 1-x O 3 (or PZT), shown on fig. 1.13, where a tetragonal distortion of the unit cell leads to the displacement of the B cations (Zr 4+ and Ti 4+ ) in their octahedral cage of negative O 2-ions. This leads to high values of polarization measured in PZT, with typically 40 µC cm -2 at room temperature for a concentration of Zr of x = 0.52 [107].

It should be noted that noncentrosymmetry of a crystal is not a sufficient condition for the presence of polar moments and all the more so for ferroelectricity. In addition, crystals exhibiting ferroelectricity also belong to the different subclasses of dielectrics that include in decreasing hierarchical order: the piezoelectrics (coupling between volatile polarization and applied stress), the pyroelectrics (a spontaneous polarization varies upon temperature changes), and the ferroelectrics.

A stronger requirement for ferroelectrics compared to other classes of polar materials is that the polarization P must be reversible under the application of the conjugate (electric) field E, as for any other ferroic orders. As such, they will also follow a typical hysteretic curve when measuring the polarization vs. applied electric field (P (E) loops) as we will detail in section 2.6.

. Theoretical frameworks and formalisms

As stated previously, ferroic materials exhibit a non-linear hysteretic response to some applied electric field when they are in the right phase. A counterpart of the FM and ferroelectric Tetragonal distortion of the lead zirconium-titanate (or PZT, for PbZr x Ti 1-x O 3 ) oxide perovskite's unit cell acoompanied by a polar displacement of the Zr 4+ /Ti 4+ ions with respect to the oxygen octahedral cage, below the Curie temperature T C . (c) Ferroelectric hysteretic loop of the polarization P versus an applied electric field E. In the initial state, average domain configuration is random and polarization starts to increase and then saturates. While sweeping the electric field in either direction, the polarization switches from one state to the other, for instance P up to P down when exceeding the coercive field E c and then goes to saturation P s . If the field is removed, a non-zero remanent polarization P r remains. Adapted from [108] (FE) phases is the behavior of paramagnetics or paraelectrics, where the order parameter (M or P) responds linearly to the external stimuli (H or E). One parameter that is considered to relate this field-dependence is the electric (or magnetic) susceptibility χ of a material. For insulating dielectric materials, the permittivity (or dielectric constant) ϵ of a material is a more convenient variable which can actually be measured and allows to describe the response of the polarization to an applied electric field via the following relation: (1.52) where χ e = ϵ r -1 is the electric susceptibility, ϵ 0 is the vacuum permittivity and ϵ r is the relative permittivity. The total permittivity of the medium is noted ϵ = ϵ 0 ϵ r .

P = ϵ 0 χ e E,
By classical electrodynamics considerations, the application of an electric field is responsible for the displacement of charges in a medium, whether these are free charges (typically electrons or holes for metallic systems) or bound charges (typically charges associated to the creation of a dipole moment, i.e. a net polarization). Note that the displacement of these charges allows to compensate the applied field and reduce the actual field inside the insulating medium: the higher the susceptibility, the better a material can polarize itself to reduce its internal field, allowing as well to store more charges at its surfaces.

For a simple dielectric, the displacement field is described as:

D = ϵ 0 E + P = ϵ 0 (1 + χ e )E = ϵE.
(1.53)

It should be noted that although we mainly considered scalar isotropic susceptibility, it is more generally a tensor for anisotropic materials. In addition, the polarization of real materials is a causal phenomenon which may have a nearly instantaneous or actually a slower response to the applied field, and thus the susceptibility and permittivity are complex functions of the field's frequency ω: ϵ(ω) = ϵ ′ (ω) + jϵ ′′ (ω), (1.54) where ϵ ′ and ϵ ′′ are the real and imaginary parts of the dielectric function respectively. The imaginary part is responsible for what is called dieletric losses and assimilated to dispersion of the electric (or electromagnetic) stimulation or simply to conduction in a medium. Furthermore, as we come to consider non-linearities of the polarization response under an applied electric field such as saturation or ferroelectricity, the polarization is sometimes expressed as a Taylor expansion with respect to the field: (1.55) where P 0 is a spontaneous polarization and is non-zero for ferroelectric materials and χ (n) are the n-th order components of the susceptibility tensor.

P = P 0 + ϵ 0 χ (1) E + ϵ 0 χ (2) E 2 + ϵ 0 χ (3) E 3 + . . . ,
Different formalisms have been devised to describe the phenomenology of ferroelectrics and specifically the phase transitions between paraelectric to ferroelectric, and some may be more relevant depending on the specific properties of a material one is looking at. Although ferroelectricity and ferromagnetism have significant differences due to the nature of their underlying interactions, one first and simple approach is to make an analogy with paramagneticferromagnetic phase transitions. [109] As the thermodynamic conditions are varied, and especially as the temperature is decreased, so as to approach the transition boundary to the ferroelectric phase (or other ferroic orders), these materials are known to exhibit higher and higher response to the applied field. The meanfield Curie-Weiss model initially describing the paramagnetic to ferromagnetic transitions, was then transposed to paralectric-ferroelectric phase transitions. As T is decreased towards the so-called Curie temperature T C , the susceptibility χ increases and tends to diverge as:

χ = C T -T C , (1.56)
where C is a constant that depends on the material. [110] In a similar framework, assuming a uniform polarization, the mean-field Landau-Ginzburg-Devonshire theory also describes relatively well how ferroelectricity appears below the Curie temperature. [111] It uses a Taylor expansion of the free energy F P with even powers of the polarization order parameter P , specifically around the transition including one low-order term with coefficient a = a 0 (T -T 0 ) that depends linearly on the temperature T :

F P = 1 2 aP 2 + 1 4 bP 4 + 1 4 cP 6 -EP, (1.57)
where, except for a, all coefficients are temperature-independent, and the last term is a contribution introduced by an externally applied electric field, so we can reproduce the P (E). This formalism comes handy as generally, the coefficients can be fitted to experiments.

For T < T 0 , this free energy expression passes from one minimum at P = 0, to two minima at ±P 0 and some energy barrier in between, as illustrated by fig. 1.14a. Actually, depending on the sign of the coefficient b, the appearance and developement of the order parameter P will either be progressive and continuous (if b > 0, which results in second-order transition) as seen on fig. 1.14b or rather abrupt (if b < 0, resulting in first-order transition), as in fig. 1.14c. The Curie temperature is generally close to T 0 although it may deviate a bit, especially for first order transitions. Some extensions of the Landau theory also include components of strain, which often plays an essential role in ferroelectricity, and the addition of long-range interactions for non-uniformly polarized bulk or thin films systems. Despite giving a rather good description of the observed phenomena, the Landau-Ginzburg-Devonshire theory still has some limits, in particular close to the transition temperature and to reinforce or complement it, other formalisms have been proposed on theoretical and conceptual grounds as well as to provide a meaningful and linking description of actual experiments.

An important contribution to the understanding of ferroelectricity has been provided by Cochran [112] and Anderson [113] whereby they noticed and formalized the importance of the lattice dynamics (phonons) in phase transitions, and in particular paraelectric-ferroelectric transitions [114]. Specifically for this case, a transverse optical phonon mode at the center of the Brillouin zone tends to first reduce its frequency as the temperature is decreased, and to finally freeze as the transition is approached, signaling an instability of the structure [115]. A similar but reversed behavior has also been observed in the ferroelectric phase, with a softening of a transverse optical phonon mode as the temperature of the system increases towards T C . [116] This evolution of the zone-center transverse optical mode to nearly vanishing frequency ω T O → 0 is directly responsible for the divergence of dielectric constant ϵ of the material at the transition, as related by the Lyddane-Sachs-Teller relation. [117,118] This also links the soft-mode theory to the more classical Landau theory since ω 2

T O ∝ |T -T C |. On the conceptual side, we note that the concept of Berry phase has also been used in the 1990's in extensive efforts to formulate a "modern theory of polarization" notably by [START_REF] Vanderbilt | Electric Polarization as a Bulk Quantity and Its Relation to Surface Charge[END_REF][119][120][121][122] which has been beautifully explained with a rather complete picture by Resta [121] and pedagogically summarized in ref.

[109] and by Spaldin [123].

It was set-up to alleviate the ill-definition of the polarization based on the simple sum of dipole moments from all unit cells within an infinite crystal, since this actually depends on the choice of the unit-cell itself. Furthermore, it becomes even less relevant in the case of crystals with (even partially) covalent bonding, where it is impossible to properly define ionic point charges in the unit cell and which electrons belong to which atom.

Something that we mentioned briefly but did not emphasize until now is how the polarization is actually expressed as a density of charge per unit area (expressed usually in µC cm -2 ), pertaining to the appearance of bound polarization charges σ P = P • n at the surfaces of a material with finite polarization (where n is the unit vector normal to each surface). Building on this consideration, a more proper way of defining this concept is actually derived by looking at the adiabatic continuous change of polarization ∆P between two stable states (or between a high symmetry structure and the lower symmetry ferroelectric or polar phase). The key point of this formalism is that it considers polarization being related to a phase argument (the Berry phase), which is cyclic modulo a lattice vector, and in fact strongly related to a material's electronic band structure. Formally, within the Berry phase theory we exposed in section 1.1.4, the parameter space to be used here is the center positions of the Wannier wavefunctions (localized equivalent of the Bloch functions): the displacement of these Wannier centers summed over all bands within a Brillouin zone give an effective displacement which can be multiplied by the Born effective charges (often higher than the formal charges of an ion, especially in ferroelectrics) to obtain the polarization [123].

An important aspect underlying the concept of polarization change, particularly utilized in the framework just presented, is the fact that a flow of bound charges (i.e. a displacement/change of dipoles) in the bulk of an insulating material will also imply some free charges moving from/to the surfaces of that material to fulfill the continuity equation. This is observable macroscopically as a current flow across the sample, as we will see experimentally in section 2.6, or can be characterized by a change at one electrode of the carrier density (or carrier sign).

. Ferroelectric domains and charge screening

Since the polarization is homogeneous to a quantity of charge per unit surface, these charges represent an electrostatic energy cost for the system to maintain long-range ordering across the entire sample as in fig. 1.15b and create an opposite depolarizing field. In the low-thickness limit, specifically in very thin films, this depolarizing field can grow sufficiently strong as to suppress the polarization.

The energy cost can however be compensated whenever the bound charges can be screened by other adjacent charges. It is thus spontaneously reduced by the formation of domains in the material, with well-defined orientation within one region (see fig. 1.15a) and most often 180°o pposed from one ferroelectric domain to the next, so that bound charges close to the domain boundaries can compensate each other. Another way of compensating for the depolarizing field is by allowing some free charges on the sample's surfaces or interfaces to come and screen the bound polarization charges. This can be done with the addition of extrinsic adsorbates at either sides, or via charge screening if electrodes are placed on the top and/or bottom interfaces or from inside when there are few intrinsic free charges (e.g. via defects, or in a doped semiconductor, or when photocarriers are generated). We have to note though, that intrinsic charges may just as well screen the polarization itself or at least make it nearly impossible to apply any field across the material and thus switch its polarization as we will see in section 1.5, which thus prevents it from being called a ferroelectric, by definition.

+ + + + + ----- + + + + -- -- Ferroelectric domains P up P down + + + + + ----- + + + + + - - - - - (a) (b) (c) (d)

. On the verge of ferroelectricity

Interestingly, there are two other families of dielectric, particularly relevant in the context of this thesis. First, some dielectrics like PbMg 1/3 Nb 2/3 O 3 (PMN) tend to show a peak of the real part of the susceptibility (or dielectric constant ϵ ′ ) that extends over a much broader range of temperatures than for the classical ferroelectrics, and is also dependent on the frequency of the electrical stimulation [125]. They also show a significant frequency dependence of the susceptibility on the frequency, particularly around and below the characteristic the dielectric constant peaking temperature T m , with higher frequency giving lower maximum of ϵ ′ as well as higher T m (see fig. 1.16a). This behavior is usually associated to a deviation from the normal Curie-Weiss law ϵ ′ ∼ 1/(T -T m ) β , with β approaching 2 for relaxors.

All of this is actually due to local polar clusters or nano-regions responding with different dynamics and forming way above T m , instead of the long-range ordering that defines ferroelectricity and appears closer to the Curie temperature T C . Because of this, these materials called relaxor ferroelectrics (or simply relaxors) usually exhibit lower macroscopic reamnent polarization than classical ferroelectrics. Well below T m , under the application of an electric field beyond a certain threshold, relaxor ferroelectrics like PMN can actually display a macro-scopic ferroelectric behavior, namely ferroelectric hysteresis. For more detailed review on the phenomenon of relaxor ferroelectrics, we invite the reader to refer to [125,126]. Because of this, they are often called as incipient ferroelectrics, but also quantum paraelectrics. The other interesting behavior that we will exploit in the chapter 3 of this thesis is the tendency of some materials to exhibit a nearly diverging electric susceptibility as the temperature is lowered, without ever passing an actual transition to a ferroelectric phase, as illustrated on fig. 1.16b. In fact, while classical ferroelectrics see their ordering destroyed above the critical Curie temperature T C , when thermal fluctuations become too important, some materials with a very low expected transition temperature actually remain paraelectric, as the transition is suppressed by quantum fluctuations and thus, the transition temperature falls down to absolute zero. The existence of such quantum critical points for various order parameters has been the focus of important research [127], and has notably been observed for the case of ferroelectricity in some oxide perovskite crystals like SrTiO 3 as well as KTaO 3 which we will both describe in more details in section 1.3 [110, 128-131].

. Transition metal oxides

The term TMOs commonly refers to a wide family of crystals composed of a mixture of oxygen anions and transition metal cations, i.e. the group of elements with partially filled d shells, to which elements of the f -blocks of lanthanides and actinides are sometimes added. The resulting binary, ternary or even higher order compounds are usually quite stable in a variety of crystalline structures (e.g. rock salt, rutile, anatase, corundum, spinel, perovskite, ...) with oxygen stoichiometry and arrangement around cations as well as cations size (and other properties) helping to stabilize one structure or another.

The valence band of these materials is usually formed by the 2p orbitals of the oxygen ions, while the conduction band is mostly composed of d (or f ) orbitals from the transition metal. However, TMOs typically possess a mixed ionic and covalent nature of the transition metal-oxygen (M-O) bonding which can introduce some hybridization between p and d orbitals, and further reflect on the band structure. In practice, the overlap between p and d orbitals dictates a large set of materials properties notably through a balance between localization and delocalization of electrons. The result can be understood in a tight-binding picture as the following: a poor overlap tends to favor electron localization on each atomic site and leads to very narrow electronic bands (1 eV to 2 eV), while a larger overlap favors a stronger hopping integral between neigboring orbitals and thus a possible delocalization of the electrons.

In the latter case, electrons arrange in the lattice to minimize on-site Coulomb repulsion energy from electron-electron interaction and this results in complex spin, charge and orbital long range orders due to what is called electronic correlations, typical of TMOs [132]. The overlap between p and d orbitals is actually a tunable parameter which can be modified by changing environmental conditions, bringing slight modifications of elemental composition or by the application of external stimuli (electric and magnetic fields, light, mechanical stress, temperature, ...), thereby affecting the macroscopic properties as well.

Indeed, the family of TMOs contains a spectrum of insulators, semiconductors or metallic materials, some materials with metal-insulator transition (MIT) (e.g. VO 2 [133][134][135]) as well as superconductors with the highest critical temperature T c at ambient pressure so far (the family of cuprates [136] Of particular interest in the oxide community and for this thesis, transition metal oxides adopting a perovskite structure with a general chemical formula ABO 3 are very tunable in many ways and first and foremost chemically, as many different elements can be incorporated in their composition. They have been a very widely studied platform in the past 20 to 30 years especially thanks to their versatility and technical advances interms of materials growth (with molecular beam epitaxy (MBE), pulsed laser deposition (PLD), sputtering, ...) rendering possible to grow thin films and stack materials with different properties together. A relatively good guess of their crystalline symmetry can usually be obtained by using the so-called Goldschmidt tolerance factor, accounting for the radius of each element in the ABO 3 structure:

t = r A + r O √ 2(r B + r O ) .
(1.58)

Materials with a Goldschmidt factor larger or smaller than 1 tend to deviate from the cubic structure to adopt the typically lower symmetry tetragonal, orthorombic or rhombohedral structures. Deviations from this may of course occur depending on other intrinsic properties of the material (magnetic or polar ordering, ...) and external parameters, such as the temperature, which may render other structures more stable. However, thanks to generally similar lattice parameter and the well-defined interfaces that can be obtained, these materials can indeed be assembled to obtain new extraodinary properties resulting from interlayer coupling between each material's properties or even from interfacial effects altogether absent from the isolated compounds. This is particularly the case for the now well-known LaAlO 3 /SrTiO Strontium titanate SrTiO 3 (STO) is a long-known non-toxic ceramic, first synthesized in the 1950's by the high-temperature Verneuil process and was later discovered in its natural form in Siberia. While it has been used in jewelry in the early years as a surrogate for diamond stones, it also quickly atttracted researchers for few very interesting properties that we will expose in this section, and was heavily studied in particular by J. Georg Bednorz and K. Alex Müller before they started working on high-T c superconducting cuprates for which they earned the Nobel prize in 1987 [150].

Thanks to a surface treatment allowing to make it particularly flat and single-terminated [151], the usage of STO developed widely in the 1990's and it started being commercially available as a substrate to grow many materials including other pervoskite oxides, and was as a matter of fact particularly central in the development of high-T c superconductors later on.

STO adopts a perovskite phase at room temperature 300 K with Ti 4+ ions occupying the centre of octahedra formed by the O 2-ions (see fig. 1.18a), while the Sr 2+ ions occupy the Asite of the perovskite structure, surrounded by 12 oxygen ions. The crystal is then a tessellation of these octahedra and cuboctahedra with mixed ionic and covalent bondings and both the crystal and electronic structure will be central in the properties of this material. Although in its room-temperature phase it adopts a cubic P m3m structure with a lattice parameter a = 3.905 Å, decreasing its temperature below 105 K, STO follows a second order transition to a tetragonal I 4 m cm structure, by progressively rotating adjacent oxygen octahedra slightly, in opposite directions along the out-of-plane c-axis: this is the so-called antiferrodistortive transition, with Glazer notation a 0 a 0 c -(see fig. 1.18b). To account for the new elementary pattern extending over more unit formulas, one has to consider a new tetragonal unit cell with a (roughly) "doubled" lattice constant along the c-axis. Although it was reported by X-ray diffraction (XRD) measurements in 1964 that STO would undergo further lowering of its crystalline symmetry to an orthorombic structure below 65 K[153], it was then proved wrong by other experiments (electron paramagnetic resonance (EPR), Raman scattering, ...) and it is since then generally admitted that STO remains tetragonal at all temperatures below 105 K [154][155][156][157][158]. We note that this cubic to tetragonal transition can also be triggered by external mechanical stimuli and is affected by defects [159] and chemical or carrier doping [160].

The now elongated c-axis can lie along three directions, leading to the formation of ferroelastic domains and this ferroelastic transition is of paramount importance for the dielectric properties of STO as it signals the onset of a huge increase in the dielectric constant ϵ down to about 4 K where it finally saturates and stays flat at more than 20 000 (see fig. 1.18c)[152]. This peculiar behavior is somewhat reminiscent of the diverging dielectric susceptibility near ferroelectric transitions and is indeed accompanied by a softening of a transverse optical phonon mode. However, in the case of STO, no actual transition to a polar phase ever occurs down to the lowest measureable temperature, in absence of stimuli. In fact, the material remains a simple paraelectric at all temperatures, even very low ones, where one would expect a ferroelectric ordering to occur, with uniform polar displacement of the Ti ion inside the octahedral O cage, in either of the two supposed metastable positions along the long axis. The phenomenon behind this suppressed transition, as we described in section 1.2.5, is in fact called quantum paraelectricity. More precisely for the case of STO here, although one expects the Curie temperature T C to occur around 35 K, the transition is suppressed first by some unknown classical mechanism until it actually enters the regime of quantum fluctuations, forbidding a stabilization of a macroscopic polarization in the material [152].

The electronic band structure of STO is mostly governed by the hybridization of Ti-3d and O-2p states, with Sr 2+ ions having only a secondary role due to their strong electropositivity.

Similar to what we previously suggested for the general case of TMOs, the valence band is then mostly formed of the O-2s and O-2p orbitals, while the conduction band is mainly cationic and formed by the Ti-3d orbitals. STO is a relatively good insulating semiconductor with an indirect bandgap of 3.35 eV (R → Γ) and an even larger direct Γ → Γ bandgap of 3.75 eV, and is expected in its nominally stoichiometric undoped form to have a nearly empty conduction band. The cubic crystal field above the ferroelastic transition already splits Ti-3d orbitals into a low energy t 2g triplet and a higher energy e g doublet (see fig. 1.18d). The former, composed of d xy , d xz and d yz orbitals, remains closer to the Fermi level and couples to neighboring identical orbitals through the O-2p orbitals that lie in between, offering some hopping path to potential carriers. We note that the lowering of symmetry to the tetragonal phase also has consequences on the crystal field, which further splits the Ti-3d orbitals t 2g (and e g ) group(s), bringing the d xy orbital higher in energy compared to the other two members of the t 2g triplet, with a z-component, although as we will see later in section 1.4, a breaking of inversion symmetry will invert this band hierarchy.

Although STO is normally an insulator, a bulk conductive state can also be easily achieved already from very low carrier concentrations 1 × 10 14 cm -3 brought by electron donors (e.g. La 3+ on the Sr 2+ sites, or Nb 5+ instead of the Ti 4+ ), acceptors (e.g. Na 1+ on the Sr 2+ sites, or Fe 3+ instead of the Ti 4+ ), or by defects (most commonly oxygen vacancies) [160,163]. What was importantly observed is that upon cooling down STO from 300 K to 4 K, the resistivity decreases by orders of magnitudes while the mobility increases up to few 1 × 10 4 cm 2 V -1 s -1 [161,164]. This increase in mobility has actually been linked to the strong increase of dielectric permittivity of the material.

Going even lower in temperatures in the sub-kelvin range, doped STO exhibits supercon- ductivity up to more than 400 mK at carrier density of 1 × 10 20 cm -2 [160,165]. As the first discovered superconducting oxide, it was one of the starting point for the later discovery of high-T c superconductors [150]. Remarkably, it is also the most dilute superconductor known to date, as this zero-resistance state could be observed even down to few 1 × 10 17 cm -2 for oxygen-deficient samples, excluding the applicability of the Bardeen-Cooper-Schieffer (BCS) theory. Finally, this material shines by the discovery of another unconventional behavior that is the non-monotonic dependence of the superconducting transition temperature with carrier density, confining this peculiar phase to a dome-shaped region (see fig. 1.19c), still in debate as of today. Of particular interest, it has been shown that the appearance of superconductivity mediated by transverse optic phonon in doped STO, was also strongly related to the aborted ferroelectric phase transition, which -as we will briefly cover in section 1.5-gave birth to new exotic concepts and intense research in understanding the proximity and mixing between (super)conducting states and ferroelectricity, particularly near the quantum critical points of STOand KTO-based compounds.

Ferroelectricity of SrTiO 3 and related compounds

The incipient ferroelectricity of materials like STO (and KTO as we will see right after) has stimulated a lot of research activity particularly in the 1960's to understand by which processes this transition was suppressed down to the lowest measurable temperatures and predictably down to the absolute zero.

One way these materials were explored was to bring them even closer to their ferroelectric phase and several ways were found to tune the dielectric properties of STO within the quantum critical phase boundaries and beyond, so it would actually undergo this transition. As shown in 1.20a, the application of a strong electric field caused, for instance some kind of quenching of the dielectric constant, but beyond a certain threshold of 200 V mm -1 , it also allowed to measure some hysteretic cycling of the polarization against the electric field, as shown in fig. 1.20b [26, 166-168]. Performing alternative current (AC) electrical measurements allowed Hemberger et al. [166] to extract the remanent polarization and third-order susceptibility term χ 3 as a function of the temperature and to draw some phase diagram of the dielectric behaviors of STO with electric field and temperature parameters (fig. 1.20c). What they concluded from these experiments was the appearance of ferroelectric correlations below 60 K, later suppressed by incoherent quantum fluctuations as the temperature is further decreased below 45 K and reappearing below 33 K due to the onset of a coherent tunneling ground state (Ti ions tunneling coherently between two positions with energy minima). In this polar phase of STO with orthorombic symmetry (very slight deformation compared to the tetragonal phase), the polarization seems to always be perpendicular to the tetragonal axis [169].

Other methods have been used to favor a ferroelectric ground state of STO at low temperatures, by applying some mechanical stress [170,171] or by achieving minute isotopic substitutions ( 16O replaced by 18 O) or isovalent ionic substitution of the Sr 2+ ions by neigbouring species in the periodic table, i.e. Ba 2+ or Ca 2+ or by Pb 2+ [172] (see fig. 1.21c). These mixed compounds are then equivalent to solid solutions of SrTiO 3 with the room-temperature ferroelectrics BaTiO 3 and PbTiO 3 or the piezoelectric but non-polar CaTiO 3 (which gave its name to the family of perovskites adopting a similar structure).

For the particular case in which we are interested for this thesis, the mixture between CaTiO 3 and SrTiO 3 may give rise to many structural and dielectric instabilities depending on their stoichiometry. Ca ions being smaller than Sr ions, the combination of both in different proportions stabilizes a spectrum of non-polar, antiferroelectric and ferroelectric phases due to "chemical" strain imposed on the structure [173]. As we mentioned before, in its pure stoichiometric phase, STO goes through only one structural cubic (C) ↔ tetragonal (T) (P m3m ↔ I 4 m cm) transition at roughly 105 K. As temperature is lowered, dipoles around these impurities start to form in the matrix of SrTiO 3 and these polar clusters grow until they can percolate at the ferroelectric T C (if the Ca concentration is sufficient) [177,178].

Starting from x = 0.0018, the compound Sr 1-x Ca x TiO 3 (or CaSTO) was discovered by Bednorz & Müller [176] to become ferroelectric (with reportedly orthorombic mm2 point group), with increasing T C as we increase the proportion x of Ca up to x r ≈ 0.016 (i.e. 1.6 %) where it then saturates around 35 K and does not evolve until the proportion of Ca becomes x = 0.12 [175,176,179]. In fact, as can be seen from 1.21b, the increased proportion of Ca 2+ ions in the range between x r = 0.016 ≤ x ≤ 0.12 smears the low-temperature structural transition to the polar phase over a broader range of temperature and this phase has been considered as a relaxor ferroelectric (or dipole glass) phase [180], while only the region 0.0018 ≤ x ≤ 0.016 is really called ferroelectric, with a polarization developing along the pseudocubic [110] and [110] directions (or ±[100] and ±[010] of the orthorombic cell) [176].

In the higher Ca 2+ -content regime, from x = 0.12 and up to x = 0.40, an antiferroelectric (AFE) (P bcm) phase is found for increasing temperatures up to 400 K in the composition Sr 0.6 Ca 0.4 TiO 3 and finally a non-polar paraelectric P nma phase is recovered above that. It seems that for the range 0.40 ≤ x ≤ 1.0, the P nma and P bcm orthorombic structures show an additional rotation of the oxygen octahedra, this time not only about the c-axis ((001) direction), but also around the (110) axis (with Glazer notation a -a -c + ), and this might extend as well in the ferroelectric region [175,179].

Interestingly, the weak saturated polarization values (P sat ≈ 0.5 µC cm -2 ) in the ferroelectric phase observed in the literature for a material with such a large dielectric constant motivated a more recent investigation on the exact low-temperature structure. After performing symmetry analysis and Rietveld refinement of XRD data at 12 K on CaSTO with 2 % and 4 % of Ca, Mishra & Pandey [181] suggested that assuming a Ic2m point group (instead of mm2) compatible with Bednorz & Müller [176] initial report, the material actually exhibited ferrielectric ordering of the polar displacements alternating between the SrO and TiO 2 planes. Using the Born effective charges formalism, with values taken from the literature, they could reconcile their claim with observed values of polarization in the literature. As this is the only report and it does not change much the considerations regarding polarization measurements, later on we will indistinctively denote this material as ferroelctric and use the abbreviated notation CaSTO or Ca("x%"):STO for convenience (with x the proportion of Ca used to replace Sr. Additionally, for what concerns this work, where only low percentages of Ca are used, up to 2 %, the relaxor ferroelectric behavior mostly affects the broadening of the ferroelectric transition without altering much the functionality associated to ferroelectric polarization, at low temperature.

. Potassium tantalate KTaO 3

Properties of KTaO 3

Potassium tantalate KTaO 3 (KTO) is in many ways similar to STO and they were both extensively studied alongside each other, in the framework of quantum paraelectricity. KTO is also an insulating material with a cubic perovskite structure (also P m3m), with a lattice constant of 3.989 Å a bandgap of 3.6 eV and a nearly diverging dielectric constant that reaches values higher than 4500 at low temperatures (fig. 1.22b). One difference arises from the alternance of positive (TaO 2 ) 1+ and negative (KO) 1-charged planes along the (001) direction as shown in fig. 1.22a.

It can also become conductive in the bulk by forming oxygen vacancies with mobilities reaching 23 000 cm 2 V -1 s -1 [182] at 4.2 K. While the conductive band in STO is mostly formed of the Ti-3d orbitals, for the case of KTO, the Ta-5d bands bring slighlty different shape to the conduction band, with smaller effective mass and thus potentially higher mobilities. Again, similarly to STO, the strong increase in Hall mobility µ H , shown in 1.22d, seems to be strongly linked to the increase of dielectric constant and a power law µ H ∝ T -3 seems to be a good and common description for both system at least before the low-temperature quantum paraelectric regime, indicating a dominating electron-phonon scattering. Below 100 K, electronic transport in KTO seems to be more largely dominated by impurities (see fig. 1.22c).

Again, the lower-lying t 2g group of the 5d orbitals from Ta (i.e. d xy , d xy and d yz ) seem to dominate the orbital character of the conduction band. However, the larger atomic spin-orbit coupling (SOC) of Ta compared to Ti, brings a larger splitting of this t 2g manifold for the electron-doped KTO, by raising the J = 1 2 states 0.4 eV higher than the J = 3 2 (instead of 0.025 eV for STO) [183][184][185]. On the contrary, because the structure of KTO remains cubic at all temperatures, there is no further crystal field splitting due any eventual effect of lowered symmetry of the lattice on the Ta environment. Additionally, no bulk superconductivity has ever been reported in bulk doped KTO despite the many other similarities with STO and the exact reasons are still not well understood as of today.

O K Ta (KO) -1 (TaO 2 ) +1 (KO) -1 (a) (b) (c) (d)

Ferroelectricity of KTaO 3 and related compounds

Importantly for us, despite the fact that KTO remains cubic at all temperatures, its similarly nearly diverging dielectric constant but paraelectric behavior all the way down to absolute zero indicate that it is also an incipient ferroelectric as for STO [129,130,186,187]. 2001) TC = A (x-xc) 1/φ ��φ= 1.9 TC Knauss et al., , Phys Rev B 55, 3472 (1997) TO→R TC→T TT→O TriebWasser, Phys. Rev. 114, 63 (1959) TO→R TC→T TT→O Boatner, Helv. Phys. Acta 50, 557 (1977) T There are also different manners to render KTO actually ferroelectric, among which stress induced by thin film growth [189], solid solutions with the antiferroelectric NaNbO 3 or the ferroelectrics LiNbO 3 [190] or more simply KNbO 3 allow again to explore a wide spectrum of crystalline phases with their specificities in terms of dielectric properties, including ferroelectricity [187].

We will give more attention here to the substitution case of Ta 5+ ions by Nb 5+ ions, more widely studied in the literature [129,130,186,187,[191][192][193][194] and which we will use in chapter 6. We show a summarizing structural phase diagram in fig. 1.23a from which we can see that the ferroelectric phase of potassium tantalate-niobate KTa 1-x Nb x O 3 (KTN) appears for concentrations of Nb as low as x c = 0.008 (0.8 %). The ferroelectric (FE) T C then increases with the concentration of Nb roughly as T C ∝ (x -x c ) 1/ϕ , with ϕ ≈ 1.9 (close to the ideal theoretical power of 2) according to Höchli et al. [130] and this behavior was found to be nearly universal for quantum ferroelectric mixed compounds (this model was then similarly applied to CaSTO later on [176]). From x = x c up to x = 0.02 [195] (or x = 0.04 according to [196]), the ferroelectric transition is displacive, soft-mode driven and bringing actual long range order [191,194,195], although some have also reported a more dipolar glassy (or quantum paraelectric (PE)) behavior, i.e. short-range polar clusters form and progressively order as temperature is decreased [190,[197][198][199] as is found in nominally pure KTaO 3 [200,201]. Within the range of small Nb content, the transition is supposedly multicritical and the material passes from the cubic P m3m to a rhombohedral structure, while for x > 0.02, three cubic-tetragonalorthorombic-rhombohedral phase transitions occur successively.

A similar dipolar glass behavior could also be observed from KTN as in CaSTO, which -as illustrated on fig. 1.23b-implies a non-zero remanent hysteresis even well above the supposed transition (at the peak of dielectric constant) for different concentrations of Nb.

KTN is commonly used in optics for the strong modulability of its refractive index and different compositions in the room-temperature ferroelectric phase can be easily found. This shows how interesting and how much potential this material can have towards applications.

. Two-dimensional electron gases at oxide interfaces

1.4.1 . Obtaining 2DEGs at the surface of SrTiO 3

Generalities

The research revolving around STO was strongly revived when in 2004, Ohtomo & Hwang [148] discovered that a conductive interface could be found between the two insulators lanthanum aluminate LaAlO 3 (LAO) and strontium titanate SrTiO 3 (STO). This was demonstrated on a heterostructure of LAO grown by pulsed laser deposition (PLD) on STO along the (001) direction, and it was first explained by the presence of a polar discontinuity at the interface between both material. Indeed, LAO(001) is a sequence of alternating (LaO) +1 and (AlO 2 ) -1 charged planes while STO is non polar along that direction. The reports that a critical thickness of 4 u.c.s (fig. 1.24c) was necessary to observe the conducting behavior at the interface, and that a TiO 2 -termination of the STO substrates was a requirement for a n-doped interface (with growth of LAO starting with (LaO) 1+ ) resulted in the suggestion of the polar catastrophe scenario as the origin of the 2DEG. To avoid the build-up of an electric-field as more and more unit cells are added, a transfer of electrons from the LAO surface to STO could be allowed to compensate for the discontinuity and bring back equilibrium [202] (see 1.24b).

Other works suggested that the creation of this two-dimensional electron system was caused by the presence of oxygen vacancies in the bulk of STO, created during the fabrication process with a high-temperature growth in very low oxygen pressure P O 2 =1 × 10 -6 mbar to 1 × 10 -4 mbar [203,204]. However, performing a post-annealing at high oxygen partial pressure P O 2 > 800 mbar would remove the bulk oxygen vacancies and limit the conducting behavior to the interface [204][205][206], potentially re-establishing a dominant contribtuion of the polar catastrophe scenario [207]. Another suggestion was that beyond a certain thickness of LAO, the built-up electric-field would favor the creation of oxygen vacancies not at the interface but at the top surface of LAO, which would then allow two free electrons to transfer to the interface in .

Later studies also demonstrated other methods to obtain a two-dimensional electron gas (2DEG) on STO, either via high-temperature epitaxial growth of other insulating polar oxides such as LaTiO 3 or LaVO 3 or by room-temperature deposition of amorphous materials like the spinel γ Al 2 O 3 [214] showing remarkably high electron mobility (140 000 cm 2 V -1 s -1 at 2 K), or even at the bare surface of STO after cleaving or exposing to UV light [215]. As we will see in section 1.4.2, this was particularly useful to study the electronic band structure of this family of two-dimensional conducting systems.

On a last note, it has been shown that electrical stimuli or the deposition of some overlayers on the LaAlO 3 /SrTiO 3 stack could affect the transport properties of the interfacial system. Indeed, the interfacial conductive properties are very sensistive to electrostatic boundary conditions such that it was possible to render the 3 u.c.-thick LAO/STO interface conducting by applying some electric-field macroscopically [213] or in a local way with a conductive atomic force microscope (AFM) tip [216], but also to tune the carrier density, the mobility or the critical thickness required to achieve the formation of a 2DEG by depositing some amorphous LAO (a-LAO), other oxides [217] or a metallic overlayer [218][219][220][221]. While some of these mechanisms simple rely on the formation and transfer of oxygen vacancies across interfaces, the latter achievement with simple metal overlayers is also made possible depending on the work function of the deposited metal with respect to the chemical potential of the oxide stack (creating a Schottky barrier if a misalignment is present in the isolated systems). For materials with a low work function, the critical thickness could be decreased down to 1 or 2 u.c., in addition to a partial oxidation of that metal, while noble metals tend to increase the critical thickness above 4 u.c..

The case of MOx/SrTiO 3

Following the general idea that the deposition of metals on top of oxides could give rise to electronic and chemical interaction, Rödel et al. [222] showed that it was actually possible to use a simple deposition process implying only a single metal instead of a binary or ternary compound to form a conducting layer at the surface of STO. This process comes from a balance between a chemical process involving the transport and interdiffusion of oxygen across length scales larger than a monolayer, or the formation of new phases, and an electronic interaction (actually a transfer of charge in this case) allowing the system to restore its equilibrium against the interfacial electric field. Finally, the overall reaction is controlled by a competition between the amount and direction of charge transfer across the interface between the metal and the oxide (governed by the difference of work functions of the materials), and the electric and chemical potential gradients, both being interrelated.

The dominant chemical reaction occuring in the process that has been pointed out here is the redox reaction in which the deposited metal Me 0 gets oxidized from the substrate while sub-oxide species of the substrates tend to be reduced. For instance, in the case of STO, the Ti IV ions of TiO 2 sub-oxide may actually result in a mixture of Ti IV , Ti III and Ti II after depositing a reactive metal on top.

The change in Gibbs free energy associated to this reaction ∆ r G = ∆ r H + T ∆ r S is determinant in whether it is favorable or not to form the new oxide from the initial species. In particular for solid state reactions, the change in entropy ∆S is negligible and ony the enthalpy change ∆H is considered. During this specific reactin, the enthalpy of formation of the new metal oxide ∆ f H O (in kJ mol -1 of O) becomes a central parameter to determine the spontaneity of that reaction under given conditions.

A tentative explanation of the redox reaction for the fabrication of 2DEGs in Metal/STO systems has been proposed based on two fundamental parameters that are the enthalpy of formation of an oxide for a given metal (its tendency to form that oxide) and the work function of that metal, and on the comparison with these parameters with that of STO, as summarized in fig. 1.25a. Based on this, three main categories can be identified. The first one on the top right quadrant of the graph in fig. 1.25a regroups mostly noble metals (Pt, Pd, Au, Ir, Ag but also Cu, Ni and Co) that have both a higher work function and enthalpy of oxide formation than the values found for STO which means that in principle they do not form oxides spontaneously when placed in contact with the STO in ambient temperature and charge transfer occurs more from STO to the metal, removing a driving froce for the redox reaction.

The second group in the second (bottom right) quadrant includes metals that spontaenosuly react with STO although their work function is usually similar or slightly higher. Thus, the oxygen scavenging by the metal is limited by the oxygen diffusion especially at low temperatures, resulting in relatively confined oxygen-deficient layers. We note the remarkable fact that some metals contained in that group such as Al, Ti, or V, although supposedly amorphous here, are also included in the complex oxide compounds creating 2DEGs on STO as well when epitaxially grown, in the same spirit as the LaAlO 3 /STO interface.

The last corner (bottom left) is that of metals having both a tendency to easily steal oxygen from STO to oxidize and cause an additional interfacial electric field favoring the diffusion of oxygen. The reaction of these metals is usually a bit less controlled, more violent in a sense, and more exothermic, resulting in reportedly less confined conducting layer in STO for these systems [224].

Experiments with different metals have been carried out in ref [225] and in the thesis of Vicente-Arche [START_REF] Vicente-Arche | Quantum Oxide Heterostructures for Spin-Charge Interconversion[END_REF], in particular with Al, Ta and Y, although other reports using a similar method with different metals appear more and more in the literature such as Gd or Eu [224,226]. Based on magnetotransport and XPS measurements, [START_REF] Vaz | Mapping Spin-Charge Conversion to the Band Structure in a Topological Oxide Two-Dimensional Electron Gas[END_REF] confirmed the mechanism of Ti reduction by Al (see fig. 1.25b) and determined the optimal deposition conditions and thickness to obtain a 2DEG with relatively high carrier density, mobility and stability in time. The latter point is made possible by ensuring that the deposited metallic overlayer is sufficiently thick initially: the first few angstrom will be oxidized in vacuum as much as it can be from the oxide substrate, while the remaining thickness should oxidize entirely from air and act as diffusion barrier for the oxygen species. At the optimal thickness (about 2 nm for Al), this dffusion process is efficiently slowed down, and the presence of a remaining metallic interlayer is avoided.

We finally note that there has also been reports of a two-dimensional hole gas (2DHG) made by a very careful deposition of only few ångström of Fe on STO, beyond which the n-type character of the carrier gas wins back, allowing to fabricate all-oxide based PN junctions and other applications relying on complementary dopings [227,228]. Electronic states found at surfaces and interfaces of oxides can be quite different from the bulk and display exotic properties. The first report of the conducting interface between the two insulators that are LAO and STO (shown in fig. 1.26a) was already a remarkable feat and other interesting phenomena adding on top of each other were discovered subsequently, with superconductivity in the 100 mK range, for instance (see 1.26c).

A residual resistivity ratio (RRR) -i.e. the ratio betweeen sheet resistances at roomtemperature and cryogenic temperature-of the order of 10 at least, is usually found in this system, in great part due to a strong increase of mobility with decreasing temperatures (which again in this 2D case seems to arise from the increase of dielectric permittivity) typically up to about 3000 cm 2 V -1 s -1 in LAO/STO (see 1.26b) and even up to few 1 × 10

5 cm 2 V -1 s -1 in the γ Al 2 O 3 /STO system at 2 K [214].
The literature has been sometimes inconsistently qualifying these systems as Fermi gases or as Fermi liquids although the former denomination is more common. We argue that given the low orbital filling and relatively weak electron-electron interactions, in first approximation the name of two-dimensional electron gas (2DEG) can be used reasonably and we will call it this way for the rest of this thesis.

The exact band structure of this buried interface has been challenging to measure directly as probes lie angle-resolved photoemission spectroscopy (ARPES) that can access electronic states with resolution in the momentum space were initially very surface sensitive and do not provide sufficient probing depth for the LAO/STO interface. This brings a limit to photoemission techniques which can only be used in the soft or hard X-ray domain (in fact, resonant X-ray ARPES was later on used to study the LAO/STO interface) which unfortunately decreases the momentum resolution one can have (see section 2.2.1).

However, the surface of in situ cleaved STO (001) revealed to be a very useful platform to study STO-based 2DEGs by UV ARPES as it was similar to DFT predictions for the LAO/STO interface, demonstrating that the band structure and electronic properties of this system arised from the host material, the STO itself [230]. As mentioned earlier, the nominal d 0 state of Ti 4+ and Ta 5+ in SrTiO 3 and KTaO 3 respectively makes these materials naturally insulators, with full degeneracy of the t 2g manifold at the Γ point and a Fermi level well below this conduction band inside the gap. It is however possible to progressively add electrons at the conduction band minimum, resulting in 3d (or 4d) transport either in the bulk or in the 2D limit at the surface.

As we show schematically in fig. 1.27a, the bulk energy dispersion along an arbitrary axis, say k y , is composed of a band with light-mass carriers which disperse strongly and a pair of two other bands with heavier mass. This is the result of a direction-dependent orbital overlap with neighboring (for the case of STO) titanium 3d xy , 3d xz and 3d yz orbitals: taking the example of the 3d xy band, the latter corresponds to a light band in the xy-plane, reflecting easy hopping of electrons within that plane, but a heavy band along the perpendicular z-direction (in fact hopping transition should vanish along this direction). Similar reasoning can be applied for the two other subbands with geometric permutations.

