Alexandre Pourret

Daniel Braithwaite

Jean-Michel Martinod

Marie-Jo Blanchard

Iulian Mattei

Bernard Salce

Michel Boujard

Karine Mony

Gérard Lapertot

Pr Shin Kambe

Pr Ōnuki

William Harima

Tatsuma Knafo

Hisashi Matsuda

Pierre Kotegawa

Rikio Settai Haen

Xavier Jehl

Julien Bertheau

Claude Chapelier

Max Hofheinz

Pierre Payet-Burin

Frédéric Poletti

Louis Jansen

François Lefloch

Silvano Defranceschi

Jean-Pierre Cleuziou

Jean-Luc Thomassin

Mathieu Taupin

Alexandra Palacio Morales

Mounir Boukahil

Benoit Voisin

Charlène Tonnoir

Alexan- Der Grimm

Andreas Pfeffer

Driss Badiane

Caroline Richard

Toai Le Quang

Natalia Ares

Jean-Eudes Duvauchelle

Amalia Fernandez Panyela

Ludovic Howald

Valentin Taufour

Eduardo Lee

Eduardo Hering

Scheilla Ramos

Liam Malone

Byron Villis

Mon

Frank Hekking

Thierry Klein

Frédéric Faure

Antoine Delon

Edinburgh Patrik Öhberg

Salford Jay Smith

Frédéric Chausse

M Farilekas

Joël Bouchitté

M Peotta

M Mazoyer

M Minet

M Grand

Ahmed Trabelsi

Guillaume Géranton

Illya Plotnikov

Keywords: Uranium compounds, strongly correlated electrons, ferromagnetism, low temperature, heavy fermions, quantum criticality, new quantum phases, resistivity, Hall effect, magnetization, XMCD, high pressure. 17

Au moment d'apposer la dernière touche à ce manuscrit, je tiens à remercier les personnes qui ont contribué à cette réalisation.

Dai Aoki who took me for an internship and then for the PhD thesis on his ERC grant. He introduced me to the job and guided me all through. He helped me a lot while letting me a large freedom, in all aspects. Working under his supervision was a great experience.

Georg Knebel qui m'a pris sous son aile, devenant ainsi mon co-encadrant en cours de route. On a travaillé ensemble sur UCoAl, et il m'a aidé sur bien des aspects. Pendant la rédaction de la thèse, qui a été une période décisive, j'ai pu compter sur sa disponibilité et son soutien, pour tous les problèmes auxquels je faisais face et toutes les questions que je me posais.

Jacques Flouquet qui a impulsé beaucoup d'énergie au labo, et s'est battu pour que je sois pris en thèse sur l'ERC. Son expertise scientifique a été d'une valeur inestimable du début à la fin.

Jean-Pascal Brison qui m'a pris sous sa responsabilité en tant que directeur de thèse officiel.

The members of the jury: Klaus Hasselbach for the role of examiner and president of the jury, Pr. Kenji Ishida and Manuel Brando for the role of reporters and for having come from so far. Their questions during the defense and the remarks on the manuscript were highly valuable inputs, and they made me feel that this PhD has been a success.

Ferromagnetic quantum criticality in the uranium-based ternary compounds URhSi, URhAl, and UCoAl

et pour tout le souci que tu t'es fait pour moi ; Maman pour m'avoir coaché et aiguillé quand j'étais dans les limbes ; mon frère pour m'avoir toujours compris.

À Radiohead, pour les nuits blanches à bosser. Aux esprits de mon monde: l'Arbre, la Bistance du Triaxe.

À mes super potes de ces années de thèse: Toby et Illya. À Marta pour avoir porté tout le stress et pour ton soutien sans faille.

Résumé en français

Dans cette thèse, on étudie la criticalité quantique ferromagnétique dans trois composés ternaires à base d'uranium, par des mesures thermodynamiques et de transport sur des échantillons monocristallins, à basse température et sous haute pression. URhSi et URhAl sont des systèmes ferromagnétiques itinérants, tandis que UCoAl est un système paramagnétique étant proche d'une instabilité ferromagnétique. Tous ont une phase ordonnée de type Ising. Dans le composé orthorhombique URhSi, on montre que la température de Curie diminue lorsqu'un champ magnétique est appliqué perpendiculairement à l'axe facile d'aimantation, et une transition de phase quantique est attendue autour de 40 T. Dans le système hexagonal URhAl, on établit le diagramme de phase pression-température pour la première fois, lequel indique une transition de phase quantique vers 5 GPa. Dans le composé isostructural UCoAl, on étudie la transition métamagnétique par des mesures d'aimantation, d'effet Hall, de résistivité et de dichroïsme circulaire magnétique des rayons X. On observe des phénomènes de relaxation magnétique intrigants, avec des sortes de marches. L'effet Hall et la résistivité ont été mesurés à des températures de réfrigérateur à dilution, sous pression hydrostatique jusqu'à 2,2 GPa, et sous champ magnétique jusqu'à 16 T. La transition métamagnétique se termine sous pression et champ magnétique au niveau d'un point critique quantique terminal. Dans cette région, il se produit une forte augmentation de la masse effective, et une différence intrigante entre champ montant et descendant apparaît dans la résistivité transverse. Ce pourrait être la signature d'une nouvelle phase, éventuellement reliée aux phénomènes de relaxation observés dans les mesures d'aimantation, et résultant de frustrations au sein du réseau quasi-Kagomé que forment les atomes d'uranium dans cette structure cristalline.

Chapter 1

Introduction

Many uranium-based intermetallic compounds are strongly correlated electron systems, showing a variety of peculiar electronic and magnetic properties. They have been extensively studied with the discovery of heavy fermion systems and unconventional superconductivity. Among the most famous examples, we could cite UPt 3 , UBe 13 , URu 2 Si 2 ... They recently attracted additional interest with the discovery of the ferromagnetic superconductors UGe 2 , URhGe and UCoGe. Finally, they are a play ground to study quantum phase transitions, where non-Fermi liquid behaviour and possibly new quantum phases may appear.

The key properties of uranium compounds

Uranium has 5f valence electrons carrying a magnetic moment dominated by the orbital part due to large spin-orbit interaction. The 5f bands are close to the Fermi surface so they participate directly or are coupled to the conduction electrons.

In addition, the 5f electrons have strong mutual Coulomb repulsion. The strong correlations give rise to high band mass and thus to a peaked density of states. In addition, the 5f electrons hybridize with the conduction electrons, and this mixing leads to the formation of quasi-bound states. This is called the Anderson-Kondo mechanism. It is a many-body effect that further enhances the density of states, and in some cases gives rise to spectacular effective mass enhancement: the heavy fermion systems.

In the subsequent sections, we introduce the basics of heavy fermion systems, by describing the correlations between f electrons and conduction electrons. After showing that this phenomenon results from competing mechanisms, and therefore in the vicinity of a quantum phase transition, we get to the subject of ferromagnetic quantum phase transitions. In particular, we describe the topic of ferromagnetic superconductors.

Finally, we replace the compounds studied in this thesis (URhSi, URhAl, UCoAl) in the family of UTX compounds, to unveil some systematic properties.

Heavy Fermions

Heavy Fermion systems are strongly correlated electron systems. They are intermetallic compounds (i.e. alloys of different metals) generally U, Ce or Yb based, in which the interactions between f electrons and conduction electrons give rise to quasiparticles of highly renormalized effective mass of about 10 -1000 times the free electron mass (hence the term «heavy» Fermions). However, these properties only appear at low temperatures (T ∼ 10 K). This manifests experimentally as a high electronic contribution to the heat capacity, and a high electron-electron inelastic scattering contribution to the electrical resistivity at low temperature. At high temperatures, these systems behave like «normal» magnetic metals. The large effective mass enhancement results from a hybridization between localized and conduction electrons through the Kondo effect.

The original picture that describes the Kondo effect is an interaction between an isolated magnetic impurity (an atom with partially field f orbital) in a see of conduction electrons, which appears as an antiferromagnetic exchange coupling. In heavy fermion systems, the Kondo effect occurs on a lattice of magnetic ions, so the Kondo lattice picture is realized.

In the single impurity Kondo effect, the magnetic impurity is completely screened by the scattering electrons with which it forms a non-magnetic singlet state in the ground state. This state is a «virtual bound state», in which the conduction electrons are almost localized by resonant scattering at the site of the impurity. A consequence of this is the formation of a narrow peak in the density of states close to the Fermi level. This Kondo resonance peak has a bandwidth of the order of k B T K . The famous signature of the Kondo effect is a minimum of resistivity at T K , followed by a logarithmic increase with decreasing temperature.

In the Kondo lattice picture, there is an overlap of the neighbouring Kondo clouds, and below T coh < T K , the coherence length of the Kondo clouds extends to a large-scale many-body band of strongly interacting quasiparticles. These interactions produce nonperturbative corrections to the free electron picture, so that the quasiparticles behave like electrons with a highly renormalized mass. The low temperature behaviour of such a Kondo lattice system differ drastically from the single impurity Kondo model, and the resistivity decreases again with decreasing temperature. At low temperature, the resistivity has a AT 2 dependence, characteristic of electron-electron scattering in the Fermi liquid picture, where the coefficient A scales with the square of the effective mass: A ∼ m 2 .

Competition with RKKY interaction: Doniach phase diagram

In metals, there is an indirect exchange interaction between local moments via the conduction electrons, called RKKY interaction (Ruderman-Kittel-Kasuya-Yosida). The RKKY interaction is oscillatory with the distance, so the coupling can favor parallel or antiparallel spin alignment.

The RKKY interaction yields a magnetically ordered ground state, while the Kondo effect leads to a paramagnetic state. Both interactions are competing. Doniach [START_REF] Doniach | The Kondo lattice and weak antiferromagnetism[END_REF] established that the energy scales for Kondo and RKKY have different dependence on the degree of hybridization. Figure 1.1 shows the famous Doniach diagram: T K ,T RKKY vs the product JN (E f ), where J is the coupling between f and conduction electrons, N (E f ) is the DOS at the Fermi level. The ground state is magnetic when RKKY dominates, and non-magnetic when Kondo dominates. This diagram illustrates the possibility to tune the system between magnetic and non-magnetic state by varying the hybridization via pressure or doping. At zero temperature, for a critical value of JN (E f ), a quantum phase transition occurs. In the frontier region, quantum critical fluctuations lead to non-Fermi liquid behaviour (NFL), and a superconducting phase sometimes appears. This picture basically applies to localized magnetic systems such as rare earths compounds, but the validity is questionable in the case of uranium compounds, which are more itinerant. Nevertheless, it qualitatively explains the typical phase diagram of these materials.

Quantum phase transitions

The physics of quantum phase transitions was successfully studied in antiferromagnetic (AF) systems, where there is a continuous change of the order parameter at zero temperature, through a quantum critical point (QCP). In contrast, when it comes to itinerant ferromagnetic (FM) systems, the QCP seems generally pre-empted by the occurrence of a first-order transition.

Antiferromagnetic quantum critical point

There are two different scenarios for the state of the system in the vicinity of a QCP (see the review of Löhneysen et al. [START_REF] Löhneysen | Fermi-liquid instabilities at magnetic quantum phase transitions[END_REF]):

• in a «conventional» QCP, the system is described in terms of quasiparticles interacting with spin fluctuations, resulting in effective mass enhancement. The Kondo temperature increases continuously and there is no change of the Fermi surface. Such AF QCPs are found in itinerant systems (= spin density wave) e.g. CeCu 2 Si 2 , CeRu 2 Si 2 ...

• in the «unconventional» QCP, the Kondo temperature goes to zero, and there is a change of the Fermi surface, which becomes larger on going from the magnetic to non-magnetic ground state due to the inclusion of the 4f states into the Fermi sea. This scenario is referred to as local criticality, or Kondo breakdown. The candidates are YbRh 2 Si 2 [START_REF] Trovarelli | YbRh 2 Si 2 : Pronounced Non-Fermi-Liquid Effects above a Low-Lying Magnetic Phase Transition[END_REF][START_REF] Friedemann | Break Up of Heavy Fermions at an Antiferromagnetic Instability[END_REF] and CeCu 6 [START_REF] Löhneysen | Non-Fermi-liquid behavior in a heavy-fermion alloy at a magnetic instability[END_REF][START_REF] Schröder | Onset of antiferromagnetism in heavy-fermion metals[END_REF].

Ferromagnetic systems: hidden QCP

In the usual case of a ferromagnetic system, the transition from paramagnetic to ferromagnetic state at the Curie temperature T C is of second-order type. In the temperaturepressure phase diagram of several weak ferromagnets, the transition temperature decreases with pressure and reaches 0 K at some critical pressure P c . At some pressure below P c , the transition becomes of first-order type, and the ordered moments disappear discontinuously. The limit between the two regimes is a tricritical point (TCP). Such type of behaviour was observed in MnSi [START_REF] Pfleiderer | Magnetic quantum phase transition in MnSi under hydrostatic pressure[END_REF], UGe 2 [START_REF] Huxley | UGe 2 : A ferromagnetic spin-triplet superconductor[END_REF], ZrZn 2 [START_REF] Uhlarz | Quantum phase transitions in the itinerant ferromagnet ZrZn 2[END_REF], and in the doped systems Co(Si 1-x Se x ) 2 [START_REF] Goto | Magnetic properties of the itinerant metamagnetic system Co(S 1-x Se x ) 2 under high magnetic fields and high pressure[END_REF] and (Sr 1-x Ca x )RuO 3 [START_REF] Uemura | Phase separation and suppression of critical dynamics at quantum phase transitions of MnSi and (Sr (1-x) Ca x )RuO 3[END_REF] where the Se-or Ca-doping plays the role of tuning parameter. The existence of a tricritical point seems to be a general rule, since it has also been observed in simple ferromagnets like Fe,Co,Ni [START_REF] Yang | Evidence for first-order nature of the ferromagnetic transition in Ni, Fe, Co, and CoFe 2 O 4[END_REF]. As a rule, a TCP occurs in the T -P phase diagram of a ferromagnet at low temperature, close to P c , where dT C /dP diverges, or at least becomes very large, in difference to a second-order phase transition. It has been demonstrated theoretically by taking into account magneto-elastic coupling effects, that the transition becomes first-order when the slope -dT C /dP is larger than a critical value which involves the compressibility and specific heat jump at T C . [START_REF] Larkin | Phase transitions of the first order but nearly of the second[END_REF].

This starting point is widely admitted among the community, but it seems not sufficient to explain the rather extended range of pressure-temperature in which first-order occurs (several orders of magnitude too low). The magneto-elastic coupling could be sufficient to explain the tricriticality in UGe 2 , provided that we take into consideration the dimensionality and the anisotropy of the system [START_REF] Vladimir | On the phase diagram of UGe 2[END_REF].