The strong confining electrical potential that restrains the electrons to the interface (due to electrostatic or electrochemical considerations) causes a band bending at the surface (or interface) and acts to lift the spatial degeneracy of the bands. With simple considerations of a triangular "wedge" potential well along the z direction, d xy band which presents a higher effective mass along the confinement axis will experience a smaller shift relative to the bottom of the triangular potential, while the d xz,yz bands will be pushed to a higher energy due to the electric field. As a result, the band hierarchy with inversion symmetry breaking finds itself reversed compared to the bulk tetragonal SrTiO 3 (with D 4h symmetry), with finally lower d xy than the d xz and d yz subbands, and a larger extension of the latter two into the STO (see figs. 1.27c and 1.27d).

The two-dimensional character of 2DEGs at the surface of STO could thus be confirmed experimentally by the observation of replicated subbands (with d xy character most visible in ARPES) as shown in figs. 1.27b and 1.27e. Indeed, ARPES measurements reveal five subbands, indicating a lighter effective mass for some of the carriers, while the two shallower bands have heavier mass. Various reports have estimated the respective effective masses from simple parabolic fits or more advanced (multiband) tight-binding fits to be about m l = 0.6m e for the light carriers and m h = 1.12m e or up to m h = 15m e for the heavier bands [START_REF] Vaz | Mapping Spin-Charge Conversion to the Band Structure in a Topological Oxide Two-Dimensional Electron Gas[END_REF]234,235].

Both the formation of multiple subbands and the higher energy of the heavy bands are signatures of the quantum confinement happening at the surface of STO. Additional clues pointing towards the bi-dimensionality of the observed electron gas could be obtained by performing polarization-dependent ARPES giving orbital selectivity due to selection rules (see section 2.2.1) Although the simplistic model described above captures relatively well the energy dispersion obtained by ARPES measurements, a more complete picture is needed to appreciate the full complexity of these systems. A more realistic quantum well description can be obtained first by solving the Poisson-Schrödinger equations and to consider an exponential shape of the confining potential. Indeed, as we saw in section 1.1.3.2, electric-fields particularly at heterointerfaces and in 2DEGs are at the source of inversion symmetry breaking and bring what is called the Rashba effect, akin to a SOC term and which breaks the spin degeneracy. In a simplified ideal linear Rashba model, the magnitude of the resulting spin splitting is in fact directly proportional to the interfacial electric-field.

(a) (b) (c) (d) (e) (f)
In more recent works, the elucidation of the band structure in AlO x /SrTiO 3 2DEGs associated to magnetotransport measurements revealed the exact nature and origin of the conduction properties and the full potential of this kind of systems for spintronics applications.

Although Ti is not a very heavy element and we should expect a rather small spin-orbit coupling, an effective (and gate-tunable) spin splitting of 2 meV to 10 meV has been reported from transport experiments at the LAO/STO interface [START_REF] Nakamura | Experimental Evidence of Cubic Rashba Effect in an Inversion-Symmetric Oxide[END_REF]236]. The non-linear dependence of this splitting with the applied gate voltage along with weak anti-localization measurements suggest that the effective Rashba effect in this system can pass from a k-linear type to a more complex mixture of k-linear and k-cubic Rashba effect from the higher-lying and heavier bands. This can be understood as originating from a non-conventional Rahsba spin-splitting emerging from the multi-orbital nature of the STO-based 2DEGs.

We can in fact see from fig. 1.28 a complex arrangements of the different subbands giving rise to complex electronic energy dispersion and Fermi surface. For low electron fillings, only the lowest d xy band is populated, and the corresponding Fermi surface is simply circular but as the chemical potential is raised, we see the added contributions of the second d xy and the rise of additional bands with higher effective mass. The interesting part here comes when the bottom of the third (orange) band is reached, where we see some orbital mixing and avoided crossings between the d xy and the d xz,yz bands, introduced by the spin-orbit coupling as shown in fig. 1.27d. This region where a second type of carriers with quite different effective mass start to be populated is called the Lifshitz point and we notice that here the successive topologically trivial and non-trivial (with band inversion) avoided crossings come out as an enhancement of the effective Rashba-type splitting in that regime of carrier densities (see figs. 1.28b to 1.28d). To be more precise, as we shall see in chapter 3, each band may contribute to the Rashba-Edelstein effect with a positive or negative contribution and points at which this effect has extrema generally corresponds to maximal splitting and to the avoided crossings. We note that the heavier subbands also have a more anisotropic character bringing a "cigar "-shaped to the Fermi surface.

Although the spin splitting of 10 meV is too small to be detected directly by ARPES with the current resolution obtainable, this complex electronic band structure combined with the highdielectric constant of STO hold together some great promises pertaining to the exploitation of electrostatic effects to tune the band filling and thus the effective spin-orbit coupling (SOC) and spin-charge interconverson in these systems.

Electrostatic gating

As we already mentioned preivously, the large dielectric constant of STO makes it a good candidate from which to fabricate devices based on a field-effect. Thiel et al. [213] showed that it was possible to tune the carrier density of STO-based 2DEGs by applying a gate voltage on a back-electrode while other works have also shown top-gating or even combined top-and back-gating modulation of the gas properties allowing to access different conducting phases of the 2DEG [237].

The first consideration pertains to the basic laws of electrostatics by which the sample can be considered as a simple capacitors contacted with two parallel plates of area A, such that its capacitance is:

C(V ) = ϵ 0 ϵ r (V ) A d , (1.59)
where V is the voltage applied across the dielectric medium, ϵ 0 = 8.854 × 10 -12 F m -1 is the vacuum permittivity, ϵ r is the relative permittivity of the medium, and d is the distance between the two electrodes (the thickness of the insulating medium). As the voltage is varied from V = V 1 to V 2 , the resulting charge accumulation or depletion on either electrode is then:

∆Q = ˆV2 V 1 C(V )dV.
(1.60)

Thus, the effect of the gate voltage on the carrier density accumulated or depleted from the capacitor plates is given by:

∆n s = 1 eA ∆Q = ϵ 0 ed ˆV2 V 1 ϵ r (V )dV.
(1.61)

The back-gate configuration consists in applying a voltage between a bottom electrode at the back of the substrate and the conductive layer from by the 2DEG on the top, with STO forming the dielectric medium, as illustrated in fig. 1.29a. For commercial substrates of 500 µm-thick STO, applying up to ±250 V (higher voltages would cause a risk of electrical breakdown of cables or in a liquid He environment) allows one to reach electric fields of 0.5 MV m -1 (more often found in kV cm -1 in the literature, i.e. here 5 kV cm -1 ). Accounting for the fact that STO's dielectric constant is both temperature but also electric field-dependent as we saw earlier in section 1.3.1, and that it drops down to ϵ ST O (E = 5 kV cm -1 ) ≈ 18 000 at 4 K, one might expect a total change of carrier density of the order of ∆n s ≈ 2 × 10 13 cm -2 at low temperatures. This is quite significant in regard to the total carrier density usually found in STO 2DEGs which typically ranges from few 10 13 cm -2 to 1.5 × 10 14 cm -2 for AlO x /SrTiO 3 2DEGs [START_REF] Vaz | Mapping Spin-Charge Conversion to the Band Structure in a Topological Oxide Two-Dimensional Electron Gas[END_REF]232].

In addition, one should also consider the effect of the gate voltage on electron mobility as a general decrease of the overall mobility can be observed in depleted regimes (with low gate voltage) particularly in the back-gating configuration. Considering both the back-gate and top-gate voltages, it seems that the back-gating acts more strongly on a change of mobility while top-gating would have the tendency to modulate more strongly the carrier density without affecting so much the mobility [237,238].

Achieving an efficient and reproducible field-effect modulation of the STO 2DEG's properties is essential in understanding these properties and to envision any potential device application. Practically, it is actually complicated to access the overdoped regime due to saturation effects offered by the system. It was in fact observed that upon increasing the gate voltage for the first time after cooling down a sample, the resistance would decrease slightly and the saturate, or even increase very slightly again with time (see fig. 1.29b). This is generally understood as the fact that the Fermi energy lies intrinsically near the top of the quantum well in the initial state and a rapid electron filling of the higher energy subband at the top of the quantum well occurs when applying a positive back-gate voltage, while the profile of energy potential is bent down. At some point, electrons start escaping from the top of the quantum well into the conduction band of the bulk STO and get trapped, causing the saturation of the carrier density. When reversing the sweep directions of the gate voltage towards smaller or negative values, one can then observe a strong increase of the resistance, not overlapping the initial bias curve. This can be overcome after the first voltage cycle, as long as the first maximal applied gate voltage is not exceeded, as can be seen from figs. 1.29 and 1.29c [239]. Effectively, the consequence is that the initial state and position of the Fermi level in the sample, defined by the fabrication methods and conditions, usually set a higher limit for the maximal carrier density attainable although it is possible on the contrary to almost fully deplete the gas.

Further cycling can also enhance reproducibility in removing hysteretic behaviors one typically observe in the "forming step" which arises from the reconfiguration of ferroelastic domains. These domains do affect the conduction path, and it is commonly known that transport is usually enhanced at ferroelastic domain boundaries [240]. Transport properties under electrostatic gating

Few more interesting properties showed by these oxide-based conductive interfaces also show specific signatures in transport measurements which we will briefly expose here and try to explain further on.

One of the first landmark in the history of properties discovered at the LAO/STO interface was the appearance of superconductivity, with a critical temperature T c also tunable by a gate voltage (see 1.30a), and reaching a maximum of 300 mK. As in the bulk case, STO 2DEGs diplay a dome-shaped dependence of the critical temperature with the carrier density (and indirectly the gate voltage) as shown in fig. 1.30d.

Magnetoresistance, which denotes a change of resistance of a material under the application of a magnetic field, is usually observed to be positive for a field applied normal to the sample's surface (or out-of-plane (OOP)) Indeed, as we shall see in section 2.4.1, in the presence of a magnetic field, ordinary magnetoresistance can be explained by the effect of the Lorentz force on charge carriers driven by an electric field E and deviated sideways in their trajectory by a perpendicular magnetic field B. Depending on the geometrical shape of the sample or particularly in presence of multiple types of carriers (i.e., with sufficiently different mobilities), one can typically observe a quadratic dependence of the resistance with the magnetic field (µB) 2 at low fields. For the case where carriers from two different bands with different effective mass contribute to transport, a deviation of the magnetoresistance from the simple parabolic shape can be observed in a higher field regime [232,241]. Here this transition would correspond to a change from only d xy transport to a situation where an added contribution from the d xz,yz bands manifests, i.e. to the passage across the Lifshitz point mentioned above. At low temperatures, as inelastic scattering processes reduce, the presence of defects at the interface (due to oxygen or cation vacancies for instance) may lead to additional transport phenomena, where elastic scattering of the carriers (electrons or holes) wavepackets by these defects give rise to quantum interference for instance. Following successive random elastic scattering events, the trajectory of these carriers has a non-zero probability to follow a closed loop in one direction or following the same but inverse path, creating a perfect interference experiment. Constructive interferences of these wavefunctions for instance will induce some localization effects, i.e. an increase of resistance, particularly strong in 2D systems and called weak localization WL [244,245]. Effectively, as a magnetic field is introduced and increased, one can observe a decrease of resistance resulting from the progressive dephasing of the carrier wavefunctions and the destruction of the interferences.

In presence of a strong spin-orbit coupling, a converse effect where the resistance at zero field (B = 0 T) is decreased was described and observed by Hikami et al. [59]: this is the weak anti-localization (WAL) effect. They suggested that the presence of spin-orbit scattering would increase the probability of destructive interferences occuring compared to the constructive ones, thereby enhancing the conductivity, until again an external magnetic field destroys the phase and leaves back the place to other magnetoresistance contributions.

These two effects are frequently observed in STO-based 2DEGs and in particular, WAL provides a rather convenient tool to extract spin-orbit scattering contributions from such sys-tems, which have shown themselves to be tunable as a function of gate voltage (see figs. 1.30b and 1.30c) [START_REF] Nakamura | Experimental Evidence of Cubic Rashba Effect in an Inversion-Symmetric Oxide[END_REF]243,246,247]. The dependence of the above-mentioned effects (superconductivity, quadratic magnetoresistance and weak-(anti)localization) on the carrier density and thus on electrostatic gating can be evidenced by performing Hall efect measurements (which we explain in more details in section 2.4.1). This effect manifests as the formation of a transverse voltage also due to the Lorentz force and reflects a charge accumulation on either sides of the sample. As we can see from fig. 1.31a, the application of a gate voltage of different magnitudes transforms the shape fo the Hall curves and we can clearly identify a transition between a regime with only one type of carriers contributing to transport (with linear Hall effect, below V G < 40 V) and a higher doped regime with another type of carriers taking part in electrical transport and adding some nonlinearity to the Hall effect (for V G above 40 V). Zooming into the low magnetic field region (see 1.31b), one can find that the Hall slopes vary a lot as the gate voltage is increased: from slopes that gradually decrease to slopes that are nearly constant and even increasing again above the critical voltage of 40 V. Extracting the carrier density (see 1.31c) shows how beyond this critical voltage the first type of carriers with mainly d xy character starts to stay more or less constant or to even decrease slightly, while the total carrier density continues to increase [232,241]. This could be explained by a progressive modulation of the confinement potential acting on the subbands splitting, as the gate voltage is varied.

More importantly, passing from a single d xy band transport to a multi-band and multiorbital d xy and d xz,yz transport as we cross the Lifshitz point seems to be the reason for the appearance of the superconducting phase, in addition to the increased spin-orbit effects at this transition as and expected from calculations [248,249] The band structure of these KTO 2DEGs is in principle quite similar to that of STO, however there are few subtleties due to the larger atomic spin-orbit coupling of the Ta and a known lower effective mass for the d xy band in particular (m * ≈ 0.23m e to 0.3m e ) [255,256]. Several studies have used KTO 2DEGs as a platform to understand the electronic properties of these conductive oxide interfaces in parallel to STO from the band structure using ARPES measurements, for instance [START_REF] Vicente-Arche | Spin-Charge Interconversion in KTaO 3 2D Electron Gases[END_REF]234,250,257,258]. Although the visualization of STO 2DEGs Rashba spin splitting remains too challenging to observe directly, the enhanced SOC present in KTO already demonstrated by WAL measurements also permitted Varotto et al. [255] to resolve his splitting and to visualize the effect of the Rashba SOC on the band structure as shown in fig. 1.32a. Another work performed in Unité Mixte de Physique CNRS, Thales laboratory during the time of this thesis showed the extraction of the effective Rashba coefficient α R ≈ -70 meV Å to -280 meV Å via bilinear magnetoresistance (BMR) measurements (based on the direct Edelstein effect (DEE), the exact measurement principle is explained in fig. 2.13 and section 3.2) and without applying any gate voltage. This larger value of effective Rashba coefficient than in STO 2DEGs is also in agreement with SOC found in ARPES measurements [START_REF] Vicente-Arche | Spin-Charge Interconversion in KTaO 3 2D Electron Gases[END_REF]255] and from previous calculations as well as spin-to-charge conversion measurements [START_REF] Vicente-Arche | Spin-Charge Interconversion in KTaO 3 2D Electron Gases[END_REF][START_REF] Zhang | Thermal Spin Injection and Inverse Edelstein Effect of the Two-Dimensional Electron Gas at EuO-KTaO 3 Interfaces[END_REF] and WAL fittings [256].

As for STO-based 2DEGs, different stimuli have allowed to tune the charge carrier concen-tration or the spin-orbit properties, either by illumination, electrostatic gating, or both [253,254,256]. Remarkably, Zhang et al. [256] showed how a LAO/KTO systems could be tuned by illumination either into a spin transport channel with spin diffusion length of 70 nm at low band filling (in the order of 1 × 10 12 cm -2 ), or into a strong spin-orbit system with a Rashba splitting of ∆ = 30 meV at medium filling (4.4 × 10 13 cm -2 ) as displayed in fig. 1.32b. We note that in addition to spin-charge interconversion, a strong orbital Edelstein effect is expected in this kind of system, which adds a paving stone on the experimental way towards the usage and manipulation of orbital currents instead of spin currents in oxides [255,[259][260][261].

On a last note, although superconductivity had never been observed in bulk doped KTaO 3 and only once at its surface by ionic liquid gating (but with so far unreproduced results), a recent discovery at the beginning of the year 2021 revived interest of the superconductivity community, as superconductivity was found in 2DEGs at the KTO (111) and (110) interfaces with amorphous LAO and EuO at temperatures one order of magnitude higher than for STO 2DEGs (see fig. 1.32c). This has now also been reproduced with the same Al deposition as described earlier [262] or with LaVO 3 as an epitaxial top layer [263]. The promise of a two-dimensional system hosting both a strong Rashba spin-orbit coupling and robust superconductivity with higher T c , now accessible at liquid helium temperature, arouses great interest for the study and discovery of topological superconductivty, Majorana quasi-particles and quantum computing [264].

. "Ferroelectric" metals

. A counterintuitive concept

As we have seen in section 1.2, ferroelectrics are normally defined by: 1. the existence of a spontaneous long-range dipole order parameter (the polarization) appearing below some temperature named the Curie temperature T C 2. the possibility to switch that polarization by applying an electric-field across the ferroelectric material.

We also discussed the fact that we can measure this polarization via the displacement current occuring when the coercive field is reached and the polarization switching actually happens, and that beyond experimental considerations, this is actually the ground on which the definition of polarization has been defined by Resta [START_REF] Resta | Theory of the Electric Polarization in Crystals[END_REF] and Vanderbilt & King-Smith [START_REF] Vanderbilt | Electric Polarization as a Bulk Quantity and Its Relation to Surface Charge[END_REF] in their "modern theory of polarization" based on the Berry phase [START_REF] Berry | Quantal Phase Factors Accompanying Adiabatic Changes. Proceedings of the Royal Society of London[END_REF].

The very nature of ferroelectrics imposes the conditon that the material be insulating. Intuitively, if there are any itinerant charge carriers, those can screen electrostatic forces responsible of the polarization in a material. Furthermore, if one were to apply an electric-field across a material with free charges, those would move within the material in response to the electric field preventing the need and the possibility for any polarization to develop -let alone to switch-in the material. This counter-indication on the coexistence of both material properties (the non-centrosymmetric polar structure and the direct current (DC) conductivity) has been formalized by Resta [265]: any metallic material should have delocalized states and the very basic definition of polarization is in contradiction with this as it requires electronic states to be localized [266]. In fact, polarization and localization can be considered as two different aspects of the same phenomenon, both related to a dimensionless complex number starting from the many-body ground-state wavefunctions. While the phase of that number (the Berry phase) gives the macroscopic polarization, the modulus is a measure of localization. Thus, for delocalized states, the polarization becomes ill-defined.

The developement of a polarization in a material at a transition temperature (i.e. pyroelectricity) is fundamentally just the result of a structural transformation lowering the symmetry of the material, with a slight displacement of ions creating a dipole within the unit cell. These small displacive transitions with no diffusion (sometimes called martensitic transformations) were pointed in 1965 by Anderson & Blount [267] (way before the modern theory of polarization was proposed) specifically in the case of the metallic V 3 Si after the report of such martensitic transition (just above the transition at 17 K to a superconducting state), implying the development of a polar axis or at least a non-centrosymmetric structure at the same time as passing from a cubic to a tetragonal phase [268]. Following Landau's theory of secondorder transitions, Anderson & Blount identified the fact that structural transitions involving only a unit-cell transformation (i.e. strain) would have to be of first-order. In other words, the only possibility for a second-order structural transition would be if an underlying primary order parameter appears to change the internal symmetry with the examples given of magnetism, superconductivity or (anti)ferroelectricity. As a consequence, Anderson & Blount thus referred to this system as a potential "ferroelectric" metal, although -as we just discussed-it was later shown that such realization should, at least in principle, not be allowed per se.

In fact, some more recent work suggested some mechanisms by which ionic polar displacements could be sustained provided that the itinerant electrons are only weakly interacting with the ions responsible for the ferroelectric "soft-mode" phonon. In other words, there could be a balance between long and short range electrostatic forces stabilizing polar structures if the elements constitutive of the conduction band orbitals are independent of the ones enduring polar displacements [269,270].

We will see how mutually incompatible properties like superconductivity and magnetism, or magnetism and ferroelectricity [271] or this new and rather counterintuitive concept of "ferroelectric" or "polar" metals stimulated some further experimental works [272,273].

. Searching for polar metallic materials

Although some report of a piezoelectric structural phase transition was made for the Cd 2 Re 2 O 7 in 2004 [274], the first actual report of a material realizing unambiguously the concept we are discussing, was in the metallic LiOsO 3 which was called ferroelectric-like metal due to its observed transition at 140 K to a non-centrosymmetric crystal with a polar axis, reminiscent of other well-studied ferroelectrics like LiNbO 3 and LiTaO 3 . More recently even, ferroelectric switching was evidenced in flakes of conducting WTe 2 [275,276].

Another approach to generate ferroelectric metals studied in parallel is based on the introduction of charge carriers in the bulk of a native bulk ferroelectric metal like BaTiO 3 where the introduction of oxygen vacancies allows the material to become metallic in the bulk for carrier densities beyond a critical value of n c ≈ 1 × 10 20 cm -3 [277] (see fig. 1.33a). In their paper, Kolodiazhnyi et al. [278] say the following: "according to Cohen [279] and Cohen [280], hybridization between the empty Ti 3d and occupied O 2p orbitals reduces the short-range repulsion and allows off-center displacement of Ti ions that drive the FE instability". What they observe is indeed some signatures of a survival of the Ti ions polar displacements by different means, including a remarkable imprint in the resistivity curve vs. temperature of each structural transitions that BaTiO 3 normally endures from cubic (C) to tetragonal (T) to orthorombic (O) and finally to a rhombohedral symmetry, in the form of anomalies (sudden jumps or kinks in resistivity, temperature-hysteretic behavior). Finally, they suggested that this ferroelectric ordering may survive up to some critical carrier density of 1.9 × 10 21 cm -3 . As we saw in section 1.3.1, some materials like STO can exhibit both a proximity to ferroelectricity as T → 0 K and conductivity or even superconductivity when a doping is induced by (oxygen) vacancies or chemical doping. In trying to understand the mecanism for superconductivity, it was evidenced that "soft" optical phonon modes could be a triggering mechanism for the pairing of electrons, in addition to being responsible for ferroelelectric transitions [112]. Researchers progressively tried to combine different methods to approach the (quantum) critical point of zero-temperature transition for the ferroelectric part on one side and of the superconductive part on the other side, since being at the edge of a phase transition may allow to understand better the origin and mechanisms of the investigate phase. This lead to the extensive work already mentioned in sections 1.3.1 and 1.3.2, on isotopic or isovalent ionic substitutions in SrTiO 3 and KTaO 3 bulk systems on one hand, combined with dilute metallicity upon carrier doping on the other hand.

In particular, Rischau et al. [281] showed in 2017 that a superconducting phase could coexist with a ferroelectric instability in Sr 1-x Ca x TiO 3-δ with also some signatures in the transport properties at the expected ferroelectric transition temperature T C (as shown in fig. 1.33b), reminiscent of what was found for the case of doped barium titanate BaTiO 3 (BTO) [282]. The transitions causing these resistivity kinks were confirmed to be of the same nature as in the insulating ferroelectric CaSTO by thermal expansion, sound velocity and Raman measurements. From the latter, it is clear that two transverse optical (TO) soft-phonon modes appear concomitantly with entrance to a ferroelectric phase, and that the results are similar for the carrier-doped or undoped samples.

Remarkably, they also saw that the ferroelectric-like transition was observable up to a certain carrier density (of n = 1.7 × 10 19 cm -3 in Sr 0.991 Ca 0.009 TiO 3-δ ) and that the superconducting transition temperature T c was enhanced in the vicinity of this carrier density compared to that of pure STO, as shown in fig. 1.33c. As the carrier density increases, the authors of the refs. [281,282] detected three different regimes: one at low carrier density with only a kink of the resistivity followed by a continued decrease as the temperature is decreased, one at intermediate carrier densities, where a Kondo-like upturn is seen below the FE T C and finally one high-doping regime where a T 2 -dependence of the resistivity is recovered as normally seen in doped SrTiO 3-δ . The low and medium carrier density regimes disturbances of the resistivity were associated to additional elastic and inelastic scattering mechanisms in the presence of aligned electric dipoles, vanishing compared to the electron-electron scattering mechanisms at higher doping so that the ρ ∝ T 2 trend is recovered [282].

. Motivations of the thesis

A remanent and full voltage-control of spin-charge interconversion would allow for a new paradigm in spintronics logics, eliminating the need to switch or even to use ferromagnets whatsoever. For this reason, using ferroelectrics to control the interfacial Rashba spin texture could bring the possibility to reduce power usage and dissipation significantly in a new generation of logic circuits

As per the definition of the Rashba effect, (i) any symmetry breaking electric field coexisiting with (ii) a sizeable atomic spin-orbit coupling would lead to the spin degeneracy being lifted, resulting -in a simple linear Rashba system-in concentric Fermi contours with spin-momentum locking of opposite chiralities (see section 1.1.3.2).

Thus, several theoretical works in the past years suggested to harness the polar character of ferroelectrics (which match perfectly criterion (i)) at the surface of materials or in heterostructure interfaces between a ferroelectric and a material with a large intrinsic spin-orbit coupling (like heavy metals). Injecting the same spin current in two similar ferroelectric Rashba systems with opposite polarization would then result in a conversion into a charge current in one direction or its opposite depending on the ferroelectric state (see fig. 1.34).

Such investigations were lead for instance at the Pd/PbTiO 3 , Pt/PbTiO 3 [283], Bi/BaTiO 3 or Ba(Os,Ir,Ru)O 3 /BaTiO 3 interfaces [284]. It was also suggested that under certain assumptions, a bulk Rashba effect could also exist, with the first example of BiTeI.

The work of Di Sante et al. [START_REF] Di Sante | Electric Control of the Giant Rashba Effect in Bulk GeTe[END_REF] in 2013, established for the first time a direct link between ferorelectric polarization and the bulk Rashba effect within the single-phase material GeTe, a narrow-gap semiconductor, already well integrated in the semiconductor industry and known for its use in phase-change memories. Later studies by Rinaldi et al. [START_REF] Rinaldi | Ferroelectric Control of the Spin Texture in GeTe[END_REF] and Varotto et al. [START_REF] Varotto | Room-Temperature Ferroelectric Switching of Spin-to-Charge Conversion in Germanium Telluride[END_REF] demonstrated that it was indeed possible to control the Rashba spin texture by somehow switching the polarization of the semiconducting material, and henceforth to control the spinto-charge conversion, paving the way for device applications [START_REF] Varotto | Room-Temperature Ferroelectric Switching of Spin-to-Charge Conversion in Germanium Telluride[END_REF]108,285,286]. This system however, has the disadvantage that the small electronic band gap allows a significant amount of carriers to persist in the bulk, and although it is not enough to kill the polarization, it strongly impedes the ease of switching the polarization from one state to the other. Varotto et al. [START_REF] Varotto | Room-Temperature Ferroelectric Switching of Spin-to-Charge Conversion in Germanium Telluride[END_REF] still found a way to switch the polarization applying very short pulses to the sample, and to check afterwards the polarization-dependent characteristic Schottky barrier height by electrical measurements, or with local conductive-AFM/piezoresponse force microscope (PFM) probe.

Finally, we note that various predictions of a ferroelectrically-controlled Rashba effect were proposed in many different materials [287][288][289] including bulk oxides like PbTiO 3 [290], BiAlO 3 [291], the multiferroic BiCoO 3 [292] and most notably strained KTO [293], but these studies focus simply on characterizing the nature of the band structure, at the valence band maximum and conduction band minimum, in undoped insulating cases. To fully exploit the Rashba properties of these systems, one needs to find a way to make these systems conductive, and falls back to the difficulty of stabilizing polarization in presence of free carriers.

In this thesis, we propose to explore a new route to control the interfacial Rashba effect by ferroelectricity, which consists in using bulk ferroelectrics and to create a 2DEG at their surface. This design approach allows to exploit the strength of both the polarization-mediated electric-field and the strong interfacial electric-field inherent to the quantum confinement of carriers in the 2DEG, which together hold the promise of a relatively large Rashba effect. The interfacial and two-dimensional nature of this conductive layer should not hamper significantly the polarization of the bulk ferroelectric material, and within the assumption of a reasonable carrier density, a spontaenous polarization should also appear below the Curie temperature of the chosen materials, not only in the bulk but also in the conductive region.

In chapter 3, we will first see how the spin-charge interconversion resulting from the direct and inverse Rashba-Edelstein effect can be characterized and controlled in a simple AlO x /STO 2DEG, which we also render ferroelectric by applying a large electric field. In chapter 4, we will then demonstrate another way of controlling remanently the spin-charge interconversion in a 2DEG made on the ferroelectric calcium strontium titanate Ca:SrTiO 3 (CaSTO). Next, we will show in chapter 5 how "ferroelectric 2DEGs" can also become ferromagnetic and display a magnetoelectric coupling, apparent in the transport properties, by utilizing epitaxially grown LaAlO 3 /EuTiO 3 /Ca:SrTiO 3 heterostructures. Finally, we will investigate the possibility to create 2DEGs in other ferroelectric materials like potassium tantalate-niobate KTa 1-x Nb x O 3 (KTN) to hopefully obtain a larger spin-orbit coupling and possibly higher ferroelectric transition temperature.

Before we proceed to the results of this thesis, the next chapter will present or remind to the reader some basic knowledge regarding the experimental techniques used or developed for this study. The top panel with the material's stack represents a general concept where the polarization of a ferroelectric material can be switched by applying a voltage between the top SOC system and a bottom electrode. Here, the suggested SOC system is a 2DEG where the Rashba effect is expected to arise, with the associated prototypical double Fermi contours of opposite spin chiral winding as seen in the second panel. The reversal of the ferroelectric polarization should ideally invert the chirality for both inner and outer contours. Using the inverse Edelstein effect (IEE), a same spin current injecting a non-equilibrium spin-up density in the Rashba system would produce a charge current of opposite direction, all of this controlled remanently by the application of a simple voltage pulse.

CHAPTER 2

Experimental techniques

In this thesis, we use mainly room temperature deposition processes and standard fabrication techniques to form 2DEGs on ferroelectric oxide substrates and use a variety of measurements techniques to study their dielectric and magnetic properties as well as the influence of ferroelectricity on spin and charge transport phenomena.

. Sample fabrication

For the purpose of this thesis, we use single crystal oxide substrates commercially available or obtained through collaborations. They were all prepared (i.e. cut to desired size and doubleside polished for optical measurements) by the crystal manufacturer SurfaceNet.

Before doing any deposition, the surface of these substrates were also prepared by standard cleaning methods using de-ionized water, acetone and isopropanol. An additional treatment in buffered hydrofluoric acid (HF) followed by annealing in oxygen-saturated atmosphere at 1000 °C for 3 h was also performed for some samples. This method is known to favor a TiO 2 termination of Ca:SrTiO 3 substrates, a necessary step to ensure better transport properties and uniformity of the 2DEG for samples grown by pulsed laser deposition (PLD).

Most samples used in this thesis were made by performing room temperature depositions by magnetron sputtering within the UHV system depicted in fig. 2.1. This system connects two UHV clusters (1 × 10 -9 mbar range) and an ensemble of experimental chambers including different pulsed laser deposition systems with integrated laser heater and high pressure reflection high energy electron diffraction, one standard and one confocal magnetron sputtering chambers, as well as a photoelectron spectroscopy chamber with X-ray and UV sources. It allows for for in situ transfer of samples between the different chambers when additonal characterization or preparation steps are needed.

. Magnetron sputtering

The magnetron sputtering technique is widely used to perform depositions and coatings of different materials, in research as well as in industry as it offers a relatively high deposition rate and a rather good conformity of small features and uniformity over large areas. [294] The principle, illustrated in fig. 2.2, is based on sputtering of a target composed of the desired base species by highly energetic ions, formed in a plasma. The sputtering chamber is typically first pumped to obtain UHV conditions. Once this is done, one can start inserting a gas in the chamber (typically an inert gas like Ar), and regulate the mass flow of gas or chamber pressure to the desired range (typically 1 × 10 -4 mbar). A large negative voltage is then imposed on the cathode (the target side) while the anode (on the sample side) is grounded, causing the former to get highly negatively charged, while the latter becomes positively charged. Beyond a certain voltage threshold corresponding to the gas breakdown voltage, electrons start to escape from the cathode and are accelerated towards the anode through the gas. During this process, electrons passing through the gas with high kinetic energy typically endure collisions with the present atoms and ionize some of them as a result, freeing even more electrons in a self-sustained cascading chain of events. Quickly, a plasma is formed in the vicinity of the target with positvely charged ions which will feel the electric field and get accelerated themselves towards the cathode. The collisions of these relatively heavy ions with the target species ejects some neutral atoms from the target and in a transfer of kinetic energy, these atoms typically fly balistically towards the substrate placed in front.

The principle of magnetron sputtering is to favor longer time of flight of electrons near the target to allow more collisions to occur and the number of ions impacting the target and thus increase the efficiency of the process. This is done by placing magnets behind the cathode itself in such a way that under the influence of the magnetic field, the electrons path will be deviated before they can reach the anode.

This phenomenon allows for efficient sputtering of metals, but is however rendered nearly impossible for insulating materials with dc voltage. On the contrary, using an RF source (usually at a frequency around few MHz) fixes this issue by avoiding high charging effects of the target, since on average the electrons diplacement is reduced, while the sputtering Ar ions are too slow to react to this changing electric field and thus follow their usual behaviour. This comes at the cost of a reduced sputtering rate for these materials, which can also be an advantage when a very low thickness of deposition must be achieved. However, it makes magnetron sputtering having the great advantage over other techniques that almost any material can be deposited without the need to reach their melting point as in evaporation techniques.

Several parameters may also be tweaked to obtain desired thin films quality or deposition rates, such as the target-to-substrate distance, amplitude of the current or rf power, heating of the substrates if necessary, or ambient pressure (higher pressures can increase the rate of the sputtering phenomenon up to a certain point but also reduce the mean free path of sputtered species travelling towards the substrate). In addition, it is possible to use either inert gas with different atomic weight, or even to perform reactive sputtering by introducing a mixture of inert and reactive gas species such as oxygen O 2 or nitrogen N 2 , which may combine with the sputtered atoms along their path or when the surface of the substrate is reached. Finally, the complexity but also the interest of the technique can even be increased when using confocal sputtering which allows the simultaneous co-sputtering of different materials to create alloys of desired stoichiometry, in a similar fashion as what can be done with molecular beam epitaxy (MBE) for instance.

For the framework of this thesis, Al, Y and NiFe depositions were performed at roomtemperature in a Plassys MP450S standard magnetron sputtering chamber with a base pressure of about 8.5 × 10 -9 mbar while most Ti and Au depositions for back-gating electrodes were obtained in the confocal sputtering chamber (similar Plassys MP450S model), with similar characteristics. Calibration of each material deposition rate (typically in the range of 1 Å s -1 ) was performed by X-ray reflectometry and atomic force microscope measurements on samples with thicknesses in the range of few tens of nanometers, by checking the step height between covered and non-covered areas of a Si/SiO 2 substrate.

For higher reproducibility of the depositions, especially since we wanted to achieve the reduction of the substrate by reactive metals in vacuum condition, a systematic pre-sputtering of the targets was performed in deposition conditions for some amount of time prior to the actual growth process.

. Device patterning

To perform advanced magnetotransport measurements, in particular for the characterization of the direct Edelstein effect charge-to-spin efficiency, some samples were patterned into Hall bar devices following the provess described in fig. 2 After spin-coating and soft-baking a positive resist (SPR700 1.0), a tabletop maskless near-UV (435 nm) optical lithography technique based on a modified projector was used to imprint the desired pattern onto the sample so that the device area could be exposed after a short step of hard-baking and chemical development in a basic solution (MF319). A thin layer (nanometer range) of the desired material (namely Al) was then deposited everywhere to create the 2DEG, followed by a lift-off in a n-methyl-2-pyrrolidone (NMP) solvent to remove the remaining resist.

To increase the visibility of the devices, two methods have been used alternatively or complementarily: either with an initial similar lithography and lift-off step with Ti/Au markers outside of the main devices area or with the addition of a thick insulating capping layer of amorphous-LaAlO 3 (a-LAO) grown by pulsed laser deposition (PLD) at room temperature, before the lift-off. The latter option has the advantage of offering an additional passivating layer to protect the 2DEG from reoxidation [295] and also of requiring only one step of lithography thereby minimizing chances of affecting it by chemical processes.

The projection lithography equipment (a Microlight3D Smart Print [296]) offers the advantages of a quick drawing of a software-defined lithography masks and the possibility to rapidly adapt and modify them if needed, to obtain features of a size down to 2 µm, and to cover the area of small samples or wafers within few minutes, with an exposure time in the range of one or up to few seconds.

. Photoemission spectroscopy

Photoelectron spectroscopy is a powerful tool to investigate electronic properties of a material. It relies on the photoelectric effect discovered in 1887 by Hertz [297] and explained as a quantization effect of light by Albert Einstein in 1905 [298]: upon exposition to some electromagnetic wave, absorption of incident photons with energy hν will excite electrons. If that excitation energy is sufficient, it may overcome the electron binding energy E B and the work function Φ of the illuminated material causing the electron escape out of their source material with a given kinetic energy E k , momentum vector k and spin orientation S, providing useful information on the electronic band structure and magnetic properties of the system under study.

In 1957, Kai Siegbahn [299] developed the technique based on incident x-ray photons which he called himself electron spectroscopy for chemical analysis (ESCA) and is also known as X-ray photoemission spectroscopy (XPS). Other similar applications have then been developed and the exact name may vary depending on the range of incident photon energy used (from few eV in the UV range to nearly 15 keV for hard x-rays) and the ensemble of properties which may be detected accessible by the associated equipment (e.g. angle, spin, space or time resolution).

. Photoemission of electrons

The photoemission process, common to all these techniques is usually described as a simplified three-step process[300], each with their associated probabilities:

• optical excitation of an electron from an occupied state to an unoccupied state;

• propagation of the electron to the surface with a certain probability of scattering;

• escape process of the electrons out of the material.

Photoexcitation

Considering non-interacting electrons, the photoexcitation is simply governed by the Fermi golden rule as a single electron is suddenly excited from an initial one-electron state |ψ i ⟩ with momentum k i and energy E i to a final state |ψ f ⟩ with momentum k f and energy E f .

For the case of electrons in a periodic crystal, the initial and final wave-functions ψ i,f correspond to Bloch states and are dependent on the wavevector k. The transition rate is therefore expressed as:

W i→f = |M if | 2 δ(E f (k f ) -E i (k i ) -hν), (2.1)
where δ is the Kronecker delta function used for conservation of the total energy during the transition, so that E f = E i + hν. In our situation, the initial state would correspond to the state with binding energy E B of the electron, i.e. the energy required to bring that electron to an unoccupied state at the Fermi level E F . In addition, the first term of eq. ( 2.1) |M if | is the matrix element of the Hamiltonian relating of the dipole interaction between the eletromagnetic vector potential A and the electron momentum, in a perturbative approach:

|M if | = ⟨ψ f (k f )|H int |ψ i (k i )⟩ , (2.2)
where H int is the electron-photon interaction Hamiltonian, equivalent in the dipole approximaton (i.e. A is constant over atomic dimensions) to:

H int ∼ eA m ϵ • p.
(2.

3)

The matrix element depends on the symmetry and overlap of intial and final states wavefunctions and is also strongly governed by the geometry of the experiment. It is thus at the core of the selection rules, including the orbital character of the Bloch states, relative orientation between the electron momentum and polarization ϵ of the incoming photon as well as its energy.

At least for typical ARPES experiments where photons in the vacuum ultraviolet (VUV) range (approximately 20 eV to 124 eV) are used, the photon momentum is usually neglected compared to the electron momentum and the latter is usually assumed to be conserved between initial and final states, modulo a reciprocal lattice vector G, k f -k i = G, i.e. we observe vertical transitions.

Summing eq. ( 2.1) for all initial and final states, (in other words, integrating over the whole Brillouin zone) yields the photocurrent intensity:

I(E i , ν, k) = i,f |M if | 2 δ(E f (k f ) -E i (k i ) -hν). (2.4)
Only electrons in occupied states can be excited, which is equivalent to introducing the Fermi-Dirac distribution function f (E, T ) = not leave the system's effective potential unaffected and a necessary broadening of the spectral function will appear, in the form of satellite peaks in the photoemission spectra.

Finally, if the interaction hamiltonian H int does not couple the photon's spin with the electron's spin, the latter will also be preserved during the optical transition.

Propagation to the surface

Once an electron is excited to the Fermi level, it still has to propagate to the surface before passing to vacuum. During this process, the electron generally has a non-zero probability to undergo elastic and inelastic scattering events due to interactions with other electrons, phonons, etc. which might affect its energy, its momentum direction or even its spin, reducing the photocurrent intensity for a given kinetic energy and k vector and adding background noise in the measured signal. The inelastic mean free path (IMFP) λ IM F P characterizes the average propagation distance between two inelastic scattering events, and follows a universal curve dependent on the kinetic energy (see fig. 2.4):

λ IM F P (E k ) = A E 2 k + B E k , (2.5)
where A and B are coefficients varying whether on whether the studied sample is of elemental nature or an organic or inorganic compound, summarized notably in ref [301][302][303][304].

The propagation to the surface actually follows a typical Beer-Lambert absorption law such that the probability factor D of transport over a distance z is:

D(E k , z) = e -z/λ IM F P , (2.6)
and is used as a multiplication factor to the photocurrent intensity. Since λ IM F P is of the order of 1 nm for VUV to soft x-ray beams, it follows that only a small fraction electrons excited at higher depths (with an incoming penetration depths of about 1 µm) can actually reach the surface and escape from it: photoemission spectroscopies are highly surface-sensitive techniques.

Photoelectron escape to vacuum

As a last step, electrons that have reached the surface may only escape to vacuum if they can overcome the work function Φ of the sample, i.e. if their kinetic energy is still higher than Φ, which is the difference between the vacuum level E vac and the Fermi level E F . Thus, the final kinetic energy of the electrons once they are out in vacuum is:

E k = hν -Φ -E B .
(2.7)

The breaking of translational symmetry along the sample surface normal, as electrons pass from the sample to vacuum, does not guarantee anymore that the momentum along the axis normal to the surface is conserved, as illustrated in fig. 2.5. On the contrary, in the reduced Brillouin zone scheme, the in-plane component of the wave-vector k ∥ is conserved. For an electron with kinetic energy E k = ℏ 2 (k ∥ + k ⊥ ) 2 /2m escaping the sample at an angle ϑ with respect to the surface normal, and an in-plane angle ϕ, the in-plane component of the momentum vector is:

k ∥ = 2mE 2 k ℏ (sin ϑ cos ϕ kx + sin ϑ sin ϕ ky ). (2.8)
Since the out-of-plane (z-)component of the momentum is not conserved, techniques like ARPES is most recommended for low-dimensional systems. However a nearly free electrons approximation with an additional term V 0 , corresponding to the inner potential of the material, can be included to account for the change in energy, and one can estimate the resulting initial out-of-plane component as:

k ⊥,i = 1 ℏ 2m(E k cos 2 ϑ + V 0 ).
(2.9)

In ARPES measurements, the component normal to the surface is always dominant over the in-plane one, allowing to use a small angle approximation cos 2 ϑ ∼ 1. Since E k depends on the incident photon energy, the value of V 0 can be estimated from fits to the measurements by performing a photon energy dependence to finally recover k ⊥,i [300,305]:

k ⊥,i = 1 ℏ 2m(hν -Φ -E B + V 0 ).
(2.10)

One can recover the binding energy E B and the approximate initial momentum vector k from measurements of the kinetic energy E k and angles of emission ϑ and ϕ. The ARPES technique allows to do this and is thus, the perfect tool to investigate the band structure of a material. Eventually, we note that the final photocurrent is also affected by the escape process, with a finite probability of transmission to vacuum T (E k , k ∥ ) and can be entirely written as function of E k and k ∥ :

I f in = I(E k , ν, k ∥ )D(E k , z)T (E k , k ∥ ).
(2.11)

. Angle-resolved photoemission spectroscopy

ARPES is a very powerful technique and has attracted a lot of attention in the past years thanks to the advancements and technological progress in terms of sources (lasers, synchrotron light sources, for instance), as well as detectors and other technical details allowing to develop more and more specialized variants of the technique such as soft X-ray ARPES, or ARPES with time, spin or even spatial resolution [300].