Another way to explain the first-order transition is the existence of non-analytic terms in the Landau free energy. Such non-analytic terms generically arise in the presence of gapless excitations (soft modes) that couple to the fluctuations of the order parameter. In the case of metallic FM-PM transition, Belitz et al. [START_REF] Belitz | First Order Transitions and Multicritical Points in Weak Itinerant Ferromagnets[END_REF] proposed that the soft modes in question are spin-triplet particle-hole excitations, which couple to the spin fluctuations. At the transition, the fluctuations become long-range, the interactions with the soft modes «dress» them with a mass. The massive modes have an energy cost, and thereby stabilize the system in either side of the transition, giving it the first-order character. This concept of mass-generation is very general. It is applicable to all metallic FM-PM transitions in dimension d > 1, even if the electron responsible for magnetism are not the conduction electrons. In this sense, this theory is above the debate of itinerant/localized FM picture.

Metamagnetic wing

Regardless of the physical mechanism producing the first order transition, a direct consequence of the first-order nature of a PM-FM transition is the metamagnetism above P c , which can be understood by simple Landau free energy considerations. This is illustrated in Fig. 1.2, with the free energy F as a function of the order parameter M in different situations of temperature T and conjugate of the order parameter H (magnetic field applied parallel to M ). The dots indicate the stable state: paramagnetic (PM) or ferromagnetic (FM).

In a second-order transition, there is only one minimum in the F (M ) expansion (Fig. 1.2-a). At the critical temperature T C , the curvature at M = 0 changes sign, and the stable M continuously changes from zero to finite values. In the PM phase close to a second-order transition, H does not induce any transition (Fig. 1.2-b).

On the contrary, in the first-order case, there are two local minima in F (M ): one at zero M and one at finite M (Fig. 1.2-c). At T C , there are two stable states separated by an energy barrier, and in the PM phase close to a first order transition, H does induce a transition (Fig. 1.2-d).

So far we only considered the two-dimensional temperature-pressure phase diagram. If we add an extra thermodynamical variable, the magnetic field H, we obtain the schematic phase diagram shown in Fig. 1.3. The tricritical point is actually the starting point of a line that extends above P c in the T -P -H phase diagram. This is a line of criti- cal endpoints (CEP) below which a first-order metamagnetic transition is observed from the PM state to a field-induced FM state. As pressure is increased, the metamagnetic field H m increases, and T CEP decreases. This first-order transition plane in the T -P -H phase diagram was named «wing» [START_REF] Robert | Proposal for Notation at Tricritical Points[END_REF]. The line of CEPs eventually reaches 0 K at a so-called quantum critical endpoint (QCEP), which is a new class of quantum criticality. The wing has been observed experimentally only in UGe 2 [START_REF] Taufour | Tricritical Point and Wing Structure in the Itinerant FerromagnetUGe(2)[END_REF][START_REF] Kotegawa | Evolution toward Quantum Critical End Point in UGe(2)[END_REF], and in UCoAl where it occurs at ambient pressure. This is the main topic of Chapter 5.

UTX compounds

UTX compounds refer to the class of ternary intermetallic compounds of Uranium with T = late transition metal (3d: Fe,Co,Ni -4d: Ru,Rh,Pd -5d: Ir,Pt) and X = p-element (3p: Al,Si -4p: Ga,Ge -5p: In,Sn). It is a class of materials which have attracted much attention because the semi-delocalized 5f electrons of the U atoms give rise to a great The ground state is paramagnetic (PM) above the critical pressure P c . The transition becomes first-order at the tricritical point (TCP). Above the critical pressure P c , there is a metamagnetic «wing» at temperatures below the line of critical endpoints (CEP), which terminates at a quantum critical endpoint (QCEP). diversity of magnetic properties.

Many of them have a magnetic ground state (ferro-or antiferromagnetic) with a reduced U moment compared to the free ion value, huge uniaxial anisotropy, and a moderately enhanced electronic specific heat compared to the bare band mass.

The majority of these compounds crystallize in either of the two crystallographic structures: the orthorhombic TiNiSi (Fig. 1.5) (or the closely-related CeCu 2 )-type and the hexagonal ZrNiAl-type (Fig. 1.4). In this thesis, we study compounds of the two families: the orthorhombic URhSi, and the hexagonal URhAl and UCoAl.

The main mechanisms affecting the magnetic properties of these compounds are the overlap of neighbouring U 5f orbitals and the hybridization of U 5f orbitals with the orbitals of the other T and X atoms, generically referred to as 5f -ligand hybridization. Both mechanisms lead to the delocalization of U magnetic moments, and the U moments will be coupled via direct 5f -5f and via indirect exchange interaction involving other non-f electrons.

Hill [START_REF] Hill | Plutonium and Other Actinides[END_REF] conjectured that for inter-U distance d U-U < 3.4-3.6 Å (the so-called Hill limit), direct overlap of the U 5f orbitals leads to the delocalization of 5f electrons and ultimately yields to the washout of magnetic order. Above the Hill limit, the picture of localized magnetic moments ordering ferro-or antiferromagnetically is normally expected.

In the Hill limit region, the 5f -ligand hybridization -which also tends to delocalize the 5f electrons -is the key that controls the magnetic properties. Indeed, the magnetic ordering seems intimately related to the degree of 5fd hybridization via the degree of d-shell filling of the transition metal. Table 1.1 lists all the UTAl compounds with ZrNiAl-type structure, sorted by increasing d-orbital filling. On top of the list are the magnetically ordered systems (FM or AF), then when decreasing the d-orbital occupancy we find PM systems with strong FM correlations (UCoAl, URuAl), and finally the nonmagnetic compound UFeAl. This tendency is also observed in the UTX compounds crystallizing in the TiNiSi-type structure (see Chapter 3).

Quite logically, the UTX materials (regardless of their crystal structure) with strongest electronic correlations (highest γ-value) are found in the borderline region of both hybridization mechanisms, where d U-U is close to 3.5 Å) and with intermediate d-filling of the T element : (URhGe, UNiAl, UCoAl, URhAl).

Hexagonal ZrNiAl-type

In the sub-group of UTX with ZrNiAl-type structure (space group P 62m) (Fig. 1.4), the coupling is generally ferromagnetic within the hexagonal planes (only exception : UNiAl q = (0.1, 0.1, 0.5)), and is either ferro-or antiferromagnetic (UNiAl, UNiGa,...) between the planes [START_REF] Sechovský | Chapter 1 Magnetism of ternary intermetallic compounds of Uranium[END_REF]. This fact does not necessarily yield a magnetically ordered ground state, but the FM correlations do play a role in URuAl (paramagnet with strong Al Rh( 1)

Rh(2) U c c U Rh(1) Al Rh (2)
Figure 1.4: Crystal structure of URhAl (ZrNiAl-type, space group p 62m), with two non-equivalent Rh positions, consisting of alternating hexagonal layers of U-Rh(1) and Al-Rh [START_REF] Löhneysen | Fermi-liquid instabilities at magnetic quantum phase transitions[END_REF]. Lattice parameters are a = 696.5 pm and c = 401.9 pm for in-plane and inter-plane distances, respectively (from [START_REF] Dwight | Developments in the Structural Chemistry of Alloy Phases[END_REF]). Credits: Nohara et al. [START_REF] Nohara | Strong Longitudinal Magnetic Fluctuations Near Critical End Point in UCoAl: A (59)Co-NMR Study[END_REF] spin fluctuations) and UCoAl (paramagnet with metamagnetic transition to a FM state, see chapter 5). For X = Al, the compounds with transition metal T = Fe, Co, Ni, Ru, Rh, Ir and Pt crystallize in the ZrNiAl-type structure. They have all been studied by Prague and Amsterdam groups (see [START_REF] Sechovský | Chapter 1 Magnetism of ternary intermetallic compounds of Uranium[END_REF] for an exhaustive review).

In the hexagonal ZrNiAl-type structure, the crystal electric field confines the electrons of large l orbitals in the basal plane, so that the hybridization mechanisms will be strongly anisotropic. As a consequence, since the strong spin-orbit coupling of the 5f states produces orbital magnetism, the moments will be locked in the c-direction, leading to Ising-like systems. It should be noted that this crystal structure (space group p 62m) does not have inversion symmetry, thus a large Fermi surface splitting is generally expected [START_REF] Fischer | Metallic Magnets without Inversion Symmetry and Antiferromagnetic Quantum Critical Points[END_REF]. Furthermore, the crystal structure forms the quasi-Kagome lattice, leading to a geometrical frustration. Table 1.1: UTAl compounds with ZrNiAl-type structure (adapted from [START_REF] Sechovský | Chapter 1 Magnetism of ternary intermetallic compounds of Uranium[END_REF])

d-filling compound lattice parameters γ (mJ T N,C ground state a ( Å) c ( Å) mol -1 .K -2 ) (K)
6s

TiNiSi-type structure

URhSi belongs to the family of UTX compounds (T = late transition metal, X= Si or Ge) crystallizing in the orthorhombic TiNiSi-type structure (space group P nma). In this structure, the U atoms are arranged in zig-zag chains along the a-axis (Fig. 1.5). There are 14 compounds in this family, showing a variety of magnetic ground states (see Fig. 1.6), which are believed to be controlled by the degree of 5f -ligand hybridization which reduces with increasing filling of d orbitals of the T elements. Magnetic order is found in the compounds with higher d. (taken from [START_REF] Aoki | Ferromagnetism and Superconductivity in Uranium Compounds[END_REF]) Crystal structure of URhSi (TiNiSi-type). The arrows on the U sites denote the direction of the magnetic moments.

Among this class of materials (shown in Fig. 1.6), the compounds URhSi, URhGe, and UCoGe are borderline cases between paramagnetic and magnetically ordered compounds, with intermediate degrees of hybridization. A consequence of this is the rather enhanced electronic correlations at low temperatures (γ ∼ 100 -150 mJ.mol -1 .K -2 ), dual localized-itinerant magnetic properties, and strongly fluctuating spins.

URhGe and UCoGe attracted much attention because of the coexistence of ferromagnetism (FM) and superconductivity (SC) [START_REF] Aoki | Coexistence of superconductivity and ferromagnetism in URhGe[END_REF][START_REF] Huy | Superconductivity on the Border of Weak Itinerant Ferromagnetism in UCoGe[END_REF]. These compounds have peculiar magnetic anisotropy profiles, c being the easy axis, a the hard axis, and b is intermediate (high susceptibility without spontaneous component). Upon application of a magnetic field along the b-axis, T C decreases and eventually reaches 0 K at H b ≈ 10-15 T. There, a field-induced SC phase was discovered in URhGe [START_REF] Lévy | Magnetic Field-Induced Superconductivity in the Ferromagnet URhGe[END_REF], and an enhancement of T SC in UCoGe [START_REF] Aoki | Extremely Large and Anisotropic Upper Critical Field and the Ferromagnetic Instability in UCoGe[END_REF]. (See the magnetization and phase diagram of URhGe, Fig. 1.7)

Ferromagnetic superconductors

In the case of s-wave pairing, magnetism and superconductivity (SC) are conflicting each other, since the spin-flip process is pair-breaking. Coexistence of antiferromagnetism (AF) and s-wave SC can be realized when the coherence length ξ 0 is much larger than the AF periodicity, so that the average value of magnetization is zero, but this situation is exceptional.

In strongly correlated electrons systems, the strong Coulomb repulsion undermines the s-wave pairing. In heavy fermion systems, high T c cuprates and Fe-pnictides, the superconductivity is unconventional: the symmetry of the SC gap is either p-wave or d-wave, and the total angular momentum is not zero.

There [START_REF] Sechovský | Chapter 1 Magnetism of ternary intermetallic compounds of Uranium[END_REF]) Schematic plot illustrating evolution of some electronic properties (type of magnetic ground state, magnetic ordering temperature, γvalue (linear coefficient of the specific heat), spontaneous magnetic moment (black part of a column), induced magnetic moment in field of 35 T (white top parts of a column) of UTSi and UTGe compounds with the orthorhombic TiNiSi-type structure or the closelyrelated CeCu 2 -type structure. PM/SF stands for paramagnetic with spin fluctuations. FM and SC): UGe 2 [START_REF] Saxena | Superconductivity on the border of itinerant-electron ferromagnetism in UGe 2[END_REF], URhGe [START_REF] Aoki | Coexistence of superconductivity and ferromagnetism in URhGe[END_REF], and UCoGe [START_REF] Huy | Superconductivity on the Border of Weak Itinerant Ferromagnetism in UCoGe[END_REF].

In UGe 2 , SC occurs under pressure, in the vicinity of the transition between two FM states (FM 2 /FM 1 ).

In UCoGe, it occurs a ambient pressure, but the maximum of T SC corresponds to the pressure where the Curie temperature T C reaches 0 K. That is, SC is observed both in FM and PM state.

In URhGe, T SC decreases and T Curie increases with pressure, indicating that pressure drives the system away from a quantum critical region. When the field is applied along the hard magnetization axis (H b) the FM fluctuations are strongly enhanced by suppressing T Curie under magnetic field. Consequently, SC reappears with spin triplet state, where there is no Pauli paramagnetic effect.

This mechanism was understood as follows. In the orthorhombic crystal structure of URhGe, the U magnetic moments order ferromagnetically along the c-direction -which is the easy magnetization axis -below T C = 9.5 K. The magnetic order is Ising-type, but with a rather pronounced susceptibility along the b-axis. When field is applied along the b-axis, at some point (H = 12 T) the magnetization reorients from the cto the b-direction. The triplet pairing mechanism is attributed to the critical magnetic fluctuations associated to this spin-reorientation process [START_REF] Lévy | Magnetic Field-Induced Superconductivity in the Ferromagnet URhGe[END_REF].

Chapter 2

Experimental techniques 2.1 Sample preparation

The compounds studied in this thesis (URhSi, URhAl, UCoAl) have been grown at our laboratory's crystal growth facility.

Crystal growth

The starting materials were prepared as follows:

• Uranium: (depleted -0.2% of 235 U) purity: 99.9% (3N). Solid bars with oxidized surface. The desired amount is cut with electro-erosion. The oxide is removed by electro-etching, in a bath of 95% acetic acid and 5% perchloric acid, with 12 V DC applied between U (anode, positive end) and a stainless steel plate (cathode, negative end).

• Rhodium: 99.99% (4N). Powder form. Very expensive (∼ $100 per gramm). It has to be melt in RF furnace first to obtain a pellet, otherwise the arc discharge would blow the powder. This process is performed under high vacuum (10 -6 Torr) which allows the material to degas.

• Cobalt: (3N) ∼ 0.1 g chunks. Etching at 85 • C (30% nitric acid, 10% sulfuric acid, 10% phosphoric acid and 50% acetic acid) and rincing with 50% nitric acid and 50% water.

• Silicon: (6N) industry-grade quality wafers.

• Aluminium: (5N) high-purity chunks. Aceton cleaning.

The total mass of material for a crystal growth is typically ∼ 6 g. The starting materials were prepared in stoichiometric proportions (all compounds studied here are congruent). We used two different furnaces to melt the materials: a radio-frequency furnace (under high vacuum), and a tetra-arc furnace (under Ar gas atmosphere).