The general principle of photoemission spectroscopy is to shine a beam with defined photon energy on a sample and to measure the photocurrent mediated by escaping electrons as a function of their kinetic energy as they pass through an electron analyzer.

Specifically for ARPES measurements, we typically use a source of VUV or soft x-ray to probe valence states near the Fermi level. While the kinetic energy can be converted to binding energy by the formula in eq. (2.7), and the out-of-plane wave-vector component k ⊥ is fixed by the photon energy, the in-plane k ∥ component is also retrieved by measuring the photoemission angle ϑ.

As a result, one can obtain the electronic energy dispersion along k ∥ at a fixed k ⊥ . Using the sample manipulator as seen on fig. 2.6b to move the sample position and angle with respect to the analyzer, energy dispersions can be mapped at different points and along different directions of interest in the reciprocal space, while orbital effects are probed by changing the incident light polarization and thus, the interaction matrix elements.

Experimental setup: measuring ARPES spectra

The ARPES experiments in this thesis were performed at the CASSIOPEE beamline in SOLEIL synchrotron equipped with a high resolution hemispherical analyzer (Scienta R4000) in regard of the sample placed in a ultra high vacuum chamber (in pressures near 1 × 10 -11 mbar) to limit the probability of collisions of the emitted electrons along their path from the sample to the entrance of the analyzer.

As shown in fig. 2.6b, the sample holder and manipulator benefits from six degrees of freedom thanks to motorized translations and rotations (X, Y, Z, χ, θ, φ) so that different areas of the sample can be probed and accessing different escape angles (and thus different Brillouin zones) is also eased. As they are collected through the entrance slit electrons are also filtered with respect to their escape angle ϑ projected in the plane of the analyzer's entry slit along a fixed axis ϑ x (corresponding to the k x direction), with an acceptance angle up to ±15°(green rectangular window in fig. 2.6d). A set of electrostatic lenses focuses, retards and also filters the electrons by their kinetic energy. Additionally, internal deflectors provide a way to shift the electrons dispersion axis by an angle ϑ y (corresponding to a k y component), which reduces the need for mechanical rotations to map the full energy dispersion along both k x and k y and avoids changes in measurement geometry thereby eliminating matrix element effects.

Once the electrons enter the hemispherical analyzer they are deviated and guided between two hemispherical plates held at a fixed voltage, and over the course of their deflected path, they spread, this time along a perpendicular axis depending on their kinetic energy E k (see fig. 2.6a). Finally, they arrive on the other side of the analyzer, pass through the exit slit and hit the detector directly mapped along two orthogonal dimensions corresponding to kinetic energy and escaping angle axes.

Thanks to the energy selectivity of the entrance electrostatic lenses, a narrow range of kinetic energy (the so-called pass energy ) can be selected and progressively varied to exploit the full resolution of the instrument: performing an energy dispersion measurement simply consists in acquiring the photocurrent intensity I(E k , k x ) along ϑ x and sweeping the pass energy. By varying the deflection angle ϑ y , one can finally recover a full 3D-map of the energy dispersion I(E k , k x , k y ) and get useful information along important symmetry directions of the Brillouin zone, and even obtain constant energy maps (section of the data at fixed selected binding energy) in post-analysis.

Experimental setup: technical details

Since photoemission spectroscopy and in particular ARPES measurement in the VUV range are surface probes, they are also very sensitive to the surface quality of the samples. To avoid any contamination and to limit degradation of the signal by adsorbates, and to preserve the 2DEGs samples from re-oxidation, samples should ideally be prepared in situ or at least transferred in UHV. CASSIOPEE beamline offers the possibility to grow different materials by MBE and to check the surface quality by low energy electron diffraction (LEED) before transferring the sample to the measuring chamber.

We finally note that experiments there can be performed down to cryogenic temperatures (about 13 K) using liquid helium which first reduces thermal noise from the measurement background and is especially convenient to investigate specific temperature-dependent phenomena that may affect the electronic properties.

. X-ray photoemission spectroscopy

In a simlar spirit as for ARPES, X-ray photoemission spectroscopy (XPS) measures the binding energy dependence of the photoemission current intensity. The source used in lab setups, such as the one connected to the cluster in fig. 2.1, is usually a fixed monochromatized Al or Mg anode emitting in the soft X-ray range with energies of 1486.6 eV (Al K α ) or 1253.6 eV (Mg K α ) respectively.

The measurement principle is the same except that instead of accounting for the emission angle, the current arriving on the detector is integrated along the dispersion angle axis, ignoring the wave-vector degree of freedom.

The range of binding energy accessed with XPS extends to few keV which allows to efficiently probe core electronic states. Each peak of intensity appearing in the spectrum should give rather detailed information about the chemical species present in the sample (or at least near its surface) as well as their environment and oxidation state for instance.

In addition to the direct photoemission phenomenon described until now, once the system is left in a highly excited core-ionized state, it can also spontaneously follow de-excitation processes either radiatively in the form of fluorescence or non-radiatively such as with the emission of Auger electrons. These are the result of the Auger-Meitner effect [307] where one electron of higher energy in outer shells de-excites itself to fill the core vacancy and by doing that transfers part of its energy to another outer shell electron which escapes the sample with some kinetic energy:

E k,Auger = E core -E B -E C , (2.12
)

where E core is the energy of the core level vacancy, E B is the energy of the first outer shell electron and E C is the energy of the second outer shell. This kinetic energy might appear in the XPS spectrum and add undesired peaks. Since the total kinetic energy depends only on the relative energy between the different states intrinsic to the atom and not on the energy of the phenomenon that intially caused the electron vacancy, changing the incident photon energy for the photoemission allows to distinguish Auger peaks with no kinetic energy change from actual photoemission peaks.

Other peaks and artefacts may disturb the analysis (satellites, energy loss lines due to plasmon excitation in metals, charging effects for insulators, peak asymmetry caused for instance by additional multiplet splitting...) which should be accounted for in the analysis to obtain proper values of the peaks' area. One should note that the different phenomena which reduce the photocurrent or shift the kinetic energy between the first step of photoexcitation and the final detection will lead to some background and charging effects that are also included in the analysis.

However, in general the spectrum from a complex material is approximately the sum of peaks of the individual components (see for example the one of STO in fig. 2.7). XPS is again a surface sensitive technique although the electrons have a slighlty larger inelastic mean free path λ IM F P ∼ 10 nm. In this way, one can perform quantitative analysis based on each peak (depending on their position, area and special features) and determine the surface chemical composition, the relative proportion of different chemical states, etc, usually with the help of experimental handbooks and reference publications [308,309].

Building on the thesis of Vicente-Arche [START_REF] Vicente-Arche | Quantum Oxide Heterostructures for Spin-Charge Interconversion[END_REF], we have used XPS in this thesis to characterize the reduction state of our STO-and KTO-based substrates before versus after depositing metals that have a high affinity for oxygen, and in particular Al. Despite a relatively low percentage of ionic substitution in our KTaO 3 :Nb substrates, we also attempted to estimate the actual concentration of Nb compared to Ta in the corresponding samples. We usually perform a survey scan to have an overview of the XPS spectrum and check the general chemical composition as well as the actual positions of the peaks of interest, and then more detailed scans can be obtained for each of these peaks.

The photoemission spectroscopy chamber attached to our lab cluster shown in fig. 2.1 endows a Scienta Omicron EA 125X hemispherical analyzer and is pumped down to 5 × 10 -10 mbar. Measurements in this thesis were performed with a Mg K α source (hν = 1253.6 eV) operating at 20 mA and 15 kV. Only XPS measurements were performed for the context of this thesis, although the chamber also comprises a VUV source based on a He discharge lamp (hν = 21.2 eV) to perform ultraviolet photoemission spectroscopy (UPS) measurements and thus examine valence states (and even some states in the conduction band if a sufficient voltage can be applied to the sample, and assuming a rigid shift of the spectrum).

In summary, photoelectron spectroscopies are powerful techniques, particularly useful to investigate chemical properties and electronic band structure at a sample surface. There are however other types of spectroscopies which may allow to access other ranges of information on a sample surface or bulk, some specifcally more mature for extracting element-specific magnetic or structural properties.

. X-ray absorption spectroscopy

X-ray absorption spectroscopy (XAS) is another type of spectroscopy technique, also based on the illumination of a sample by a monochromatic X-ray beam which thus renders possible the excitation of core electrons. A few differences with photoelectron spectroscopies should nonetheless be noted. Instead of scanning the kinetic energy of the escaped electrons, the incoming photon energy is continuously varied to obtain the spectra as an absorbed intensity vs photon energy.

The same theoretical background for the photoexcitation process described in section 2.2.1 can be used here and from the selection rules, one can understand that for some experimental configuration like the geometry and light polarization as well as for a given photon energy, matching the necessary energy for a core electron to be promoted to one empty valence state, an increase of absorption should be noticed. This gives again some indication on the elemental composition and physical as well as chemical environments.

In principle, X-ray absorption spectroscopy (XAS) is performed with a priori knowledge on the sample chemical composition, allowing to select the range of energy to scan to observe the transitons associated to one particular element. Due to the high chemical selectivity and relative sparsity of the excitation events, one typically needs to perform the experiment in a synchrotron facility where both high tunability of the photon energy and high beam intensity can be attained, and thus higher signal to noise ratio.

. Detection modes

The absorption of a beam is generally characterized by the Beer-Lambert law by measuring the transmitted beam intensity after it passed through the sample and comparing it with a similar measurement made without the sample in the optical path: (2.13) where I out is the transmitted intensity out of the sample, I 0 is the incident beam intensity, κ is the absorption coefficient of the medium and l is the length of the path the beam is crossing inside the absorbing medium.

I out (ν) = I 0 (ν)e -κl ,
While in principle it is possible to measure the beam intensity coming out of the sample, it also involves the entire substrate to contribute to the signal which might be undesired for the study of thin films, surfaces and interfaces. There are actually other indirect methods usually employed to observe the absorption process in XAS experiments, depending on the experiment geometry, on the sample transparency and other properties to investigate. For the purpose of this thesis, mainly two modes were used: the total fluorescence yield and total electron yield.

Total fluorescence yield

Secondary effects of the removal of an electron from a core level exploiting relaxation processes are often good alternatives to the direct measurement of transmitted signal. One example of relaxation may come from one electron of an outer shell being deexcited to fill the vacancy and in doing so, emitting a photon with a characteristic energy of the element and electronic levels, with lower energy than the incident photon.

Placing a diode in regard of the sample surface, one can collect the re-emitted light either selectively in energy for the specific elemental fluorescence line(s) or as a whole, in which case we call it total fluorescence yield (TFY). Because the fluorescence photons may again be absorbed by the studied material, one can usually use this technique to probe up to 10 nm to 50 nm below the sample surface.

Total electron yield

In a similar fashion that an electron can jump to the lower energy vacant state after photoexcitation to produce a fluorescence photon, it can also contribute to the emission of another electron in an Auger process as decribed in section 2.2.3. If one measures the electrons coming out of the sample during absorption undistinctively of their kinetic energy, the Auger electrons will add up to the primary photocurrent and one can retrace a similar absorption spectrum as obtained from direct absorption or total fluorescence yield (TFY) measurements. Typically, a wire is bonded to the surface of the sample to offer the best conductive path possible to ground and the total electron yield (TEY) is measured through an electrometer placed outside the measurement chamber.

Similar to what we stated in section 2.2.1, the effects of scattering on electrons limit the probing depth of this technique by only few nm (up to about 15 nm depending on the incident photon energy) [311].

As we can see, each measurement method emerges either from the photoexcitation process or specific relaxation mechanisms and provides different characteristic features in the spectra as well as information on specific regions of the sample.

. XAS spectra

A signficant difference pertaining to XAS is that instead of probing the occupied states as in photoemission spectroscopy, the technique gives more insight into the final unoccupied states to which core electrons are excited. For each element present in the sample, at a specific energy, the absorption cross-section suddenly rises as electrons around that element are then allowed to transition, creating a clear absorption edge in the measured spectrum.

In the pre-edge region, the absorption level is supposedly vanishing, although it can be nonzero as a residual absorption contribution from electrons with lower binding energies may still exist. As soon as the binding energy of the core state under study is reached (hν = E B ), one starts to see the sudden rise in absorption and complex peaks structures and oscillation patterns may appear in the spectra of correlated systems. The study of this X-ray absorption nearedge structure (XANES) is used to determine oxidation state, to deduce the three-dimensional structure and to probe the unoccupied density of states. For higher photon energies, the electrons have enough energy to escape to the continuum or follow different relaxation processes and the observed extended X-ray absorption fine structure (EXAFS) is particularly used to understand not only the density of unoccupied states but also the local structure and to study coordination chemistry.

In the frame of this thesis, in chapter 5 we will use XAS to study some titanates (namely EuTiO 3 and Ca:SrTiO 3 ), which are first row transition metal oxides with 3d 0 nominal ground state configuration for the Ti IV ions and 6s 0 4f 7 for the rare-earth Eu II ions.

The nomenclature for the edges actually reflects from which core states electrons are excited. We note each edge with a letter that corresponds to the principal quantum number of the initial state, i.e. K for n = 1, L for n = 2, M for n = 3, etc. (see an example for 3d transition metals in fig. 2.8a) For the L-edge, we can see from the example XAS spectrum of Pb in fig. 2.8b that it is actually composed of three sub-edges, denoted as L 1 , L 2 and L 3 . L 1 corresponds to the excitation of electrons from 2s states, while L 2 and L 3 refer to the spin-orbit split 2p 3/2 and 2p 1/2 , respectively: when a p electron is promoted to an unoccupied state (the 3d empty states being the most probable, via dipole transitions), it results in two different effective transition energies, and thus two absorption peaks (this is also true in XPS where we typically see two different binding energies for the 2p 3/2 and 2p 1/2 states of transition metals).

The electric dipole transition rules and Hund's rules restrict considerably the final states for which transition probabilities are non-vanishing or non-negligible: for each initial state, final states that can be reached are different, making XAS of 2p states an extremely sensitive technique. What is interesting for us is thus the L 2,3 and M 4,5 edges which correspond to Because electrons are excited to higher shells, XAS also probes the empty density of states of a system and can provide detailed information on the local electronic structure (e.g. orbital occupation and symmetry) and magnetic properties of the studied element. Effectively, there are a number of additional contributions like charge transfer, spin-orbit coupling, crystal and ligand fields, Zeeman splitting, etc. which modulate the final density of states, transition probabilities and/or lift the orbital or spin degeneracy.

. X-ray magnetic circular dichroism

While dichroism was initially referring to the change of color of some material absorbing light in one direction or another, nowadays this term is more generally used to describe the dependence of absorption spectra on light polarization.

The existence of X-ray magnetic circular dichroism (XMCD) was first explored theoretically in 1975 in a work by E.A. Stern [314,315] in which dichroism was predicted at the M 2,3 edges of ferromagnetic nickel (Ni). It took another ten years however, before the first signature of a magnetic dichroism with linearly polarized light was actually observed based on atomic multiplet simulations [316], at the Tb M 4,5 absorption edges in the ferrimagnetic garnet Tb 3 Fe 5 O 12 [317] and soon after with circularly polarized light at the Fe K edge in an iron foil. X-ray magnetic circular dichroism (XMCD) is of course not limited to XANES [318], nor to the hard x-ray range and a huge dichroic signal was thus quickly shown at the Ni L 2,3 edges for a ferromagnetic film of nickel [319].

XMCD is thus a magneto-optical phenomenon obtained by taking the difference between two XAS spectra taken with different helicities of the incoming X-ray beam.

XMCD mechanism

A two-step model (schematized in 2.9a), initially suggested by Schütz et al. [320] and further extended by Stöhr & Nakajima [321] is used to describe the phenomenon. The first step consists in the incoming photon transferring its ±ℏ angular momentum (for left-handed C+ or right-handed C-photon, respectively) to an electron, due to conservaton of angular momentum. If the initial core states are spin-orbit split, either or both spin and orbital moment of the photoelectron can be affected by this photon angular momentum.

For example, at the L 2,3 edges, the angular momentum can be transferred partly to the spin due to spin-orbit coupling. The sign of the photoelectron's spin polarization then depends on the helicity of the incident X-ray photon and on the excited edge due to spin-orbit coupling (l + s for the L 3 edge and l -s at the L 2 edge): it is reversed if the helicity is changed, or between L 3 and L 2 . Note that as a result, for a given helicity, the transition probabilities depend strongly on l due to the selection rules.

In the second step, as the photoelectron is promoted to the valence shell, the density of empty states will then act as a filter or detector for the photoelectron carrying a spin and orbital moment. As a consequence, an imbalance in spin or orbital moment in the final states will give rise to a dichroic signal (with the example of Co in fig. 2.9b), of which the amplitude is proportional to the scalar product between the magnetization M of the sample and the helicity vector k of the incident light (the effect is maximal when both are aligned). We remark that this dichroism can be observed either by changing the light polarization helicity or simply by switching the magnetization (applying a sufficient opposite magnetic field).

The resulting XMCD spectra can be exploited to extract the contributions of the spin and orbital moments to the total magnetic moment related to a specific element and state, with the help of a powerful framework called the sum rules, provided by Thole et al. [322] and Carra et al. [323,324]. We provide a quick illustrative example in fig. 2.10 where we denote the XMCD intensities (the area of each peak) for the L 3 edge as A and for the L 2 edge as B which clearly have opposite sign, here as signature of the opposite spin-orbit coupling of p 3/2 (l + s) and p 1/2 (l -s) levels.

If we sum the contributions to the dichroism signal of all electrons that can be excited from L 2 and L 3 , it appears that the spin contribution is eliminated and the orbital moment is thus proportional to the sum A + B. Another powerful rule links the spin moment to the difference A -2B with a simple multiplication factor related to the interaction cross-section (the square of the matrix element for the specific orbital and spin moment between initial and final states).

Although these sum rules have some limitations due to some assumptions discussed in [325], and some arbitrariness in defining the limit of A and B, their validity for the L 2,3 absorption edges in a wide range of 3d, 4d and 5d transition metals as well as for M 4,5 edges of actinides seems well established.

Not only can one extract the respective spin and orbital angular momentum for each studied element of the measured compound, but for elements with a ferromagnetic character, one can also measure magnetic hysteresis cycles corresponding to that element by measuring step by step the XAS and subsequent XMCD for different values of the field. Typically, the photon energy is fixed at a value which maximizes the dichroic signal (obtained previously in a XMCD spectrum) and the field is swept or changed step-by-step or with a sign-reversal and inversions of the light helicity as well, between each step. The hysteresis cycle is then reconstructed from measurements taken in similar conditions. The field and helicity reversal technique allows to reduce spurious effects of drift and changes in the incident beam and the sample under illumination.

. X-ray linear dichroism

On a similar principle as for the XMCD studied in the previous subsection, we also briefly mentioned another type of dichroism, i.e. X-ray linear dichroism (XLD) that also occurs in the X-ray range but this time depending on the linear polarization of the incident photons. This phenomenon doesn't have to be due to magnetism (neither does circular dichroism) and in fact we use it in this thesis to study interfacial structural properties of our samples rather than for the magnetic properties.

As we already suggested, transition matrix elements governing the photoexcitation process are sensitive to the local environment of the absorber ions. In this way, a change in the local symmetry or relative positions of surrounding ions may lead to a change of crystal (or ligand) field energy splitting and thus to a modulated anisotropic response to light depending on its

polarization.

A good example for the context of this study is the XAS and X-ray linear dichroism (XLD) spectra at the L 2,3 edge for the Ti ions in SrTiO 3 -based 2DEGs.

In the cubic perovskite structure of SrTiO 3 , the oxygen octahedral cage surrounding the Ti ions already splits 3d states with 10D q ≈ 2.5 eV, regrouping the d xy,yz,xz into lower energy t 2g levels and the the d x 2 -y 2 ,3z 2 -r 2 into e g levels. Passing from the cubic structure to a tetragonal structure further reduces the symmetry and lifts the orbital degeneracy even more by separating the orbitals with only in-plane components (x and y) from those pointing in the out-of-plane (z) direction (see fig. 2.11).

The dipolar interaction between the light polarization and the electrons in each orbital (and thus the transition probablity) is reinforced or weakened depending on their relative orientation as illustrated in fig. 2.11b: while in normal incidence both orthogonal linear polarizations of the light (0°and 90°) will probe a and b in-plane axes of the material, there will be a noticeable difference in a grazing incidence, if the light polarization is mostly parallel to the elongation c axis of the crystal, or to the in-plane a and b axes (i.e. one will probe more or less of the in-plane or out-of-plane orbitals).

The lowering of symmetry in SrTiO 3 can be achieved simply by passing below the structural transition temperature at 105 K, by inducing compressive or tensile strain but also by applying an electric field and exploiting some electrostriction effect.

In the case of most STO(001) 2DEGs, and in particular the prototypical LAO/STO system, it is also known that the strong interfacial inversion symmetry breaking results in the conduction bands close to the interface being formed mainly of the Ti lower lying d xy orbitals while higher d yz,xz orbitals can be progressively filled by electrostatic gating (see section 1.4).

It was shown by Salluzzo et al. [328] that the step-by-step addition of LAO monolayers caused a crystal field splitting of few tens of meV to appear already from the first u.c., causing a XLD signal (in grazing incidence) to grow, up to the critical thickness of 4 u.c. where it saturates, as the conductive layer appears at the surface of STO (with an example shown in fig. 2.11c. They excluded a strong contribution from oxygen vacancies and suggested that a rumpling of the TiO 2 vs. SrO planes (i.e. a polar distortion) was formed. The orbital ordering seen in these 2DEGs would thus originate not only from the confining potential due to the presence of charges itself (and a mixed Ti 3+ /Ti 4+ valence or mixed 3d 0 /3d 1 configuration), but also probably in a response to the polar discontinuity with LaAlO 3 .

Other examples of XLD attributed to a polar structure particularly in ferroelectrics have been shown in different papers [329][330][331][332], where the dichroism signal could even be changed depending on the polarization direction.

XAS (and the related dichroism phenomena) is thus a promising technique for us to detect signs of magnetism or any potential change in the structure of STO in the 2DEG region due to ferroelectricity.

. Electrical transport measurements

A range of magnetotransport characterization has been performed in this thesis to investigate the local charge and spin transport properties of our samples as well as their dependence on the application of a gate voltage. In this section, we will describe the fundamental principles and technical details used for the standard resistivity, magnetoresistance and Hall measurements as well as for the in-plane magnetoresistance measurements from which we can try to detect characteristic contributions of the charge to spin conversion by the direct Edelstein effect (DEE) effect.

. Basics of magnetotransport measurements

In this thesis, basic characterization of the transport properties have been performed on the full-plane and patterned samples to extract the sheet resistance, carrier density and mobility.

Sheet resistance

The concept of sheet resistance is convenient for the measure of how thin films behave electrically when subjected to a current flow. It applies particularly for conducting layers that can be considered uniform over the whole thickness and where essentially the conduction takes place in a two-dimensional system along the plane. It is thus some kind of extension of the resistivity, abstract from the thickness of the conductive layer. Formally, in the case of a 3D conductor with cross-section A = tW , width W , thickness t and length L, the well-known resistance formula R is:

R = ρ t L W = R s L W , (2.14)
where ρ is the bulk resistivity of the material, and R s is the so-called sheet resistance, homogeneous to a resistance and expressed in Ω or more often in Ω □ -1 . The symbol □ symbolizes how the resistance of a square two-dimensional conducting layer of any size will have the same value as its sheet resistance. Thus, the simple link between both quantities R s and R is just a scaling factor corresponding to the aspect ratio of the conducting channel W L .

Measurement configurations

An important technical notion in electrical transport measurement is that of the fourprobes technique. It is widely used to remove the error one would get from the series resistance inherent to any contact, and it can be quite significant compared to the sample's own resistance, especially for point contacts on semiconductors.

Effectively, as illustrated on fig. 2.12a, one injects a fixed current to pass between two contacts and measure the voltage difference between two points typically placed along the current path (local transport). Since the current is forced in the main channel it is independent of the series resistances of the current contacts and within the assumption that no current flows through the voltage probes to the measurement apparatus, the voltages measured there are also abstracted from any contribution from the contact resistance.

Additionally, some common artefacts due to thermoelectric effects or offset from the current source or measurement equipment can be removed by changing the polarity of the injected current and acquiring the corresponding voltage, which results in the resistance: Hall bar devices are usually a good method to measure transport properties as the main channel makes well-defined paths for the flow of current, and the transverse arms provide the additional voltage contacts, needed to get rid of the contact resistance and to measure the Hall effect. One can simply use Ohm's law between opposite contacts, for instance injecting current I 12 between contacts 1 and 2 and then measuring a longitudinal voltage between 3 and 5 (V 35 ) and a transverse voltage between 6 and 5 (V 65 ), we get: Obtaining the longitudinal sheet resistance is as simple as explained just above, i.e. using the aspect ratio of the current channel as a scaling factor.

R = V (I + ) -V (I -) I + -I - , (
R
Hall bars have the advantage that one can measure both the longitudinal and transverse voltage at the same time, since all the needed contacts are there, and the well-defined geometry and current paths usually limits some of the spurious effects found in other measurement configurations. However, one has to pay attention to few geometrical details when fabricating such a device, to optimize the signal quality, reduce errors in the measurements due to contacts misalignement, or current flow in the voltage branches for instance. This also requires usually to affect the sample's shape by patterning it either by lithography techniques or deposition through a mask, for instance, which both limits the reuse for some other experiments and also comes with a chance of alteration or degradation of the initial physical properties. Finally, using this kind of contacting configuration also requires to have some prior knowledge of the sample's exact dimensions and distances between the contacts to extract any transport property.

Another method abstract from the latter limitations was proposed in 1958 by van der Pauw [333] as a way to estimate the average resistivity for samples of nearly any shape, provided they have homogeneous transport properties in terms of thickness, that they have no geometrical "hole" (or insulating areas in the middle) and that the contacts are small and located on the perimeter.

Figure 2.12c displays an example of van der Pauw configuration on a square sample as typically used in this thesis. Contacts are placed in the corner and while the current is injected for instance from contact 1 to contact 2 (I 12 ), the voltage is measured along the opposite edge, in this case between contacts 3 and 4 (V 34 ). The resistance is then again: R 12,34 = V 34 I 12 .

( .18) With permutations between the contacts, one can compute the average resistance values along both directions, say x and y, of the sample, resulting in: (2.19a) and R yy = R 13,24 + R 24,13 2 .

R xx = R 12,34 + R 34,12 2 ,
(2.19b)

Van der Pauw showed that the following relation could be drawn between the resistances in both directions:

exp - πt ρ 3D R xx + exp - πt ρ 3D R yy = 1, (2.20)
where again t is the thickness of the sample (or across which conduction happens) and ρ 3D its resistivity. As a result, one can rewrite eq. ( 2.20) to isolate ρ 3D such that:

ρ 3D t = π ln(2) R xx + R yy 2 f, (2.21)
where again we can recognize the sheet resistance R s on the left side of the equation, and f is a special correction (van der Pauw) factor that takes values between 1 and 0.2 and is related to the ratio q = Rxx Ryy (or q = Ryy Rxx , whichever is the largest) by the equation: 2) .

q -1 q + 1 = f arccosh{ 1 2 exp ln(2) f } ln(
(

2.22)

To obtain the actual value of f one may either use an approximation of the above formula for values of q close to 1, or solve for f using iterative methods.

Assuming contacts are mirrored with respect to symmetry axes of the samples if any, and using the reciprocity theorem R 12,34 = R 34,12 , one may be allowed to reduce the number of measurement considerably. We note particularly that, if the sample shows some transport isotropy along the main axes of measurement, q = Rxx Ryy = 1 and the sheet resistance equation is further simplified:

R s = ρ 3D xx t = π ln(2) R xx .
(2.23)

Similar to the Hall bar case, one can also measure transverse voltage, however it cannot be done simultaneously along with the longitudinal voltage. Indeed, to do this, one has to physically exchange two contacts, say 1 and 3, so that the current is injected along one diagonal and the voltage measured along the other, between 2 and 4.

This becomes particularly important to perform magnetotransport and obtain both the (longitudinal) magnetoresistance and the Hall effect (transverse).

Magnetotransport

Any carrier with charge q moving within a plane at a velociy v in a uniform perpendicular magnetic field B experiences a force F L = qv × B called the Lorentz force which deviates its trajectory. The charge carrier's path will then form closed orbits of radius r = mv |q|B scaling inversely with the magnetic field strength and at a frequency ω = |q|B m , m being the carrier's mass. If additionally the carrier is subject to an electric field E, the total force it experiences can be written as:

F L = q(E + v × B).
(

2.24)

What this means is that upon injecting a current in the plane of a two-dimensional conducting sample, while charge carriers should propagate along the main direction of the electric field (for instance, E = E x x) corresponding to the imposed current (J = J x x), in fact carriers will start to be deviated along the ŷ axis by the magnetic field B = Bẑ and accumulate on one side while being depleted from the other side. In reponse to this charge accumulation, an opposite electric field component along ŷ also builds up, until it brings back equilibrium against the magnetic force (making Lenz' law effective): this is called the Hall effect. In the steady state, one can thus observe a transverse potential drop called the Hall voltage V H , with no effective current flowing along ŷ and here, we will define the 2D transverse resistivity as:

ρ 2D xy = ρ 3D xy t = V + H -V - H I --I + .
(2.25)

Similarly to the approximation we have done for the longitudinal resistance, we consider ρ 2D xy valid for the resistivity of the two-dimensional sheet with negligible thickness, and we note it later on as R xy .

As the magnetic field increases, the transverse voltage build-up should increase linearly with a slope called the Hall coefficient, which in 2D systems can be defined as:

R 2D H = R xy B .
(2.26)

This coefficient is a powerful tool used to determine the doping type (p-type with holes or p-type with electrons) as well as the carrier density and mobility. Indeed, if we are in presence of holes, the slope will be positive, while it should be negative for electron transport and the sheet carrier density (the number carriers per unit area in cm -2 ), n s = n 2D is expressed as:

n s = 1 qR 2D H . (2.

27)

At zero magnetic field and assuming that the conduction is only or dominantly achieved by one type of carriers, the sheet resistance is simply the inverse of the longitudinal sheet conductivity σ -1 xx = (n s qµ) -1 , where the mobility µ (in cm 2 V -1 s -1 ) is thus:

µ = 1 |q|R s n s .
(2.28)

As we have seen, the resistivity cannot in all circumstances be expressed as a scalar, especially in the presence of a magnetic field and one has to use tensorial formalism to express Ohm's law E = ρ 2D J, where E and J are respectively the electric-field (in V m -1 ) and current density (in A m -1 ) vectors in the plane and ρ 2D the 2D resisitivity expressed as:

ρ 2D (B) = ρ xx ρ xy ρ yx ρ yy = R s B qns -B qns R s , (2.29)
where, from the Drude model, µ = eτ m is the mobility of the carriers, m is their effective mass and τ is the scattering time. Note that due to Onsager's reciprocal relations, the resistivity matrix is anti-symmetric, with ρ xx = ρ yy and ρ xy = -ρ yx which explains the last part of eq. (2.29).

The conductivity is also a tensor which we can obtain by inverting the matrix from of the resistivity tensor:

σ 2D (B) = nsqµ 1+(µB) 2 nsqµ 2 B 1+(µB) 2 -nsqµ 2 B 1+(µB) 2 nsqµ 1+(µB) 2 = σ xx σ xy -σ xy σ yy , (2.30)
where we used the relation eq. ( 2.28) to replace R s . It can also be noted that the diagonal elements of the resistivity matrix (a.k.a. the sheet resistance) in eq. ( 2.29) are independent of the magnetic field, while the conductivity is not. In fact, the magnetoresistance in a single band isotropic system should ideally be quasi-inexistent, although some variations can actually observed in reality and the magnetoresistance is generally expressed as a function of the ratio between the magnetic field strength and zero-field resistivity as: M R = f ( B ρ 0 ) (Kohler magnetoresistance).

The situation changes when one considers a system with multiple conduction channels or types of carriers with different mobilities, contributing to transport. This can then be approached as a network of parallel conductors, which means that the total conductivity is simply the sum of each channel's conductivity:

σ 2D tot (B) = N j σ 2D j (B), (2.31)
where σ j is the conductivity of the j-th channel.

In the same way we did from eq. (2.29) to eq. ( 2.30), one can again revert to resistivity by inverting the conductivity matrix. In the special case where there are only two types of carriers N = 2, each with density n j , charge q j and mobility µ j the longitudinal sheet resistance becomes simply:

R s (B) = σ xx (σ xx ) 2 + (σ xy ) 2 (2.32a) = (a 1 (1 + b 2 2 ) + a 2 (1 + b 2 1 ))(1 + b 2 1 )(1 + b 2 2 ) (a 1 (1 + b 2 2 ) + a 2 (1 + b 2 1 )) 2 + (a 1 b 1 (1 + b 2 2 ) + a 2 b 2 (1 + b 2 1 )) 2 .
(2.32b)

where a j = n j q j µ j and b j = µ j B, and the transverse resistance becomes:

R xy (B) = σ xy (σ xx ) 2 + (σ xy ) 2 = - B (n 1 µ 2 1 + n 2 µ 2 2 ) + (µ 1 µ 2 B 2 )(n 1 + n 2 ) e((n 1 µ 1 + n 2 µ 2 ) 2 + (µ 1 µ 2 B 2 )(n 1 + n 2 )) , (2.32c)
for which we assumed both types of carriers were electrons q j = -e. This time, the contributions of both channels introduce some mixing of their respective conductivity, which then results in a dependence of the resistance in b 2 (or (µB) 2 ). With sufficiently different mobilities, one would be able to see some transition regime between the low-field and high-field regions (depending on whether b j = µ j B ≪ 1 or b j = µ j B ≫ 1 for each type of carrier), with one type of carrier or the other dominating charge transport. Particularly of importance, in the high-field limit, the formula for the transverse resistance R xy ≈ 1 B(n 1 q 1 +n 2 q 2 ) similar to eqs. (2.26) and (2.27) tells us that if both types of carriers are of the same sign, the total carrier density can be extracted from the slope at high magnetic field.

In this thesis, we use results in eqs. (2.32b) and (2.32c) to fit experimental transport data and extract the carrier densities and respective mobilities.

Experimental setup

Most of the magnetotransport data presented in this work were performed in a Quantum Design Dynacool physical property measurement system (PPMS) capable of applying a maximum magnetic field of 9 T and of reaching the base temperature of about 1.8 K. Most of the basic transport measurements were performed by applying a dc current either with the Resistivity Module option of the PPMS or with an external Keithley 2400 sourcemeter unit, after bonding the samples in the van der Pauw configuration with Al microwires using a Kulicke & Sofa 4523AD wedge bonder.

For measurements requiring the application of a gate voltage, an additional Ti/Au backelectrode was sputtered onto the back of the sample, which was then glued with Ag paste onto a piece of Al foil or of a Ti/Au-coated (by sputtering) MgO crystal, itself glued on the sample holder. Some bonding was then made onto the conductive "buffer substrate" so a gate voltage could then be applied between the sample bottom electrode and the ground lead of the current source, using a Keithley 2450 sourcemeter as a current limited voltage source (with a compliance of ≤ 1 µA).

. Nonreciprocal transport from the Edelstein effect

Nonreciprocal transport is a phenomenon that causes a directional propagation of particles (electrons, photons, magnons, ...) or current in a medium. In other words, given a twoterminal system displaying nonreciprocal response, the (optical, electrical, ...) conductivity σ of that medium will show a different value whether the stimuli comes from the right or from the left terminal:

σ (k, ω) ̸ = σ (-k, ω) (2.33)
where k is the wavevector of an electromagnetic wave and ω is its frequency (see fig. 2.13a). This may occur in a variety of systems exhibiting broken inversion symmetry Î, one typical example being the rectification effect of a p-n junction, but the broken Î does not necessarily imply the existence of nonreciprocal phenomena. However, the additional breaking of timereversal symmetry T in microscopic systems typically causes them to exhibit a macroscopic nonreciprocal response, as is seen for the magnetochiral effect [334].

In the case of electrical transport, where we can replace the k vector by a current I (or current density j), a heuristic approach leads to the following expression for the resistance [335]:

R = R 0 1 + βB 2 ± γBI (2.34)
where the B 2 -dependent term is the usual magnetoresistance, and the sign of the third term is dependent on the direction of the current.

In particular for the direct Edelstein effect (DEE), as we saw in section 1.1.5, when applying a current along one direction +x in a Rashba system (i.e., with SOC and broken Î), a trans- verse spin polarization appears along ±ŷ depending on the spin chirality of the Rashba-type Fermi surface. Upon reversing the current direction, i.e. injecting it towards -x, the sign of that "magnetization" (spin accumulation) will also be reversed. While this phenomenon does not affect directly the directionality of the transport properties, removing T by applying an external magnetic field for instance, triggers this kind of phenomenon, as we explain in details in section 3.1.3. In this case, the external in-plane magnetic field B interacts constructively or destructively with the effective field B j (or magnetization M j ) generated through the DEE, whether they are parallel or antiparallel. In other words, the presence of B yields different values of resistance, whether the current I is injected in one direction or another, at 90°or 270°from B. We can use a corrected form of eq. (2.34) to express this vectorial relation:

R = R 0 1 + βB 2 + γB × I (2.35)
The resulting change of resistance is thus linearly dependent with the applied field and injected current, but its amplitude is usually rather small in comparison to the other terms, and particularly to the base resistance at zero magnetic field R 0 = R (B = 0): for LaAlO 3 /SrTiO 3 2DEGs, at B = µ 0 H ∥ = 9 T and j = 40 A m -1 , the observed values reach up to maximum 1 % of the sheet resistance [336].

We perform this kind of measurement by injecting a fixed current amplitude along the main channel of Hall bar patterned on the desired sample that we place on a rotating sample holder inside a cryostat, which allows us to rotate the sample while keeping the applied magnetic field in the film plane which in the reference frame of the sample is equivalent to a rotation of the field with respect to the current density vector (see figs. 2.14c to 2.14f).

. Low-noise transport measurements

To maximize the amplitude of the signal of interest described in the previous subsection and furthermore the signal-to-noise ratio (SNR), it is not only required to increase the magnetic field and current density as much as possible, but one needs to use low noise measurement techniques and appropriate measurement hardware. As for classical transport measurements (see section 2.4.1), we can perform dc electrical measurements by applying a dc current using a low noise current source (e.g. a Keithley 6221) and measuring the longitudinal resistance with a high precision voltmeter (e.g. with a Keithley 2182A Nanovoltmeter). In this case, we acquire the resulting voltage for both directions of the current and the directional component of the resistance R asym is obtained by taking the half difference between both states, while the other symmetric components R sym are obtained as usual by taking their average: R asym = 1 2

V + I + - V - I -, R sym = 1 2 V + I + + V - I -.
(2.36)

Here, we should note that the equation for R sym is equivalent to eq. ( 2.15), provided that there is no current offset and that its amplitude is actually the same for both polarities, that is to say:

I + = -I -= I.
One inconvenient with this technique is that both symmetric and directional components are initially mixed in the same signal and that any thermoelectric effect (for instance due to the contacts) might appear dominant and disturb the disentanglement and further analysis. It is also subject to most sources of noise in a large frequency range, which can dominate the small asymmetric signal components, although part of that noise can be reduced with power linecycle (50 Hz) filtering and statistical averaging.

An alternative technique is to use synchronous detection of electrical transport responses with a lock-in amplifier (LIA): an ac current I (ω, t) = I max sin(ωt) is injected in the device and used to modulate the signal at a chosen reference frequency ω = 2πf . The resulting oscillating voltage V (ω, t) = ZI (ω, t) across the sample is measured by the LIA which is able to [337]:

• filter out any undesired signals (and noise) away from the reference frequency while keeping signals that are coherent with it;

• compute the dephasing of the output voltage signal and separate the in-phase X and quadrature Y (90°dephased) signal components;

• do both of the above at any integer multiple (harmonic) of the base reference frequency ω, i.e. extract signals generated at nω.

The latter point comes of particular interest for transport measurements involving higher order harmonic generation that is typical in materials showing some nonreciprocal response or simply breaking one or both of Î or T symmetries [334]. Specifically for the situation studied in this subsection, the impedance Z can be replaced by a simple resistance term R and the total voltage V (ω, t) obtained at the input of the LIA in response to a current I (ω, t) is the following:

V (ω, t) = RI (ω, t) = R 0 1 + βB 2 + γB × I (ω, t) • I (ω, t) = R 0 1 + βB 2 I (ω, t) ± R 0 γBI 2 (ω, t) , (2.37)
which, replacing I (ω) by its full expression leads to:

V (ω, t) = R 0 1 + βB 2 I max sin(ωt) ± R 0 γBI 2 max sin 2 (ωt) = R 0 1 + βB 2 I max sin(ωt) ± 1 2 γBI 2 max [1 -cos(2ωt)] , (2.38)
where we can finally identify three harmonic components at nω, with n = 0, 1, 2:

V dc (t) = ± 1 2 R 0 γBI 2 max (2.39a) V ω (t) = 1 + βB 2 R 0 I max sin(ωt) (2.39b) V 2ω (t) = ∓ 1 2 R 0 γBI 2 max sin 2ωt + π 2 .
(2.39c)

The dc component is a rectification term as a consequence of the modulation of the resistance itself when the injected current induces an instantaneous reversal of the spin density generated by the Edelstein effect. With the LIA, we will discard this rectified dc term but we will be able to retrieve separately the first and second harmonic signal amplitudes containing all the information we need about the symmetric and asymmetric components, respectively. From these equations the formulas to obtain the resistance are easily obtained as:

R ω = √ 2⟨V ω ⟩ I max = R 0 1 + βB 2 ,
(2.40a)

R 2ω = √ 2⟨V 2ω ⟩ I max = ∓ 1 2 R 0 γBI max .
(2.40b)

Care must be taken to use properly scaled (root-mean-square (RMS) or peak) amplitudes to retrieve the right value of the resistance. In addition, one can see that the second harmonic resistance should be multiplied by two to recover the proper value of the unidirectional term seen in eq. ( 2.34). More generally, any term of the resistance that depends linearly with the input current will appear in the resulting voltage in the second harmonic with a phase shift of π 2 radian (or 90°), i.e. in the quadrature component. Specifically for the bilinear (with the current and the magnetic field) magnetoresistance effect considered here, one can define the figure of merit γ = A BM R R 0 IB (also called η in the literature) which allows to better compare results from one system to another.

. Spin-to-charge current conversion using spinpumping ferromagnetic resonance

To characterize the spin-to-charge conversion efficiency of our systems, we have performed spin pumping experiments using ferromagnetic resonance (FMR) in a RF resonant cavity. We explain hereafter what is the general working principle, how the experimental setup is actually operated and spurrious effects that might appear.

. Ferromagnetic resonance

Magnetization dynamics

As we briefly saw in section 1.2.1, in a ferromagnetic material, strong exchange interaction tends to align the magnetic moments in the same direction, giving rise to collective behaviors and a total macroscopic magnetization M = M s m, aligned with a magnetic field H ef f .