In the case of RF furnace (Fig. 2.1), the material is contained in a copper crucible, and it is heated by induction from an external coil. The coil and the crucible are watercooled. The crucible consists of an assembly of winglets which are separated by a gap. This construction is designed to avoid Eddy currents and thus minimize the heating of the crucible. High vacuum is obtained with an ion pump. The melt pellet is selfsustaining and hence it is quite unstable. This is why the RF furnace was used only to melt the starting material and let it degas. The single crystal was pulled using the tetra-arc furnace.

In the tetra-arc furnace, four torches (tungstene, 2% La doped) are positioned at ∼ 5 mm from the pellet (see Fig. 2.2). Electrical arcs ionize the gas between the torches and the pellet, which has electrical continuity to the crucible. The crucible is made of copper, and it is water-cooled. The crucible can rotate for homogeneous heating.

The single crystal is obtained by immersing a cold tip (tungstene) into the melt, and then slowly pulling upwards. The material crystallizes on the tip, and forms a long-shape ingot. Typical pulling rate of 15 mm/h is kept constant, while the position of the torches and the arc intensity control the thickness of the ingot. Approximately 1 cm below the tip, the thickness of the ingot is locally reduced to 1 mm, and expanded again. This way, if there were several crystal grains forming at the beginning, this «necking» process singles out one orientation, thus favouring a single-crystal ingot building up afterwards. The ingot can be very fragile at the necking, and it sometimes break (as shown in Fig.

2.2).

Crystal orientation

The single crystal and orientation is checked with X-ray Laue photograph (Fig. 2.3). A sample is positioned in the X-ray beam (polychromatic), and the diffraction pattern is recorded in a back-scattering geometry. Historically, the diffraction pattern was recorded on argentic film, which necessitated long exposure time (∼ 30 min) and developpement process in a dark room. The film was recently1 replaced by a CCD camera, thus reducing the exposure time to ∼ 30 s.

A clean diffraction pattern is the fingerprint of a single-crystal phase. We use the software OrientExpress to index the diffraction peaks and find the orientation of the crystal. In Figure 2.3 we show a Laue photograph of a single crystal of UCoAl. We recognize the hexagonal pattern. The spot close to the center is the c direction. 2 The software then provides the possible combinations of rotations φ x , φ y , φ z to orientate the crystal in a particular direction.

The sample is set on a goniometer and cut with an electro-erosion machine, which is a wire saw with a current flowing between the wire and the sample. It is performed in a petrol bath, to avoid spreading radioactive dust.

Safety considerations related to Uranium

In our crystal growth facility, the activity on uranium is controlled. Special rooms are dedicated to the handling of Uranium materials. These areas are supervised by the CEA radioprotection team. Access is restricted. As a staff working with uranium material, we have to attend special safety training, and we receive regular medical attention.

Natural uranium contains 3 isotopes : 238 U (99.275%), 235 U (0.72%) and 234 U (0.0054%), which are all radioactive. They are alpha-emitters, and they produce a series of shortlived alpha and beta-emitters, in addition to the gamma emission accompanying every decay.

The uranium used for our research is depleted uranium (0.2% of 235 U), which is a by-product of 235 U enrichment processes for the nuclear industry. The total activity of depleted U (15 kBq/g) is reduced compared to the natural composition (25 kBq/g), mainly owing to the reduction of the 234 U abundance, which is responsible for half of the activity in the natural U [31].

Because of its short penetration length, the alpha emission is potentially hazardous on ingestion or inhalation, while the beta and gamma emissions are potentially hazardous on external exposure.

Given the small quantities of material that we use, the external exposure is not a concern. At a distance of a few cm from the source, the activity of a few grams of U vanishes in the ambient radioactivity level.

On the contrary, internal exposure (by ingestion or inhalation) is a matter of serious concerns. For this reason, special care is taken to avoid producing dust: cutting and polishing are performed in a petrol bath, methodical cleaning of the furnace and tools is performed at every step of the processes.

The samples going out from the facility (to perform experiments in another building or outside the CEA) require an official declaration, and are subject to the radioactive material transport regulations.

Hydrostatic pressure techniques

Piston-cylinder cell

The piston-cylinder cells are used for hydrostatic pressures up to ∼ 3 GPa (1 GPa = 10 kbar). The advantage of this type of cells is the rather large sample space. The model shown in Fig. 2 Pressure is determined by the superconducting transition temperature of lead. Two Cu wires (ø25 µm, ∼ 20 turns) are wound around a piece of lead (ø200 µm). The superconducting transition temperature is determined by AC susceptibility measurement.

The dimensions of the setup shown in the pictures (Fig. 2.4) are ø3 mm and 6 mm high. The preparation requires some dexterity and self control! A setup like this can only be used once, since once it has been used at high pressure, the plug is deformed and cannot be used again.

Pressure cell for magnetic measurements

A special pressure cell was used for magnetic measurements. It is made of a Cu-Ti alloy, which has a low magnetic response compared to other strong alloys such as NiCrAl or CuBe. This cell has been developped by Yusei Shimizu in 2009 at Hokkaido University. It is based on the models developped by Kobayashi et al. [START_REF] Kobayashi | Nonmagnetic indentertype high-pressure cell for magnetic measurements[END_REF] and Tateiwa et al. [START_REF] Tateiwa | Miniature ceramicanvil high-pressure cell for magnetic measurements in a commercial superconducting quantum interference device magnetometer[END_REF]. This is a cell of very small dimensions (ø8.8 mm) which is designed to fit in a com- It consists of a Cu-Ti tube and locking nuts, two zirconia anvils, and a hybrid CuBe-NiCrAl gasket (see Figs. 2.5 and 2.6). This construction is called Indenter-type: the plateau of the anvils is smaller than the diameter of the hole in the gasket.

We used Daphne oil 7373 as a pressure transmitting medium. Pressure was determined from the superconducting transition of Pb. The background -mainly due to the NiCrAl gasket -is punctual, and thus yields a simple dipolar signal which adds up to that of the sample. The magnetization of the sample is then obtained by simply substracting the magnetization of the cell, which is measured independently. The signal of the sample is of the same order of magnitude as the background. Figure 2.7 shows the magnetization of the empty cell (background), and with a sample of UCoAl inside. We can see the signal of the metamagnetic transition at 0.7 T (here, P = 0).

The pressure is determined by the detecting the superconducting transition of a piece of lead. The signal of the Pb is negligible compared to that of the sample, but just enough to detect an anomaly at T SC . To detect the transition, it is necessary to apply field, while at the same time this decreases T SC . Figure 2.8 shows the magnetization versus temperature at around 4 K, at various applied field close to 0. There is a clear anomaly associated with the SC transition of Pb. The anomaly grows with field (diamagnetic response of a superconductor), while T SC decreases. The maximum of T SC is inferred to be 7.18 K, which corresponds to zero pressure. The maximum occur at a finite value of applied field, which compensates the remanent field.

Diamond anvil cell

We use diamond anvil cells for high pressure studies (typically up to ∼ 10 GPa).

The main difficulty is the small sample space. This type of pressure cell has been used during this thesis for AC calorimetry measurements. The pressure chamber consists of a hole drilled in a stainless steel gasket, clampled between two diamond anvils (Fig. 2.9-b). Liquid argon is used as a pressure transmitting medium.

The anvils are octogonal truncated pyramids, with a 700 µm wide plateau. First, the gasket is clamped once between the anvils, in order to get a crater in which the thickness of the gasket is reduced ∼ 100 µm. A ø350 µm hole is drilled at the center, and edges gently smoothed.

In order to allow electrical connections to squeeze in-between the gasket and the anvil, an insulating coating is made of an alumina-Stycast paste plated on the gasket. This operation follows a precise recipe: the density of the paste, the amount of paste deposited in the crater, the squeezing force, the baking temperature and time -all these are key parameters to obtain a hard enough, ∼ 30 µm thick coating. The connections are realized with gold stripes obtained from ø13 µm gold wires that are flattened between two glass plates with a hydraulic press. The final thickness is of the order of ∼ 1 µm.

In such a setup, the initial dimensions of the pressure chamber are ø350 µm by 130 µm. The sample size is typically ø170 µm by 60 µm thick. It is obtained by polishing, or cleaving. In the setup for AC calorimetry measurements, the thermocouple was realized by spot-welding two wires (Au-AuFe) on the sample (Fig. 2.9-a). The wires are squeezed onto the incoming stripes on closing the cell (Fig. 2.9-c).

The cell is closed in liquid argon, which serves as a pressure transmitting medium. A few ruby grains are trapped together with the sample. The pressure is determined by the fluorescence peak of ruby (which is pressure-dependent).

Changing the pressure in-situ

The AC calorimetry measurements were performed in a 4 He bath cryostat specially adapted to change the pressure in situ [START_REF] Salce | Versatile device for low temperature in situ generation of forces up to 25 kN: Application to hydrostatic pressure experiments[END_REF]. It is a cryo-pneumatic system fed with He gas and coupled to a force-amplifying lever stage, producing a force of several kN onto the pressure cell. It allows to change the pressure at low temperature. Hence this system allows to save a lot af time and liquid Helium consuption because it is not necessary to warm up the experiment to change the pressure. This consequently allows for very fine mapping of a phase diagram under pressure.

Pressure was increased in the pressure cell by pressurizing the bellow with He gas while monitoring the ruby fluorescence peak, until the desired pressure was reached. The bellow was then partially pumped in order to mechanically decouple the pressure cell from the thermal expansion of the gas inside the bellow, the pressure in the cell being maintained by the force-multipling lever system.

AC calorimetry

AC calorimetry is a technique which allows heat capacity measurement under pressure, while the relaxation method would be impossible due to the coupling of the sample to its environment. It is not a quantitative measurement since it is impossible to separate the sample contribution from the addenda, and to evaluate the power applied to the sample. But it provides a semi-quantitative information, which means that it is possible to compare curves between one another, provided that they are obtained in the same experimental conditions. In particular, it allows to determine the high pressure phase diagrams by a thermodynamical bulk probe. Here the AC calorimetry technique was used to draw the temperature -pressure phase diagram of URhAl (Chapter 4).

Principle of AC calorimetry

The sample is inside a diamond anvil pressure cell (DAC), which is transparent to visible light. The sample is heated by a laser which is chopped at a frequency ω in the kHz range. On the other side of the sample, a thermocouple measures the temperature oscillations T AC . The amplitude R and the phase θ of the thermocouple signal are obtained by lockin detection synchronized with the chopper. The amplitude of the signal is linked to the temperature oscillations T AC through the thermoelectrical power of the thermocouple:

R = S th T AC (2.1)
The basic model for AC calorimetry have been proposed by Sullivan et al. [START_REF] Sullivan | Steady-State, ac-Temperature Calorimetry[END_REF], and it is presented in Fig. 2.10. It is a first-order system, where the heat capacity C (sample + thermocouple) absorbs the power P (t) = P 0 (1 + sin(ωt)), and relaxes to the bath

T AC = P 0 κ B +iωC (2.2)
In order to maximize the importance of C in comparison to κ B in the signal, we should work at high frequencies. The main difficulty of the method in the pressure cell is that the power P 0 absorbed by the sample and the thermal leak κ B are not known.

A more accurate model [START_REF] Derr | Valence and magnetic ordering in intermediate valence compounds: TmSe versus SmB 6[END_REF] takes into account the coupling κ s between the thermocouple and the sample, which respectively receive a fraction of the power aP 0 and (1 -a)P 0 . In this case, there is a cut-off frequency at ω = κs C above which the thermocouple starts to decouple from the sample. This typically limits the working frequency to the kHz range (in our experiment, we chose ω/2π = 1710 Hz). Under such conditions, we can consider that:

C ∝ P 0 S th R (2.3)
Therefore the signal is inversely proportional to the total heat capacity C. At a phase transition, the specific jump of the sample will be detected as a jump in 1/R. In the phase, the signature of a specific heat jump depends on the frequency, but in the region ω ≈ κs C it will appear as a positive jump in θ. In order to get semi-quantitative data, the power received by the sample must be the same all along the experiment. But the defocusing of the laser beam throughout the optical path inevitably varies when working the mechanical system to change the pressure. The ad hoc solution consists in setting the laser power at the beginning of every scan so that to yield a DC temperature rise of 0.3 K above the bath temperature.

Another source of uncertainty is the thermoelectrical power of the Au:AuFe thermocouple, which is not known under pressure. We suppose it has low pressure dependence in our study. The thermocouple response also varies from one thermocouple to another. Because our thermocouple had not been calibrated, we applied a model reponse (Fig. 2.11) which is a fit obtained out of several thermocouple calibrations. As we can see, the temperature dependence of the thermocouple is particularly strong below 10 K.

Strain gauge dilatometry

Strain gauges are used for dilatation measurements. A strain gauge is a flexible foil supporting a long thin conducting path forming a zig-zag pattern. The strain gauge is glued on the surface of a sample (Fig. 2.13) to follow its deformation (=strain) 3 . The strains along the conductive pathway induce variations of the resistance (R = ρl/σ).

The design of a strain gauge is such that it is unidirectional: it is sensitive to strains along the direction of the long thin parallel sections of conductor, while the short thick turns have a negligible contribution to the total resistance.

The response to a strain is:

∆R R = κ (2.4)
where R is the resistance of the gauge and κ is the gauge factor (generally κ ≈ 2). We used standard commercial millimeter-size strain gauges (Kyowa), made of Ni-Cr alloy. The gauge must be tightly bound to the material, using a special cement (Kyowa PC-6).

The strain gauge is set in a Wheatstone bridge (Fig. 2.12) along with a «dummy» gauge, which is placed in the same conditions, so that to compensate the drift with temperature, field, pressure. This is called active dummy method. The dummy gauge is sticken on a piece of silicon, which has very low thermal expansion below 40 K. The other half of the bridge (outside the cryostat) consists of two simple resistors R 0 (which are chosen for their temperature stability), and a variable resistor (actually a decade box of resistors, which is more stable) mounted in parallel with one of the two resistors, to balance the bridge. When the bridge is balanced (V ≈ 0), the signal is proportional to the variation of the straing gauge:

∆V V 0 = RR 0 (R + R 0 ) 2 ∆R R (2.5)
The resistors are normally chosen with value R 0 ≈ R (standard values are 120 Ω or 350 Ω), so that the bridge factor reduces to 1/4, and we have the simple relation for the strain = ∆l l :

∆l l ≈ 4 κ ∆V V 0 (2.6)
which is independent of the bridge resistances.

The measurements are performed with a lock-in detection at ∼ 17 Hz. Typical value of the bridge excitation voltage is V 0 = 0.1 V, so that the power dissipated by the strain gauge is < 0.1 mW. A dilatation of 10 -4 will thus produce a signal of 5 µV.

This method can be used down to T ∼ 5 K. At lower temperatures, the thermal drift of the strains is very large and the compensation cannot be achieved.

Chapter 3 URhSi

Introduction

URhSi attracted our attention because it is very similar to the ferromagnetic superconductor URhGe, with a FM ground state below T C ≈ 10 K and a spontaneous moment of ≈ 0.5 µ B along the c-axis (see e.g. [START_REF] Prokeš | Magnetic specific heat and magnetoresistance of URhSi[END_REF]). The main target of our study was to test whether URhSi can also be tuned to a ferromagnetic quantum phase transition by applying magnetic field perpendicular to the easy magnetization axis (c-axis), as it is the case in URhGe [START_REF] Aoki | Ferromagnetism and Superconductivity in Uranium Compounds[END_REF].