Starting from this, a first model of the precession dynamics of the magnetization in a ferromagnetic material was proposed by Landau and Lifshitz in 1992 [338, 339]. The Landau-Lishitz equations -as they were coined-are differential equations that describe the precession mechanism of a magnetization as the system is brought out of equilibrium, where a torque τ = M × H ef f starts to act on the magnetization:

∂M ∂t = -γM × H ef f , (2.41)
where γ = gµ B ℏ is the gyromagnetic ratio, g is the electron g-factor (taking values between 2 and 2.2 in ferromagnets), µ B is the Bohr magneton and ℏ the reduced Planck constant. The effective field H ef f regroups contributions from the external magnetic field H 0 , the demagnetizing field H D and the anisotropy field H an :

H ef f = H 0 + H D + H an , (2.42)
The simple model of eq. ( 2.41) describes a perpetual motion of the magnetization (fig. 2.15a) which, of course is not achieved in reality. Some relaxation processes like electron-phonon, electron-magnon or magnon-phonon scattering mechanisms will cause a loss of angular momentum. Effectively, the amplitude of the precession cone will be attenuated until the system comes back to equilibrium with M parallel to H ef f (see fig. 2.15b). Gilbert [340] added a damping torque to the eq. ( 2.41) to describe these losses, resulting in the Landau-Lifshitz-Gilbert (LLG) equation:

∂M ∂t = -γM × H ef f + α M s M × ∂M ∂t , (2.43)
where α is a dimensionless damping constant (Gilbert damping) and characterizes how fast the system goes back to its equilibrium which occurs over a time scale of t ≈ 1 αγH ef f . Equation (2.43) is that of a damped harmonic oscillator, characterized with a resonance frequency ω res . In other words, an external magnetic field oscillating at the resonance frequency would strongly amplify the precession or even maintain it: this is the ferromagnetic resonance (FMR) condition, first described by Griffiths [341] and which can be found by solving the LLG equation. The detailed resolution of eq. ( 2.43) can be found in [342]. The Zeeman effect lifts the spin degeneracy of eigenstates as a magnetic field H 0 is applied. This splitting increases linearly with the filed and given an excitation photon with energy hν (or ℏω), dipolar magnetic transitions are allowed if the system reaches the resonance condition with H 0 = H res . Reused from [START_REF] Vicente-Arche | Quantum Oxide Heterostructures for Spin-Charge Interconversion[END_REF] Ferromagnetic resonance

In practice, in ferromagnetic resonance (FMR) experiments, the resonance condition is usually reached by applying a static magnetic field, which establishes a uniform single magnetic domain configuration, while a small transverse RF magnetic field at a frequency ω sustains an out-of-equilibrium state. The DC magnetic field splits eigenstates of opposite spin direction due to the Zeeman effect with an energy splitting of E Zeeman = gµ B H 0 . RF excitation with suitable energy ℏω res can then give rise to magnetic dipole transitions between the spin-split levels as illustrated in fig. 2.15c. This excitation condition is met at a frequency:

ω res = -γH ef f = gµ B ℏ H ef f .
(2.44)

However, eq. ( 2.44) doesn't take into account the link between the demagnetizing field H D and the total magnetization and the consequential effect on the effective field H ef f and the dynamics. Doing so results in the following equation named the Kittel formula:

ω res = gµ B ℏ (M s + H 0 + H an )(H 0 + H an ), (2.45)
where M s is the (static) saturation magnetization along the applied dc magnetic field. Most often, the FMR spectrum is obtained by fixing the excitation frequency ω and sweeping the DC magnetic field H 0 around H res (the resonance field), easier experimentally. Concretely, the magnetic excitation response of the material will lead to variations of the RF power absorption especially around the resonance.

Spin accumulation and spin pumping

SP-FMR, which is a reciprocal effect of spin torque, is one of the most popular techniques for spin-to-charge conversion experiments. Demonstrated for the first time by Johnson & Silsbee [343], the injection of a spin current from a ferromagnetic material at its resonance into a non-magnetic (NM) material has been more extensively studied and modeled by Tserkovnyak et al. [344,345]. It is only in 2006 that Saitoh et al. [346] and Costache et al. [347] showed for the first time the detection of this spin current as it could be converted in Pt to a charge current by the inverse spin Hall effect (ISHE).

The absorption of the spin current by the adjacent non-magnetic (NM) will act as a new damping term ∆α SP , adding to the intrinsic Gilbert damping (α):

α = α 0 + ∆α SP , (2.46)
which corrects the Landau-Lifshitz-Gilbert (LLG) equation to the following:

∂M ∂t = -γM × H ef f + α M s M × ∂M ∂t + ∆α SP M s M × ∂M ∂t .
(2.47)

Note that the extra-damping ∆α SP reflects the presence an additional relaxation channel for the magnetization, thanks to the spin injection in the NM and the efficiency of the injection is thus, related to this term. One can estimate this parameter by performing similar experiments with and without the additional relaxation channel, i.e. with and without the NM layer.

The instantaneous polarized spin-current generated by a precessing ferromagnet and injected into the adjacent NM layer, derived in refs [345,348,349], depends on the precession motion of the magnetization:

j pump s,0 (t) = - ℏ 4π Re g ↑↓ M 2 s M(t) × ∂M(t) ∂t 2e ℏ , (2.48)
from which we can see that the instantaneous spin current is always orthogonal to the magnetization and its derivative, leading to a spin accumulation at the interface with the adjacent NM material. We also see that eq. ( 2.48) depends on the spin-mixing conductivity g ↑↓ expressing the FM/NM interface spin transparency, i.e. how well spins can propagate across the interface. It is thus of paramount importance to characterize the efficiency of spin injection, and the extra damping ∆α SP is of course depending on it. Assuming that the NM material is a good spin-sink and there is no reflection back to the FM layer (in other words, the thickness of the NM material t N M is much smaller than its spin diffusion length λ s ), the following linear relation can be established between ∆α SP and g ↑↓ :

∆α SP = gµ B 4πM s t F M Re g ↑↓ .
(2.49)

Finally, the DC spin current can be evaluated by integrating eq. ( 2.48) over one period of the RF signal, i.e. one full cycle of the magnetization precession [350,351] and it is crucial to estimate the final spin-to-charge conversion efficiency.

. Spin-pumping experiments in RF cavity

Ferromagnetic resonance (FMR) measurements are conventionnally achieved in two different ways: either by using a stripline or a microwave cavity. While the former allows to tune the frequency of the experiment over a wide range, the second is restricted to a narrow range around the cavity's resonance frequency. It consists in a hollow electromagnetic resonator tuned in the RF range which can yield high amplification of the magnetic field (quantified by the quality factor Q), orders of magnitudes higher than with striplines. This is then ideal as it enhances not only the signal-to-noise ratio (SNR) but also the power absorption by the FM, and thus the spin current injection.

Cavities also offer the advantage that the sample can be placed strategically at the center of the cavity where the magnetic field component of the RF excitation is maximal while the electric field component vanishes, which avoids some spurrious effect as we will see later on.

Figure 2.16 displays a schematic visual of the FMR cavity setup used for our spin-to-charge conversion experiments. The sample is held suspended inside the cavity between the coils of two electromagnets used to apply the DC magnetic field H 0 in the plane of the sample. A microwave radiation of fixed frequency is generated by a Gunn diode and sent through a circulator which directs the RF intensity to and from the cavity: while part of the power is absorbed by the cavity, the rest is reflected back to a detector. Since the absorption of power is used to quantify the spin-current injection at the ferromagnetic resonance, a proper calibration should be performed beforehand so that the cavity is well-coupled, i.e. out of the resonance, the reflected power measured at the detector should be similar to the power actually sent to Measurement configuration for SP-FMR experiments using a resonant cavity. A DC magnetic field is applied via electromagnets between which the sample is placed. The latter is glued onto a holder which allows the connections of two electrical contacts to the top of the sample at extremeties, and one contact on the back, to apply gate voltages. The RF field is sent through a circulator to the cavity and fed back through it to the detector. Reused from [START_REF] Vicente-Arche | Quantum Oxide Heterostructures for Spin-Charge Interconversion[END_REF] the cavity. The measurement is then performed as briefly explained earlier: the DC magnetic field is swept around the resonance field while measuring the power absorption response.

Experimental setup

SP-FMR experiments shown in this thesis as well as their analaysis, have been performed through a collaboration by the group of Laurent Vila and Jean-Philippe Attané at CEA Spintec (in Grenoble, France). Their setup consists of a Bruker ER 4118X-MS5 cavity with resonance frequency around 9.68 GHz and a Bruker ESP300 X-band CW spectrometer. After being cut in slabs of about 0.4 mm × 2.4 mm the samples are glued onto thin printed circuit board (PCB) sample holders with a connection to the back-gate electrode (V G on fig. 2.16) made with silver paste, and with some Al wire bondings at either extremities of the sample. The RF magnetic field is applied in the longitudinal direction of the sample, while the DC magnetic field is applied in the transverse direction, both lying in the film plane.

The connection to the sample allows one to detect the DC voltage generated during the experiment, due to the conversion from the injected spin current j s to a charge curent (density) j c , alongside other spurious voltages. The resulting voltage across the sample length could then be measured between V + and V -with a Keithley 2182A nanovoltmeter along with the application of a gate voltage between V G and V -with a Keithley 2400 sourcemeter. The charge current and measured voltage are simply related by:

j c = V sym Rw , (2.50)
where V sym is the symmetric component of the voltage, R is the sample resistance measured independently and w is the sample width. An example of SP-FMR measurement is shown in fig. 2.17 where a spin current is pumped at the resonance of a 15 nm-thick CoFeB layer, into an adjacent Pt layer of similar thickness (Pt is a well-known reference material for spin-charge interconversion, with a spin Hall angle of about 6 % [352, 353]). Panel (a) shows the FMR signal of the CoFeB layer, while the second panel shows the measured charge current I c (obtained using Ohm's law) while sweeping the DC magnetic field. I c shows a symmetric peak, of maximal intensity at the resonance of the FM where the spin injection is the most efficient. This peak is also reversed upon changing the sign of the magnetic field, and thereby changing the spin-polarization direction, while the conversion handedness intrinsic to the material is conserved, which results in a negative vs. positive charge current. We should also note that the current is normalized by the square of the RF magnetic field h 2 rf , since the spin current injected is proportional to the RF intensity. This allows for comparisons between different spin-to-charge conversion experiments.

A more detailed explanation of the experimental conditions and optimization of the setup can be found in the thesis of Noel [354].

. Spurious effects

Due to the geometry of the experiment, the voltage measured at opposite edges of the sample may also capture magnetoresistance effects coming from the ferromagnetic layer. Therefore, one should also consider these additional components, of which we are briefly explaining the properties, along with some solutions to avoid them.

Spin rectification effects

Microwave radiation sent to the cavity has two components which may also generate two contributions in the ferromagnetic material: the RF electric field in the cavity e rf , producing a current in the sample and the RF magnetic field h rf exerting a torque on the magnetizaton and inducing its precession. The time-dependent oscillation of the magnetization also induces a time-dependent magnetoresistance of the sample. In addition, the RF current circulating in the sample due to e rf couples with the oscillating resistance and a non-vanishing DC voltage is obtained [355]. These non-desirable effects due to the existence of anisotropic magnetoresistance (AMR), AHE or planar Hall effect (PHE) add up to the spin-charge conversion output signal due to ISHE or inverse Edelstein effect (IEE).

To avoid them, the minimization of the electric field can be achieved by placing properly the sample in one node of that electric field, at the center of the cavity. In addition, a disentanglement of the symmetric and asymmetric components of the charge current signal allows to further distinguish spin-charge conversion phenomena from residual spin-rectification effects: the first are usually found in the symmetric part while the spin rectification is typically dominating the asymmetric one. To accurately assess and avoid spin-rectification effects which even in the best conditons have been reported in CoFeB [356], a complete angular dependence should be performed [355].

Thermal effects

The power absorption of the cavity-sample system at the FMR is accompanied by a nonuniform thermal dissipation (a thermal gradient) in the sample. Various thermoelectric effects may then appear in the detected votage, due to spurious thermal spin and charge contributions like the anomalous Nernst effect (ANE) or spin Seebeck effect (SSE). These two effects are generally difficult to identify because of their similarity with the spin-to-charge conversion (SCC) mechanisms (they are found in the symmetric component).

However, a method to exclude these thermal effects has been proposed by [357]. The first step consisting in doing measurements with a similar sample but inserting a thick insulating layer between the FM and the NM which blocks the injection of spin current and in this way allows to observe only the ANE coming from the ferromagnetic layer (since both the ISHE/IEE and the SSE are related to the spin current). On the other side, the relative time scales of the spin-to-charge conversion (SCC) which is a fast process happening in the range of nanoseconds, and the thermal effects happening over the range of seconds, allows to discriminate between both effects. Concretely, for measurements with a slow sweep of the magnetic field, the SSE would increase the output intensity, while fast FMR spectra acquisition would leave the peak intensity mostly unaffected, apart from the SCC origin.

. Ferroelectric characterization

As described in section 1.2, ferroelectric materials provide several signatures of their phase transition or their intrinsic properties accessible via different observables (dielectric permittivity, polarization-switching current, piezoelectric response, soft-mode phonon, ionic displacement, inversion symmetry breaking, ...) and thus, through various types of direct or undirect physical measurements (dielectric measurements, PFM, transmission electron microscope (TEM), optical second harmonic generation (SHG), ...). For the scope of this thesis, we will focus primarily on macroscopic polarization switching in electrical measurements.

. Electrical measurements

As we have seen in sections 1.2.2 and 1.2.3, the polarization response of an insulating material to an electric field offers a rather direct and visual method to characterize the ferroelectric (or paraelectric) nature of that material and obtain interesting parameters associated to it such as the remanent polarization or the coercive field.

We also noted that any change in polarization ∆P with respect to time (P is expressed as a density of charge per unit surface) was in fact homogeneous to a current density j P (t).

We can actually derive the expression for the corresponding current from the Maxwell equations. Starting from the first part of eq. (1.53), the total displacement current within the material is given by:

j = ∂D(t) ∂t = ϵ 0 ∂E(t) ∂t + ∂P(t) ∂t .
(2.51)

While polarization and displacement field are not directly measurable quantities (and polarization is generally ill-defined on its own, as explained in section 1.2.3), currents well below than picoamperes are easily measured by modern instrumentation [358][359][360] and can be related to the current density by I = jA s (A s being the sample area).

A typical setup to characterize a dielectric sample is thus based on the Sawyer-Tower circuit [361]: a voltage is applied on one electrode, swept and cycled between two opposite extrema (typically using a triangular signal shape) while the current from the other electrode to ground is acquired or directly integrated over time (typically using a reference capacitor, in early circuits).

In the case of a simple capacitor sample with a linear dielectric response, and a polarization axis perpendicular to its surfaces, eq. (1.52) states that P = ϵ 0 χE, which results in the well known capacitor equation

I C (t) = C s dV (t)
dt , where C s = ϵ 0 ϵr d is the capacitance of the sample and d is the sample's thickness so that E = V d . For a triangular voltage signal, the current in that circuit is just a constant (ûp to a sign), and the integrated current (the polarization) shows the expected linear relationship with the electric field (see figs. 2.18c and 2.18d).

On another hand, when an electric field is cycled across a ferroelectric with uniform polarization, what happens is the following: two characteristic current peaks with opposite sign occur at well-defined and (generally, but not necessarily) symmetric values of the electric field ±E c (see fig. 2.18a). Indeed, as we have seen in section 1.2.3, the polarization is actually described with additional terms such as a spontaneous polarization P 0 , which we may also call remanent polarization P r , (as well as higher order non-linear terms, which we leave aside for the moment), such that when this polarization changes, we have a switching current:

j P = ∂P ∂t .
(2.52)

From eq. ( 2.52), we can finally obtain the polarization change in a time interval ∆t by integrating the current density over that interval: 2.53) and recover the classical polarization hysteresis of a ferroelectric vs. the electric field. One must note that on top of classical dielectric and ferroelectric contributions, additional currents may intervene since no dielectric is perfectly linear nor a perfect insulator. In extreme cases, resistive or diode characteristics can be easily identified in quasi-static current-voltage (IV) loop and are typical of leakage in doped crystals or thin films. The corresponding C(V ) characteristics which may appear as closed loops should not be mistaken for ferroelectric loops [362,363]. For higher frequencies and/or sufficiently high amplitudes of electric field, dielectric losses may also occur in some materials even for samples not showing dc conductivity.

∆P = ˆ∆t 0 dtj P (t), (
We may thus generalize the equation governing this type of experiment by modelling the sample as a parallel resistor-capacitor (RC) circuit with the addition of the polarization switching term:

I(t) = A s dP (t) dt + C s dV (t) dt + V (t) R s , (2.54)
where R s is the equivalent sample resistance in parallel with its capacitance C s . To isolate the actual polarization switching from other undesired contributions in measurements, different strategies exist. One can simply remark that once the polarization of a ferroelectric is switched in one direction, assuming it is uniform across the sample, if we apply again an electric field in the same direction, only non-switching current comes into play. A very commonly used technique exploiting this is to apply pulses in a PUND sequence as illustrated in fig. 2.19. First, a preset pulse of negative amplitude is applied to initialize the polarization in the "down" state. Then, a positive (P) pulse switches the polarization up while also provoking displacement current and leakage. The same pulse U is applied a second time but this time only the last two effects occur. This means that the current of the second pulse U can be substracted from the total current measured during pulse P, leaving only the polarization switching current as intended. The same procedure as in P and U can then be applied for the two successive negative N and down D pulses with opposite polarity, and the current can then be integrated for each branch after correction, to finally reconstruct the full loop. This tech-nique is largely applicable in thin films as long as the polarization is not completely obfuscated by ohmic contributions as in strongly doped semiconductors.

P V t U N D Initialization (a) (a) I V (b) P V t U N D Initialization (c)
In this thesis, all measurements are performed with a Radiant Precision Multiferroic II, a dedicated commercial device which can perform many types of electric measurements on dielectrics. The associated software offers in particular an option similar to PUND called Remanent Hysteresis (RH) which we exploit for some samples to extract the actual remanent polarization independently from the large displacement polarization of SrTiO 3 which has a remarkably large dielectric constant in the low temperature range we are interested in. The Remanent Hysteresis consists in applying pre-pulses of alternatively same or opposite polarity than the beginning of the following voltage sweep but instead of applying only four monopolar measurement pulses, it actually performs four full hysteresis loop, i.e. two starting with positive voltage, and two with negative voltage. The data used to reconstruct the four sub-branches of the P (E)-loop is only taken from the first half of each hysteresis loop as illustrated in fig. 2.19c.

CHAPTER 3

Electric control of spin-charge interconversion in AlO x /SrTiO 3 2DEGs

3.1 . Tunability of spin-charge interconversion in the literature

. Experimental review of spin-charge interconversion

The perspective of conceiving efficient devices based on spin currents relies on the understanding and mastering of three essential bricks: the generation and detection of spin currents, the transmission of spin information and the manipulation of spins and spin currents.

While the direct detection of a spin current is delicate, measuring a charge current is on the contrary less of a challenge, usually. In this sense, a variety of techniques has been used to characterize both the possibility to inject a spin current in a system, the efficiency of conversion, as well as the capability of a medium to carry spin currents over long distances, mainly by using electrical voltage and/or current injection and detection.

The diversity in experimental geometries (direct stacking between the injector/detector and the propagation/conversion medium, vertical and lateral spin valves, single system nanodevice) reflects the multiplicity of injection and detection methods, materials constraint and phenomena considered (see 3.1). Some of the first observations of spin current were made possible by optical spin injection [START_REF] Ganichev | Conversion of Spin into Directed Electric Current in Quantum Wells[END_REF] or by a detection via the magneto-optical Kerr effect [START_REF] Kato | Observation of the Spin Hall Effect in Semiconductors[END_REF][START_REF] Sih | Spatial Imaging of the Spin Hall Effect and Current-Induced Polarization in Two-Dimensional Electron Gases[END_REF]. At the basis of spintronics, it is also known that a spin-polarized current can be obtained by simply passing a charge current through a ferromagnet, but also that a spin (polarized) current can be detected by using a FM as a detector (reminiscent of the polarizer/analyzer optical analog), and finally that it can affect the magnetization of a FM layer. Other popular techniques which emerged more recently such as spin-pumping via the spin Seebeck effect (illustrated in fig. 3.1a) or via FMR as described in section 2.5, allow the injection of pure spin currents in a system under study. Finally, some experiments even allow to avoid the use of any ferromagnet or external light source, implying for instance the double conversion from an electrical signal to a spin signal back to an electrical signal, and -via the Hanle effect-help characterizing the conversion efficiency and spin diffusion length in a given system (see fig. 3.1d).

Among the most commonly used and studied materials, Ta, W and above all Pt are references in terms of bulk SHE-based spin-charge interconverters with a well established spin Hall angle of θ SHE ≈ 6 % to 8 % for Pt and we should note that Ta is known to have a conventional negative spin Hall angle (opposite to that of Pt) of about -0.37 % [364]. It is thus common to see simple stacks of FM/HM (HM = Pt, Ta) in the literature as reference systems.

Spin-to-charge conversion (SCC) has also been demonstrated in various bulk materials like Table 3.1: Summary of spin-charge interconversion efficiencies obtained from the literature in different systems.

semiconductors exhibiting the spin Hall effect (SHE) but also in Rashba-type interfaces between single compounds with large SOC like Ag/Bi or in topological insulators such as [Bi compounds].

Transition metal oxides (bulk and interfaces) also have their share in the picture of spin-charge converters and can actually exhibit quite large efficiency values of λ IEE up to 190 nm reported very recently at the LaTiO 3 /SrTiO 3 interface from SP-FMR measurements [379]. Despite their usually lower Rashba coefficient, the fact that the spin diffusion length is significantly higher than in metals, for instance, allows to achieve much higher conversion efficiencies. A (nonexhaustive) summary table of selected values from the literature can be found in table 3.1 for various systems (for more exhaustive specialized works, the reader can refer to Galceran et al.

[383] for van der Waals materials or Chen & Yi [384] and Trier et al. [385] for oxides).

. The effect of gating on spin-charge interconversion

Spin-charge interconversion phenomena are intrinsic to the electronic band structure of a material and the amplitude and sign of conversion are thus system-dependent. In Rashba systems it is however known that the amplitude of the Rashba effect can be modulated by the application of a gate voltage (see section 1.1.3.2), as demonstrated by weak anti-localization (WAL) measurements in LAO/STO for instance [243].

With the help of SP-FMR measurements in LAO/STO 2DEGs, a complex dependence of the SCC on the gate voltage was demonstrated with a remarkably large value of up to 6.8 nm at 125 V in Lesne et al. [11]. Thanks to Hanle experiments with the magnetic field applied along different directions, it was also shown that the 2D spin Hall effect could have a dominant contribution to spin-charge interconversion phenomena [380,386,387], with a maximum near the Lifshitz point (see fig. 3.2a), i.e. when then higher and heavier d xz,yz band start to be populated. In addition, a demonstration of the charge-to-spin conversion was shown in the same samples, using the bilinear magnetoresistance (BMR) to extract the Rashba coefficient as a function of the gate voltage as we will explain in the next section [336] and is shown in figs. 3.2b and 3.2d. Later, other SP-FMR measurements in a AlO x /STO 2DEG showed even higher value of the inverse Edelstein length, up to 28 nm with also a complex dependence on the gate voltage shown in fig. 3.2e. With the help of tight-binding calculations associated with Boltzmann transport computation of the Edelstein tensor, it was suggested that each pair of subband could contribute positively or negatively to the total Edelstein effect, which could explain qualitatively the nonmonotonic behavior of λ IEE with the gate voltage [START_REF] Vaz | Mapping Spin-Charge Conversion to the Band Structure in a Topological Oxide Two-Dimensional Electron Gas[END_REF]235]. Indeed, as each subband pair starts to be populated, the total Edelstein tensor value passes through maxima and minima depending on the contribution sign of the new band being populated. Other important contributions arise when avoided topologically trivial or non-trivial band crossings are encountered.

The higher values of conversion efficiencies obtained in AlO x /STO 2DEGs compared to LAO/STO at least in spin-pumping experiments may be explained by a better injection through the AlOx layer (less back-scattering) than through LAO. The wide range of values and the numerous extrema seen in fig. 3.2e are also well explained by the complexity of the band structure deduced from ARPES and via tight-binding calculations, with topologically trivial and non-trivial avoided crossings showing up as each band is filled up with electrons (see figs. 3.2c, 3.2f and 3.2g). There might also be differences in the confining potential between both types of 2DEG, affecting the relative shift (energy splitting) between the different subbands, thereby modifying the exact values taken by the effective Rashba coefficient.

Signatures of charge-to-spin and/or spin-to-charge conversion controllable with a gate voltage have also been investigated from simpler magnetotransport experiments. Over the last years, several groups have reported an angle-dependent (anisotropic) behavior of the in-plane magnetoresistance in different systems and doping regimes sometimes with several angular harmonics [388,389] and most importantly some unidirectional term, often attributed to the presence of an intrinsic or non-equilibrium magnetization and tunable with a gate voltage [390,391]. Other systems like topological insulators have also benefited from significant attraction with also tunable charge-spin conversion allowed by a control of the balance betweeen surface states [368] and some tentative models have been suggested to describe all these experimental observations [368,392,393].

. Bilinear magnetoresistance: a new probe for the Rashba-Edelstein effect

Anisotropic and unidirectional magnetoresistances in systems with spin-momentum locking

Anisotropic magnetoresistance (AMR) denotes a dependence of the resistance of a system with the relative angle between an applied magnetic field and charge current. It is generally thought to arise from angle-dependent scattering due to magnetic impurities (or spin-polarized bands in a ferromagnet) in presence of a sizeable SOC and the term normally refers to ferromagnetic systems. Interestingly, an AMR phenomenon has also been observed in non-magnetic Rashba two-dimensional electron systems in semiconductors [390] or at oxides interfaces [394,395] when a magnetic field B is applied in the plane of the sample, sometimes with a quadratic dependence on B. In the most simple case, it was then observed to be maximal when the latter field is either parallel or antiparallel to the current density vector and minimal when both are orthogonal to each other.

In the case of STO 2DEGs, although it has been suggested that ferromagnetism could occur from Ti sites at the interface, more recent models suggest that the AMR more generally arises because of anisotropic carrier lifetime simply related to the Rashba spin texture [396,397]. Under the application of an in-plane magnetic field, the Zeeman effect may amplify or reduce back-scattering probabilities, depending on the orientation of the field with respect to the applied current, by forcing (anti-)alignement of the spins along the different subbands of Rashba contours and thus by opening/closing relaxation channels. Indeed, scattering should be enhanced only when the initial and final states have the same spin orientation and when the transition implies only a small variation of the momentum vector k. Boudjada et al. [397] have applied this theory to the specific case of STO 2DEGs for a doping regime where both lower subbands with mostly d xy orbital character (except along main axes k x and k y ) and larger Fermi surface and higher subbands with dominant d xz,yz character and smaller Fermi surface are present (see also fig. 3.2). They showed that the largest scattering path should be from the most circular (d xy ) contour to the more strongly anisotropic d yz Reused with permission from [START_REF] Vaz | Mapping Spin-Charge Conversion to the Band Structure in a Topological Oxide Two-Dimensional Electron Gas[END_REF].

subband (for a current along x). Additionally, they could show that this relaxation channel was the largest (high resistance) for the magnetic field (anti-)parallel to the current (or k vector) and quenched (low resistance) when they are orthogonal, leading to the observed π-periodic angular dependence.

For sufficiently large magnetic fields, higher order angular harmonics are also expected to come into play in this system depending on the balance with Rashba field strength and (anti-)alignement pattern of the spin texture between subbands as the magnetic field B is rotated, in accordance with previous experimental observations. This is true for sufficiently large electron dopings, beyond the Lishitz point, i.e. when the d xz,yz bands play a significant role and a strong crystalline (/orbital) anisotropy is added [388].

On the other hand, some groups have started to notice the superposition of a unidirectional term to the AMR in systems with spin-momentum locking, that was dependent on the current sign and antisymmetric with reversal of the field and with extrema when the field and current are orthogonal [391,394]. Quickly, this non-reciprocal phenomenon was attributed to the Edelstein effect and exploited to characterize the spin-orbit strength of the system at hand.

A theoretical model for UMR and AMR in non-magnetic Rashba systems

As we briefly expressed in section 2.4.2, the generation of a non-equilibrium spin density transverse to an injected charge current, through the direct Edelstein effect and the resulting pseudo magnetic field B j may be characterized by non-reciprocal transport phenomena specifically when applying an in-plane external magnetic field B. These two fields interact in such a way that a change in resistance is observed depending on the relative orientation between the current and the external magnetic field, and can be expressed as a cross product between the two.

Such unidirectional magnetoresistance (seen as 2π-periodic oscillations of the resistance as a function of the angle between j and B in the xy-plane) has been reported by different groups in LAO/STO and several other systems before such as the InAs [390], BiTeBr and Ge(111) [369] Rashba interfaces or in the topological insulators Bi 2 Se 3 [365], for instance. Following a phenomenological description, Guillet et al. [369] derived a simple model equation and used this unidirectional magnetoresistance (UMR) term divided by the amplitude of the anisotropic magnetoresistance, quadratic in B, to extract some estimate of the Rashba coefficient α R from transport measurements at the Ge(111) surface. Vaz et al. [336] and Dyrdał et al. [393] proposed a microscopic model for Rashba 2DEGs in the simple case of transport by a single pair of circular Fermi contours. They considered scattering from randomly distributed impurities at points r i , introduced as a short-range potential V (r) with second statistical moment ⟨V (r)|V (r ′ )⟩ = n i v 2 0 δ(r -r ′ ), where v 0 is the potential of a single impurity. Taking two Zeeman-like terms into account -one for the applied magnetic field and one for the pseudo magnetic field from the Edelstein effect-, they deduced a modification of the relaxation rate Γ(b) (or relaxation time τ b ) dependent on the magnetic field and nonequilibrium spin density S y :

Γ(b) = ℏ 2τ b = Γ 0 1 + 3( J b y S y 4Γ 2 0 + b 2 8Γ 2 0 ) , (3.1)
where ε F is the Fermi energy,

Γ 0 = n i V 2 0 m *
2ℏ 2 is the scattering rate with zero magnetic field, b = (b x , b y ) is the magnetic field expressed in energy units, J = -2π ℏ m * so that the DEE field (in energy units) is b j = J S y .

The result is the addition of two terms to the diagonal resistivity:

ρ xx = ρ 0 xx + 3π 4 h e 2 α R τ 0 |e| j x b sin(ϕ) ε 2 R + ε 2 F + ε F τ 0 h b 2 cos(2ϕ) ε 2 R + ε 2 F , (3.2
)

where ρ 0 xx = ρ xx (B = 0) = h e 2 Γ 0 E F ε 2 R +ε 2 F
and ε R is the Rashba energy as defined in section 1.1.3.2.

The first new resistivity term is linear both in current density j and in magnetic field B (i.e. bilinear ) and depends on the sine of the angle ϕ between the two (2π-periodicity as expected), while the second term is simply quadratic in magnetic field and oscillates as a cosine of 2ϕ (π-periodic).

Each term of eq. ( 3.2) can be rewritten as a normalized magnetoresistance term, considering M R = [ρ xx (B) -ρ xx (B = 0)]/ρ xx (B = 0) so that bilinear and quadratic magnetoresistances become:

BM R = A BM R j x j sin(ϕ), (3.3)
QM R = A QM R cos(2ϕ), (3.4) with the amplitudes:

A BM R = 3 2 π gµ B |e|ℏ α R τ 2 ε F jB, (3.5
)

A QM R = 3 4 gµ B ℏ 2 τ 2 B 2 .
(3.6)

From these last two equations, we see that both terms scale quadratically with the scattering time of the system. In addition, the BMR term scales linearly as expected with both the injected current density (along x) and the magnetic field, and changes sign if either of these two variables is reversed, while the quadratic magnetoresistance (QMR) term also scales quadratically with the (in-plane) magnetic field.

We also note that A BM R is proportional to the Rashba coefficient α R which means one could extract the Rashba coefficient directly by measuring the amplitude of the unidirectional magnetoresistance and checking its dependence on B and j when both are transverse to each other. Although one could use the QMR to estimate the scattering time and use it in the BMR amplitude equation, it is also possible to further simplify the calculations of α R by reducing the number of system-dependent parameters and removing the scattering time, simply by taking the ratio between the BMR and QMR amplitudes:

A BM R A QM R = 2πℏ |e|gµ B α R ε F j B .
(3.7)

Here, mainly universal constants and experimental parameters remain, and provided that ε F and g the electrons g-factor of the system are known, this ratio is a measure of the Rashba spin-orbit coupling strength.

We should emphasize that this model relies on the assumptions of a simple linear Rashba system where only a single pair of bands (or at least band pairs with only one orbital character) contributes to transport. This model is a close transposition of the original model from Dyrdał et al. [393] initially intended to explain a mechanism for the BMR observed in TIs without hexagonal warping (at low Fermi energies) [365], and Vaz et al. [336] could apply it to the case of LAO/STO at different gate voltages and used the above equations to extract an effective Rashba parameter matching relatively well values obtained by tight-binding calclulations even in the multiband regime, as well as estimations from WAL measurements in the literature.

Similar models have also been rather successfully applied to the LaAlO 3 /SrTiO 3 (110) interface [398] and to other systems like the AlO x /KTaO 3 2DEG [START_REF] Vicente-Arche | Spin-Charge Interconversion in KTaO 3 2D Electron Gases[END_REF] or PdCoO 2 [399].

In summary, this magnetotransport technique and model seems to be particularly convenient to extract the Rashba coefficient in oxide 2DEGs and estimate the charge-to-spin conversion efficiency in a wide-range of gate voltages (i.e. carrier densities) even across the Lifshitz transition and not relying on the existence of a weak (anti-)localization and complex fitting procedures.

. Electrical control of charge-to-spin conversion in SrTiO 3 Rashba 2DEGs

While both spin-to-charge conversion (SCC) and charge-to-spin conversion (CSC) have been extensively studied in 2DEGs formed at the LaAlO 3 /SrTiO 3 interface, only SCC has been measured as a function of the gate in their AlO x /SrTiO 3 counterpart to date. As mentioned in section 1.4, fabricating AlO x /SrTiO 3 2DEGs is an inexpensive and scalable room temperature fabrication technique compared to the high-temperature growth necessary with crystalline LaAlO 3 (c-LAO). The intrinsic properties are also slightly different, owing to the difference in the mechanisms responsible for the 2DEG formation, with higher carrier density and overall lower mobilities observed in AlO x /STO in the ground state (i.e. even before gating).

These variations of properties among others should allow one to access a different region of the SrTiO 3 electronic band structure and possibly to obtain higher conversion efficiencies as was demonstrated in SP-FMR experiments by [START_REF] Vaz | Mapping Spin-Charge Conversion to the Band Structure in a Topological Oxide Two-Dimensional Electron Gas[END_REF]. In this section, we employ ourselves to demonstrate how the converse effect (charge-to-spin conversion) can also be achieved in this system and harnessed to extract the Rashba coefficient at different gate voltages.

. Sample preparation

To characterize the presence of unidirectional and anisotropic magnetoresistances in AlO x / SrTiO 3 and determine its significance in terms of the charge-to-spin conversion (CSC) via the Rashba-Edelstein effect (REE), samples have been fabricated with Hall bar patterns following the procedure described in section 2.1.2. Specifically, samples with a nominal thickness of 117 2.0 nm of Al were then exposed to air and capped by roughly 30 nm of a LAO grown in a PLD chamber at room-temperature to enhance visibility of the devices and their stability in time.

For experiments under back-gating conditions, samples were contacted after a depositition of Ti and Au on the back, and sticking them onto the appropriate sample holder with a backgate electrode as explained in section 2.4. The high potential plug (+ side) of the sourcemeter used to apply the gate voltage was connected to the back of the sample while the low potential (-side) was connected to a contact on the measured device, usually the current drain contact (I-) by convention, defining the common ground of the system.

. Basic magnetotransport characterization

To begin with, we perform a basic characterization of the transport properties of the prepared devices. Figure 3.3a shows the temperature dependence of the sheet resistance of one 20 µm-wide device with the usual metallic behavior typical of AlO x /SrTiO 3 2DEGs, displaying a value of ∼3.1 kΩ □ -1 300 K and a residual resistivity ratio (RRR) of about 10. Backgate dependence of the sheet resistance at 2 K, with repeated voltage cycling after initialization between ±150 V, showing some overlap between cycles with a bit of hysteresis, and a non-monotonic behavior with a maximum around 20 V when going from positive to negative voltage.

After cooling down to 2 K, the back-gate voltage V g is first increased to the upper boundary of interest for the following experiments so that the carrier density is saturated and then cycled few times so that any trace of hysteresis in the transport coming from the displacement of ferroelastic domain boundaries can be eliminated. In fig. 3.3b, an example of cycling curve between -150 V and +150 V shows how the sheet resistance becomes rather stable from one cycle to the next after the gate initialization, although a small counterclockwise hysteresis seems to remain. Remarkably, a non-monotonic dependence on sheet resistance is observed as the gate is varied with a maximum near 20 V. This behavior is rather unusual in STO 2DEGs as usually R s only increases as V g is decreased and the gas is depleted more and more.

We note that we have occasionally observed the same behavior across different devices of the same sample and also over different samples, even with a different type of capping of sputtered Al 2 O 3 instead of PLD-grown a LaAlO 3 , with a slightly stronger (counterclockwise) hysteretic behavior also present which we assign to charge (de)trapping at interfaces or in STO.

To understand better the reason behind the nonmonotonic gate-dependent effect in this system, we also perform longitudinal and transverse resistance measurements in DC as a function of the out-of-plane magnetic field µ 0 H z up to ±9 T for various gate voltage within the same boundaries of ±150 V. First, we can see from fig. 3.4b how at large negative gate voltages the longitudinal resistance signal is dominated by a negative magnetoresistance contribution. As the gate voltage is increased, one can see that the classical B 2 Lorentz magnetoresistance contribution starts to dominate at higher magnetic fields, until at the maximum voltage V g = 150 V it constitutes most of the signal. We note however, that at low field (less than 1 T), a small dip of the resistance (not easily seen at the scale used on fig. 3.4b) emerges from 100 V and up, characteristic of weak-antilocalization phenomena.

On the other hand, we can see from the Hall curves in fig. 3.4a how in the depleted regime (negative gate voltages), the Hall curve seems fully linear and the slope decreases slightly as the gate voltage is increased which indicates an increasing carrier density as expected. Beyond a certain voltage, a nonlinearity clearly appears reflecting the introduction of carriers with a different mobility, i.e. the population of electronic states with a different character, probably passing from d xy -dominated charge transport to a mixture with also carriers that have d xz,yz orbital character, as should be the case for STO 2DEGs.

Based on this hypothesis, we apply a simple linear fit at high field (between 8 T to 9 T) which we use to obtain an initial guess of the total carrier density and Hall mobility. We then also incorporate this initial guess as a regularizing term when fitting to a more complete twobands model as described in section 2.4.1 to extract the density and mobility of both types of carriers (in this case electrons) as shown in figs. 3.4c and 3.4d.

At -150 V, we obtain a carrier density of about 9.87 × 10 13 cm -2 with a mobility of 146 cm 2 V -1 s -1 . As V g is increased, the carrier density n s ≈ n 1 increases slowly as expected, up to about 1.07 × 10 14 cm -2 at 20 V while the mobility of the majority carriers decreases slightly to 104 cm 2 V -1 s -1 , probably due to an increase in the Coulomb repulsion. At even higher gate voltages, the rise of the carrier density of the second type n 2 is significantly faster although their concentration remains at least two orders of magnitude lower than the majority carriers, reaching up to 1.5 × 10 12 cm -2 at +150 V. On the contrary, the majority carrier density n 1 stagnates around 1.05 × 10 14 cm -2 with small variations in the fitted values probably resulting from fitting errors (not evaluated here). We argue that the increase of mobility µ 1 may also come from fitting uncertainties to compensate a decrease in carriers n 1 or from the filling of an additional band with similar orbital character (similar mobility) as for the carriers already present but higher energy due to the confinement. Finally, the mobility of the second type of carriers µ 2 is also one order of magnitude larger than that of the majority carriers µ 1 which despite the low n 2 should enhance significantly the contribution of the second type of carriers to transport.

In fact, we may explain the non-monotonic trend of the sheet resistance in fig. 3.3b based on the relatively slow increase of the majority carrier density n 1 at low voltages combined with the initial decrease of mobility, probably due to Coulomb repulsion within the lowest highly filled d xy band, and leading to an increase of the resistance instead of the decrease usually observed. At higher voltages, as the second type of carriers with much higher mobility starts to contribute more and more to charge transport, and the total carrier density increases significantly, the overall resistance then decreases.

In addition, the higher carrier density necessary to see the Lifshitz transition compared to LAO/STO could be explained by a difference in the 2DEG formation mechanism, that could cause the splitting between t 2g levels to be higher and a larger extension of the gas into the STO due to different electrostatic boundary conditions. The higher carrier density and formation of oxygen vacancies being the underlying mechanism here may also explain the lower mobilities for AlO x /SrTiO 3 2DEGs.

. In-plane angle-dependent magnetoresistance: existence of bilinear and quadratic magnetoresistances

Following another cooldown and gate initialization, this time up to ±200 V, with the sample installed on a rotating sample holder with the magnetic field kept in-plane, we perform angle dependence measurements of the resistance at 200 V and at fixed magnetic field B = µ 0 H ∥ and AC charge current density j injected along the main channel of the Hall bar. Using a Stanford Research 860 lock-in amplifier, we extract successively the first and second harmonic longitudinal voltage responses of the device. showing unidirectional 2π-periodic magnetoresistance and (d) and (f) the corresponding fits of their amplitude as a function of field and current density, respectively. Field-dependent measurements were performed with j = 40 A m -1 and current-dependent measurement with a field of 9 T. Some points on (f) at low current densities correspond to curves not shown on (e) for clarity. Error bars correspond to standard error obtained from the fit of the sinusoidal amplitudes.

Figure 3.5a shows the angular dependence of the magnetoresistance obtained from the first harmonic R ω xx , after removing a drift component linear in time (attributed to charging effects after V g is changed), shifting so that the resistance in the configuration where j and B are parallel is zero (at ϕ = 0°and 180°) and normalizing by the zero field resistance as per the definition in section 3. 1.3 and Vaz et al. [336].

As can be seen, the signal shows indeed a sin(2ϕ) dependence on the angle with increasing amplitude, which we can fit as a quadratic function of the magnetic field as it increases from 0 T to 9 T, following eq. (3.4). Using eq. (3.6) and assuming the g-factor to be equal to 2 as was reported in LAO/STO at high gate voltages [243], we obtain a momentum scattering time τ m ≈ 1.6 × 10 -14 s which goes up to 6.4 × 10 -14 s for g = 0.5 (as used in ref. [336]). This is similar to values obtained from a classical transport model with the data obtained in the previous section, considering transport dominated by (majority) carriers of channel 1, with τ m ≈ 5.5 × 10 -14 s to 1.2 × 10 -13 s at the highest gate voltage (for a d xy -only mass m * = 0.6m e [START_REF] Vaz | Mapping Spin-Charge Conversion to the Band Structure in a Topological Oxide Two-Dimensional Electron Gas[END_REF] or an estimated m * = 1.3m e used in ref. [336], respectively).

In addition, as expected due to the Edelstein effect, we obtain a 2π-periodic signal from the second harmonic signal which we fit to extract the amplitude and plot as a function of the magnetic field in fig. 3.5d and as a function of the current density in fig. 3.5f. We see that this signal is mostly linear in both current density and magnetic field, i.e. the bilinear magnetoresistance (BMR) term from charge-to-spin conversion. Note that here we chose to use the same sign convention as [336] where a negative sine function is associated to a positive amplitude and as a consequence, the positive sine function obtained at a gate voltage of 200 V yields a negative BMR amplitude.

. Controlling the BMR and QMR signals with electric gating

As the Rashba spin-orbit property and spin-charge interconversion efficiencies are known to vary significantly under the application of a gate voltage, we also perform measurements of the BMR and QMR signals at different V g values. In a third cooldown, this time again with initialization between ±150 V at 2 K, we measure simultaenously the first and second harmonic voltages with two Stanford Research 830 LIA at fixed magnetic field B = 9 T current of 150 µA in a device with 20 µm-wide channel, i.e. a current density j = 7.5 A m -1 .

In the first place, we focus on the resistance obtained from the second harmonic signal. After removing a linear (time drift) contribution and an out-of-plane field contribution, we can see from fig. 3.6a that a nearly sinusoidal signal is kept across the entire range from +150 V to -150 V, with a clear change of amplitude and even a change of sign of the BMR, passing from negative to positive amplitude (with the same convention as above in fig. 3.5) as we lower V g . The complete dependence of the BMR amplitude A BM R is displayed in fig. 3.6b from which we can identify a cross-over area around V g ≈ 0 V to 20 V. We also measure the second harmonic resistance at a fixed angle of 90°, or in other words, keeping the current and magnetic field transverse to each other while sweeping the latter between ±9 T. Figure 3.6c confirms the linear dependence of the second harmonic magnetoresistance with the magnetic field (the BMR) with slopes of opposite sign between V g = -150 V and +75 V.

As shown from the literature in section 3.1.2, it is known that the complex band structure of STO 2DEGs can give rise to varying amplitude and sign of the Edelstein tensor [START_REF] Vaz | Mapping Spin-Charge Conversion to the Band Structure in a Topological Oxide Two-Dimensional Electron Gas[END_REF] and of the Rashba coefficient [243,336] for each subband and for the effective overall values of the system. Checking upon eq. (3.5), we see that the BMR amplitude should depend on multiple system parameters that can be varied. First, the g factor has been reported to vary between 0.5 (in depleted regime) and 2 (at high gate voltage) in LAO/STO 2DEGs (extracted from WAL fits) [243] and cannot explain either of the strong amplitude variations nor the sign change. In the single-band model of Vaz et al. [336], A BM R is also inversely proportional to the Fermi energy ε F = ℏ 2 πns m * which thus scales linearly with the carrier density for a 2D system Error bars obtained from the fits of the sinusoidal amplitude are all at least 2 orders of magnitude smaller that the corresponding obtained value. (c) Sheet resistance from the second harmonic channel as the in-plane magnetic field is swept between +9 T and -9 T transverse to the current, at V g = -150 V (in orange) and +75 V (in green). The lines are a guide to the eye.

and should anyway remain positive by definition. Assuming a constant effective mass for the whole dispersion, it means that A BM R should actually decrease as the gate voltage is increased and the carriers are accumulated in the 2DEG. Even considering variations of the effective mass from a d xy -only to a d xz,yz -dominated transport, calculating the effective mass of each subband separately from the tight-binding model of Johansson et al. [235] yields m * dxy ≈ 0.65m e and m * dxz,yz ≈ 1.2m e (m e being the electron mass) which is still not a sufficient variation to explain on its own the change observed in BMR amplitude. Fianlly, we turn to the fact that A BM R is expected to scale with the square of the scattering time. Since we know that the overall mobility µ increases a bit with the gate voltage (mainly due to the second type of carriers) the momentum relaxation time, related to µ by τ m = (m * µ)/|e|, increases accordingly and could explain a fast increase of the BMR amplitude in absolute value, but not in sign. In fact, one has to look at the individual contributions from each subband to the overall effective Rashba coefficient α R which may indeed result in a positive or negative Edelstein efficiency.

To complete our vision of the in-plane magnetotransport and of the charge-to-spin conversion phenomenon, we also show in fig. 3 similar gate voltages matching with fig. 3.6b. Note that here the values are normalized with respect to the highest value measured during each sweep and expressed as percentages. We notice an increased amplitude of the sinusoid at high gate voltages, and more surprisingly an unexpected nearly rigid shifted angle dependence, different between the high voltage regime above 100 V and lower voltages. Indeed, the expected maxima at 0°and 180°is recovered only near 150 V while in most of the voltage range accessed in the experiment, the maxima are shifted down by approximately 30°. Although this transition seems to coincide with the introduction of the second type of carriers, we do not have a clear explanation for the phenomenon and a more extensive characterization of the in-plane magnetoresistance should be carried out as a function of the magnetic field and gate voltage for a better understanding. Furthermore, extracting the sinusoid amplitudes as a function of V g of this AMR (3.7b) shows a surprising nonmonotonic variation which, according to our model should only be attributed to variations of the scattering time since it is the main varying parameter in eq. (3.6).

We note that while the peculiar geometry of the experiment (with the device rotated 45°with respect to the principal crystallographic axes) should have a negligible influence on the QMR as long as the Fermi contours are isotropic, it may significantly affect the available relaxation channels as different bands are populated by the application of a positive gate voltage.

Nonetheless, neglecting these aspects and assuming this amplitude corresponds fully to the QMR amplitude (in fig. 3.7b) we use it to compute the Rashba coefficient α R from a reduced set of system parameters. In fig. 3.7c, we thus plot an estimation of α R using eq. (3.7) under few assumptions: g is considered constant over the entire range of gate voltages g = 0.5, and we follow the approximation of a single pair of bands imposed by the model of [336]. Despite this additional assumption being one of the biggest sources of uncertainty, since it requires prior knowledge of the effective mass and we know the latter should vary across the band structure, we choose a fixed conservative value of m * ≈ 1.3m e allowing us a direct comparison with the results of Vaz et al. [336].

Variables

Values*

A BM R (-0.532 ± 0.002) % A QM R (1.817 ± 0.007) % τ m,Hall 5.5 × 10 -14 s α R -0.758 eV Å η -2.1 × 10 -5 m/(AT) λ EE 6.33 nm *with parameters: V g = 150 V, B = 9 T, j = 7.5 A m -1
Table 3.2: Summary of the system parameters extracted from magnetotransport measurements used for or obtained from calculations of the charge-to-spin conversion efficiency.

As can be seen, because the QMR amplitude is much smaller than previosuly reported for LAO/STO while the BMR is of similar amplitude, the estimated amplitude of the effective Rashba coefficient in our system goes up to -760 meV Å at high positive gate voltage, which is way higher than that calculated or reported in any STO 2DEG to the best of our knowledge. This inconsistency with the literature may simply show that we are reaching the limit of the model used here to extract α R , although its large value is somewhat compensated by a short scattering time so that the equivalent Edelstein length is in a similar range to what has been obtained before by SP-FMR experiments in other STO 2DEGs.

In comparison, fitting weak (anti-)localization (W(A)L) with a Maekawa-Fukuyama model may provide estimations of the inelastic and spin-orbit fields as well as the g-factor [START_REF] Maekawa | Magnetoresistance in Two-Dimensional Disordered Systems: Effects of Zeeman Splitting and Spin-Orbit Scattering[END_REF]243] which we are lacking here, but it involves many parameters and only provides an absolute value estimation of α R . On the contrary, even though one should take our quantitative analysis of the Rashba coupling with a grain of salt, the change of sign of the Rashba coefficient should be well captured by the experiment and is confirmed in our case to occur around V g = 20 V, as the heavier carriers increase visibly their contribution to charge transport.

We may try to relate these new experimental observations to the evolution of the Edelstein efficiency with the chemical potential (or gate voltage), predicted from tight-binding calculations [235] and observed through spin-to-charge SP-FMR experiments and BMR by [336] in AlO x / STO 2DEGs. Two possible corresponding cross-over regions with which our dataset could match are identified. The first region corresponds indeed to the population of the first states with d xz,yz orbital character, from the orange pair of bands (5 and 6) in fig. 3.2 which have a negative Rashba contribution. The second possible region positioned way above the Lishitz point actually starts above the topologically non-trivial avoided crossing between the orange and green bands, with a band inversion, leading to a decrease of the total Edelstein efficiency.

We note however, that with the state of our knowledge, it is difficult to pinpoint precisely which one is really matching, here. Indeed, the fact that we mainly observe a linear Hall effect at low gate voltage tends to favor the first hypothesis. On the other hand, if we compare our estimation of the carrier density evolution with V g with that obtained by [336], it seems that the appearance of the non-linearity in the Hall effect and actual significant increase of the total carrier density beyond 1 × 10 14 cm -2 should occur in a range of gate voltage already corresponding to the second region, above the topological avoided crossing.

In addition, one can wonder why we observe a nearly monotonic variation and only one sign cross-over of either of the BMR amplitude A BM R or Rashba coefficient α R and not the complex gate dependence with multiple sign changes observed in spin-to-charge conversion experiments. Even considering the reduced range of gate voltages accessed during our experiments, the concentration of electric field lines from the back of the sample to the small device area could be expected to increase the effective electric field to accumulate or deplete carriers in the 2DEG. It is in fact commonly known that the application of a gate voltage does not necessarily results in a rigid shift of the chemical potential across the band structure. One possibility explaining our results could then be from a modification of the 2DEG confinement potential profile with this electric field, thereby modifying the splitting between different subbands and in general the electronic band structure. Furthermore, we have seen in section 1.4.2 that the different orbitals may extend over different lengths into STO and as such feel different electric field intensities. One should also remember that the Rashba coefficient is itself dependent on the electric field felt by the electrons at the interface, which is larger in a device than in a uniform slab as we just explained. Finally, a simple explanation may come from the difference in the materials stack used here, where the thicker Al deposition to form the 2DEG and the thick insulating capping layer might interact differently than with the 0.9 Å of Al and the ferromagnetic NiFe overlayer used for SP-FMR.

While all these remain open questions for the moment, we can in any case think of exploiting the functional property of sign-changing Rashba coefficient and spin-charge interconversion to design applications.

. Towards all electrical logic gates based on spincharge interconversion

As recalled several times, the direct Edelstein effect (DEE) is the generation of a nonequilibrium magnetization upon injection of a charge current in a 2D conductive Rashba system (or along the surface of a topological insulator), which de facto imposes an electric field across the system, parallel to the current. This magnetization is uniform across the area subject to this electric field and can diffuse in other directions, following the drift-diffusion equations described in section 1.1.2. In the case where we place a ferromagnetic material on top, this can cause the phenomenon known as spin torque, where the FM's magnetization may be affected by the injected spin (diffusion) current, and which is widely studied and already envisioned for memory applications.

Here, we propose another type of device, in the case where the spin diffusive current j s flows in a transverse arm and can propagate over a certain distance before it is converted back into a charge current. This distance is defined by the spin diffusion length λ s and in this particular case also by the related inverse Edelstein length λ IEE . If we place another arm at a distance d, parallel to the first channel where the charge current j c in was initially injected, we expect to obtain the generation of another charge current j c out through the inverse Edelstein effect (IEE) and proportional to the spin density that actually propagated over the distance of the transverse spin-diffusion channel.

Here we thus propose to exploit the new logic functionalities offered by the possibility to control the Rashba coefficient amplitude and sign by a gate voltage in a H-shaped device equivalent to some sort of ferromagnet-free all-electrical lateral spin valve, similar to already suggested or realized devices based on SHE in semiconductors by Choi et al. [400] and Wunderlich et al. [401]. Although our demonstration relies on a back-gating configuration, since a top-gate modulation of the charge carrier density was already demonstrated with even higher effficiency, we can imagine that a similar functionality could be achieved by applying a gate voltage from the top. Going even further, the combination of top-and back-gate voltages may allow to access the desired regime of operation to easily pass from one sign of conversion to the other with usual voltages accessible in modern semiconductor-based circuits, and to optimize the spin diffusion length in the transverse spin channel.

(a) � V G <0 � � � � V G >0 � � � � V G <0 � � � � � � V G >0 � J in c < 0 : 0 J in c > 0 : 1 V G < 0 : 0 J out c > 0 : 1 J out c < 0 : 0 V G > 0 : 1 J out c < 0 : 0 J out c > 0 : 1 J c1 =0 � J c2 >0 � � � � J c1 >0 � J c2 >0 � � � J c1 =0 � J c2 =0 � � J c1> 0 � J c2 =0 � � � (c) � IEE , � SHE V J 1 J 1 +J 2 0 J 2 or J c 1 = 0 : 0 J c 2 > 0 : 1 J c 1 = 0 : 0 J out c > 0 : 1 J out c < 0 : 1 J c 2 > 0 : 1 J out c < 0 : 1 J out c
> 0 : 0 The XNOR function can be realized by using the charge current in the first arm and the gate voltage at the intersection as inputs (V G , j c in ). (c) The NAND function can also be realized by using two charge currents (I 1 , I 2 ) from upstream devices to accumulate/deplete carriers on the gate contact at the intersection. Depending on the total current (and the resulting equivalent gate voltage), the activation function is triggered or not.

In fig. 3.8a, we show the example of one such H bar device where a single top-gate contact is placed at the intersection between the first current injection arm and the transverse spin diffusion channel (deposited patent FR2202874 [402]). Depending on the applied gate voltage V G , one can tune the direction (or the amplitude) of the magnetization that diffuses in the transverse arm which in turn governs the direction (or the amplitude) of the output charge current j c out from the inverse Edelstein effect. Assuming a discretized binary set of inputs (V G , j c in ) where V G = 0 corresponds to a 0 and and V G > 0 corresponds to a 1, and the sign of the input charge current j c in dictates the binary value of the second input (e.g. j c in < 0 ↔ 0 and j c in > 0 ↔ 1), the device shown here realizes an exclusive NOR (or XNOR) logic function as seen from the truth table in fig. 3.8a.

Going further, we also suggest another type of logic gate based on the same principle where the charge current output j c out,n of one, two or more devices can all be connected in parallel to the top-gate contact and thanks to Kirchoff nodal rule, push or pull charges from it and thus increase or decrease the electric field felt by the electrons in the 2DEG (i.e. effectively changing the equivalent gate voltage V G ). An example is shown in fig. 3.8c, where the logic function is fully determined by two current inputs (I 1 , I 2 ) on the gate contact and a simple NAND function is realized. This kind of logic gate is universal which means that any other type of logic function can be realized by chaining and connecting multiple gates together in a specific way.

Although we have chosen to define some digital logic gates, one can also go beyond our simple presentation of the concept and use this platform as an analog device with mutiple levels of conversion efficiency and directionality allowing to combine several devices together and perform additive functions or even choose to either exploit the intrinsic nonlinearity of the system with the gate voltage or to operate in a quasi-linear regime.

The devices and applications that can be directly envisioned from the results obtained in this section are purely nonvolatile ones, lacking the capability of retaining information on its own, although in principle a ferroelectric material could be introduced as the gating element and controlled in the same fashion as described until now. In the next section, and building on the demonstration of electric-field-controllable spin-charge interconversion, we demonstrate the possiblity to add a remanence functionality to the system by making STO ferroelectric.

. Non-volatile electric control of spin-to-charge conversion in AlO x /SrTiO 3 2DEGs

As stated in section 1.3.1, the instability of STO towards ferroelectricity has been exploited in several ways, either to study physics at the edge of the transition, or to actually exploit the ferroelectric phase and the functional memory effect it offers.

One route we have mentioned is by application of a large electric field across a (nominally) pure SrTiO 3 crystal at cryogenic temperatures (below 40 K). Using this approach in our case leaves us with the possibility to use a known platform for charge transport and spin-charge interconversion, while adding a non-volatile functionality directly embedded into the host material of the 2DEG. The results in this section are published in ref. [381].

. Sample preparation

To perform the following experiments, we have used standard commercially available substrates of STO from CrysTec, with a size of 5 mm × 5 mm × 0.5 mm that we thinned down to allow the application of greater electric fields with our available equipment (going up to ±200 V maximum).

To obtain the best surface possible, we first applied a surface treatment by immersing the substrate in a buffered hydrofluoric acid (NH 4 F HF7:1) solution for 30 s followed by a water bath to stop the reaction and a 3 h annealing at 1000 °C All this allows to obtain clean stepterraces on the surface with TiO 2 -termination (possibly maximizing the effect of Ti reduction by a metal depositon later on). To limit the surface contamination or damage by later processes, and thus to ensure the best 2DEG quality, we chose to deposit the reactive metal (Al) layer on top as a first step.

The samples were then covered with a protective resist coating on the Al side and then polished from the back side on diamond pads to make them as thin as desired. A back-electrode could then be deposited on the back-side either with Ti/Au or with a transparent indium tin oxide (ITO) electrode (allowing optical measurements to be performed, if needed), after which the samples were finally cleaned from the resist coating and other residues by a gentle acetone and isopropanol cleaning, in mild ultrasounds.

A similar process was employed to prepare samples for SP-FMR experiments, with a difference in the Al thickness (0.9 nm instead of 1.8 nm) and the addition of a 20 nm-thick NiFe layer and a 2.5 nm capping of Al oxidizing spontanesouly in air (see ref. [START_REF] Vaz | Mapping Spin-Charge Conversion to the Band Structure in a Topological Oxide Two-Dimensional Electron Gas[END_REF]). The thickness after polishing was also slightly different, with (250 ± 20) µm for SP-FMR samples instead of (200 ± 20) µm for ferroelectric and magnetotransport characterizations.

. Ferroelectric behavior of SrTiO 3 and its influence

on the interfacial 2DEGs Starting from simple AlO x /STO 2DEG samples, we first tried to induce the ferroelectric phase in the bulk of the substrate, using the metallic contact on the back as a positive electrode and the 2DEG as a reference top electrode. After cooling the samples in a cryostat down to 2 K, and checking from the sheet resistance that the sample properly displays a metallic behavior (see fig. 3.9), we perform a series of polarization loops between different boundaries of electric field and at different frequencies, as described in section 2.6. We show in fig. 3.10a polarization loops performed at 2 K at a frequency of 1 kHz. At low electric fields, we observe initially a simple linear dependence of the polarization with the electric field, typically expected for a normal dielectric material. Going beyond an electric field of 7 kV cm -1 however, a hysteresis develops associated with switching current peaks visible as well in the I(E) curve. The saturation polarization is about 4 µC cm -2 , consistant with earlier reports from Hemberger et al. [166] and Weaver [167].

We go on by measuring similar polarization loops as a function of temperature (fig. 3.10b) and plot the corresponding remanent polarization in fig. 3.10c. We see that the open hysteresis cycles visible at low temperature, close progressively and that the curvature originating from the saturation of the polarization disappears as we increase the temperature, until a quasi-linear dependence of the polarization with the electric field is recovered above 50 K to 60 K. The remanent polarization corroborates the presence of the Curie temperature T C near 50 K, and fig. 3.10c coincides remarkably well with data from Hemberger et al.

[166] and more generally with the literature, confirming the possibility to induce a ferroelectric state in SrTiO 3 .

It is worth noting however that in our samples the threshold electric field to induce this ferroelectric (FE) state seems higher than reported previously. Because the exact mechanism of this transition requiring some kind of wake-up field is not fully understood, we do not have a clear explanation for this small discrepancy either. Once the FE state is activated though, it remains stable as long as the temperature is kept below T C even if the polarization is cycled within smaller electric field boundaries as shown in fig. 3.11a. Interestingly, cycling similarly the sheet resistance with the electric field also reveals a hysteretic dependence with two distinct resistance state at zero electric field. The resistance decreases as expected when increasing the back-gate voltage and increases again as the voltage is lowered back to the negative extreme, except here the forward and backward curves show a hysteresis, opposite to what is normally observed in STO 2DEGs after cycling multiple times the resistance. The strong correlations observed between figs. 3.11a and 3.11b already points towards a "ferroelectric" control of the charge transport properties. Performing Hall measurements in both polarization states indeed shows a difference in carrier density ∆n s = 5.45 × 10 12 cm -2 , only two times smaller than the theoretical value ∆n s = 2P r /e = 1.13 × 10 13 cm -2 (using a remanent polarization of P r = 0.9 µC cm -2 ), which is a notable efficiency compared to the literature [403,404].

. "Ferroelectric" control of spin-to-charge conversion

Building on the findings that a ferroelectric behavior could be induced in STO and be utilized to modulate the carrier density of an interfacial 2DEG, and with the knowledge that the Rashba coefficient evolves with the chemical potential, SP-FMR experiments were carried by collaborators in SPINTEC laboratory (in Grenoble, France), at low temperature and under the application of strong electric fields as well. Measuring the voltage response across the 2DEG a sample of NiFe/Al/STO with a Ti/Au back-electrode, as the DC magnetic field is swept around the FMR, we can obtain the charge current resulting from spin-to-charge conversion using eq. (2.50).

In the initial pristine state after cooldown to 7 K, the voltage drop obtained at resonance corresponds to the production of a positive charge current of 1.2 A m -1 mT -2 as displayed in black in panel A of 3.12b (or black star A in 3.12a).

After applying high enough voltages to induce the ferroelectric transition, up to ±200 V (i.e., electric fields of ±8 kV cm -1 ) [START_REF] Hemberger | Quantum Paraelectric and Induced Ferroelectric States In[END_REF]169], and performing a first initilization cycle as usually done for (non-ferroelectric) STO 2DEGs, up to +200 V, then -200 V and back to +200 V, we observe a hysteretic cycle of the charge current generated from SP-FMR as shown in fig. 3.12a. The charge current produced at ferromagnetic resonance have opposite signs when a voltage of +200 V or -200 V is applied as seen at points B, D and F. After applying the maximum voltage, the normalized current produced at ferromagnetic resonance goes as high as ±8.8 A m -1 mT -2 , a value beyond what was previously obtained in LaAlO 3 /SrTiO 3 and AlO x /SrTiO 3 2DEGs (around 5 A m -1 mT -2 ). Here, the effective spin mixing conductance G ↑↓ ef f is evaluated to 1.2 nm -2 to 3.2 nm -2 , leading to an injected spin current j 3D

s ranging from 100 MA m -2 T -2 to 240 MA m -2 T -2 , with mean value of 160 MA m -2 T -2 .
From this, we can estimate a spin-to-charge current conversion efficiency λ IEE ≈ ±60 nm also one of the highest in the literature, one or two orders of magnitude higher than in metallic Rashba interfaces [START_REF] Rojas Sánchez | Spin-to-Charge Conversion Using Rashba Coupling at the Interface between Non-Magnetic Materials[END_REF] or topological insulators [367].

Even more remarkably, the charge current at remanence (C and E) also has opposite signs which has been observed across several samples of thinned down STO (see fig. 3.13a) but never on a 500 µm-thick STO substrate, explained by the presence of a critical field to induce the FE state. The non-volatile electric control of the spin-charge conversion is further evidenced by fig. 3.12c where we successively pole the sample up and down (applying pulses of ±200 V during 500 ms and going back to 0 V) and measure the charge current at remanence. Finally, ) ) In fig. 3.13b, we show that after applying ±200 V pulses at 7 K, the spin-pumping current remains essentially the same over hours, demonstrating the fact that if any relaxation process occurs, it is too slow to be seen here and thus that a non-volatile electric control of spin-charge conversion can be achieved in AlO x /STO-based 2DEGs. Going back to room temperature, the effect vanishes, however a spin-charge conversion signal similar to panel A of fig. 3.12b can be recovered over different cooldowns before applying any gate voltage, showing repeatability in addition to being reproducible (as seen previously from fig. 3.13a).

Temperature (K) (a) (b) (c) (d)
μ 0 (H-H res ) (mT) After V g > 0 After V g < 0 After V g > 0 After V g < 0
Several mechanisms may be invoked to explain the observation of a hysteretic charge current generated from the inverse Edelstein effect. While the ideal reversal of the local electric field could revert the Rashba coupling as well, it is more probably originating from effects of the electronic structure, since the multiorbital band structure of STO 2DEGs is known to produce opposite signs of the effective Rashba SOC constant as seen in previous sections [START_REF] Vaz | Mapping Spin-Charge Conversion to the Band Structure in a Topological Oxide Two-Dimensional Electron Gas[END_REF]. In addition, although further calculations for the FE STO shoud be performed, we can expect a substantial modification of the electronic band structure under the presence of switchable polar displacements of cations and anions compared to the paraelectric case as was predicted for KTaO 3 for instance [293], potentially occasioning the appearance of additional avoided crossings and a super-efficient spin-charge conversion phenomenon. Moreover, it has been recently suggested that polar moments with large horizontal components in the vicinity of the interface could be induced at negative gate voltages and reduced by going back to positive gate voltages [405], which could also explain some disparities in hysteretic behavior between the bulk polarization and the spin-to-charge conversion. Finally, fig. 3.13d shows an R(V g ) loop measured in a two-probes configuration on the NiFe/ Al/STO sample, superimposed with the charge current dependence with gate voltage I c (V g ). Both have rather similar shapes indicating a similar origin for both hysteretic behaviors. We note however, a different shape compared to that of the R s (V g ) in fig. 3.11b, which can be explained by the difference in the sample electrostatic properties due to the thick metallic NiFe layer in the spin-pumping sample, or by a different shape and thickness, or even by the difference in measurement method (two probes in fig. 3.13d vs. four probes in fig. 3.11b). One additional reason could come from the fact that the SP-FMR samples are cut from plain samples which induce some additional defects, modifying the coercivity.

. Summary

In this chapter, we have shown how it was possible to fabricate some simple devices based on AlO x /SrTiO 3 two-dimensional electron gases whose spin-orbit properties and charge-to-spin conversion efficiency were characterized exploiting the bilinear magnetoresistance (BMR) and quadratic magnetoresistance (QMR) phenomena. The application of a back-gate voltage to these systems allowed us to demonstrate a change of sign in the conversion efficiency which we attributed to a change of sign of the effective Rashba coupling coefficient α R based on the model of Vaz et al. [336] and Dyrdał et al. [393] and in agreement with previous theoretical predictions and experimental reports. Then, a suggestion spintronics logic device was presented, relying exclusively on a spin-orbit material and without any ferromagnet, utilizing the gate tunability of the spin-charge interconversion with a gate voltage to perform simple logic operations.

Finally, we demonstrated how ferroelectricity could be induced in STO substrates to add non-volatility in the electrical control of transport properties and of spin-to-charge conversion in AlO x /STO 2DEGs. One can also imagine logic device units with embedded memory relying on the electric polarization of a ferroelectric material, and in which a ferromagnetic layer would only serve as a spin injector. Such a device named ferroelectric spin-orbit logic (FESO) device has been proposed and patented by the UMPhy and collaborators in SPINTEC [406], and is described in more details in the thesis of Noel [354].

CHAPTER 4

Ferroelectric control of transport properties in AlO x /Ca:SrTiO 3 2DEGs

While we have seen in the previous chapter how SrTiO 3 -based 2DEGs are a very powerful platform for spintronics applications to which an electrical nonvolatility ingredient could be added, the necessity to apply very large electric fields to wake up this functionality makes it a bit impractical. Instead, in this chapter we devise another way to obtain an imprint of ferroelectricity first on transport properties and then on spin-charge interconversion.

. Obtaining 2DEGs at the surface of Ca:STO

To achieve this, we start from a very similar system and use Ca-substituted SrTiO 3 (further noted Ca(x%):STO) substrates, instead of the pure SrTiO 3 , which allows to obtain a ferroelectric behaviour in the ground state of the samples as we go below the ferroelectric Curie temperature T C , without the need for a wake-up procedure. The susbstrates are commercially available single crystals from SurfaceNet with dimensions of 5 mm × 5 mm × 0.5 mm.

We actually choose crystals with 1 % or 2 % of Ca. While the former are expected to show ferroelectricity with T C ≈ 27 K, the second are within the relaxor ferroelectric phase region just above the phase boundary at x ≈ 1.6 %. This brings however a slightly higher nominal transition temperature to collective ordering of polar displacements, at 35 K and we may hope to observe a more robust ferroelectric behaviour at the working temperature of 2 K where we perform most of our magnetotransport experiments, and thus stronger modulation of the transport properties.

We prepare different series of samples on substrates with a (001) surface orientation, with slightly different methods or substrate properties depending on the experimental needs. While it is possible to obtain 2DEGs similarly to the case of pure SrTiO 3 with the simplest procedure, just by sputtering a reactive metal like Al at room temperature, we find that the preparation of the substrate and surface quality seem to play a more important role in the final transport properties. We take even more care for samples destined for highly surface-sensitive experiments such as XPS and ARPES. Because of this, some samples are pre-annealed in a ultra high vacuum (UHV) chamber, using an infrared laser heater to reach a nominal temperature of 800 °C for 1 h in an oxygen-rich atmosphere with partial pressure p O 2 ≈ 300 mbar to 500 mbar. A layer of 1.82 nm of Al is subsequently deposited in situ by magnetron sputtering at room temperature. An example is shown in fig. 4.1 where we display the reflection high energy electron diffraction (RHEED) pattern taken just after the pre-annealing step (fig. 4.1a) typical of highly crystalline 2D surfaces. In figs. 4.1b and 4.1d, we show the X-ray photoemission spectra of the Ti-2p core levels obtained in situ before and after Al deposition. The Ti ions are visibly reduced from a pure Ti 4+ valence state to a mixture of mainly Ti 4+ and Ti 3+ with also the possible presence of a minor Ti 2+ peak in the photoemission spectrum. The area of the Ti 3+ component with respect to the total area for Ti 3+ and Ti 4+ amounts to 38 %. It is consistent with the presence of a 2DEG at the AlO x /CaSTO heterointerface possessing a carrier density in the range of 1 × 10 14 cm -2 at room temperature, similar to the AlO x /STO case [222,225]. As explained in section 1.4, the main scenario is that the deposited reactive metal oxidizes by pumping oxygen from the oxide substrate, thus reducing the topmost atomic layers. Once the sample is exposed to air, what is left of the metallic overlayer should be oxidized from the air into AlO x and acts as a protective capping layer for the 2DEG. As can be seen from the AFM image fig. 4.1c, the topography of the surface after deposition remains that of a typical annealed STO sample with terraces and steps of about 4 Å of height.

Another popular spectroscopic tool to characterize the presence of a 2DEG at the surface of an oxide is angle-resolved photoemission spectroscopy (ARPES), which we also performed at the CASSIOPEE beamline in SOLEIL synchrotron. The facility allows us to prepare and measure the sample in situ, first by pre-annealing for 2 h in UHV and then depositing 1 Å of Al by MBE with the substrate at room temperature, before finally transferring to the measurement chamber.

In figs. 4.2a and 4.2b, we first show the energy dispersion along k y obtained with an incident photon energy of 27 eV and a LH polarization, at 15 K and 55 K respectively, in other words below and above the expected ferroelectric transition temperature of T C = 26 K. No significant difference from the light nor the heavy bands could be distinguished in these experimental conditions that could be attributed obviously to the emergence of polarization in the sample whether between the two temperatures or compared to other STO 2DEGs [START_REF] Vaz | Mapping Spin-Charge Conversion to the Band Structure in a Topological Oxide Two-Dimensional Electron Gas[END_REF]222,407].

Higher quality dispersions were measured at 15 K at Γ 102 (see fig. [START_REF] Vaz | Mapping Spin-Charge Conversion to the Band Structure in a Topological Oxide Two-Dimensional Electron Gas[END_REF]231,408] or in AlO x /STO samples also grown in CASSIOPEE [222,409], without any noticeable difference. Ideally, further ARPES studies with better resolution, possibily spin-resolution and in situ electric gating would give deeper insights into the Rashba properties of such a 2DEG grown on a ferroelectric substrate.

To simplify the fabrication process we also prepare non-annealed samples using simply cleaned substrates with acetone and isopropanol following a standard procedure as described in section 2.1. Those can then be compared with the pre-annealed samples.

Similar to the work of Vicente-Arche et al.

[225], we also prepare one sample with a different metallic overlayer, replacing Al by Y which is expected to yield a slightly different profile of the electron confining potential at the interface on the STO side, and a lower carrier density [START_REF] Vicente-Arche | Quantum Oxide Heterostructures for Spin-Charge Interconversion[END_REF]225].

Finally, we also prepare patterned samples with simple Hall bar devices to perform more advanced magnetotransport characterization such as the BMR and QMR measurements, using the same method as in the previous chapter, namely the deposition of metallic markers outside the devices which are themselves composed of approximately 20 nm to 35 nm of amorphous LaAlO 3 and 2 nm of Al on Ca(x%):STO with x = 1 % and 2 %.

Similarly, for all experiments requiring the application of a back-gate voltage, we deposit a metallic back-electrode of Ti and Au. A summary table of all the samples used in this chapter is presented in table 4.1.

. Ferroelectric characterization

To investigate the ferroelectric properties of AlO x /Ca:STO samples, we start by characterizing the Raman scattering properties of a substrate alone and of an entire sample with the 2DEG, both with 1 % of Ca. These measurements were performed mostly by collaborators from LMPQ, Université Paris Cité with Maximilien Cazayous and details of the experimental setup and related work in SrTiO 3 compounds are shown in the thesis of Pierre Hemme [410].

In the configuration used for these experiments, the probing depth is about 60 µm, which means that although the measurements are done in a reflection geometry on a sample with the 2DEG already formed, and we are targetting the top surface with the laser, a large part of the signal comes anyway from the bulk of the substrate, well beyond the 2DEG region. However, because of electrostatic boundary conditions, a small difference in the Raman spectra may be observed between the bare substrate and the sample with an AlO x overlayer and thus a 2DEG on the top surface, when crossing the ferroelectric transition temperature. In fig. 4.3a, we show Raman spectra acquired at different temperatures from 10 K to 40 K, in a window of Raman shifts from about 0 cm -1 to 80 cm -1 . We observe a peak with nearly invariant intensity and Raman shift around 56 cm -1 in the range of low temperatures used for this experiment. This peak is known in the literature to correspond to the A 1g mode associated to the antiferrodistortive structural transition at 105 K in SrTiO 3 .. Another interesting feature here, particularly at 10 K, is the peak at 15 cm -1 probably corresponding to the transverse optical TO 1 mode. It is a sign of the ferroelectric ordering, as it arises from the breaking of inversion symmetry, allowing the activation of a soft infrared optical phonon mode.

We also note the presence of another adjacent peak at 7.5 cm -1 . It seems as though the TO 1 peak progressively shifts towards lower energy and decreases in amplitude as the temperature is increased, until it merges with the other peak at 7.5 cm -1 at 25 K and finally vanishes above this temperature. Although one could see this temperature evolution as a transfer of the areal weight from the 7.5 cm -1 peak to another separate peak at 15 K, it seems more probable from a comparison with the existing literature [281], that the somewhat broad peak of the TO 1 mode is indeed softening (lowering its energy) with increasing temperature which results in a visual artefact falsely enhancing the otherwise constant 7.5 cm -1 peak.

Some confirmation of the latter intuition can be obtained from the temperature-dependent Raman spectra of the AlO x /CaSTO sample with our 2DEG in fig. 4.3b, where we detect a much higher relative amplitude of the supposed TO 1 peak compared to the A 1g peak. Indeed the low energy TO 1 peak reduces quite visibly in intensity and energy as the temperature is increased step-by-step from 10 K to 30 K at which the peak actually vanishes. This is then the expected soft phonon mode behavior associated to the appearance of ferroelectric ordering in a material. In fig. 4.4, we plot the relative intensity of the peak associated to this TO 1 mode with respect to that of the A 1g -which, as we have seen, remains mostly unaffected at the temperatures of interest-, and normalized to the maximum intensity measured at 10 K. From these temperaturedependent measurements, we conclude that the ferroelectric order is present below 30 K, with the strongest change in the soft-mode intensity occuring near 25 K.

To complete the characterization of ferroelectricity in our samples, we use the AlO x /CaSTO samples to perform a series of polarization measurement while cycling the electric field applied between a back-electrode and the 2DEG (see methods in section 2.6).

To begin with, fig. 4.5 shows the ferroelectric characterization of sample C with 1 % of Ca (see table 4.1). Figure 4.5a shows the current measured across the sample while the electric )

Electric field (kV.cm -1 ) field is swept between ±1.4 kV cm -1 (corresponding to a voltage of ±70 V) at a temperature of 2 K. While a conductive behavior seems absent or negligible, we notice some contribution from a capacitive behavior with a constant background for both sweeping directions, and the presence of two peaks characteristic of polarization switching, with a coercive field near ±250 V cm -1 . Integrating this current over time yields the polarization plotted in fig. 4.5b, which displays a hysteretic behavior with a saturated polarization of ±2.2 µC cm -2 and a remanent polarization of about 0.5 µC cm -2 .

K K K K K K K K K K K K K K K K K K k K K K K K K (b) (a) (c) (d)
Performing the same measurements with increasing temperature, we see in fig. 4.5c that the opened loops at low temperature become progressively flatter until a simple linear dielectric behavior is recovered above 36 K. This is rather well transcribed in the temperature dependence of the remanent polarization in fig. 4.5d with an extended tail visible up to 40 K although a strong decrease is already observed at lower temperature around 22 K.

The fact that a non-zero remanent polarization can still be extracted at higher temperatures than the expected Curie temperature is reminiscent of the fact that even in this range of Ca substitution, Ca:STO may host polar clusters well above the temperature at which its dielectric constant is maximal. Given the relatively large electric field applied here to perform the measurements, we may indeed expect to observe some "remanent" polarization in our electrical measurements above T C . Extrapolating the slope from the range of strongest decrease in remanent polarization between 20 K to 22 K down to the temperature axis yields a temperature of about 26 K, more in phase with the transition temperature observed in the literature and from our Raman measurements.

. Influence of ferroelectricity on transport properties

. Transport properties across the ferroelectric transition

We now turn to a basic characterization of transport properties of the 2DEGs obtained at the surface of Ca(x%):STO.

In fig. 4.6a, we show the temperature dependence of the sheet resistance measured while cooling down sample A with x = 1 % of Ca (see table 4.1) in is pristine state, i.e. before applying any gate voltage, between 40 K and 2 K (the full temperature dependence up to 300 K is shown in the inset). We see that the sample indeed displays the typical metallic behavior observed for similar 2DEGs on non-ferroelectric STO substrates. On closer inspection however, we also notice an unusual "kink" with a minimum of sheet resistance near 30 K, followed by an increase of the sheet resistance for lower temperatures.

Such a "kink" in the temperature-dependent resistivity (ρ(T )) was interpreted as a fingerprint of the emergence of ferroelectricity in n-type bulk Ca: STO [281] and n-type strained STO thin films [411] as mentioned in section 1.3.

The derivative of the sheet resistance with respect to the temperature (blue curve in fig. 4.6a) reveals a transition peaking at a temperature of around 26 K. Performing magnetotransport measurements, we obtain quasi-linear Hall curves at 2 K allowing us to simply extract an electron density of about 8.4 × 10 13 cm -2 at 2 K (see fig. 4.6b). Same as for the sheet resistance, when the temperature is increased, we can distinguish two regimes in the carrier density evolution below or above 26 K with correspondingly different slopes.

Using the temperature dependence of the sheet resistance and carrier density in figs. [START_REF]IPCC. Climate Change 2022: Mitigation of Climate Change isbn[END_REF].6a and 4.6b and assuming a single-band transport, we can also extract values of the mobility as a function of temperature (see fig. 4.6c).

Above T C , since the decrease of carrier density is accompanied by a continuous decrease of resistance as the sample is cooled down, we thus have a large increase of mobility in compensation up to a maximum appearing near 29 K. Below 29 K, the mobility then seems to decrease a bit and to mostly saturate or to increase slightly. Charge transport is thus somehow affected either by the introduction of new scattering mechanisms or a change of the band structure and/or of the confining electric field associated to the emergence of ferroelectricity.

These are then seemingly the first imprints of the ferroelectricity that we can observe in the transport properties, before applying any external electric field. From longitudinal resistance measurements in the van der Pauw configuration as a function of the out-of-plane magnetic field, we see in fig. 4.6d how the magnetoresistance also changes behavior with temperature. At 2 K, the resistance follows a sharp dip below 1 T and then decreases for higher magnetic field, although we seemingly expect another increase at high field, beyond what we could access with our setup. In fact, it seems that the signal is dominated mostly by weak anti-localization (WAL) at low field and weak localization (WL) in the intermediate range, indicative of the 2D nature of the conducting layer and the presence of a strong spin-orbit coupling affecting electrical transport. As the temperature is increased, the WAL contribution quickly decays, leaving only the weak localization (WL), and the usual Lorentz contributions at higher fields. Although the magnetoresistance curves seems to recover a mostly B 2 -type of dependence and slightly decrease in amplitude at 9 T above T C , it is difficult here to pinpoint directly which of the magnetoresistance contributions are caused naturally by the 2DEG as in AlO x /STO samples at low temperature or by the introduction of ferroelectricity on top of this. .7a displays the temperature dependence of the sheet resistance for sample A both while cooling down and subsequently warming up, both at the same rate of 4 K min -1 . Although we have seen that ferroelectricity seems to have an impact on the sheet resistance, causing an upturn at low temperatures, it is clear by comparing both curves here that an unexpected hysteretic behavior can be observed. Taking their derivative (in the inset of fig. 4.7a) reveals a major kink for both curves occuring either around 26 K (for the cool down) or around 85 K (for the warm up). This behavior could be reproduced consistently across various samples CaSTO-based 2DEG samples obtained during this thesis.

It is also somehow reminiscent of the report from Biscaras et al. [239] where the application of a positive gate voltage across LaTiO 3 /SrTiO 3 leads to electrons getting trapped by the defects of the STO crystal. They suggested that only an increase of the temperature above some corresponding activation energy could allow the release of these trapped electrons, with a first threshold near 70 K observed by different groups and a second threshold at higher temperature although there seems to be less consistency on the latter value [213,239,412]. Although no gate voltage is applied here between the cool-down and warm-up steps, the emergence of ferroelectricity at low temperature could favor the trapping of carriers by different manners below the transition. Note that we also did not consistently observe a second higher-temperature remarkable change across all measured samples. This hysteretic behavior may thus originate from other mechanisms than suggested here, but since it was not central to the topic of this thesis, we did not investigate further these observations. In fig. 4.7b, we show a comparison of sheet resistance temperature dependences for various samples, all measured while cooling down. We can more or less distinguish for all samples the metallic behavior and how the initial decrease of resistance is affected at low temperature, at least by a kink a bit below 30 K for samples with 1 % of Ca and near 35 K for those with 2 % of Ca. Following these accidents, the sheet resistance of sample samples seems to continue dropping, while for others it starts increasing and appears to reach some saturation at the lowest measured temperature of 2 K except for sample F for which we have grown a thicker Y layer instead of Al which thus probably affect the redox reaction in the 2DEG formation process and results in slightly different transport properties. We note that overall, the individual behavior of these 2DEGs are reminiscent of the transport properties observed in bulk-doped Ca:SrTiO 3-δ by Rischau et al. [281] depending on the carrier concentration.

. Controlling transport properties by ferroelectric switching

In this section, we investigate the influence on the transport properties of ferroelectric switching by the application of an electric field.

In fig. 4.8, we show experiments performed on sample C (see table 4.1) at 2 K comparing polarization and sheet resistance measurements as a function of the electric field, for different values of maximum electric fields after a gate initialization (or forming step) up to V g,max = ±70 V (i.e. E max = ±1.4 kV cm -1 ).

Although polarization loops performed at 200 Hz with increasing electric field extrema remain centered around the same point at least in this range of electric fields, in fig. 4.8a, we shift them for the sake of clarity. We see that they all display a hysteretic behavior with the electric field, with a shape typical of ferroelectric polarization loops. We note a slight asymmetry with a somewhat deformed shape for positive electric fields, particularly in the upwards sweeping direction.

We confront these results with sheet resistance vs. electric field cycling curves in Figure 4.8b, performed within similar boundaries via a LIA technique, at the much slower rate of few mHz that our FE tester and setup allows. Similarly, we shift each curve only for clarity. We see an expected modulation of the sheet resistance R s with an increase as the back-gate voltage is ramped towards negative values and a decrease when going to positive back-gate voltage. Remarkably, these curves also display a hysteretic behavior with two clear remanent states at 0 V depending on the P up or P dn state. This behavior mimics that of the polarization loops and after applying ±1.4 kV m -1 , maximal values for the remanent sheet resistance and remanent polarization are ∆R s = 72 Ω □ -1 and 2P r = 0.85 µC cm -2 respectively. We also observe an asymmetry between positive and negative electric fields with a nearly saturating behavior in the negative range and a more open loop up to the maximum voltage applied.

Although we do not show it here, going to gate voltages higher than about 80 V across our 500 µm-thick samples tends to alter the shape and shift both the polarization and sheet resistance cycling curves towards positive electric bias. We generally observe a faster degradation of the conduction properties of such samples compared to simple and well-known pure SrTiO 3based 2DEGs when going to higher voltages although it is unclear whether this phenomenon is solely due to trapping of charges, pinning of FE domains or a more profound degradation of the samples by electric-field activated processes. For the sake of reproducibility and to keep -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 1.0 symmetric voltage boundaries, for the rest of this thesis, we keep the boundaries of electric-field used between ±75 V maximum for any CaSTO-based 2DEG sample.