Context

URhSi was first reported in 1988 [START_REF] Troć | Magnetic properties of UT(Si,Ge) series[END_REF], with X-ray powder diffraction establishing the orthorhombic structure (first believed to be CeCu 2 -type, very close to TiNiSi-type) and magnetic measurements on polycrystalline samples establishing the FM ground state.

In the 90's, specific heat measurements showed a large jump at the FM transition at T C ≈ 10 K, and different γ values of ∼ 130 -160 mJ.mol -1 .K -2 are found in the literature [START_REF] Prokeš | Magnetic specific heat and magnetoresistance of URhSi[END_REF][START_REF] De Boer | Development of the ground state within the UTSi system[END_REF], with still a significant magnetic contribution to C/T at ∼ 1 K. In spite of the uncertainty on the electronic contribution to the specific heat, the authors could infer a rather small integrated value of the magnetic entropy S m ≈ 0.1 -0.2R ln 2, suggesting the itinerant nature of the magnetic moments. Furthermore, de Boer et al. [START_REF] De Boer | Development of the ground state within the UTSi system[END_REF] have reported an upturn of C/T on cooling below 0.6 K, most likely due to the nuclear contribution.

A large negative magnetoresistance of URhSi around T C was reported, both in a polycrystal [START_REF] Tran | Large magnetoresistance of the ferromagnetic intermetallics URhSi and URhGe[END_REF] and in a single crystal sample with field applied along c [START_REF] Prokeš | Magnetic specific heat and magnetoresistance of URhSi[END_REF]. This was attributed to the rapid suppression of magnon scattering.

There has been a controversy about the value of the spontaneous moment (0.11 µ B [START_REF] Prokeš | Magnetism in URhSi[END_REF], 0.27 µ B [START_REF] Tran | Large magnetoresistance of the ferromagnetic intermetallics URhSi and URhGe[END_REF]), which was partially settled in 2003 with magnetization and neutron diffraction experiments on single crystal samples [START_REF] Prokeš | Magnetic structure of URhSi single crystal[END_REF], who found 0.58 µ B along the c-axis. Even then, there remained an ambiguity concerning the magnetic anisotropy since these magnetization experiments found a sizeable spontaneous component of ∼ 0.1 µ B along a-and b-axes, suggesting a tilt of µ off the easy axis, while neutron data [START_REF] Prokeš | Magnetic structure of URhSi single crystal[END_REF][START_REF] Prokeš | Magnetization densities in URhSi studied by polarized neutron diffraction[END_REF] only found collinear moments. One should note that all these experiments were performed on the same sample.

In our laboratory, new single crystal M -H measurements were performed using a rotator, since the detection of a magnetic component in the hard a-b plane is very sensitive to the field angle. Figure 3.1 shows the magnetization curves at 2 K along the principal axes. The results seem to lift the controversy : there is no spontaneous moment along a, and although a small spontaneous moment (0.03 µ B ) still remains along b-axis, these results seem to be basically consistent with the results of neutron experiments. It indicates that URhSi has a uniaxial magnetic anisotropy, but unlike URhGe (Fig. 1.7) there is no intermediate axis. It should be noted that the first nearest U atoms form zig-zag chains along the b axis in URhSi, while the zig-zag chain in URhGe is along the a axis, which may explain the different susceptibility along b.

Experimental results

Sample preparation

Crystal growth :

Several crystal growths were performed in order to get the best sample quality as possible. Starting material was depleted U (99.9% -3N), Rh (3N) and Si (5N) in stoichiometric proportions. The components were melt in a tetra-arc furnace under Ar gas atmosphere, or in a radio-frequency furnace under ultra-high vacuum. Several ingots were pulled by Czochralski method with pulling rate of 15 mm/h. The single crystal was checked by X-ray Laue photograph, and the high-temperature phase with orthorhombic TiNiSi-type structure was obtained (there is also a low-temperature phase of URhSi [44]).

Resistivity

Fig. 3.2 shows the resistivity versus temperature measured from 300 K to 2 K on a barshape sample of URhSi, with current along a-axis. The residual resistivity ratio of the sample is ∼ 3, which is poor but no better samples were obtained by other groups (to our knowledge).

The resistivity is rather flat down to the Curie temperature T C = 10.5 K, below which it drops dramatically. At low temperature, the data can be fitted with a ρ = ρ 0 + AT 2 up to ≈ 7 K, then it deviates on approaching T C . The obtained coefficient is A = 2.0 µΩ.cm/K 2 . This contrasts with the results of Prokeš et al. [START_REF] Prokeš | Magnetic specific heat and magnetoresistance of URhSi[END_REF] who found a better fit with a T 5 3 law which is typical of spin-fluctuations close to FM instability. This T 2 temperature dependence of resistivity is typical of FM metals. If local magnetic moments are considered, as it is almost the case for T > T K , in a lattice of Heisenberg spins coupled with conduction electrons by exchange interaction J, perturbative development leads a magnetic contribution: ρ mag (T ) ∝ |J| 2 , which is temperatureindependent above T C . In the FM state, typical spin-wave dispersion relations yield

ρ mag (T ) = A mag T 2 [45].
Below T C , the pseudo-parabolic temperature dependence of ρ which changes brutally at T C therefore indicates that the magnon scattering contribution accounts for a large part of the A coefficient.

Specific heat

Heat capacity was measured in a PPMS (Quantum Design) by the relaxation method, between 2 K and 50 K, using ≈ 10 mg samples with a flat surface and Apiezon N grease for thermal contact. Heat capacity of URhSi is shown in a C/T vs T 2 plot (Fig. 3.3). The heat capacity shows a sharp peak at 10.5 K which is attributed to the PM-FM transition. At low temperature, C p /T vs T 2 is successfully fitted by a straight line between 2 and 5 K according to the law:

C p = γT + βT 3 (3.1)
with the value of the Sommerfeld coefficient γ ≈ 170 mJ • mol -1 • K -2 , and the coefficient β = 1.95 mJ • mol -1 • K -4 (which accounts for both lattice and magnetic contributions), in good agreement with the literature [START_REF] Prokeš | Magnetic specific heat and magnetoresistance of URhSi[END_REF]. Above T C , C p /T vs T 2 can be also fitted by equation 3.1 with slightly lower γ ≈ 150 mJ • mol -1 • K -2 , and

β = 0.46 mJ • mol -1 • K -4 .
With the γ value below T C and the A coefficient of resistivity determined before, the Kadowaki-Woods ratio A/γ 2 ≈ 70 µΩ.cm.(mol.K/J) 2 is almost one order of magnitude larger than for typical heavy fermion systems (10 µΩ.cm.(mol.K/J) 2 ). This is probably because of the large magnon scattering contribution dominating the A coefficient.

The change of the β coefficient is ascribed to the magnetic contribution below T C . Theoretical models predict a T 3/2 law for ferromagnetic magnons, logarithmic contribution for spin fluctuations in weak itinerant ferromagnets [START_REF] Ikeda | Quenching of spin fluctuations by high magnetic fields[END_REF], and exponential behaviour in the case of gapped magnons [START_REF] Hessel | Electron-magnon interaction and the electrical resistivity of Tb[END_REF], but none of these model could be singled out, owing to the lack of knowledge of the other contributions. The change of γ on either side of T C is not surprising since electronic correlations are probably affected by the onset of magnetic order.