1.2 1.4 1.6 1.8 2.0 2.2 ±10 V ±20 V ±30 V ±40 V ±50 V ±60 V ±70 V
Following the observation of a ferroelectric imprint on the R s vs. E loops, we perform magnetoresistance and Hall effect measurements in different polarization states. Figure 4.9a shows the magnetic field dependence of sample B sheet resistance at 2 K after gate initialization between ±70 V, successively performed at +70 V, positive remanence (0 V), -70 V, and finally negative remanence (0 V). We can see again how the overall resistance is modulated by the application of gate voltage with also two distinct remanent states. In fig. 4.9b, where we show relative magnetoresistance, we can see better how although it stays negative in all four gating states mentioned above, the magnetoresistance changes shape in each case. Although the amplitude of the negative magnetoresistance seems higher here than previously observed for smaple A in fig. 4.6d at 2 K, we notice a similar behavior reminiscent of W(A)L-dominated signals up to to the maximum aboslute field of 9 T. The difference between the magnetoresistance curves at positive or negative remanence (coming from +70 V or -70 V respectively) in fig. 4.9b is however more tenuous, especially in the low-field region below ±3 T. This value is more than 50 % of the value expected from the field effect when considering the remnanent polarization of the Ca:STO (2P r /e = 1.075 × 10 13 cm -2 ). The corresponding efficiency is among the highest reported in the literature [403,404].

P up P dn V g = -70 V V g = +70 V V g = 0 V after +70 V V g = 0 V after 70 V
As we may expect, the higher carrier density is achieved for the P up state accompanied by a seemingly slightly lower mobility for both majority and minority carriers and one order of magnitude difference between them.

Thus, although the change in magnetoresistance seems small at first sight, the change in carrier density and rise of the minority carriers would tend to indicate that a modulation of the 2DEG properties across or near the Lifshitz point is possible. This is very promising as it would potentially allow to control the Rashba-Edelstein properties in a non-volatile way simply by applying a transient electric field across the sample to switch its ferroelectric state and could be enhanced by the modulation of carrier mobility and thus momentum relaxation time to which the spin-orbit scattering time is usually correlated.

. Ferroelectric control of charge-to-spin conversion in AlO

x /Ca:SrTiO 3

To verify that we can indeed achieve a nonvolatile electrical control of the charge-to-spin conversion through the Edelstein effect, we prepare samples with Hall bar patterns, allowing better geometrical definition of the transport properties and the injection of higher current density necessary for a high signal-to-noise ratio (SNR) especially for the detection of bilinear magnetoresistance (BMR) (which should depend linearly with the current density). The fabrication method is similar to that of section 3.2 and described in section 2.1.

. Basic transport properties

Measurements have been performed on both samples H and I with 1 % and 2 % respectively, and the expected BMR and QMR signals could indeed be observed in the 1 % case, similar to AlO x /STO. However, due to the absence of a sufficiently clear effect of ferroelectricity in our measurements at remanence, we focus in this section on the data acquired in sample I only, with 2 % of Ca (see table 4.1) with slightly higher T C . In fig. 4.10, we show the temperature dependence of the sheet resistance measured during two temperature sweeps from 300 K to 2 K. After the first cooldwon (CD1) and performing gating experiments (not shown here) up to 200 V, we do not observe a significant hysteresis in electrical cyclings of either the polarization or the sheet resistance. Following this observation, to recover a virgin state, the sample is warmed up to room temperature taken out of the cryostat and thus exposed to air and ambient light for few minutes for closer inspection of the sample mounting. The sample is then inserted back into the cryostat and cooled down for the second time (CD2).

We first note that during both CD1 and CD2, the sheet resistance displays a rather sharp kink near 33 K followed by an increase of resistance similar to what we observed previously for Ca(2 %):STO. Some other incidents can be observed in the first derivative of both curves (see fig. 4.10), particularly near 180 K, a value close to the reported cubic-tetragonal transition temperature of Ca:STO at 170 K for this composition [181].

It is striking that the sheet resistance R s is decreased by approximately a factor of three (at 300 K) between CD1 and CD2. Although warming up the sample above a certain threshold temperature and/or exposing it to light should ensure that all carriers trapped by applying a positive gate voltage at low temperature are released back, it seems as though an additional unintended effect is present and we attribute this decrease of resistance to the addition of carriers in the sample by photoexcitation, contributing to the 2DEG conductivity.

As a result, the R s (T ) curve of CD2 features an additional local minimum near 85 K and several bumps above the Curie temperature of 33 K can also be identified from the derivative in fig. 4.10. Although further investigation should be pursued to understand the underlying physics and effects of illumination at different temperatures on the sample's transport properties, we decide to go on with the measurements as the most significant change we might expect here seem to come from the carrier density.

For the following data presented in this section, we thus present an overview of some preliminary magnetotransport data acquired with the magnetic field in the plane of the sample and in a back-gating configuration, using a similar experimental setup as in section 3.2. Here we use a Keithley 6221 AC current source combined with a Zurich Instruments HF2LI lockin amplifier (LIA) which allows us to measure simultanesouly the longitudinal and transverse resistance both at the first and second harmonic of the reference frequency (i.e., the applied current frequency).

. Characterizing the in-plane magnetotransport in

AlO x /Ca:SrTiO 3

Following the second cooldown CD2 and before applying any gate voltage, we perform a series of magnetotransport experiments with the field applied in the plane of the sample while varying the angle between the injected current and the applied magnetic field B = µ 0 H.

First, we show in fig. 4.11 an example of few curves measured at +9 T for different current densities. We notice at first sight that the average resistance value in fig. 4.11a obtained form the first harmonic signal first decreases as the current is increased for low currents, and then starts to increase again above j = 10 A m -1 . This also coincides with a change of the angle dependence with maxima at 0°and 180°for the low current regime and a progressive inversion visible from 8 A m -1 so that at higher currents, the maxima are at 90°and 270°.

On the contrary, the second harmonic resistance (from which we substracted the average background resistance and slope before plotting in fig. 4.11b) shows the presence of a simple sinusoidal dependence with a monotonic increase of its amplitude. We show in fig. 4.11c how this amplitude seems to increase quasi-linearly at low current and then tends to deviate with lower values as the current density exceeds 10 A m -1 . To get a broader overview over the entire range of current densities explored with positive and negative magnetic fields, we use height map plots which which helps to keep track of the different features in each set of conditions even for complex angular dependences and with the least data processing possible.

In fig. 4.12, we thus show the angle and current-density dependence of the first and second harmonic longitudinal resistance signals normalized to the highest value for each sweep, and measured first at +9 T (figs. 4.12a and 4.12d) and then at -9 T (figs. 4.12a and 4.12d).

We notice from figs. 4.12a and 4.12b measured at 9 T and -9 T respectively, how a complex evolution of the angle dependence emerges as the current is increased from 10 µA to 300 µA (i.e. the current density is increased from 1 A m -1 to 30 A m -1 ). For the lowest current densities of only few A m -1 , the angle dependence follow a cos(2Φ) function expected from the QMR model of Vaz et al. [336] and observed previsouly in STO 2DEGs. As j is increased, an additional component turns in with the same periodicity but shifted by 90°, and already dominates the signal around 10 A m -1 for +9 T and about 15 A m -1 for the opposite field. For slightly higher current densities itt also seems that an additional feature brings an asymmetry between the new local maxima at 90°and 270°although it is unsure how to deduce a clear trend from it. To eliminate the field asymetry, since we are mostly interested in detecting even functions of the magnetic field and also want to get rid of sample misalignement errors leading to out-of-plane field component, we plot in fig. 4.12c the half-sum (the average) between equivalent data measured at opposite fields. Although drift contributions present for some angle sweeps are not eliminated by this method, it confirms the general trend just described with a progressive change from a cos(2Φ)-dominated signal to a sin 2Φ -π 2 -dominated signal, which stays mostly so up to the highest current density used albeit with apparently smaller amplitude at j = 30 A m -1 . Note that we also observe a strong decrease of the average first harmonic resistance with this current density, possibly indicating some heating effect of the device.

On the other hand, for the second harmonic shown in figs. 4.12d and 4.12e, we observe a sin(Φ) component for most of the current density range explored for both ±9 T although with opposite polarity between positive andb negative magnetic field. Although here we can easily track the appearance and disappearance of features in the angular-dependence, we do note that using normalized values for the second harmonic prevents from identifying the proper amplitude evolution as plotted before in 4.11c. Indeed, altough substracted from fig. 4.11b a significant offset is also observed in the second harmonic signal on top of the sinusoidal component and it increases with the current, too. Next, we also plot the angular and field dependence in the same fashion as for the current dependence, both as simple angular dependences in fig. 4.13 and with height maps in fig. 4.14. For this field dependence however, we note that measurements were performed at a fixed current density of 25 A m -1 which is already in a regime where the sample magnetotransport is affected significantly, possibly due to heating.

We can see from fig. 4.13a (or figs. 4.14a and 4.14b), measured at positive and negative fields, that the first harmonic signal has a nearly flat and constant angular dependence at 0 T and as the field is increased in absolute value, two maxima can be identified at 90°and 270°w ith a strong imbalance towards either 90°or 270°depending on the field sign, similar to the high current density regime observed previously.

The average of these signals in fig. 4.14c between the positive and negative fields eliminates this asymmetry with respect to the field and thus restores a pure sin 2Φ -π 2 dependence with maxima at equivalent levels at 90°and 270°and growing amplitude with the magnetic field. Further investigations should thus be pursued in the future to disentangle the various possible contributions to first and second harmonic signals from the QMR and BMR respectively, or from additional phenomena especially from spurious thermal effects. We should as well perform similar measurements at lower current densities in the intermediary regime (10 A m -1 to 20 A m -1 ) and further in the low current regime below 5 A m -1 at the cost of a worse signal-to-noise ratio (SNR).

. Ferroelectric control of the bilinear magnetoresistance

In this section, we present a brief snapshot of the gate control of the in-plane magnetotransport, with the unique non-volatile electrical control functionality offered by the ferroelectricity of the system. First, we show in fig. 4.15 the gate voltage cycling of the sheet resistance performed during CD2 after the first magnetotransport measurements shown in previous subsection. We plot together the first gate initialization cycle (in blue) between ±50 V and subsequent loops (in green) at a slower rate and with finer steps. This demonstrates how after few voltage cycles, the resistance follows a repeatable hysteretic behavior with the gate voltage with two different states at remanence, as we have ssen previously for unpatterned Ca:STO samples with 1 % of Ca. We then perform a series of magnetotransport measurements at different voltages between ±50 V along the cycle, all with a current density of 7.5 A m -1 and at opposite fields of ±9 T.

We show in fig. 4.16 the resulting gate dependence of the first and second harmonic resistances, from which we first remove a slope component resulting from resistance drift after each change of gate voltage, still present despite a stabilization time of at least 15 min at each step. We also substract the average background resistance to show just the amplitude evolution as a function of the gate voltage, of which we plot the sequence in fig. 4.16a.

One can see a clear modulation of the amplitude of both first (fig. 4.16b) and second (fig. 4.16c) harmonic signals as the gate voltage is cycled with similar angular dependences for equivalent conditions, but with slight differences whether we come from positive voltage to negative or the contrary. Surprisingly, the first harmonic resistance in fig. 4.16b conserves a cos(2Φ)-dominated dependence for all gate voltages except at the positive maximum of +50 V where it acquires the sin 2Φ -π 2 dependence seen before for medium to high current densities. If the emergence of this second term with maxima at 90°and 270°were attributed to a thermal contribution only, we would rather expect a significant increase of its effect on the first harmonic signal as the voltage is changed to lower or negative values thereby increasing the resistance and thus the Joule effect. Here, we may thus be in presence of another contribution, perhaps from a complex dependence of the available relaxation channels with the gate voltage (and thus the ferroelectric state and the Fermi surface) [396,397]. Focusing on the second harmonic resistance signal shown in fig. 4.16c, we generally observe a sinusoid of higher amplitude at positive remanence (V g = 0 V, coming from the positive side of gate voltages) than at negative remanence (V g = 0 V, after applying a negative voltage). We illustrate this in fig. 4.17 by plotting an average of angle sweep curves acquired in similar conditions (average for same magnetic fields and half difference for opposite fields) both in the positive (P up , fig. 4.17a) vs. negative (P dn , 4.17b) remanence states.

Importantly, the amplitude of this signal is higher in the P up state where the total resistance is actually lower than in the P dn state. Note that here we plotted the values in units of resistance, i.e. even before normalizing to the total resistance of the sample at zero field for the corresponding gate voltage and polarization state, which means that the modulation effect can be attributed to a factor independent from the simple change in resistance. For instance, it seems to eliminate the possibility of this second harmonic signal being (solely) due to thermal effects, since we would otherwise expect it to be higher for a higher resistance state (due to stronger Joule effect), in other words higher in the P dn state than in P up . Despite the need for further analysis of the full dataset and further characterizations, it seems reasonable to attribute this contribution to the interaction of the non-equilibrium spin density generated by the Rashba-Edelstein effect (REE) with the external magnetic field, observed similarly in other STO-based 2DEGs such as the non-ferroelectric AlO x /STO (see section 3.2), i.e. the BMR. In this case, we would thus have the first demonstration of the ferroelectric control of charge-to-spin conversion by the Rashba-Edelstein effect (REE) in an oxide 2DEG.

Although here we might have only achieved a small modulation of the BMR amplitude (and thus expectedly of the Rashba coefficient amplitude), it is plausible that a larger exploration of the gate voltage space and initialization boundaries could allow us to reach a point where the modulation would be stronger thereby turning on or off the conversion or as in the case of AlO x /STO even include a change of sign. Indeed, doing a gate initialization between different voltage boundaries would allow to access different ground states of carrier density and thus, different points in the band structure when poling up or down, hopefully closer to the remarkable avoided crossing points with maximal Edelstein effect or to its full sign reversal.

This achievement would again open up new perspectives in the field of spintronics and for in-memory logic devices applications, where non-volatility would be achieved not by ferromagnetism but by ferroelectricity, for instance with the realization of FESO devices. the Kondo effect [420], negative magnetoresistance or anomalous Hall effect (AHE) [388] are usually thought to arise from localized magnetic moments (either intrinsic or extrinsic) with with the itinerant electrons interact. However, in general, ferromagnetic ordering arising specifically from these interfaces remains elusive and the hysteretic magnetotransport data reported by Brinkman et al. [420] are still scarce in the literature, to date [424].

In addition to the intrinsic ferromagnetism, several groups have tried to enhance the ferromagnetic ordering by adding dopants into the bulk of the STO substrates, in LAO or by inserting a magnetic buffer layer like europium titanate EuTiO 3 [425].

Europium titanate EuTiO 3 (ETO) is very similar to STO in many ways, starting with the same lattice parameter of 3.90 Å, and a similar electronic structure with O-2p valence states separated by about 3.0 eV from the conduction band formed mainly of Ti-3d states. One significant difference however, lies in the presence of an additional Eu-4f narrow band below the Fermi level, shortening the band gap to 0.93 eV, much smaller than that of STO and LAO.

While it has a G-type antiferromagnetic ordering below its Néel temperature T N ≈ 5.5 K in its bulk form [140], it can be tuned into a ferromagnetic material by electron doping [426,427] or lattice strain [428][429][430][431][432]. De Luca et al.

[425] showed using XMCD that δ-doping of the LAO/STO heterostructure with 2 u.c. of ETO grown epitaxially could induce a ferromagnetism of the EuTiO 3 below 7.5 K, translated into a Kondo scattering effect on the transport properties, still present and preserved for thicknesses of LAO beyond 4 u.c. (see fig. 5.1a). More remarkably, Stornaiuolo et al. [433] later demonstrated not only that a superconducting phase could emerge from the ferromagnetic one at low temperature (with a T c around 125 mK) but also that an anomalous Hall effect was introduced and could be varied by applying a gate voltage (as shown in fig. 5.1b). A transition was also visible in the temperature dependence of the sheet resistance as a function of the gate, passing from the already observed Kondo scattering at low gate voltages to a drop of resistance below the onset of the ferromagnetic T M C around 7.5 K (see fig. 5.1c). This change of transport behavior seemed to coincide with the Lishitz transition, i.e. with the population of the d xz,yz states. As shown in 5.1d, these transport signatures combined with new XMCD data confirmed that Ti ions within the conductive interface were also adding their own contribution to the total magnetic moment of the system and that the 2DEG was indeed showing signs of an electrically-tunable itinerant ferromagnetism.

. Designing a multiferroic 2DEG

Building on the previous work of our collaborators from Naples in the group of Marco Salluzzo (from CNRS-SPIN) and Daniela Stornaiuolo (from Università di Napoli, Federico II), in this section we devise a method harnessing the properties of a spin-polarized 2DEG at the LAO/ETO/STO interface with the ferroelectric control of electronic and spin-orbit coupling offered by making STO ferroelectric as seen in chapter 4 and section 3.3.

With the help of another collaborator Julien Varignon from the Université de Caen, we perform some density functional theory (DFT) calculations to support our design strategy and predict the properties that we might expect at the interface.

Starting from a slab consisting of 10 u.c. of STO oriented along the (001) direction, and 2 u.c. of ETO capped by 6 u.c. of LAO, a ferromagnetic arrangement is imposed for the Eu 2+ cations and a small compressive strain of 1.6 % is applied onto STO to make it ferroelectric with a polarization along the (001) axis. We then lock the STO atomic positions of the first four unit cells of the slabs and consider two cases with polarization pointing either towards the interface (P up ) or away from it (P dn ). We then compute the electronic structure of this system.

In figs. 5.3b and 5.3c, we plot the projected density of states (DOS) layer-by-layer for either polarization states. We obtain a n-type carrier gas at the interface between STO/ETO and LAO as inferred by the projected DOS. In these calculations, the 2DEG thus appears as a consequence of a polar catastrophe [202] although one cannot exclude that oxygen vacancies would contribute as well in reality [211]. Although the resulting DOS at the Fermi level is sizeable for P up , it is smaller but finite for P dn .

Due to the presence of the ferromagnetically ordered Eu 2+ spins, Ti cations experience a spin-dependent potential which produces asymmetries between the spin-up and spin-down channels for the d-states. It results in a spin-polarized DOS and a net magnetic moment on Ti cations as well, which spreads over four TiO 2 layers in the STO below the interface with ETO/LAO. As can be seen from fig. 5.3d, the Ti magnetic moment is maximal for the layer sandwiched between two Eu 2+ layers below the interface with LAO and indeed extends into the first STO unit cells.

Figure 5.3e displays the relative displacement of the Al/Ti (B cation) with respect to oxygen ions along the growth axis across the structure. In LAO, the displacements are finite and large due to the electric field created by the polar interface with ETO but they are equivalent for both P up and P dn within error bars. In STO, the polar displacmeents related to Ti O rumpling ∆z Ti-O are also present but they are different depending on the polarization state as a consequence of ferroelectricity. Far away from the interface, the two polarizations and associated polar displacements although not shown here would be exactly opposite. As we get closer to the interface, they are both biased towards negative value due to the 2DEG confining electric field caused by the polar discontinuity which causes a polar shift of Ti ions by about 0.1 Å to 0.15 Å in a non-ferroelectric LAO/STO system. Importantly here, in the 2DEG region, ∆z Ti-O takes two different values for P up or P up indicating that the 2DEG possesses two different remanent polarization states, i.e. that it is ferroelectric.

According to these theoretical considerations, the 2DEG region may thus host concurrently and counter-intuitively both ferromagnetic and ferroelectric orders [271]. Additionally, an interesting prediction of fig. 5.3d is also that the magnetic moments on Ti cations are different for P up and P dn . Thus, ferroelectric switching should offer a direct control of the magnetization in the multiferroic 2DEG. Our calculations therefore predict that our 2DEGs are not only multiferroic but also magnetoelectric. This fact may originate from the different band bending induced at the STO/ETO and ETO/LAO interfaces for the two polarization states (see figs. 5.3b and 5.3c): P up (P dn ) brings a positive (negative) contribution to the electric field present at the TiO 2 LaO interface, thereby increasing (decreasing) the band bending. Consequently, the Fermi level crosses more Ti 3d states and further away from the interface for P up than for P dn state where it is limited to three TiO 2 layers below the interface.

. Sample preparation

In the next sections, the theoretical hypotheses drawn from DFT calculations shall be verified by realizing a similar heterostructure as used in the literature and for the simulated slab.

We have shown already in chapter 4 that a 2DEG could be easily obtained at the surface of calcium strontium titanate Ca:SrTiO 3 (CaSTO) by a room-temperature Al deposition process via magnetron sputtering, and that a strong imprint of the ferroelectricity would be present in the charge transport [434] and spin-orbit properties. Tuvia et al. [435] also showed shortly after that a 2DEG could also be obtained by growing LAO on top of CaSTO, very similarly to pure STO, with the addition of two superconducting transition temperatures T c depending on the polarization state of the substrate (but still in the 100 mK range).

Here, we use stacks of 10 u.c. of LAO and a 2 u.c. buffer of ETO grown sequentially onto a Ca(1 %):STO substrate with (001) surface orientation. After a BEO HF etching and annealing to obtain a TiO 2 -terminated surface (see 2.1), the samples provided by the group of Marco Salluzzo were grown by pulsed laser deposition (PLD) from a sintered Eu 2 Ti 2 O 7 target, in a 8 × 10 -5 mbar O 2 atmosphere at 720 °C [433]. In situ monitoring of the growth was perfomed by reflection high energy electron diffraction (RHEED). As can be seen from the RHEED oscillations, a layer-by-layer 2D growth mode is achieved for both thin films and a smooth step-terrace configuration is still observable from the AFM image of the surface after the deposition. As for previous sections, experiments requiring the application of a gate-voltage were performed in a back-gating configuration either after sputtering of Ti/Au (20 nm/50 nm) on the back side or simply by contacting the sample's backside with silver paste onto an electrode of Ti/Au deposited on a piece of MgO crystal, to ensure a good and uniform gating of the sample.

. Combining ferroelectric and magnetic orders in the 2DEG

In this section, we use a combination of spectroscopic tools followed by magnetotransport measurements to characterize the different properties of interest in our samples, i.e. ferroelectric and magnetic orders.

. Can the 2DEG itself be ferroelectric?

We start by measuring the X-ray linear dichroism (XLD) at the Ti-L 3,2 edge on a typical sample prepared as explained in the previous section. As explained in section 2.3.4, this elementspecific technique is obtained from the difference of X-ray absorption spectra (XAS) measured in either linear horizontal or vertical polarization of the incident photons and gives information on any changes in the local crystal field splitting. We perform these experiments by contacting the 2DEG from the top so we can measure the total electron yield (TEY) which is particularly sensitive to the first 2 nm near the surface which means the signal measured here corresponds to the interfacial region including ETO and the first atomic planes of STO, i.e. where the 2DEG is located.

We perform these measurements after successively applying ±50 V pulses in situ to switch the polarization of the sample either up or down. For practical reasons, we disconnect the total electron yield (TEY) measurement setup when applying a gate voltage and vice cersa so that absorption spectra can only be measured at 0 V, i.e. at electrical remanence. In fig. 5.5a, we show an example of the XAS signals obtained at the Ti L 3,2 edge from the TEY with both linear horizontal (LH) and linear vertical (LV) light polarizations, with the sample in the P up (in red) or P up state (in blue). We can see that how changing either the photon polarization from LH to LV or the sample's ferroelectric polarization state from P up to P dn can affect the absorption spectrum especially for the e g final states, but not only.

The XLD signals collected at 2 K for both polarization states P up (in red) and P dn (in blue) are shown in fig. 5.5b in the top part. The XLD spectra look similar to previous ones reported in LAO/STO heterostructures [328] (see section 2.3.4) but are visibly modulated by the remanent polarization direction, with the main changes seen in features B and D corresponding to the Ti e g states. Atomic multiplet simulations reproducing the data show that the splitting of the e g states is (80 ± 5) meV for P dn and (95 ± 5) meV for P up consistent with the larger ∆z Ti-O values computed by DFT for P dn compared to P up . The corresponding TiO 6 octahedra configurations are sketeched in fig. 5.5c. These results thus indicate that the positions of the offcentered Ti ions with respect to the O ions depend on the voltage history of the sample thereby demonstrating the presence of switchable dipoles in the 2DEG and confirming its ferroelectric character.

More quantitatively, since in LAO/STO an orbital splitting of the e g states by 100 meV corresponds to a built-in polarization of P built-in ≈ -7 µC cm -2 [328, 436] , we can estimate that the actual spontaneous ferroelectric polarization in the 2DEG is roughly P FE-2DEG ≈ (0.5 ± 0.1) µC cm -2 . The effective total polarization in the 2DEG is then P built-in +P FE-2DEG ≈ -6.5 µC cm -2 in the P up state and P built-in -P FE-2DEG ≈ -7.5 µC cm -2 in the P dn state.

. Coexistence with ferromagnetism

In addition to the two electrically switchable and remanent polar states, to selectively probe the magnetic response of the elements present in the interface region, we perform XAS experiments at both Ti L 3,2 and Eu M 5,4 edges, under an in-plane magnetic field and with the two right-and left-circular polarizations from which we compute the X-ray magnetic circular dichroism (XMCD) signal.

Figures 5.6a to 5.6d display XAS and XMCD at 5 T for Ti and Eu measured again from the TEY. The XAS spectrum measured at the Ti L 3,2 edge is typical of Ti in a valence state close to 4+. A small fraction of Ti 3+ (a few percent) is however expected to be present owing to the population of the Ti t 2g states. Figure 5.6b displays the normalized difference between XAS collected with either polarizations and thus a clear dichroic signal more than 2 %, indicating the presence of Ti magnetic moments in the sample near the interface. The shape of the XMCD is reminiscent of that measured by Stornaiuolo et al. [433]. Extracting the value of the Ti moment from the XMCD signal is not trivial because sum rules do not apply well to early 3d transition metals. However, comparison with ref [433] suggests a total moment of around 0.025 µ B /Ti. Figure 5.6e shows the dependence of the XMCD signal at 457.6 eV (corresponding to the Ti L 3 edge) as a function of the in-plane magnetic field: a clear open hysteresis cycle is observed, consistent with a ferromagnetic behavior probed along an easy axis magnetization axis. The XAS and XMCD at the Eu M 5,4 deges are also presented in figs. 5.6c and 5.6d and are compatible with a Eu 2+ valence. The large XMCD and the hysteresis loop displayed in fig. 5.6f show that ETO is ferromagnetic as in the electron-doped ETO [426,427] rather than antiferromagnetic as in the bulk [140]. This seems to indicate, as predicted by DFT, that a non-zero density of states in ETO might be filled by electrons and thus contribute to charge transport.

. Interplay and ferroelectric control of magnetism: a magnetoelectric 2DEG

We now turn to the electrical characterization of the bulk ferroelectric properties and magnetotransport of the 2DEG. ). An example of the reconstructed loop obtained at 2 K in fig. 5.8b shows how we can extract an approximate value of the remanent polarization. Note that, while these loops correspond to the whole CaSTO, the remanent values are in the range of those estimated solely for the 2DEG region from XLD data and simulation.

Applying both simple P (V g ) and Remanent Hysteresis loops with increasing temperature, we observe a sustained hysteretic behavior up to about 35 K, extract the remanent polarization vs. temperature P R (T ) curve (fig. 5.7c) and deduce the ferroelectric Curie temperature T E C from the point at which the absolute remanent polarization slope ∂P R (T ) ∂T is maximal which coincides again with the value of 27 K. The broadening towards higher temperatures may be a sign of remaining polar nanoclusters known to occur in bulk CaSTO [176] and sustained especially by the applied electric field in our case. the magnetic field), we substract the slope obtained by a linear fit in the mid-field range near 3 T to 5 T, slightly adjusted for each temperature. Finally a simple smoothing is applied before presenting the data shown in figs. 5.10a and 5.10b. In these two figures, we plot the AHE at different temperatures for P up (fig. 5.10a) and P dn (fig. 5.10b). Its shape matches that of XMCD cycles measured at the Ti L 3 edge with the magnetic field perpendicular to the sample plane (i.e., along the hard magnetization axis). The AHE decreases with temperature and vanishes rather abruptly around 20 K (see fig. 5.10g) which most likely corresponds to the magnetic Curie temperature T M C . The AHE is typically slightly higher in the P dn state and its shape also seems to depend on the ferroelectric state as we will discuss in the next section. These data thus point to a ferroelectric control of the AHE and thus of the magnetization, which indicates the presence of a magnetoelectric coupling in our 2DEGs.

In addition, figs. 5.10c and 5.10d show the magnetoresistance (MR) measured with a perpendicular magnetic field as a function of temperature at positive and negative ferroelectric remanence. At high T (e.g. 50 K), thte MR shows a small positive quasi-parabolic dependence, typical of Lorentz MR and commonly observed in oxide 2DEGs. The MR scales as (µB) 2 with µ the electron mobility, which we estimate to about 70 cm 2 V -1 s -1 at 50 K, in agreement with values deduced from the sheet conductance and carrier density extracted from the ordinary Hall coefficient. As temperature decreases, the MR develops a negative field dependence that resembles what is found in magnetic systems with some degree of spin disorder [139], and particularly what has been seen in doped (ferromagnetic) EuTiO 3 [438,439]: as magnetic field increases, spins rotate to align with the external magnetic field, reducing the resistance.

The low temperature MR shows an inflection at some magnetic field, which can be better seen from the derivatives in figs. 5.10e and 5.10f and manifests as a pronounced dip at a field B * . The inflection may signal the approach to saturation of the magnetic lattice. Consistent with this interpretation, the dip in the derivative disappears as T increases. Plotting 1/B * as a function of temperature in fig. 5.10h yields a transition temperature that matches the T M C deduced from the T dependence of the AHE amplitude.

From the temperature dependence of the R s (B) data for both ferroelectric remanent states, we can compute the electroresistance ER = (R Pup -R P dn ) at zero magnetic field and at 9 T, Figures 5.11c to 5.11e illustrate the cyclability of the anomalous Hall effect and magnetoresistance upon setting different polarization states. Next, we apply successively gate voltages of +50 V, 0 V, -50 V and 0 V to respectively pole the ferroelectric state upwards, set it in its positive remanent state P up , pole the ferroelectric state downwards and finally set it in its negative remanent state P dn . For each state, we have measured the AHE and the MR (see figs. 5.11d and 5.11e) which both increase when going from more positive to more negative polarizations. Remarkably, the variations are reproduced from one cycle to the next, attesting of the magnetoelectric coupling.

. Discussion

While the multiferroic and magnetoelectric characters of the 2DEG seem rather well supported by the data, some results deserve further investigation. For instance, although our DFT calculations suggest that the projected out-of-plane magnetic moment carried by Ti ions (and thus magnetization M ) in the 2DEG should be higher for P up than for P dn , experimentally we find the AHE amplitude to be stronger for P dn than for P up . These results are not necessarily incompatible since the magnetization easy axis seems to lie in the sample plane and not perpendicular to it and the AHE reveals different slopes depending on the polarization state, compatible with a slightly modified anisotropy (see fig. 5.12a). In addition, in ETO heterostructures, the AHE is known to depend non-monotonically on the saturation magnetization arising from topological points in the band structure and even showing a sign change with doping [439]. From XLD data, we find that in addition to changing the carrier density in the 2DEG, the polarization direction actually modifies substantially the crystal field splitting, which indicates a non-trivial polarization dependence of the electronic band structure, as also suggested by DFT calculations.

Another puzzling finding is the high T M C ≈ 20 K, which is higher than that found in strained or doped ETO, or in the non-ferroelectric LAO/ETO/STO (with T M C peaking around 10 K). The different electronic structure of the ferroelectric 2DEG compared to non-ferroelectric STO 2DEGs or doped ETO, as well as enhanced correlations in this low-dimensional system, may collaborate to yield this increase. As can be seen from fig. 5.12b, field-dependent XMCD measured at the Eu M 5 edge with the field out-of-plane, i.e. in similar conditions as the AHE doesn't show any sign of detectable hysteresis either, contrary to in-plane data displayed in fig. 5.6f. Lebedev et al. [442] argued recently that AHE need not be strictly a sign of ferromagnetism but could arise simply from a gate-dependent spin-orbit coupling (SOC) combined with a paramagnetic or super-paramagnetic response of the system to an applied magnetic field, possibly explaining the absence of magnetic hysteresis in their AHE data in LaAlO 3 /GdTiO 3 /SrTiO 3 . Although again we cannot exclude -and we actually expect-a variation of the SOC with the ferroelectric polarization state in our samples, we do obtain clear ferromagnetic hysteresis cycles from XMCD on both Eu and Ti cations with the field in-plane. Thus, it seems reasonable to think that ferromagnetic order is indeed the cause of the AHE, which then may indeed vary by controlling either the SOC strength and/or the magnetization strength and/or easy axis. In addition, we also obtained magnetoresistance curves with the magnetic field in-plane which do possess a small but measurable hysteresis (see figs. 5.12c and 5.12d), with a coercive field of about ±0.4 T in a similar range to that obtained from XMCD at the Ti L 3 edge, bringing a conclusive argument allowing us to call the 2DEG ferromagnetic.

Further interesting work -although requiring intense efforts-could be to perform XLD and XMCD as a function of temperature and possibly gate voltage to further correlate spectroscopic data specifically from the 2DEG region with the observed transport properties and macroscopic polarization. A somewhat simpler step could be to perform in-plane magnetotransport measurement again as a function of temperature and gate voltage, keeping track of the magnetic hysteresis.

. Summary

In summary, we have designed and prepared LaAlO 3 /EuTiO 3 /CaSTO heterostructures harbouring a two-dimensional electron gas that displays coupled ferroelectricity and magnetism coexisting with a metallic behaviour. Our results not only marry the fields of multiferroics and oxide interfaces but also provide a fascinating new physics playground for non-volatile spin-orbitronics [385] and nonreciprocal phenomena [334]. They call for further theoretical calculations and experiments to address the interplay of the multiferroic behaviour with Rashba SOC. For instance, because of their intrinsic inversion symmetry breaking, MF-2DEGs are expected to display Rashba-type Dzyaloshinskii-Moriya interaction (DMI) [443] leading to chiral spin textures such as skyrmions which here should be controllable by ferroelectricity [444]. Low-temperature imaging techniques (optical, X-ray, near-field) allowing the application of a voltage (at least switching pulses) and measurement of magnetic properties, would in that case be ideal tools.

. Sample preparation

For the framework of this chapter, samples are prepared using non-commercial KTaO 3 :Nb crystals provided by the collaborator Mario Maglione from (ICMCB -CNRS, Université de Bordeaux) with different nominal concentrations of Nb. The few crystals with various irregular shapes and sizes were cut in few pieces when possible and polished by the company SurfaceNet. The resulting small substrates with (001) surface orientation could then be used for further process to obtain 2DEGs and study both transport and ferroelectric properties.

After a simple cleaning of the surface in acetone and isopropanol, we proceed to the deposition of Al following the preliminary work of Vicente-Arche [START_REF] Vicente-Arche | Quantum Oxide Heterostructures for Spin-Charge Interconversion[END_REF], with the addition of an intermediary annealing step in a UHV chamber for some samples, with the initial intent to improve the surface quality and thus the 2DEG's conduction properties. Some samples also have an additional capping layer of Al deposited ex situ after a first exposition to air to allow full oxidation of the first layer.

At the end of the fabrication process, all samples show two-point conductivity on the top surface with resistance in the range of few kΩ, and insulating behavior (R 2W > 60 MΩ) between the top surface and bottom surface, indicative of the presence of a 2DEG as previously observed by Vicente-Arche et al. [START_REF] Vicente-Arche | Spin-Charge Interconversion in KTaO 3 2D Electron Gases[END_REF] in AlO x /KTO samples.

Additionnally, for each sample, we check the surface quality by AFM after the initial cleaning step and at the end of the process, after the Al depositions, and also acquire the RHEED pattern before and after annealing, when relevant.

A summary of all prepared samples is presented in table 6.1. An example of surface characterization is shown in fig. 6.1 for sample B and C obtained from the same substrate of KTN with 1.5 % of Nb. In particular, we find that for these preannealed samples, the annealing temperature and pressure conditions play an important role in the resulting surface quality and -as revealed later by gating experiments-bulk insulating properties of the samples, which is why most of them are then pre-annealed at 800 °C with an oxygen partial pressure p O 2 = 0.3 mbar. Indeed, we notice from the AFM image in fig. 6.1e that for sample B annealed in vaccum at 600 °C, some important defects in the form of circular holes appear at the surface that are not present for sample C. Although we do not have a clear explanation, the difference between both samples could arise from the release and sublimation of species from the substrate for low ambient pressure, most likely forming potassium deficiency and/or oxygen vacancies. In that case, this behavior could be limited or suppressed by the introduction of oxygen in the chamber.

Further systematic studies and chemical analysis with a probe sensitive enough could help supporting this speculation. However, the observation of a slight brown/gray color postannealing particularly for the last two samples D and E prepared together, seems reminiscent of what has been observed for vacuum-annealed STO, although here we do not report bulk conductive behavior for any of the samples C, D and E annealed in oxygen (see following sections). In addition, although the RHEED pattern of sample C annealed at higher temperature (figs. 6.1c and 6.1d) appears to be more characteristic of a 2D surface (with a streak pattern instead of dots) after annealing, we also notice for both samples the appearance of intermediary streaks signaling a surface reconstruction with higher periodic ordering. Even though strain marks are visible from the topography in AFM images, it does not seem plausibly leading to a significant change in this regard that could explain the obtained RHEED patterns.

. Ferroelectric characterization

Using similar methods as in previous chapters described in section 2.6, electrical measurements of samples B, C, D and E were performed between the bottom and top surface to characterize ferroelectric properties of the crystals.

In this section, we show an excerpt of these measurements for the case of sample C with a KTaO 3 :Nb(1.5 %) substrate which will also be the focus of the next section. In fig. 6.2, we show an example of the basic electrical characterization performed on sample C at 2 K under the application of a triangular waveform with maximum amplitude V max = 75 V (i.e. E max ≈ 1.7 V cm -1 since the sample is about 450 µm-thick) and a frequency of 10 Hz. We notice a peculiar behavior of polarization loops in fig. 6.2a with a hysteresis loop "pinched " around low voltages. We can also see from the corresponding current-voltage characteristic in fig. 6.2b the presence of 4 peaks instead of two as normally expected for ferroelectric polarization loops. In particular, the two additional asymmetric peaks occur each in one sweep before the voltage sign is switched, i.e. at slightly positive (resp. negative) voltage for decreasing (resp. increasing) voltages.

This behavior is similar to that of an "antiferroelectric" (AFE) system with a ground state antipolar alignment of electric dipoles along two intertwinned networks or in different regions, thus offering a partial compensation of the total polarization at zero electric field. Under the application of an external electric field however, both networks or regions will see their macroscopic polarization progressively aligned once the coercive field of the corresponding subnetwork/region is exceeded. When the field is ramped back to zero, the polarization is again diminished as dipoles go back to their anti-aligned position.

Here we do not expect such a behavior in a nominally purely ferroelectric compound and to the best of our knowledge, this is the first report of such antiferroelectric-like polarization loops in a KTN single crystal. We should emphasize that in our case, at 2 K these loops are also trongly asymmetric and that the polarization is not perfectly "cancelled" for zero applied voltage but instead we observe a macroscopic remanent polarization P r = 0.5 µC cm -2 . In addition, the saturation polarization is in a similar range to what was reported by Knauss et al. [188] for similar Nb concentrations and similar value of applied electric field. Since it seems rather unlikely that a new antiferroelectric phase would be found in the bulk of KTN, we thus conjecture that the conductive region created near the interface acts as the host of polar displacements although the determination of the exact polarization arrangement and dynamics of the entire sample cannot be drawn simply from these preliminary measurements. To start elucidating the above-described phenomenon, we perform more electrical measurements in different conditions by varying one parameter at a time. In fig. 6.3, we first show a frequency dependence of the polarization loops performed at 2 K with a triangular waveform of amplitude V max = 40 V. We see that in the low frequency range, the shape and characteristics of these P (E) loops are mostly unchanged with the pinch near zero electric field as seen previously. Above 20 Hz however, the loops seem to enlarge and to become more square with also a slightly decreasing saturation polarization. As a consequence of the enlargement, the polarization at zero field and the apparent coercive field thus increase significantly. We interpret this as a difference in the response dynamics of either polar subnetworks/regions: as the frequency is increased, the voltage sweep becomes too fast for one of the subnetworks/regions to relax so that both regions seem to revert their polarization simultaneously. For reference, we also show a polarization loop at the maximum frequency of 2000 Hz, showing how the sweeping rate simply becomes too fast for the system to respond properly to the applied voltage altogether. The frequency dependence here is probably strongly influenced first by the low charging time constant due to the high resistance of the "top electrode" at 2 K, by the fact that the electrode is part of the switched material itself and finally by the asymmetry between top and bottom electrodes.

A similar observation as in fig. 6.3 can be drawn from fig. 6.4 where we vary the maximum applied voltage while keeping a frequency of 10 Hz and a temperature of 2 K. There, we notice that for the highest voltage boundaries, the pinched region looks slightly less pronounced indicating again that the ground state polarization cannot be fully recovered within a sufficiently short time; although here we should also emphasize on the presence of a increasing bias towards higher voltage as V max is increased. Finally, we show in fig. 6.5 a temperature dependence of the polarization cycling with the electric field for V max = ±75 V both at 10 Hz (figs. 6.5a and 6.5b) and 100 Hz (figs. 6.5c and 6.5d). We notice how the hysteresis cycles have a clear different shape between measurements at 10 Hz and 100 Hz as seen previously. Strikingly, for 10 Hz, fig. 6.5a shows that as temperature is increased, the cycles become more and more pinched at low applied voltage, reducing significantly the effective remanent polarization already at 10 K. Above this temperature, a nonlinear dielectric behavior is kept with a slightly more pronounced opening for positive voltages until the paraelectric behavior seems to be finally recovered for 28 K and above. This evolution is reflected in fig. 6.5b where we plot the average remanent polarization (i.e. the average between polarization values in down and up sweeps at zero applied voltage) which decreases sharply from 2 K to 10 K and remains mostly constant over 10 K to finally decrease to negligible values.

For the comparison, we also show the temperature evolution for loops performed at 100 Hz in fig. 6.5c where we observe mostly ferroeelctric-like hysteresis cycles up to 25 K above which temperature, the linear dielectric characteristics are again recovered. Thus, we observe a simpler decrease of the average remanent polarization in fig. 6.5d with values becoming indeed close to zero near 25 K to 28 K. Note that this value is very slightly higher than what can be found in the literature for a concentration x Nb = 1.5 %, as it should be around 20 K [130, 188, 445], but it is aso consistent with the report of Knauss et al. [188] of surviving non-linear dielectric behavior and non-zero remanent polarization even above the Curie temperature (defined as the temperature of maximal dielectric constant).

Although not shown here, we note that a somewhat similar electrical dependence of polarization and current could be observed in sample B with same percentage of Nb although a higher leakage current was present, impacting also the shape of the polarization loops as a result. Sample D (with 5 % of Nb) seems to display ferroelectric-like polarization loops at low temperature (near 2 K), passes through a similar phase as sample B and C with a pinchedshape hysteresis loop in intermediary temperatures near 15 K and goes back to a ferroelectric behavior before it crosses its Curie temperature of about 40 K. We should also point out that although the nominal Nb concentration of sample D is 5 %, which should give a T C ≈ 55 K, the observation of a lower transition temperature indicates the need to better characterize the actual stoichiometry of crystals that were provided. Finally, with the new observations of antiferroelectric-like polarization loops, the investigations on these samples would benefit from additional measurements and theoretical calculations to better understand the polarization distribution across the bulk and interface.

. Transport properties of AlO x /KTN interfaces

In this section, we review basic transport properties of the conductive AlO x /KTN interfaces and show preliminary results of an unusal and complex and hysteretic gate dependence of these transport properties. For all samples listed in table 6.1, we perform simple measurements of the sheet resistance as a function of temperature. In fig. 6.6a, we show the corresponding sweeps obtained while cooling down each sample from 300 K to 2 K. We notice how most of them have a metallic behavior at high temperature below 300 K, with decreasing resistance as the temperature is ramped down, except for sample E with 8 % of Nb. In fact, below some temperature which seems to depend on the Nb concentration, their resistance starts increasing strongly so that at 2 K, they all have a sheet resistance in the order of few 10 5 Ω □ -1 to 10 7 Ω □ -1 , much higher than the range usually observed for AlO x /STO or even simple AlO x /KTO 2DEGs, which are metallic down to the lowest temperature (and even superconducting for the case of AlO x /STO).

We can see however, that there seems to be an additional break of slope in the insulating phase, again at a temperature that depends on the concentration of Nb. Using a similar approach as for AlO x /Ca:STO 2DEGs, we extract a transition temperature from the derivative for each sample where the kink of highest temperature occurs and obtain values matching relatively closely the ferroelectric Curie temperature expected from literature or measured in the previous section by polarization measurements. Note that sample E (with 8 % of Nb) also displays abrupt changes in the resistance derivative curve below 100 K which may coincide with successive transitions to lower symmetry structures. In fig. 6.7, we show a completed phase diagram of KTN, showing the relatively good agreement of the values extracted by our methods with those of the literature. et al., Phys. Rev. Lett. 39, 1158(1977) T C Girshberg et al., , J. Phys.: Condens. Matter 13 (2001) T C = A (x-x c ) 1/φ ��φ= 1.9 T C Knauss et al., , Phys Rev B 55, 3472 (1997) T O→R T C→T T T→O TriebWasser, Phys. Rev. 114, 63 (1959) T O→R T C→T T T→O Boatner, Helv. Phys. Acta 50, 557 (1977) T (C/T/O)→R T C→T/O Höchli et al, Phys Rev B 20, 266, (1979) T of the gate voltage. We start by running Hall and magnetoresistance measurements at 2 K successively under the application of a gate voltage V g = +75 V, 0 V, -75 V and 0 V. We can clearly see from fig. 6.12 how the Hall curves pass from a negative slope at +75 V to a positive slope as the voltage is brought back to 0 V, a behavior enhanced at -75 V. Finally, a negative slope is again recovered at negative remanence going back to 0 V, while keeping a strong nonlinearity still indicative of a mixed transport. These measurements thus reinforce the initial assumption on the presence of both holes and electrons in the conducting region probed at the AlO x /KTN interface although we have not performed fitting of these curves with a multiband model at the time we write this version of the manuscript. Additionally, we identify in fig. 6.12b various contributions to the magnetoresistance curves such as the Lorentz B 2 -like dependence at -75 V and at negative remanence, with the addition of a WL-like contribution in the low field region. More remarkably at -75 V and positive remanence, the MR saturates and even starts to decrease at high field. In any case, all measurements at this temperature show a positive magnetoresistance.

Following measurements in fig. 6.12, we repeat Hall and MR measurements as a function of temperature but this time at alternating positive and negative remanence from ±75 V for increasing temperatures up to 22 K. At positive remanence (fig. 6.13a, the Hall effect transitions rather smoothly from a hole-dominated transport at 4 K -as was observed at 2 K in fig. 6.12a-, to an electron-dominated transport already at 8 K and seems to nearly saturate above 14 K with an estimated electron density of about 3.7 × 10 13 cm -2 consistent with the observation in fig. 4.6b. For the negative remanence (fig. 6.13b), only a negative slope of the Hall curve is observed through the temperature range explored here, but we do see a significant jump between 10 K and 14 K. This temperature dependence may be compared with the remanent polarization measured for increasing T at low frequency in fig. 6.5b where values seemed to drop as the polarization hysteresis cycles took a shape closer to antiferroelectric-like polarization loops although it remains unclear how both should be linked.

Magnetoresistance curves at positive and negative remanence in figs. 6.14a and 6.14b respectively do not display any clue of a transition as clear as for the Hall effect. We should emphasize though, that instead of the positive magnetoresistance seen previosuly at 2 K, only an overall negative magnetoressitance is observed from 4 K and up to 22 K. The shape of the magnetoresistance curve at 4 K and negative remanence could indicate the presence of a weak anti-localization behavior which normally subsists only at low temperature, but would signal a strong spin-orbit coupling in this system. Although we do not expect magnetism in AlO x /KTN contrary to the system studied in chapter 5, we cannot fully exclude that other contributions are present here. Complementary measurements could allow to detect the presence of interfacial magnetism or elucidate the role of other potential mechanisms.

In a third cooldown, we finally perform a more complete gate dependence of the Hall and magnetoresistance at 2 K. We apply successively 0 V, +50 V, +100 V, +50 V, 0 V, -50 V, -100 V, -50 V and 0 V and plot the resulting signals in figs. 6.15 and 6.16b (each curve is lettered from A to I according to the marking introduced in fig. 6.16a).

We see in fig. 6.15 that at 2 K and for all voltages, the MR remains positive with smaller amplitude for higher applied gate voltages with clear weak (anti-)localization effects probably in addition to the Lorentz MR and a hysteretic dependence as a function of the gate voltage.

Finally, in fig. 6.16b we show another example of the complex hysteretic gate dependence of the Hall effect in the system under study with multiple crossing from a positive to a negative slope and strong non-linearity particularly in the low-field region although it should be noted that measurements artefacts and noise cannot be excluded for some curves shown here. A series of measurement at lower field or with a different sweeping rate of the magnetic field could alleviate these issues and clarify the possible presence of features like anomalous Hall effect (AHE). The complex gate and temperature dependences of the slope and the inferred presence of holes observed in this section for this new system may seem surprising, as this seems to be the only report of hole transport in a 2DCS at oxide interfaces after Anh et al. [227] demonstrated tunability of the hole conductivity at the Fe/STO interface by varying the thickness of Fe in the range of few angstrom. Furthermore, no sign of bands close to the Fermi level that could create a hole pocket has been either predicted nor observed from angle-resolved photoemission spectroscopy (ARPES) in the pure KTaO 3 2DEGs.

We do note that Kolodiazhnyi [277] also reported hole conductivity in the bulk "ferroelectric metal " BaTiO 3-δ in the high-doping regime obtained through the formation of oxygen vacancies. They explained this behavior by the change of symmetry occuring at low temperatures particularly when BTO adopts a rhombohedral structure: the crystal field then splits the Ti-3d subbands with a small gap of few meV and the lowest subband becomes nearly filled as the temperature is lowered and k B T becomes sufficiently small so that electrons cannot thermodynamically cross the crystal-field gap to populate the higher subbands. Applying a similar scenario to the case of Ta-4d subbands of our system, we could imagine that in its ferroelectric phase (also with a rhombohedral structure), the bulk KTN could both affect the carrier density in the 2DCS and change slightly the effective gap due to the crystal field, allowing to pass between the different regimes of electrons/holes/mixed conductive states in a complex manner. These results thus call for additional experimental work using other probes such as ARPES as well as theoretical calculations.

. Summary and perspectives

It is clear that a nonvolatile electric control of the carrier doping type in a two-dimensional conductive system (2DCS) would offer great promises for electronics applications to create reconfigurable transistors, for instance. More broadly, in this chapter, we started exploring the AlO x /KTN conductive interface combining a nominally ferroelectric material and our knowledge on oxide 2DEG fabrication. We obtained remarkable polarization dependences with the electric field applied across the sample and with temperature indicating the competition of different polar orders in samples with different Nb concentrations.

Additionnally, although we did not achieve metallicity down to low temperature as is normally observed for STO-and KTO-based 2DEGs, we could successfully extract a ferroelectric transition temperature from measurements of the sheet resistance R s (T ). Eventually, we could report the presence of both electron-and hole-dominated transport regimes, tunable hysteretically by the application of a gate voltage with opposite carrier signs at remanence.

Although some more work should be done to decypher the complexity of the abovementioned properties and to try to obtain metallicity at lower temperature, further envisioned perspectives could also be to explore the formation of two-dimensional conductive system (2DCS) at the surface of KTN with a (111) orientation, building on the reports of superconductivity at the surface of KTO( 111) and (110) and those mentioning an enhancement of superconducting critical temperature in bulk ferroelectric STO-based compounds.

Conclusion and outlook

This thesis presents a collection of results demonstrating how charge and spin transport properties at oxide interfaces can be electrically tuned both in a volatile and non-volatile way by introducing ferroelectricity as a new ingredient.

In chapter 3, we used a simple lithography technique to pattern 2DEG Hall bars at the AlO x / SrTiO 3 interface exploiting the previously demonstrated oxido-reduction reaction between Al and SrTiO 3 [START_REF] Vaz | Mapping Spin-Charge Conversion to the Band Structure in a Topological Oxide Two-Dimensional Electron Gas[END_REF]222,225,409]. We used these devices to characterize the low temperature charge-to-spin conversion efficiency as a function of the back-gate voltage V g via the direct Edelstein effect (DEE). Based on these results and applying the model of Vaz et al. [336] and Dyrdał et al. [393] for the so-called bilinear magnetoresistance (BMR), we showed how both the amplitude and sign of the effective Rashba coefficient α R could be modulated by the gate voltage. We thus proposed an all electrical, cascadable, ferromagnet-free and universal logic gate device based on spintronics, with a double conversion (from charge to spin and back to a charge current) in which we suggest to exploit the bidirectionality of the charge-spin interconversion depending on the applied gate voltage to perform simple logic operations. This kind of device is a good example of how versatile and powerful spintronics could be in replacing modern transistors for logic operations and we could go even further by using a ferroelectric as the gate dielectric to perform in-memory computing.

In the second part of chapter 3, we then exploited STO's capability of adopting a ferroelectric behavior by applying large electric fields [START_REF] Gränicher | Induzierte Ferroelektrizitat von SrTiO3 bei sehr tiefen Temperatur und uber dieKalterzeugung durch adiabatic Entpolarisierung[END_REF][START_REF] Hemberger | Quantum Paraelectric and Induced Ferroelectric States In[END_REF]. Thanks to this, and using spin pumping by ferromagnetic resonance, we could actually demonstrate not only a hysteretic dependence of the sheet resistance with the applied electric field but also a particularly efficient non-volatile electric control of spin-to-charge conversion in AlO x /STO samples, with opposite signs of conversion for positive vs. negative remanent states. This also opened up the road for another type of spintronics devices called ferroelectric spin-orbit logic (FESO) where the memory element used to store information does not rely anymore on a ferromagnetic element like in the the magnetoelectric spin-orbit logic (MESO) device proposed by Manipatruni et al. [22,[START_REF] Manipatruni | Beyond CMOS Computing with Spin and Polarization[END_REF], but only by a ferroelectric, removing the need for a perfect interfacial magnetoelectric coupling.

Building on our knowledge of STO 2DEGs, we explored other mechanisms allowing to push the incipient ferroelectric that is STO, into a ferroelectric phase, namely by using substrates with a partial isovalent substitution of Sr 2+ ions by Ca 2+ ions [176]. Similarly as for the AlO x / STO case, we could obtain the formation of a 2DEG at the surface of Ca:STO after the deposition of a thin layer (from few angstrom to few nanometers) of Al. We have shown how the ferroelectric properties of the bulk could have an influence on the transport properties of the interfacial 2DEG with the presence of two clear remanent states with different resistance and carrier density, promising for the modulation of spin-orbit properties which are intrinsic to the band structure. We also started investigating the in-plane magnetotransport which, through measurements of the BMR, already seems to show a ferroelectric control of the charge-to-spin conversion amplitude, although more investigations are still required.

We then devised a way to introduce an equilibrium spin polarization in the Ca:STO-based 2DEG, combining the fields of multiferroic materials and oxide 2DEGs together. Using epitaxially grown LaAlO 3 /EuTiO 3 /Ca:SrTiO 3 heterostructures from collaborators in Naples, and characterizing them by synchrotron techniques like X-ray magnetic circular dichroism (XMCD) and X-ray linear dichroism (XLD) via total electron yield (TEY) measurements, we could demonstrate the presence of both ferromagnetism and of switchable polar moments in the 2DEG region, making the latter a multiferroic and metallic system, despite the usual counter-indication.

We could also show through transport measurements the presence of both switchable anomalous Hall effect (AHE) and negative magnetoresistance (MR) with different amplitudes depending on the polarization state, and cyclability. Further simple in-plane magnetotransport measurements seemed to confirm the presence of a ferromagnetic order with in-plane easy-axis. We thus achieved to obtain a magnetoelectric two-dimensional electron gas, in which it could be particularly interesting to study non-colinear spin textures due to the presence of ferromagnetism and the strong interfacial spin-orbit coupling inherent to these systems.

Finally, we proposed to explore alternative oxide systems for the realization of ferroelectric 2DEGs, with possibily higher Curie temperature and even larger spin-orbit coupling, which would allow larger spin-charge interconversion efficiencies.

We thus used a cousin system of STO, namely potassium tantalate KTaO 3 (KTO), which can also be rendered ferroelectric by introducing a small proportion of Nb 5+ in place of Ta 5+ [130], and which was independently used to also obtain 2DEGs [263], in particular using the same Al deposition technique as for STO. We succeeded in the realization of a conducting interface in the resulting potassium tantalate-niobate KTa 1-x Nb x O 3 (KTN) compound by this method and used it to perform ferroelectric characterizations by polarization vs. electric field measurements which unveiled an unexpected antiferroelectric-like behavior at low temperature. We suggested that this behaviour could be due to the presence of two main regions of the sample (perhaps from the bulk KTN and the interfacial region) with opposite polarizations partially cancelling each other when the electric field is removed.

Peforming transport measurements in our samples revealed the arisal of a second order metal-to-insulator transition at different temperatures, depending on the Nb concentration and signatures of structural transition including the emergence of ferroelectricity in the temperature dependence of the resistance. Furthermore, gate-dependent transport measurements at low temperature proved the possibility to also achieve a hysteretic ferroelectric control of the interface's resistance in this system. Finally, we saw the emergence of a positive Hall coefficient at low temperature and positive remanence instead of the otherwise negative value, indicating the possibility to pass from a hole-dominated transport regime to the electron-dominated one, more commonly observed at conductive oxide interfaces. Based on previous similar observations in bulk-doped BaTiO 3-δ [277], we proposed that the transition to a lower-symmetry crystal structure could influence the crystal field in such a way that the carriers subbands of the two-dimensional conductive system (2DCS) would arrange in such a way that a change of polarization (and thus crystal field and carrier density) could be enough to pass from one regime to the other. This opens up wide perspectives in terms of applications, where one could imagine to design electrically reconfigurable transistors and other PN junctions in a non-volatile way.

To summarize, we have shown in this thesis different ways to generate and control both a non-equilibrium spin density and the ferromagnetic order in ferroelectric or multiferroic oxide 2DEGs respectively. It would be particularly interesting to study the influence of ferroelectricity on the two-dimensional superconducting properties found in some of these oxide 2DEGs as it was done in their bulk counterparts [281,411]. Additionally, researchers have already started considering these systems as a new platform for the search of topological superconductivity and Majorana particles [264] especially after the discovery of superconductivity at the interface of KTaO 3 (111) with other oxides, with a critical temperature one order of magnitude higher than in the STO 2DEG counterpart.

Finally, while the demonstration of efficient spin-charge interconversion in the systems we studied are already encouraging to pursue a technological transfer to the electronics industry, the investigations of spin-charge interconversion phenomena at higher temperatures in these systems or using other ferroelectric materials would definitely help solving a major challenge in the field, helping to reduce considerably the power consumption of information and communication technologies, by using new computing paradigms in which data storage processing are brought together.

  xiv

Figure 1 . 34

 134 Figure 1.34 Ferroelectric control of interfacial Rashba spin texture and Rashba-Edelstein effect

Figure 3 :

 3 Figure 3: Modulabilité du phénomène de magnétorésistance bilinéaire avec une tension de grille arrière dans un gaz 2D d'électrons AlO x /SrTiO 3 . Mesures de transport effectuées via la technique de détection synchrone à différentes harmoniques, pour un courant sinusoidal d'amplitude I ac = 150 µA. (a) Dépendance en angle du signal harmonique de rang 2 pour diverses tensions de grille et sous un champ magnétique de 9 T dans le plan de l'échantillon. Les points colorés correspondent aux données mesurées après soustraction de contributions d'arrière-plan, tandis que les courbes ajustées par rapport au modèle théorique sont représentées par des lignes continues. Seules quelques coourbes sont montrées et décalées verticalement pour un soucis de clarté. (b) Amplitude de la BMR extraite des ajustements paramétriques des dependances en angle, en fonction de la tensionde grille. Les barres d'erreur sont obtenues lors des ajustements, correspondent à l'erreur résiduelle et sont toutes au moins deux ordres de grandeur en dessous des valeurs d'amplitude. (c) Résistance planaire extraite du signal à la seconde harmonique en fonction du champ magnétique placé dans le plan de l'échantillon et orthogonal au courant, rampé entre +9 T et -9 T. Le signal est mesuré à V g = -150 V (en orange) et +75 V (en vert). Les lignes sont uniquement à titre indicatif.

Figure 5 :

 5 Figure 5: Contrôle des propriétés de transport pa rla ferroélectricité dans des échantillons de AlO x /Ca:SrTiO 3 à une température de 2 K. (a) Dépendance avec le champ magnétique de la résistance planaire, à +70 V, 0 V (rémanence positive), -70 V et 0 V (rémanence négative). (b) Magnétorésistance relative (en %) obesrvée dans chacun des états décrits ci-dessus obtenues à partir des courbes de (a). (c) et (d) Courbes d'effet Hall obtenues à rémanence (0 V) après avoir appliqué +70 V ou -70 V, respectivement. Pour ces deux figures, la ligne continue en rouge est une courbe ajustée selon un modèle de transport à deux bandes électroniques.

Figure 6 :

 6 Figure 6: Démonstration du contrôle réversible de l'effet Hall extraordinaire et de la magnétorésistance négative dans le gaz 2D multiferroïque LaAlO 3 /EuTiO 3 /Ca:SrTiO 3 . (a) Dépendance en température à 0 T et 9 T de l'électrorésistance, c'est-à-dire la différence de résistance entre l'état up ou down de polarisation de l'échantillon. (b) Dépendance en température de l'électro-magnétorésistance calculée à partir des données de (a). (c) Train d'impulsions de tension appliquées à l'échantillon après une initialisation de la tension de grille arrière à 2 K. (d) Effet Hall extraodinaire (AHE) et (e) magnétorésistance (MR) mesurés pour chacune des tensions indiquées sur la (c).

Figure 7 :

 7 Figure 7: (a) Dépendance de la résistance planaire d'un échantillon AlO x /KTaO 3 :Nb avec la tension de grille appliquée à l'arrière de l'échantillon. Les lettres colorées indiquent les mesures faites à 0 V, +50 V, +100 V, -50 V and -100 V le long du cycle d'hystérèse. (b) Cyclabilité de l'effet Hall mesuré à 2 K, indiquant un changement de signe des porteurs de charge majoritaires (électrons vs trous).

Figure 8 :

 8 Figure8: Time evolution of microprocessors-related trends to improve the performances since the 1970's. Reproduced from[START_REF] Rupp | Microprocessor Trend Data/50-Years Processor Trend[END_REF]

Figure 1 . 2 :

 12 Figure 1.2: Illustration of different types of currents. (a) Charge current with equal contributions between the spin-up and spin-down channels. (b) Spin-polarized current due to an unbalance between one spin population with respect to the other. (c) Pure spin current originating from equal and opposite flow of carriers from either spin populations.

Figure 1 . 3 :

 13 Figure 1.3: Simplistic illustration of the atomic spin-orbit coupling. (a) In its own reference frame, the electron feels an induced magnetic field B from the positively charged nucleus orbiting around it.This magnetic field interacts with the spin magnetic moment of the electron so that depending on the relative direction between the orbit around the nucleus and the spin, L and S may be (b) parallel or (c) antiparallel, leading to two energy-split states. Reused from[START_REF] Vaz | Spin-to-Charge Current Conversion in SrTiO3-based Two-Dimensional Electron Gases[END_REF].

Figure 1 . 4 :

 14 Figure 1.4: Illustration of the effects of time-reversal and space inversion symmetries. (a) Centrosymmetric cubic perovskite unit cell, invariant under spatial inversion symmetry operation Î. (b) Time-reversal symmetry operation T reverses both the velocity (and momentum k) and the spin. (c) Electronic energy dispersion of a time reversal and space inversion symmetric system. (d) Under the application of a magnetic field, the Zeeman effect splits electronic energy dispersion along the energy axis, lifting the spin degeneracy and thereby breaking time reversal symmetry.

Figure 1 . 5 :

 15 Figure 1.5: (a) An electric-field along the z direction confines electrons in the xy-plane. As a consequence, the electronic energy dispersion in the three dimensional (k x , k y , E) space passes from (b) a spin degenerate paraboloid to (c) a more complex dispersion with two sub-dispersions that are spin-dependent.

Figure 1 . 6 :

 16 Figure 1.6: Illustration of the linear Dresselhaus and Rashba effects and their consequences on the band structure, assuming a parabolic dispersion. (a) Energy dispersion for quasi free electrons with a Rashba SOC term, splitting the spin degeneracy by shifting bands in opposite directions for all in-plane k. If the Fermi level is placed above the spin-subbands crossing, the resulting Fermi contours typically look like (b) with spin-momentum locking and opposite spin winding along the inner versus outer contours. The linear Dresselhaus-type SOC gives slightly different spin texture with a spins parallel to each k vector instead of perpendicular as for the Rashba case.

Figure 1 . 7 :

 17 Figure 1.7: Illustration of the (a) ordinary and (c) anomalous Hall effects, with (b) Au or (d) Co transverse resistance measurements.In the case of the ordinary Hall effect, the transverse resistance is continuously linear in magnetic field, while for the anomalous Hall effect, a low-field regime with a strong slope appears, with saturation beyond some value of magnetic field. Reused with permission from[START_REF] Vaz | Spin-to-Charge Current Conversion in SrTiO3-based Two-Dimensional Electron Gases[END_REF][START_REF] Vicente-Arche | Quantum Oxide Heterostructures for Spin-Charge Interconversion[END_REF].

Figure 1 . 8 :

 18 Figure 1.8: Extrinsic ((a) skew scattering and (b) side jump) and intrinsic mechanisms related to spin-orbit, giving rise to the AHE and spin Hall effect (SHE). Reused with permission from [74].

Figure 1 . 9 :

 19 Figure 1.9: Illustration of the (a) direct and (b) inverse spin Hall effects in a system with strong spin-orbit coupling (SOC). (a) Upon injection of a charge current, electrons of opposite spins will be deviated transverse to the injected charge current resulting in a spin current (or a spin accumulation on the edges of the sample) with polarization orthogonal to both charge and spin currents. (b) Injecting a spin current in a SOC system conversely creates a charge current transverse to it. Reused from [91].

Figure 1 . 10 :

 110 Figure 1.10: Illustration of the spin Hall effect arising in Rashba systems with simple parabolic energy dispersion in p x -p x space. (a) A corresponding Fermi contour, cutting the dispersion where both the inner and outer subbands are filled. Green arrows represent the p = ℏk vector while red arrows show the spin direction orthogonal to p. (b) and (c) Under an electric field E x , each Fermi contour shift between two scattering events at t = t 0 and a p-dependent torque acts on the spins to realign them with the expected direction, orthogonal to the new momentum, giving rise to a spin current along y, polarized along the z-axis.

Figure 1 . 11 :

 111 Figure 1.11: Illustration of the (a) direct and (b) inverse Rashba-Edelstein effects. (a) The injection of a charge current along -x shifts both Rashba contours in the +k x direction, and generates a spin accumulation along y. (a) Injecting a spin density oriented along y shifts inner and outer contours in opposite direction, resulting in a net charge current along -x.

Figure 1 . 12 :

 112 Figure 1.12: The four ferroic orders. Reused with permission from [106]

Figure 1 . 13 :

 113 Figure 1.13: Properties of ferroelectric materials. (a) Diagram of the hierarchical relationship between the different subclasses of dielectric materials, from piezoelectrics, to pyrelectrics, to ferroelectrics. (b)Tetragonal distortion of the lead zirconium-titanate (or PZT, for PbZr x Ti 1-x O 3 ) oxide perovskite's unit cell acoompanied by a polar displacement of the Zr 4+ /Ti 4+ ions with respect to the oxygen octahedral cage, below the Curie temperature T C . (c) Ferroelectric hysteretic loop of the polarization P versus an applied electric field E. In the initial state, average domain configuration is random and polarization starts to increase and then saturates. While sweeping the electric field in either direction, the polarization switches from one state to the other, for instance P up to P down when exceeding the coercive field E c and then goes to saturation P s . If the field is removed, a non-zero remanent polarization P r remains. Adapted from[108]

Figure 1 . 14 :

 114 Figure 1.14: Ferroelectric-paraelectric phase transitions. (a) Schematic potential landscape of materials passing through paraelectric-ferroelectric second-order phase transition around the temperature T C . Temperature dependence of the electric susceptibility and polarization for (c) first-and (b) secondorder ferroelectric transitions.

Figure 1 . 15 :

 115 Figure 1.15: Formation and dynamics of domains in ferroelectrics. (a) Schematic overview of the presence of domains in a ferroelectric crystal. (b) Ideal single domain configuration in a ferroelectric crystal and subsequent bound surface charges. (c) 180°domain configuration in a ferroelectric, lowering the overall energy of the crystal. (d) Example of bound charges screening with metallic electrodes allowing to decrease the depolarization field. Partially reproduced from [124]

Figure 1 . 16 :

 116 Figure 1.16: Special cases of systems close to ferroelectricity. Temperature dependence of the electric susceptibility for the case of (a) relaxor or (b) inverse susceptibility of quantum paraelectrics.

  such as YBa 2 Cu 3 O 7 [137, 138] or Bi 2 Sr 2 Ca n-1 Cu n O 2n+4+x ), but also materials with magnetic ordering (manganites [139], rare-earth titanates [140, 141], yttrium iron garnet Y 3 Fe 5 O 12 (YIG) [142]), (anti)ferroelectricity (e.g. BaTiO 3 [110], HfO 2 or PbZrO 3 [143, 144]) or even combinations of different ferroic orders such as the prototypical room temperature multiferroic (magnetoelectric) BiFeO 3 [145, 146].

Figure 1 . 17 :

 117 Figure1.17: The various degrees of freedom of oxides can be fully exploited and combined in thin films by interfacing these materials with one each other, leading to exotic phases of matter. Reused with permission from[147]

Figure 1 . 18 :

 118 Figure 1.18: General properties of strontium titanate SrTiO 3 (STO). (a) Unit cell (u.c.) of STO in its cubic phase, with a typical oxide perovskite structure and alternating neutral (SrO) 0 and (TiO 2 ) 0 planes along the (001) direction. (b) Antiferrodistortive structural transition occuring for STO around 105 K as it passes from a cubic to a tetragonal unit cell: a 0 a 0 c -rotations of the octahedral oxygen cage around the B cation (Ti) require to redefine the unit cell whereby its volume is doubled. (c) Dielectric permittivity ϵ dependence on the temperature for STO: as temperature goes down, ϵ increases strongly and almost diverges until it reaches a maximum of about 23 000 at 4 K. Reused with permission from [152]. (d) Illustration of the energy splittings of the Ti 3d states brought successively by cubic or tetragonal crystal field, and the addition of inversion symmetry breaking.

Figure 1 . 19 :

 119 Figure 1.19: Conductive properties of bulk oxygen-vacancy or Nb doped STO. (a) Resistivity and (b) mobility dependence on the temperature showing a metallic behavior with remarkably high mobility at low temperature, reaching up to few 1 × 10 4 cm 2 V -1 s -1 . Reproduced with permission from [161]. (c) At very low temperature, STO becomes superconducting even at very low carrier concentration, while the maximum critical temperature T max c

Figure 1 . 20 :

 120 Figure 1.20: Making nominally pure and stoichiometric STO ferroelectric with the application of electric fields. (a) As an electric field is applied, the low-temperature dielectric constant ε ′ of STO is quenched and the temperature at which ϵ ′ reaches a maximum is increased. (b) Applying a sufficiently high electric field allows one to observe hysteretic cycles of the polarization evocative of ferroelectriciy, with remanent polarization vanishing around 50 K to 60 K. (c) Summarizing phase diagram of (quantum) paraelectric vs ferroelectric transitions of STO in the presence of electric field. Reused from [166]

Figure 1 . 21 :

 121 Figure 1.21: Ferroelectricity in STO made possible by strain or isovalent cationic substitution. (a) STO shows a quantum critical point and can be tuned at the edge or out of its quantum critical phase for instance, by the application of strain. Reused with permission from [174]. (c) Replacing Sr by a fraction x of Ca makes STO ferroelectric already from x = 0.25 % while T C increases with x. Ca or Pb may also be used similarly to induce ferroelectricity with different concentration dependence. Reused from [175] (b) Temperature evolution of the dielectric constant of calcium strontium titanate Ca:SrTiO 3 (CaSTO) for various concentrations of Ca. Reused from [176]

Figure 1 . 22 :

 122 Figure 1.22: General properties of potassium tantalate KTaO 3 (KTO). (a) Crystal structure of KTO with alternating charged KO -and TaO + planes along the (001) direction. (b) Temperature-dependence of KTO's dielectric permittivity showing similar quantum paraelectric behavior as for STO. (c) Bulk resistivity of reduced KTO (i.e. electron-doped through oxygen-vacancies) showing typical metallic behavior, with (d) relatively high carrier mobilities at low temperature, as well. Reused from [182]

  .,Phys. Rev. Lett. 39, 1158 (1977) TCGirshberg et al., , J. Phys.: Condens. Matter 13 (

Figure 1 . 23 :

 123 Figure 1.23: Dielectric/ferroelectric properties of the solid solution (KTaO 3 ) 1-x (KNbO 3 ) x (potassium tantalate-niobate KTa 1-x Nb x O 3 (KTN)) with: (a) its continuous phase diagram as a function of temperature and the relative concentration of Nb with respect to Ta and (b) example polarization loops performed on KTN at different temperatures and with different concentrations as well. [188]

Figure 1 . 24 :

 124 Figure 1.24: Characteristics of the 2DEG at the LAO/STO interface. (a) and (b) Scenario of the polar catastrophe explaining the origin of the conducting behavior at the interface: the diverging build-up of an electric field is avoided by a transfer of charge from LAO's top surface to STO. Reused from [202]. (c) Thickness-dependent conductivity, showing the existence of a critical thickness of 4 unit cell (u.c.) necessary to observe the metallic behavior. Reused from [213]. (d) Conductive atomic force microscope (AFM) image of the LAO/STO interface with or without annealing, demonstrating the confined nature of the carrier gas. Reused from [205].

Figure 1 .

 1 Figure 1.25: (a) Enthalpy of formation of the most stable oxides for various common metals as a function of their work function, near room temperature. Dashed horizontal and vertical lines in the center correspond to the enthalpy of oxide formation of STO and its work function, respectively. These lines are empirically values corresponding roughly to the energy required to form one oxygen vacancy in STO and to the electronic affinity of n-type TiO 2 -terminated STO. (b) X-ray photoemission spectroscopy (XPS) spectrum of an AlO x /SrTiO 3 sample before vs. after deposition of Al by sputtering, where one can see the reduction of Ti. (a) Reused with permission from [223] and (b) from[START_REF] Vaz | Mapping Spin-Charge Conversion to the Band Structure in a Topological Oxide Two-Dimensional Electron Gas[END_REF]

Figure 1 . 26 :

 126 Figure 1.26: Basic transport properties of the LAO/STO interface. (a) Sheet resistance and (b) Hall mobility of few LaAlO 3 /SrTiO 3 samples grown at different partial pressures of oxygen. Reproduced with permisison from [148]. (c) Sheet resistance of a LAO/STO sample in the range of few 100 mK, showing superconductiving behavior which can be killed by an out-of-plane magnetic field of few 10 mT. Reproduced with permission from [229].

Figure 1 . 27 :

 127 Figure 1.27: Band structure and electronic properties of the STO 2DEGs. (a) Schematic illustration of the band masses based on geometrical considerations of the orbitals. (b) Rough schematic of the different subbands of STO 2DEGs with three light subbands and two heavy subbands with strong anisotropy. (c) Confinement potential and quantum well profile of the LAO/STO 2DEG, showing how the d xy subbands at the bottom of the quantum well lie mostly close to the interface, while d xz,yz subbands extend further and are closer to the top of the quantum well. (d) Energy dispersion of the different subbands with their dominant orbital character, without SOC (top) and after introducing SOC (bottom) which causes orbital mixing and avoided crossings. (e) and (f) ARPES data showing the energy dispersion along the main reciprocal lattice axes. (f) also shows the absence of dispersion along k z , evidencing the two-dimensional confinement of the electron gas. (a) Reused with permission from [230], (b) and (e) from [231], (c) and (d) from [232], and (f) from [233].

Figure 1 . 28 :

 128 Figure 1.28: Experimental (a) Fermi surface for the 2DEG formed at the AlO x /SrTiO 3 interface, around the Γ 11 point and energy dispersion along (b) k x , (c) the [110] direction or (d) a specially chosen direction, showing a topological avoided crossing. Solid lines with different colors are superimposed and represent the different subbands obtained from a fit with a tight-binding model. Reused with permission from[START_REF] Vaz | Mapping Spin-Charge Conversion to the Band Structure in a Topological Oxide Two-Dimensional Electron Gas[END_REF]

Figure 1 . 29 :

 129 Figure 1.29: Gate voltage initial cycling of the LaXO 3 /SrTiO 3 2DEGs: (a) Configuration of the gating experiment; (b) Gate initialization starting from positive gate voltages and back to the opposite maximum voltage; (c) Similar but, this time starting towards negative voltages.As long as the maximum (positive) voltage is not exceeded, the resistance will ovelap on the same path in either directions, without much hysteresis. Otherwise, the following curves are modified, with a higher resistance than previously observed. Reused with permission from[239]

Figure 1 . 30 :

 130 Figure 1.30: Gate tunability of the transport properties of LAO/STO 2DEGs: (a) superconducting T c ; (b) magnetoresistance showing a weak (anti)-localization behavior of different extent depending on the applied gate voltage; (c) spin-orbit and inelastic scattering times, as well as the effective Rashba SOC coefficient; (d) coincidence of the resistance gate voltage dependence and the superconductive T c , hte latter showing a dome-like dependence on the carrier density. (a) and (d) Reused with permission from [242], (b) and (c) from [243]

Figure 1 .

 1 Figure 1.31: (a) Transverse resistance of a LAO/STO 2DEG as a function of the gate voltage and displaying a nonlinear behavior beyond 50 V approximately. (b) Zoom on the low-field region, where it is clear that the low-field slope evolves non-monotonically. (c) Evolution of the extracted carrier densities in a two-channel transport model as a function of the applied gate voltage. Reused with permission from [232].

  .

  1.4.3 . 2DEGs based on KTaO 3 Very similarly to the case of STO, a variety of methods allowing to fabricate 2DEGs at the (001) surface of potassium tantalate KTaO 3 (KTO) has been demonstrated, either by a cleaving method [250], by growing other epitaxial polar oxides on top like LaTiO 3 [251], LaCrO 3 [252], LaVO 3 [253], or by depositing amorphous materials like LaAlO 3 [254] or simple metals like Al in a similar fashion as for Metal/SrTiO 3 2DEGs[START_REF] Vicente-Arche | Spin-Charge Interconversion in KTaO 3 2D Electron Gases[END_REF][START_REF] Zhang | Thermal Spin Injection and Inverse Edelstein Effect of the Two-Dimensional Electron Gas at EuO-KTaO 3 Interfaces[END_REF][START_REF] Vicente-Arche | Quantum Oxide Heterostructures for Spin-Charge Interconversion[END_REF].

Figure 1 . 32 :

 132 Figure 1.32: Properties of KTaO 3 -based 2DEGs. (a) Energy dispersion of an AlO x /KTO 2DEG along the k y direction from ARPES measurements measured on CASSIOPEE beamline in SOLEIL. (b) Experimental variations of the spin-orbit and dephasing fields and lengths, as well as the spinorbit splitting ∆ for different carrier densities, obtained in illuminated KTO 2DEG samples. (c) First demonstration of superconductivity in EuO/KTaO 3 (111) 2DEGs. (a) Reused from [255], (b) from [256] and (c) from [183].

Figure 1 . 33 :

 133 Figure 1.33: Examples of ferroelectric insulating oxides, made conducting in the bulk by creation of oxygen vacancies (a) in BaTiO 3 or (b) in CaSTO. (c) In the latter case, the ferroelectricity enhances the superconducting critical temperature T c compared to samples of conducting STO for similar carrier densities, particularly in the vicinity of the ferroelectric/paraelectric quantum critical point. (a) Reused with permission from [242] and (b) and (c) from [281].

Figure 1 . 34 Figure 1 . 34 :

 134134 Figure 1.34 Ferroelectric control of interfacial Rashba spin texture and Rashba-Edelstein effect

Figure 2 . 1 :

 21 Figure 2.1: Panoramic view of the complex ultra high vacuum (UHV) system used for material growth (mainly in the sputtering chambers) and in situ characterization (mainly by XPS), where the sample passes from chamber to chamber via transfer arms in two clusters, both linked by a tunnel and with independent load-lock chambers. The system is part of the fabrication equipments at the Unité Mixte de Physique CNRS, Thales, Université Paris Saclay laboratory.

Figure 2

 2 Figure 2.2: (a) Schematic view of the magnetron sputtering technique. Reused with permission from [74]. (b) Example picture of a plasma glow discharge generated from a (Au) sputtering target in a confocal chamber.

Figure 2 . 3 :

 23 Figure 2.3: Step-by-step schematic representation of the patterning procedure to fabricate Hall bars measured in this thesis, based on a simple lift-off process. The prepared substrate (a) is spin-coated with a photoresist and (b) and the area where the device should be formed is exposed to light via a projected mask. (c) The photoresist in the exposed area can then be removed in the development step so the substrate becomes uncovered and (d) the materials of interest can be deposited to form the 2DEG and add any necessary capping layer. (e) Finally, the remaining resist and undesired deposited materials can be removed from the sample surface, leaving only the patterned devices.

Figure 2 . 4 :

 24 Figure 2.4: Universal curve of the electrons inelastic mean free path (IMFP) as a function of their kinetic energy (bottom axis) or the corresponding photon energy (with the assumption that the work function is about 4 eV. Coloured areas illustrate the usual definition of electromagnetic waves, from lower to higher energy: in the visible spectrum (light gray), in the VUV range (dark gray) and soft X-ray (salmon pink) range. Reused with permission from [300]

Figure 2 . 5 :

 25 Figure 2.5: (a) Top panel: General principle and geometry of the angle-resolved photoemission spectroscopy (ARPES) technique. Electrons emitted upon illumination with a photon hν from the material via the photoelectric effect escape with angles θ (or ϑ in the text) and ϕ are collected by the analyzer. The bottom panel illustrates the momentum conservation parallel to the sample surface k ∥ , while the out-of-plane component k ⊥ is not conserved. (b) Conservation of energy during the photoemission process such that: E kin = hν -Φ -E B where E kin (or E k in the text) is the kinetic energy of the photoelectrons, E B is the binding energy, Φ is the material's work function, N (E) is the density of states and E F and E vac are the Fermi and vacuum level respectively. Reused with permission from [300]

Figure 2 . 6 :

 26 Figure 2.6: (a) Working principle of the hemispherical analyzer. (b) Sample manipulator used on CASSIOPEE beamline in SOLEIL, with mechanical translation and rotation axes. (c) Reference frame for the angle coordinates of the photoemitted electron. ϑ is the angle between the normal to the surface and the electron momentum, while ϑ x and ϑ y are the two orthogonal in-plane projections. (d) Top view of the in-plane projections, with the axis of ϑ x corresponding to the dispersion direction of the analyzer. (a) Reused with permission from [306]. (c) and (d) Reproduced from [108].

Figure 2 . 7 :

 27 Figure 2.7: Example XPS wide spectrum of a bare SrTiO 3 substrate. Reused with permission from [310]

Figure 2 . 8 :

 28 Figure 2.8: (a) Conventional naming of the absorption edges depending on the origin orbital of the absorbing electron. Reused from [312] (b) Example of a large absorption spectrum of Pb, where we can see multiple absorption edges crossed successively. The inset shows a zoom on the region of the L edge with apparently 3 sub-edges. Reused from [313] (c) and (d) Example of narrower absorption spectra at the Co L 2,3 and Tm M 4,5 edges.

Figure 2 . 9 :

 29 Figure 2.9: (a) Schematic illustration of the L 2,3 transitions in a ferromagnet. The selectivity resulting from the handedness of the light circular polarization, the electron's spin and the available final states results in a dichroic signal, as is exemplified in (b). (a) Reused with permission from [312] and (b) from [141]

Figure 2 .

 2 Figure 2.10: (a) Illustration of the X-ray absorption spectroscopy (XAS) of some transition metal at the L 2,3 edge and the corresponding spectrum, with linearly polarized beam. (b) Illustration of the absorption of a ferromagnetic transition metal at the L 2,3 edge and the corresponding XMCD spectrum from which the spin moment can be extracted by a substraction A -2B of the peaks area. (c) Similar to (b), but this time the splitting between the A and B peaks is due to an orbital moment on the final states. Reused with permission from [326].

Figure 2 .

 2 Figure 2.11: . (a) Principle of X-ray linear dichroism (XLD) at the L 3,2 edge, where a linearly polarized X-ray beam excites electrons from the 2p spin-orbit split states into the 3d states which are also split by the crystal field. (b) Illustration of the interaction between the orientation of the linearly s-or ppolarized X-ray beam and the p and d orbitals, depending on the incidence angle of the beam (grazing or normal incidence). The two bottom panels examplify this interaction in two cases of tetragonal unit cell: either elongated c > a or vertically compressed a > c. (c) Example XAS of a LaAlO 3 / SrTiO 3 (001) sample at the Ti L 2,3 edge with 4 main features corresponding to the pair of spin-orbit split initial states (largest splitting of about 6 eV), and the e g vs. t 2g crystal field split (∼ 2.5 eV) and with additional further distinguishable sub-peaks present due to the tetragonal crystal-field splitting (few 10 meV) Reused with permission from [327]

2 . 15 )

 215 To perform transport measurements in this thesis, we have mostly used either a classical Hall bar geometry (fig. 2.12b) or the van der Pauw configuration (fig. 2.12c).

Figure 2 .

 2 Figure 2.12: (a) Bus bar configuration in four-probe configuration, with all contacts placed in a row. (b) Hall bar configuration where a current is injected along the main channel and longitudinal and transverse voltages can be measured at the same time from branches placed along the device. (c) Standard van der Pauw configuration of the four-probe measurement for a square sample, where the contacts are placed in the corners to measure the sheet resistance (or magnetoresistance), and (d) the Hall effect, which can be done by exchanging two neighboring contacts of the standard van der Pauw geometry. Reused from[START_REF] Vicente-Arche | Quantum Oxide Heterostructures for Spin-Charge Interconversion[END_REF]

Figure 2 . 13 :

 213 Figure 2.13: Conceptual illustration of non-reciprocal effects, where spatial inversion and time-reversal symmetries are broken, unlocking new transport phenomena with directionality effects. (a) A system with inputs (e.g. a current) a 1 (from the left) and a 2 (from the right), where time-reversal is broken by a magnetic field B will show different responses (e.g. voltages) b 1 and b 2 for similar stimulation in opposite directions, unless B's direction is reversed. (b) Example of a sample exhibiting the Edelstein effect, whereby a spin density is uniformly created in opne direction or another depending on the injected current direction. (c) When a magnetic field is applied, the spin density interacts constructively or destructively, resulting in an decrease or an increase of the resistance.

Figure 2 .

 2 Figure 2.14: (a) and (b) Electrical rack of the PPMS used in UMPhy laboratory, with instrumentation from third-party vendors, including current sources (8) and nanovoltmeter (3), sourcemeters (4, 6 and 7) and various lock-in amplifiers (1 and 2). (c) Schematic representation of the in-plane rotation with corresponding field and current geometry. (d) and (e) Rotation setup of the Quantum Design Dynacool PPMS: the sample holder sits at the bottom of a cane controlled by a step-motor externally. Mechanical gears transmit the rotation from the cane rotation indexer at the top, to the sample holder. (f) In-plane rotating sample holder with a patterned sample attached with silver paste onto a back-gate aluminum foil

Figure 2 .

 2 Figure 2.15: (a) Magnetization precession cone at the ferromagnetic resonance (FMR). This can be the particular case of an (unrealistic) undamped magnetic oscillator. (b) Magnetization precession cone for a damped oscillator. With time, the magnetization goes back to an equilibrium position. (c) The Zeeman effect lifts the spin degeneracy of eigenstates as a magnetic field H 0 is applied. This splitting increases linearly with the filed and given an excitation photon with energy hν (or ℏω), dipolar magnetic transitions are allowed if the system reaches the resonance condition with H 0 = H res . Reused from[START_REF] Vicente-Arche | Quantum Oxide Heterostructures for Spin-Charge Interconversion[END_REF]

Figure 2 .

 2 Figure 2.16: Measurement configuration for SP-FMR experiments using a resonant cavity. A DC magnetic field is applied via electromagnets between which the sample is placed. The latter is glued onto a holder which allows the connections of two electrical contacts to the top of the sample at extremeties, and one contact on the back, to apply gate voltages. The RF field is sent through a circulator to the cavity and fed back through it to the detector. Reused from[START_REF] Vicente-Arche | Quantum Oxide Heterostructures for Spin-Charge Interconversion[END_REF]

Figure 2 .

 2 Figure 2.17: (a) Illustration of the spin pumping by ferromagnetic resonance experiment: the magnetization of a ferromagnet is excited with combined DC and RF magnetic fields. (b) At the resonance, a spin current propagates in adjacent layers, resulting in (c) a symmetric charge current response when a material with strong SOC like Pt is present, which can be reversed simply by changing the external DC magnetic field (and magnetization direction). (a) Reused with permission from [74] and (b) and (c) from [108].

Figure 2 . 18 :

 218 Figure 2.18: Electrical measurements of polarization via current-voltage loops. (a) Typical currentvoltage characteristic of a ferroelectric material with associated switching peaks at similar positive and negative coercive field value and (b) associated polarization loop obtained by integrating the current with respect to time, resulting in a typical square hysteresis. (c) Current-voltage characteristic of a simple dielectric material with (d) the expected linear contribution to the polarization loop, which is typically found in ferroelectric materials in addition to the polarization switching hysteretic shape. (e) Current-voltage characteristic of a resistor following Ohm's law and (f) the associated polarization-loop contribution which opens a fake hysteresis cycle independent of the real polarization switching.

Figure 2 . 19 :

 219 Figure 2.19: Electrical measurements of polarization via pulse sequences. (a) Typical Positive-Up-Negative-Down (PUND) voltage sequence. (b) Application example of the Positive-Up-Negative-Down (PUND) principle and advantages for a leaky diode-like ferroelectric sample typical of what can be observed in some thin films. (c) Pulse sequence of the Remanent Hysteresis mode provided with the Radiant Precision Multiferroic controlling software.

Figure 3 . 1 :

 31 Figure 3.1: Example spin-charge interconversion measurement configurations. (a) Pure spin current injection by the spin Seebeck effect.A thermal gradient along the z direction across a ferromagnetic insulator coupled with the application of an in-plane magnetic field causes the dffusion of non-equilibrium magnons and thus, the creation of a spin current which can be measured after it is converted into a charge current in the SOC material. (b) A lateral spin valve geometry where a spin polarized current is injected from a ferromagnet into a non-magnetic material. The spin current propagates along the main channel until it is sensed by another ferromagnetic element polarized in one direction or another by an external magnetic field. (c) A similar spin valve geometry where the spin current in the main channel is converted back into a charge current by a SOC material and detected as a non-local voltage at a certain distance of the injector. (d) Ferromagnet-free configuration where the spin-current is both generated and converted back into a charge current by the SOC medium.

Figure 3 . 2 :

 32 Figure 3.2: Spin-charge interconversion in SrTiO 3 2DEGs observed (a) from the Hanle effect and (e) from spin pumping by ferromagnetic resonance measurements. (a) Nonlocal measurement of the Hanle effect with field in-plane along x, i.e. the direction of the injected charge current, at different gate voltages (left panel) and spin Hall angle (right panel) extracted from Hanle fittings as a function of gate voltage, showing a maximum slighlty below the Lifshitz point. The dominant mechanism identified by geometrical considerations is the 2D spin Hall effect. Reused with permission from [387]. (b) Rashba coefficient as a function of gate voltage obtained from bilinear magnetoresistance (BMR) measurements in LAO/STO. (c) Energy dispersion of a STO 2DEG obtained from tight binding calculations, with close up views around the Lisfhitz point and Fermi contours at different energies and (d) theoretical Rashba coefficient with contributions from the different subband pairs and the total effective coefficient (in black). Reused with permission from [336]. (e) Inverse Edelstein length extracted from SP-FMR measurements in a AlO x /SrTiO 3 and as a function of the gate voltage, showing several maxima and minima corresponding to different points in the band structure, and an absolute maximum of λ = 28 nm. (f) Energy dependence of the Edelstein tensor κ xy with positive/negative contributions from each pair of Rashba subbands (top panel) and the resulting total value (bottom). The color code is the same as in (b), (c) and (d). (g) Fermi lines at various energies showing the progressive population of different subbands with topologically (non)-trivial avoided crossings corresponding to extrema of κ xy .Reused with permission from[START_REF] Vaz | Mapping Spin-Charge Conversion to the Band Structure in a Topological Oxide Two-Dimensional Electron Gas[END_REF].

Figure 3 . 3 :

 33 Figure 3.3: Basic transport characterization of an a LaAlO 3 /Al/SrTiO 3 device. (a) Temperature dependence of the sheet resistance showing a metallic behavior with a typical residual resistivity ratio (RRR) of about 10. (b)Backgate dependence of the sheet resistance at 2 K, with repeated voltage cycling after initialization between ±150 V, showing some overlap between cycles with a bit of hysteresis, and a non-monotonic behavior with a maximum around 20 V when going from positive to negative voltage.

Figure 3 . 4 :

 34 Figure 3.4: Basic magnetotransport of the same device. (a) Hall effect and (b) magnetoresistance at different gate voltages and with magnetic field µ 0 H z applied out-of-plane. At low gate voltages, the magnetoresistance displays a weak-localization behavior while a B 2 magnetoresistance dominates again at higher voltages, with a small dip at very low field indicating some weak-antilocalization. The Hall curves are mostly linear in the depleted regime and as the voltage is increased, the high-field slope seems to first decrease a bit and then increase again with a strong nonlinearity appearing as carriers are added into the 2DEG. (c) Carrier density and (d) Hall mobility for each carrier type extracted from a fit of the Hall data with a two-bands model.

Figure 3 . 5 :

 35 Figure 3.5: (a) Angle dependence of the longitudinal resistance obtained from first harmonic and normalized by the sheet resistance R s for magnetic fields increasing (from blue to red) up to 9 T. (b) Amplitude of the first harmonic signal as a function of the in-plane magnetic field amplitude µ 0 H ∥ showing a quadratic dependence. (c) and (e) angular dependences of the second harmonic signal R 2ω xx

Figure 3 . 6 :

 36 Figure 3.6: Gate dependence of bilinear magnetoresistance obtained from the second harmonic transport measurements with I ac = 150 µA. (a) Angle-dependence of the second harmonic signal at fixed gate voltages, applying 9 T in the plane of the sample. Colored circles represent the measured data, after removing some background contributions, while the fits are indicated by continuous lines. A reduced selection of curves with a step of 20 V are shown and shifted by an offset depending on the gate voltage for clarity. (b) BMR amplitude extracted from the fits to the angle dependencies as a function of the gate voltage. Error bars obtained from the fits of the sinusoidal amplitude are all at least 2 orders of magnitude smaller that the corresponding obtained value. (c) Sheet resistance from the second harmonic channel as the in-plane magnetic field is swept between +9 T and -9 T transverse to the current, at V g = -150 V (in orange) and +75 V (in green). The lines are a guide to the eye.

-Figure 3 . 7 :

 37 Figure 3.7: (a) Angle and gate-dependence of the anisotropic magnetoresistance, obtained from first harmonic measurement at 9 T and normalized by the maximum resistance obtained. The color scale reflects the value of this relative magnetoresistance (in percents) with zero indicating the highest resistance point along an angle sweep. (b) Gate-dependence of the AMR amplitude extracted from (a), assuming a simple sinusoidal function with the angle shift parameter left free. (c) Gate-dependence of the Rashba coefficient as obtained from eq. (1.26), assuming the AMR signal to originate purely from QMR.

Figure 3 . 8 :

 38 Figure 3.8: Logic device based on spin-charge interconversion tunable by a gate placed at the intersection between a charge current signal arm and a transverse spin diffusion channel. (a) The XNOR function can be realized by using the charge current in the first arm and the gate voltage at the intersection as inputs (V G , j c in ). (c)The NAND function can also be realized by using two charge currents (I 1 , I 2 ) from upstream devices to accumulate/deplete carriers on the gate contact at the intersection. Depending on the total current (and the resulting equivalent gate voltage), the activation function is triggered or not.

Figure 3 . 9 :

 39 Figure 3.9: Sheet resistance versus temperature curve of Al(1.8 nm)//STO sample used in this section, performed after all the above-mentioned sample preparation steps.

Figure 3 .

 3 Figure 3.10: (a) Polarization vs. electric field curves measured at 2 K between ±2.5 kV cm -1 (green curve) and ±10 kV cm -1 (red curve). While a simple dielectric behavior is obsverde in the initial state at low fields, a hysteresis is visible in the second curve after the activation threshold has been passed. (b) Electric-field dependence of the polarization as a function of the temperature, showing progressively less open curves as the temperature is increased, until a quasi-linear dieletric behavior is recovered above 50 K. (c) Remanent polarization as a function of temperature, extracted from (b) at zero electric field.

Figure 3 .

 3 Figure 3.11: (a) Polarization vs. electric field curves measured at 2 K for decreasing electric field boundaries. Curves are shifted by 2 µC cm -2 for clarity. (b) Electric field cycling of the sheet resistance between similar boundaries after ferroelectric state initialization. Curves are shifted by 3 kΩ for clarity.

Figure 3 . 12 :

 312 Figure 3.12: Demonstration of non-volatile electric control of spin-to-charge conversion. (a) Dependence of the charge current generated with the gate voltage showing a hysteretic dependence. (b) Corresponding charge current signals obtained by SP-FMR in the initial state (A) and successively at +200 V (B), positive remanence (C), -200 V (D), negative remanence (E) and back to +200 V (F), showing a sign reversal of the charge current with the sign of the gate voltage but also at remanence between either up or down states. (c) Charge current signals obtained after successive pulses of ±200 V during 500 ms demonstrating repeatability of the switching. (d) Similar measurements allow to plot the difference between the currents respectively at positive or negative remanence as a function of temperature, showing a maximum of charge current around 30 K and a following decrease, correlated to the ferroelectric transition.

fig. 3 .Figure 3 .

 33 fig.3.12d displays the temperature dependence of the difference ∆I c between both states. ∆I c is large below 30 K and vanishes around 50 K coinciding with the Curie temperature of STO previously observed.

Figure 4 . 1 :

 41 Figure 4.1: Fabrication steps of the AlO x /Ca:STO samples with 1 % of Ca. (a) Reflection high energy electron diffraction (RHEED) pattern of a substrate after a 1 h-long annealing at 800 °C in a vacuum chamber with a partial pressure of O 2 p 02 = 300 mbar. (b) X-ray photoemission spectrum (XPS) of the Ti-2p core levels of a bare Ca:STO substrate after annealing in oxygen showing nearly exclusively Ti 4+ valence states. (c) Atomic force (AFM) image of the surface after deposition of 1.8 nm of Al and exposition to air, showing cleanliness of the sample's surface. (d) XPS of the Ti-2p core levels after the in situ Al deposition and before exposing the sample to air. Fits from the Casa XPS software reveal the presence of additional features compared to (b), identified as Ti 3+ -and Ti 2+ -originated peaks.

  4.2c), with linear vertical (LV) photon polarization and an energy of hν = 47 eV. The energy dispersion (in fig. 4.2c) and Fermi surface (in fig. 4.2d) are reminiscent of previous reports of 2DEGs created at the bare STO surface

Figure 4 . 2 :

 42 Figure 4.2: Angle-resolved photoemission spectroscopy (ARPES) data of a AlO x /Ca(1 %):STO 2DEG prepared by molecular beam epitaxy and measured in situ at CASSIOPEE beamline in SOLEIL. Energy dispersion measured along k y with linear horizontal polarization across the ferroelectric transition temperature at (a) 55 K and (b) 15 K with an incident photon energy hν = 27 eV. (c) Energy dispersion measured along k y with linear vertical polarization at 15 K with an incident photon energy hν = 49 eV (d) k x k y Fermi surface of the 2DEG obtained at 15 K around the Γ point of the 102 Brillouin zone.

Figure 4 . 3 :

 43 Figure 4.3: Raman spectra of (a) a simple Ca(1 %):STO substrate and (b) of a AlO x /Ca:STO sample obtained from 10 K to 40 K showing the nearly constant peak corresponding to the A 1g phonon mode, and the appearance of a peak at low Raman shifts, below 30 K attributed to a TO 1 phonon mode. Dotted lines are guides to the eye.

Figure 4 . 4 :

 44 Figure 4.4: Temperature dependence of the relative intensity of the TO 1 -associated peak with respect to the A 1g peak, extracted from figs. 4.3a and 4.3b. Solid lines are guides to the eye.

Figure 4 . 5 :

 45 Figure 4.5: Electrical characterization of the bulk polarization in a AlO x /Ca(1 %):SrTiO 3 sample. (a) Electric-field dependence of the current flowing across the sample for an electric-field swept at 200 Hz between ±1.4 kV cm -1 at 2 K and (b) the corresponding polarization loop, obtained by integrating the current with respect to time. (d) Remanent polarization as a function of temperature from 2 K to 50 K. Curves are shifted by 1 µC cm -2 for clarity.

Figure 4 . 6 :

 46 Figure 4.6: Basic transport properties of the 2DEG hosted at the surface of sample A across the structural transition of CaSTO. (a) Temperature dependence of the sheet resistance during cool down and its derivative (in inset) showing a kink around 26 K. (b) Temperature dependence of the carrier density extracted from linear fits to the Hall measurements and (c) the related mobilities calculated from the carrier density and sheet resistance. (d) Temperature evolution of the magnetoresistance showing different regimes as a function of the out-of-plane magnetic field.

Figure 4 . 7 :

 47 Figure 4.7: Sheet resistance of various Ca:STO-based 2DEGs. (a) Temperature dependence of the sheet resistance of sample A measured while cooling down (blue) or warming up (down). The derivative of these two curves in the inset shows dominating peaks either around 26 K or 85 K depending on the sweep direction. (b) Comparison of the temperature dependence of the sheet resistance for various samples from table 4.1, while cooling down.

Figure 4

 4 Figure 4.7a displays the temperature dependence of the sheet resistance for sample A both while cooling down and subsequently warming up, both at the same rate of 4 K min -1 . Although we have seen that ferroelectricity seems to have an impact on the sheet resistance, causing an upturn at low temperatures, it is clear by comparing both curves here that an unexpected hysteretic behavior can be observed. Taking their derivative (in the inset of fig.4.7a) reveals a major kink for both curves occuring either around 26 K (for the cool down) or around 85 K (for the warm up). This behavior could be reproduced consistently across various samples CaSTO-based 2DEG samples obtained during this thesis.

Figure 4 . 8 :

 48 Figure 4.8: Electric-field cycling of the polarization and of the sheet resistance with different electric field boundaries. (a) Polarization vs. electric field cycling performed with increasing boundaries at 2 K at a frequency of 200 Hz. Curves are shifted by 1.25 µC cm -2 for clarity. (b) Sheet resistance cycling with the electric field after a gate forming procedure at 2 K up to ±70 V, with a cycling frequency in the mHz range. Curves are shifted by 150 Ω □ -1 for clarity.

Finally

  we show in figs.4.9c and 4.9d the transverse resistance under an out-of-plane magnetic field displaying nearly but not perfectly linear Hall effect at fig.4.9c positive or fig.4.9d negative remanence. We fit the data (in blue) to a two-channels parallel conduction

Figure 4 . 9 :

 49 Figure 4.9: Electrical switching of magnetoresistance and Hall effects at 2 K. (a) Magnetic field dependence of the sheet resistance at +70 V, 0 V (positive remanence), -70 V and 0 V (negative remanence). (b) Relative magnetoresistance in each state computed from the curves in (a). (c) Hall effect curve measured at positive remanence (0 V) after having applied +70 V. (d) Hall effect curve measured at negative remanence (0 V) after having applied -70 V. In (c) and (d), the red solid line is a fit to the data (in blue) with a two-bands model

Figure 4 .

 4 Figure 4.10: (a) Temperature dependence of the sheet resistance of a 10 µm-wide a LAO/AlO x / Ca(2 %):STO device during two successive cooldowns CD1 and CD2. (b) First derivative of the sheet resistance curves from (a).

Figure 4 . 11 :

 411 Figure 4.11: Current dependence of the in-plane AC magnetotransport performed at +9 T with an injected current of frequency f = 77 Hz in a 10 µm-wide device. (a) Angular dependence of the first harmonic longitudinal resistance signal measured at +9 T and at different current densities. (b) Angular dependence of the second harmonic longitudinal resistance signal measured at +9 T performed at different currents and (c) the corresponding estimated amplitudes calculated as the half-difference between the maximum and minimum resistance from (b).

Figure 4 . 12 :

 412 Figure 4.12: Current dependence of the in-plane AC magnetotransport performed at ±9 T with an injected current of frequency f = 77 Hz in a 10 µm-wide device. Data are normalized to the highest value for each angle sweep and expressed as a percentage. (a) and (b) Angle and current dependence of the first harmonic longitudinal resistance signal measured at +9 T and -9 T respectively and (c) their average. (d) and (e) Angle and current dependence of the second harmonic longitudinal resistance signal measured at +9 T and -9 T respectively and (f) their half-difference.

Figure 4 . 13 :

 413 Figure 4.13: Field dependence of the in-plane AC magnetotransport performed at positive and negative magnetic fields with an injected current of frequency f = 77 Hz and amplitude 250 µA in a 10 µm-wide device. (a) Angle and field dependence of the first harmonic longitudinal resistance signal measured from +9 T to -9 T. (b) Corresponding angular dependences of the second harmonic longitudinal resistance signal measured from +9 T to -9 T and (c) the corresponding amplitudes calculated from the half-difference of the maximum and minimum resistance values R 2ω xx .

  Finally, fig. 4.13b (or figs. 4.14d and 4.14e) show a sin(Φ) dependence of the second harmonic signal with again opposite polarity between positive and negative magnetic field values. Here, the offset present in the second harmonic signal (again substracted from fig.4.13b) remains mostly constant for all magnetic fields and we recognize a linear dependence of the sinusoidal amplitude of R 2ω xx in fig.4.13c. An important remark here, is that the sign of the sinusoidal angular dependence is reversed to what we observed in fig.4.11b where we plotted curves up to 20 A m -1 only, but it is consistent with the angluar dependence seen in the height map in fig.4.12f for current densities of 25 A m -1 and above.

Figure 4 . 14 :

 414 Figure 4.14: Magnetic field dependence of the in-plane AC magnetotransport performed at positive and negative magnetic field with a fixed current amplitude I ac = 250 µA (current density j ac = 25 A m -1 ) and frequency f = 77 Hz in a 10 µm-wide device. Data are normalized to the highest value for each angle sweep and expressed as a percentage. (a) and (b) Angle and magnetic field dependence of the first harmonic longitudinal resistance signal measured at +9 T and -9 T respectively and (c) their average. (d) and (e) Angle and magnetic field dependence of the second harmonic longitudinal resistance signal measured at +9 T and -9 T respectively and (f) their half-difference.

Figure 4 . 15 :

 415 Figure 4.15: Gate voltage dependence of the sheet resistance of the AlO x /Ca(2 %):STO device used in this section. The blue curve is measured during the first forming step (gate initialization) with steps of 10 V and at least 15 s per step and the green curve is measured right after with steps of 2.5 V and at least 60 s between each step. Measurements are performed in AC with a current I ac = 20 µA of frequency f = 77.0 Hz and an LIA integration time of 0.3 s.

Figure 4 . 16 :

 416 Figure 4.16: Gate dependence of the in-plane AC magnetotransport performed at +9 T with an injected current of frequency f = 77 Hz of amplitude 75 µA in a 10 µm-wide device. (a) Sequence of gate voltages used during each angular dependence measurement. (b) Angular dependence of the first harmonic longitudinal resistance signal measured at each of the gate voltages in (a). (c) Angular dependence of the second harmonic longitudinal resistance signal measured at each of the gate voltages in (a). A slope and average resistance background are substracted from the raw angular dependence data in (b) and (c).

Figure 4 . 17 :

 417 Figure 4.17: Angle dependence of the second harmonic resistance at (a) positive remanence and (b) negative remanence at an absolute field of |B| = 9 T. Curves are averages from multiple angle sweeps performed in similar conditions -i.e. at positive vs. negative remanence-and antisymmetrized with respect to the magnetic field -i.e. taking the half-difference of data at +9 T vs. -9 T.

Figure 5 . 1 :

 51 Figure 5.1: Gate-tuneability of ferromagnetism in LaAlO 3 /EuTiO 3 /SrTiO 3 2DEGs. (a) Temperature dependence of the sheet resistance under different gate voltages showing transition from Kondo-like scattering at low gate voltages to a regime with strong resistance downturn at 7.5 K coinciding with the Eu 2+ ferromagnetic ordering for higher gate voltages . (b) Tuning of the Hall effect with gate voltage with apparent nonlinearity at low magnetic field for positive gate voltages. (c) Evolution of the extrapolated anomalous Hall resistance as a function of gate voltage (left panel) or temperature (right panel). The effect seems to vanish below 10 K. (d) Field dependence of the normalized AHE after a substraction of the slope at 6 T from the Hall curve, for different gate voltages and superimposed with the field-dependent normalized out-of-plane Eu magnetization extracted from Eu XMCD data at zero gate with field perpendicular to the surface. Figures reused with permission from [433].

Figure 5 . 2 :

 52 Figure 5.2: Conception of a multiferroic 2DEG based on the LaAlO 3 /EuTiO 3 /Ca:SrTiO 3 heterostructure.

Figure 5 . 3 :

 53 Figure 5.3: Theoretical predictions of the multiferroic LAO/ETO/CaSTO heterostructure obtained from density functional theory (DFT) calculations. (a) Schematic representation of the slab tested in the electronic structure simulations consisting of (LAO) 6 /(ETO) 2 /(STO) 10 with 20 Å of vacuum on top to avoid interaction between periodic replica of the slabs. The STO is made ferroelectric by applying a 1.6 % compressive strain onto STO. (b) and (c) Layer-decomposed projected density of states (DOS) on O 2p, Ti 3d and Eu 4f states as a function of the polarization state pointing (b) towards the interface (P up ) or (c) away from the interface (P dn ). Positive and negative values stand for spin up and spin down, respectively. The Fermi level is indicated by dashed blue lines. (d) Computed magnetic moments on Ti cations for the two polarization states. (e) Relative displacement of the Ti cations with respect to the O anions along the growth direction.

Figure 5 . 4 :

 54 Figure 5.4: (a) Reflection high energy electron diffraction (RHEED) intensity as a function of time during the preparation of a LAO/ETO/CaSTO heterostructure. (b) AFM image of the sample after growth. Each terrace is separated from its neighbours by a step height of 4 Å.

Figure 5 . 5 :

 55 Figure 5.5: Dependence of the interfacial crystal field on the electrical remanent state in a LAO/ ETO/CaSTO. (a) X-ray absorption spectrum of a sample under both horizontal and vertical linear polarizations, at 2 K and remanence after applying a +50 V pulse, disconnecting the gate contact and reconnecting the TEY signal cable. The inset shows a closer view of the spectra near the lower energy absorption peak due to e g state. (b) Experimental X-ray linear dichroism signal for both P up and P dn remament states (top) and XLD signal from atomic multiplet simulations (bottom). (c) Sketches of the TiO 6 octahedral cages corresponding to the simulations in (b), evidencing different off-centered Ti O displacements ∆z Ti-O between P up and P dn consistent with the DFT calculations.

Figure 5 . 6 :

 56 Photon Energy (eV)

Figure 5 .

 5 7 illustrates the strong coupling between ferroelectricity and transport properties in our MF-2DEGs. As visible on the inset of fig.5.7a, the 2DEG displays the expected metallic temperature dependence of the sheet resistance R s upon cooling from 300 K. As seen in the previous chapter for simple AlO x /CaSTO 2DEGs, near 35 K the data shows a minimum below which the resistance increases again. The actual maximum of the kink in the slope occurs a bit below, i.e. around 27 K which correlates nicely with the transition to the ferroelectric state as observed previously in electron-doped bulk Ca(1 %):STO [281] and in other 2DEGs based on Ca(1 %):STO [435, 437] as seen in chapter 4.

Figure 5 . 7 :

 57 Figure 5.7: Ferroelectric and transport properties. (a) Temperature dependence of the sheet resistance. (b) Polarization vs voltage loop at 2 K. (c) Temperature dependence of the remanent polarization. (d) Gate voltage dependence of the 2DEG sheet resistance at 2 K. (e) Temperature dependence of the difference in carrier density between the two ferroelectric polarization remanent states, obtained from Hall effect measurements.

Figure 5 .

 5 Figure 5.7b presents the back-gate voltage dependence of the polarization (P (V G ) cycles, see inset illustrating the measurement configuration), which exhibit a clear hysteresis loop, associated with the presence of broad switching peaks in the current-voltage loop I(V G ) (see fig.5.8a). To extract only the contribution from the switched (or remanent) polarization, i.e. to eliminate the non-remanent linear dielectric polarization, we use the Remanent Hysteresis mode of the ferroelectric tester Radiant Precision Multiferroic II (see section 2.6). An example of the reconstructed loop obtained at 2 K in fig.5.8b shows how we can extract an approximate value of the remanent polarization. Note that, while these loops correspond to the whole CaSTO, the remanent values are in the range of those estimated solely for the 2DEG region from XLD data and simulation.Applying both simple P (V g ) and Remanent Hysteresis loops with increasing temperature, we observe a sustained hysteretic behavior up to about 35 K, extract the remanent polarization

Figure 5 . 8 :

 58 Figure 5.8: (a) Gate voltage dependence of the current corresponding to the P (V G ) cycle in fig. 5.7b. (b) Reconstructed Remanent Hysteresis loop obtained with the Radiant Precision Multiferroic II at 2 K.

Figure 5 .-Figure 5 . 9 :

 559 Figure 5.7d shows the gate dependence of the sheet resistance: after gate initialization, R s displays a clockwise hysteretic dependence with the gate voltage that is reproducible from one cycle to the next, a feature absent in equivalent 2DEGs grown on non-ferroelectric STO [239]. R s is lower when P points towards the interface with the 2DEG due to an electron accumulation. The data, thus indicate that the electronic properties of the 2DEG are strongly modulated by ferroelectricity. The carrier density n extracted from the ordinary Hall effect (with fits at high magnetic field) also depends on the remanent polarization state: with n = 2.65 × 10 13 cm -2 for P up and n = 2.45 × 10 13 cm -2 for P dn at 2 K, it yields a difference ∆n = 1.973 × 10 12 cm -2 . The dependence of ∆n with temperature shown in fig. 5.7e follows a similar trend as P R as expected with the strongest decrease near T E C = 27 K, and ∆n eventually vanishes near 35 K. At 2 K, this ∆n actually corresponds to about 45 % of the value of the switched remanent polarization of CaSTO, i.e. 2P R /e = 4.38 × 10 12 cm -2 with P R = 0.35 µC cm -2 .Furthermore, although the Hall effect in our 2DEG is mostly linear, closer inspection reveals a nonlinear contribution at low temperature and low magnetic field, ascribed to an anomalous Hall effect (AHE) contribution, a priori due to the ferromagnetic order. In fig.5.9, we show the different steps of the data analysis process used to extract the AHE contribution: after an anti-symmetrization of the data with respect to the vertical axis (since R xy should be odd with

Figure 5 .

 5 Figure 5.11: (a) Temperature dependence of the electroresistance at 0 T and 9 T. (b) Temperature dependence of the electro-magnetoresistance calculated from (a). (c) Sequence of gate voltages applied to the sample at 2 K after gate initialization. (d) Anomalous Hall effect and (e) magnetoresistance measured at at the voltages indicated in (c).

Figure 5 . 12 :

 512 Figure 5.12: Illustration of the magnetic anisotropy coming out in transport measurements and its control by ferroelectricity. (a) Anomalous Hall effect in the P up and P dn states, where the P up curve is rescaled for ease of comparison between the slopes. (b) Anomalous Hall effect comparison with the X-ray magnetic circular dichroism at Eu M 5 edge measured with the field out-of-plane. (c) Magnetoresistance curve measured with the field in the plane of the sample, parallel to the current. (d) A close-up view of the low-field region of (c), displaying a clear hysteretic behavior in the magnetoresistance.

Figure 6 . 1 :

 61 Figure 6.1: Surface characterization of (AlO x /)KTN samples at different stages of the fabrication process. reflection high energy electron diffraction (RHEED) pattern of sample B (a) before and (b) after annealing at 600 °C in vacuum for 30 min. reflection high energy electron diffraction (RHEED) pattern of sample C (c) before and (d) after annealing at 800 °C with an oxygen partial pressure p O 2 = 0.3 mbar for 30 min. (e), (f) and (g) atomic force microscope (AFM) image of the samples surface. (a) Surface of the KTN crystal with 1.5 % of Nb divided in two pieces and used for samples B and C, of which the surface at the end of the fabrication process is shown respectively in (e) (sample B) and (g) (sample C).

Figure 6 . 2 :

 62 Figure 6.2: Voltage dependence of the (a) polarization and (b) corresponding current vs. voltage (I(V )) loop measured across sample C's thickness at 2 K with a frequency f = 10 Hz.

2 )Figure 6 . 3 :

 263 Figure 6.3: Polarization vs. voltage loops at 2 K performed between ±40 V with different sweep frequencies.

2 )Figure 6 . 4 :

 264 Figure 6.4: Polarization vs. voltage loops performed at 2 K and fixed frequency of 10 Hz for increasing voltage boundaries V max .

Figure 6 . 5 :

 65 Figure 6.5: Temperature dependence of the polarization loops of sample C. Voltage cycling of the polarization for increasing temperature at (a) 10 Hz and (c) 100 Hz, and (b) and (d)corresponding remanent polarization vs. temperature.

6. 3 . 1 .Figure 6 . 6 :

 3166 Figure 6.6: (a) Temperature dependence of the sheet resistance of AlO x /KTN interfaces with various Nb concentrations. (b) First derivative of the logarithm of the sheet resistance curves from (a)

Figure 6 . 7 :

 67 Figure 6.7: Phase diagram of the KTa 1-x Nb x O 3 compound as a function of temperature and Nb concentration x from a compilation of the literature and data extracted from this work, T C being the ferroelectric Curie temperature and T R the transition to a rhombohedral phase (estimated for the case of 8 % of Nb from the major kink visible in fig. 6.6b).

Figure 6 . 10 :

 610 Figure 6.10: Out-of-plane magnetic field dependence of (a) the relative longitudinal magnetoresistance and (b) transverse resistance R xy for increasing temperatures from 4 K to 100 K.

Figure 6 . 11 :

 611 Figure 6.11: Temperature dependence of the sheet carrier density extracted from the high field slope of Hall curves in fig. 6.10b, assuming single band transport.

Figure 6 .

 6 Figure 6.12: (a) Hall and (b) magnetoresistance curves at 2 K for a positive applied voltage of +75 V, at positive remanence (0 V), for a positive applied voltage -75 V and at negative remanence (0 V).

Figure 6 . 13 :

 613 Figure 6.13: Temperature dependence of the Hall effect at (a) positive and (b) negative remanence after application of +75 V or -75 V respectively.

Figure 6 . 14 :Figure 6 . 15 :

 614615 Figure 6.14: Temperature dependence of the magnetoresistance at (a) positive and (b) negative remanence after application of +75 V or -75 V respectively.

Figure 6 .

 6 Figure 6.16: (a) Gate voltage dependence of the sheet resistance, repeated from fig. 6.9c, with letter markings for measurements at 0 V, +50 V, +100 V, -50 V and -100 V along the hysteresis cycle. (b) Gate voltage cycling of the Hall effect performed at 2 K after CD3 and the corresponding gate initialization shown in (a).

  .3.

	(a)	(b)	(c)
	Spin coating + exposition	Photoresist development
		Photoresist		Photoresist
	Substrate	Substrate		Substrate
	(e)	development Photoresist	(d)	capping reactive metal + Deposition of
	2DEG Hall bar
	Substrate		Substrate

  where the longitudinal resistance R 12,35 can also be written R xx and the transverse resistance R 12,65 can also be written R xy .

		12,35 =	V 35 I 12	,	(2.16)
	and	R 12,65 =	V 65 I 12	,	(2.17)

Table 4 . 1 :

 41 Summary of the Ca(x%):SrTiO 3 -based samples used in this chapter.

	140

Table 6 . 1 :

 61 Al(1.8 nm)/Al(1.8 nm) † 30 min, p O 2 = 0.3 mbar, T = 800 °C Al(0.9 nm)/Al(2.0 nm) † 30 min, p O 2 = 0.3 mbar, T = 800 °C †The additional layer of Al corresponds to a capping layer deposited after a first exposition of the sample to air. Summary of the KTaO 3 :Nb(x%)-based 2DEG samples used in this chapter.

	Samples Letter Full name	x N b %	Depositions	Pre-annealing
	A	PLAS1868	2	Al(2.1 nm)	-
	B	PLAS1984	1.5	Al(1.8 nm)	30 min, UHV (10 -9 mbar), T = 600	°C
	C 1.5 D PLAS2043 PLAS3269A 5	Al(0.9 nm)/Al(2.0 nm) †	30 min, p O 2 = 0.3 mbar, T = 800	°C
	E	PLAS3269B 8

J'aimerais d'abord adresser de nombreux remerciements à mes directeurs de thèse Manuel Bibes et Agnès Barthélémy de m'avoir donné l'opportunité de travailler avec eux et accordé leur confiance, et d'avoir été des guides dans le monde de la recherche. A Manu, merci pour ton intérêt et ton enthousiasme constants sur ce projet, tes efforts pour toujours faire avancer les choses coûte que coûte, ta capacité inspirante à proposer de nouvelles idées et possibles chemins parallèles et enfin pour la patience que tu as sûrement dû développer avec moi... A Agnès, merci pour ton soutien chaleureux, et le temps pris pour d'intenses discussions scientifiques, particulièrement dans des périodes intenses pour moi et pour toi.Je souhaite remercier les directeurs de l'Unité Mixte de Physique CNRS/Thales, Frédéric Petroff et Frédéric Nguyen Van Dau, et leurs successeurs Paolo Bortolotti et Vincent Cros et saluer leurs efforts pour perpétuer le haut niveau de la qualité de recherche du laboratoire tout en maintenant une certaine cohésion au sein de la grande famille qui le constitue. Je veux aussi saluer tous les membres du laboratoire, permanents et non-permanents, qui ont contribué à leur manière au maintien d'une organisation redoutable et d'un lien si important, surtout pendant et au sortir du premier confinement qui a affecté la vie de tous en 2020, mais aussi avant et après lors de nombreux événements et réunions, pour toujours mémorables. Parmi ces personnes, je veux mentionner l'ensemble de l'équipe des ingénieurs qui n'ont eu de cesse d'essayer de rendre le travail expérimental non seulement possible mais surtout de meilleure qualité, plus facile, plus sûr et dans une atmosphère humaine et chaleureuse: en particulier, merci à Cécile Carretero, Eric Jacquet, Anke Sander, Sophie Collin et Florian Godel. Je dois adresser toute ma gratitude à l'équipe secrétariale et tout particulièrement à Anne Dussart pour son travail plus que remarquable, et son aide en toutes circonstances. I also want to thank Juan Trastoy for taking the responsibility of being my tutor, listening, helping me to express my comments, concerns and actually understanding them.

1+e (E-E F )/k B T in the model. This can be noticed during the measurements as a rather sharp decrease in the momentum-averaged energy distribution curve at some energy corresponding to the so-called Fermi edge, although some effects (including the temperature) can cause some broadening of this edge. Experimentally it can usually still be found by taking the point of steepest slope or strongest peak in the first derivative of the intensity with respect ot the energy.In addition, there are few things we should note concerning the final excited state of the total system, with one less electron. In correlated systems, the removal of one electron will

Acknowledgements xxiii

Introduction

Acknowledgements

Cette thèse n'est pas seulement le résultat d'un travail acharné pendant les quatre dernières années, c'est aussi une étape importante sur un chemin de vie largement influencé par de nombreuses rencontres, et ce manuscrit n'aurait peut-être pas vu le jour sans le soutien et les contributions de toutes ces personnes.

CHAPTER 5

Multiferroic 2DEGs in LaAlO 3 /EuTiO 3 /Ca:SrTiO 3 heterostructures

While so far we have only investigated the incorporation of ferroelectricity in oxide 2DEGs and the generation of a non-equilibrium magnetization, the obtention of a conductive interface with magnetic ordering in the ground state is also of paramount importance for the field of spintronics.

In this chapter, we merge the fields of multiferroics [413] and oxide interfaces [147] by incorporating ferroelectricity and magnetism into a 2DEG based on STO. This work echoes research on 2D van der Waals materials in which ferroelectricity [414,415], ferromagnetism [416,417] and, most recently, multiferroicity [418] were introduced, albeit in this latter case while keeping the material insulating.

The main results of this chapter are published in ref. [419].

. Magnetism in oxide 2DEGs

A first report of ferromagnetic traces in LAO/STO 2DEGs was made in 2007 by Brinkman et al. [420] who showed butterfly -shaped (hysteretic) magnetoresistance (MR) curves as a function of an applied magnetic field as well as a low temperature upturn of the sheet resistance which they attributed to Kondo scattering on magnetic impurities. Several groups have tried to reproduce and further study the ferromagnetic behavior in this LAO/STO system, with different techniques implying direct magnetometry of bulk samples or scanning superconducting quantum interference device (SQUID) measurements for instance revealing localized magnetic moments organized inhomogeneously in ferromagnetic patches [421][422][423]. More broadly, it seems unclear whether ferromagnetism in this systems originates from oxygen vacancies or A or B cation (i.e. Sr or Ti) defects in STO. Traces of magnetism affecting the 2DEG transport properties via as expected. Interestingly, ER at 0 T and 9 T almost overlap at high temperature but are different below about 20 K, which would correspond to T M C . The difference in the electroresistance between 0 T and 9 T corresponds to an electro-magnetoresistance (EMR) effect, i.e. an effect reflecting the influence of the ferroelectric remanent state on the magnetoresistance, or reciprocally to the influence of magnetic field on the electroresistance. The electro-magnetoresistance (EMR) decreases with increasing temeprature and isappears when the first ferroic ordering temperature is reached, here T M C . The observation of an EMR effect is reminiscent of the behavior found in multiferroic tunnel junctions [440,441].

CHAPTER 6

Ferroelectric control of transport properties in AlO x /KTaO 3 :Nb two-dimensional conducting systems So far we have been focusing on the addition of ferroelectricity as a new ingredient brought into the well-known STO 2DEGs, by making STO ferroelectric itself. However, one can think of other approaches, especially with the two main concerns of increasing the spin-orbit coupling strength and the ferroelectric Curie temperature. With these two concerns in mind, the control of Rashba spin-orbit properties by ferroelectric switching and the obtention of a large spincharge interconversion efficiency would render these platforms more seducing for an integration into modern electronics.

Some obvious approaches could be on one side to start from other known 2DEG systems hosted by materials with larger intrinsic SOC (e.g. with large atomic SOC from the sub-species), and on the other side, to try to obtain 2DEGs at the surface of known ferroelectrics with higher Curie temperature.

In this chapter, we propose an intermediary solution mixing both approaches by using potassium tantalate KTaO 3 (KTO) with an isovalent partial substitution of Ta ions by Nb (see section 1.3.2). Indeed, as mentioned in section 1.4, it is also known that 2DEGs can be obtained by various manners at the surface of KTO or by interfacing it with other oxides, in a similar fashion to STO. In addition, a small fraction of Nb is already sufficient to bring KTN into its ferroelectric phase which -contrary to Ca:STO-is then reportedly found up to the full substitution rate (i.e., pure KNbO 3 ) with continuously increasing Curie temperature (e.g. at x Nb ≈ 30 %, T C ≈ 300 K) [129].

We thus present some preliminary results on the dielectric properties of such compounds after the deposition of Al as a top layer, creating a conductive interface of which we also investigate the transport properties.

. Gate-dependent magnetotransport of AlO x /KTN

We focus once again on sample C and turn to its interfacial transport properties particularly under the application of a gate voltage, and try to characterize the effects of the polarization vs. electric-field dependence seen in section 6.2. Following data reported in this section are acquired during different cooldowns after which the method of the initial gate forming step is performed at 2 K as for the STO-based 2DEGs. In fig. 6.9, we show example curves of such measurements performed for each cooldown which we will refer in the following as CD1 (of which the R s (T ) curve is shown in fig. 6.8), CD2 and CD3. Note that CD1 and CD2 are performed during the same measurement session at one day interval, while CD3 is performed one month later, after storage of the sample in a N 2 -saturated atmosphere.

In the sheet resistance derivative vs. temperature fig. 6.8, in addition to the similar "kink" as in fig. 6.6a, we seem to distinguish additional incidents at lower temperatures, including one near 14 K.

We can see how all curves seem to show some hysteretic dependence of the sheet resistance with the applied gate voltage, reminiscent of what we could observe in Ca:STO 2DEGs. We notice however a significant asymmetry between positive and negative voltages, especially for cycling curves from CD1 and CD2 which follow a very similar trend: they both show a significant opening for positive voltages while the resistance almost overlaps itself when the voltage is swept back and forth in the negative range and goes close to saturation. While the third curve of CD3 is also asymmetric, the opening on the negative voltage side is only slightly thinner. Note that in all three cases, a significant difference in resistance is present between the -100 up and down sweep directions at 0 V meaning that an electrical remanent control of transport properties can be achieved in this system as seen previously for AlO x /CaSTO.

Following the gate initialization of CD1, we perform a series of basic magnetotransport measurements. In figs. [START_REF] Moore | Cramming More Components onto Integrated Circuits[END_REF].10a and 6.10b, we show a temperature dependence of the magnetoresistance and Hall effect respectively from 4 K to 100 K, performed successively, just after having applied 75 V at 2 K and gone back to 0 V (the sample is thus in a positive remanence state when the measurements are started). In fig. 6.10a, the sample shows a negative magnetoresistance for all temperatures up to about 35 K where it seems to recover a quadratic behavior with very small amplitude. At 4 K, there also seems to be traces of WAL although very faint compared to the dominant signal over full range of magnetic field.

The temperature dependence of the Hall effect displayed in fig. 6.10b is also surprising as the slope seems to be non-monotonic. For some temperatures at 28 K and below, these curves bear characteristics not only of electron transport but also reveal the possible presence of holes contributing to transport.

Despite these nonlinearities and complex behavior, we fit the slope of each curve at high field and assuming a pure electron-mediated transport extract the corresponding carrier density, plotted in fig. 6.11. Although it clearly results in some values looking a bit erratic, we obtain electron densities of the order of few 10 13 cm -2 consistent with that obtained in AlO x /KTO 2DEGs and we seem to distiguish a trend of decreasing number of electrons (in absolute value) as temperature increases up to about 40 K above which it saturates or increases again slightly.

To ascertain the presence of holes in transport as preliminarily conjectured from temperaturedependent Hall effect, the sample is brought back to 2 K (CD2) and after the gate initialization (shown in fig. 6.9b) we perform additional magnetotransport characterization as a function