Thermal expansion with fine tuning of field direction

The main motivation of our study was to test if URhSi could be driven to a ferromagnetic instability by applying a magnetic field along b, like the closely related material URhGe, since they have many similarities. We chose to track T C by measuring thermal expansion -which is a thermodynamical property -along the main crystallographic directions using strain gauges. The strain gauges were glued on the surfaces of a cubical single crystal sample of URhSi with dimensions 0.8 × 1 × 1.2 mm 3 . We measured the thermal expansion using a Wheatstone bridge with a «dummy gauge» glued on a piece of silicon in order to compensate for the temperature drift of the strain gauge.

Furthermore, it is important to have a precise alignement of the sample with respect to the field direction, since a tiny field component along the c-axis is known to kill the transverse spin fluctuations that are the driving force for FM instability in URhGe and UCoGe. In order to achieve the condition H ⊥ c, the sample was mounted on a 1-axis rotator so that to have in situ control of the field angle in the b -c plane.

Results

Thermal expansion of URhSi along the main crystallographic directions, at zero field, is shown in Fig. 3.4. A linear trend and an offset were substracted from the curves so that they match to zero above the Curie temperature. It is reasonable to do so because:

• the offset depends on the balance of the Wheatstone bridge

• the temperature drift of a strain gauge response being large in this temperature range, it is very unlikely to achieve perfect compensation with the dummy gauge.

The thermal expansion changes radically at T C . On entering the FM phase, URhSi expands along the c-and a-directions and contracts along the b-direction. As infered from Fig 3.4, the jumps of thermal expansion coefficients α are negative along a and c axes, and positive along the b axis. Thus the thermal expansion in volume will be negative, which is consistent with the fact that T C decreases with pressure [START_REF] Tatsuma | Magnetic Anisotropy and Field Induced Effective Mass Enhancement in URhSi[END_REF], because of Ehrenfest relation. At least for the b-and c-directions, the thermal expansion seems to scale with magnetization. The data is not relevant below ∼ 5 K, which is the lower limit for the strain gauge technique. Magnetic field was applied along the b-direction. Temperature scans were performed at various angles by moving the rotator step by step, repeating the procedure for each strain gauge and every field value. The effect of field angle is illustrated in Fig. 3.5, showing thermal expansion curves ∆L b /L b at 9 T for various field angles around the perpendicular configuration. The curve with sharpest transition corresponds to the situation where H ⊥ c. When the field angle deviates from the H ⊥ c situation, the transition broadens and T C increases. In this example, with a field of 9 T along the b-axis, a misorientation of 0.25 • gives a field projection along the c-axis H c = 0.04 T which is already too much to observe the decrease of T C . Such extreme field angle dependence was also observed in URhGe [START_REF] Lévy | Magnetic Field-Induced Superconductivity in the Ferromagnet URhGe[END_REF] and UCoGe [START_REF] Aoki | Extremely Large and Anisotropic Upper Critical Field and the Ferromagnetic Instability in UCoGe[END_REF], where a misalignement of less than 1 • kills the fluctuation-induced superconductivity. This fact stresses the necessity of having precise control over the sample orientation.

Figure 3.6 shows the temperature dependence of thermal expansion coefficients along the b and c axes, in the optimum angle configuration found by working the rotator to obtain the sharpest and lowest transition temperature. The most important result of these thermal expansion experiments is the fact that T C decreases with H b. From T C = 10.5 K at zero field, T C decreases to 9.5 K at 9 T, which was the maximum field available for this experiment. The evolution of T C versus H b is summarized in Fig.

discussion

The decrease of T C is related to enhanced magnetic fluctuations along b-direction. The evolution of T C up to 9 T is comparable to the case of URhGe, though the decrease is slower in URhSi. The decrease of T C is theoretically explained by V. P. Mineev [START_REF] Mineev | Magnetic field dependence of pairing interaction in ferromagnetic superconductors with triplet pairing[END_REF] on the basis of Ginzburg-Landau theory, when the field is applied perpendicular to the easy magnetization axis. According to this theory, the decrease of T C is described by:

∆T C ∝ -H 2 (3.2)
Our data can be fitted with this equation (see Fig. our data and the maximum of the derivative of ρ(H), namely H max ∂ρ/∂H . Reciprocally, the fit of our data using Eq. 3.2 yields an extrapolated value at 0 K : H(0) ≈ 43 T, in excellent agreement with the maximum of magnetoresistance. It would be interesting to follow the magnetoresistance anomaly using a rotator, to clarify whether T C connects to H max ρ or with H max ∂ρ/∂H . Nevertheless, it seems already clear that the high-field anomaly in magnetoresistance is connected to T C at low field. This is also the case in URhGe, but not in UCoGe. In the case of UCoGe, a similar anomaly in magnetoresistance occurs at ≈ 50 T at low temperature, and is connected at high temperature to a maximum in magnetic susceptibility rather than to T C : (T max χ ≈ 35 K) (T C ≈ 3 K) (see Fig. 3.9-b). On the contrary, there is no such χ max in the case of URhSi and URhGe.

Conclusions

We obtained single crystals of URhSi with which we performed resistivity, heat capacity, and thermal expansion measurements.

Our resistivity and heat capacity data confirm the existence of a FM transition at 10.5 K. A Fermi liquid behaviour is observed in below ∼ 6 K, with a large magnon contribution to the resistivity. Our specific heat data follows the law C/T = γ + βT 2 in the FM phase, with a γ value of ≈ 170 mJ • mol -1 • K -2 , which is higher than the one extrapolated from the PM phase (150 mJ • mol -1 • K -2 ), in agreement with the literature. This is ascribed to a significant magnetic contribution to the specific heat at 2 K.

The thermal expansion measurements The main result of our study on URhSi is the decrease of T C by applying a magnetic field along b, and the high sensitivity to the field orientation. We can therefore think of URhSi as a system driven towards FM instability by spin fluctuations, like URhGe and UCoGe: a field along b enhances the fluctuations, but a small component of H along c stabilizes the moments in the easy direction and kills the fluctuations.

Finally, we show that the high-field magnetoresistance anomaly measured by W. Knafo along the b axis is in the continuity of T C , thus suggesting that URhSi is tuned to a FM instability at H b ≈ 43 T. Further efforts should be devoted to improve the quality of the crystals of URhSi, since the RRR ∼ 3 at the present stage is too low to expect any superconductivity.

Chapter 4 URhAl

URhAl belongs to the UTX family (T = transition metal , X = p-element), and has the layered hexagonal ZrNiAl-type structure (Fig. 1.4), which is the most common structure in the UTX family. The inter-U distance d U-U = 3.63 Å is in the upper Hill limit region, and the ground state is FM below T C ≈ 27 K, with a moment M 0 ≈ 0.9 µ B per U-atom along the c-axis with a huge uniaxial anisotropy [START_REF] Veenhuizen | Magnetic properties of URuAl and URhAl single-crystals[END_REF].

Since Rh is located just below Co in the periodic table, and given the larger lattice constants compared to UCoAl, one can naively infer that URhAl corresponds to the negative pressure case of UCoAl. Therefore, the suppression of T C by applying pressure can be expected. Pressure studies of the lattice parameters of URhAl at room temperature [START_REF] Havela | U ternaries with ZrNiAl structurelattice properties[END_REF] have reported a linear compressibility κ a ≈ 2.8 × 10 -3 GPa -1 along the a-axis, and negligible along the c-axis. Reported pressure dependence of T C by AC susceptibility measurements on URhAl [START_REF] Cuong | Composition and volume dependence of magnetism in URu (1-x) Rh x Al and URuAl (1-y) Sn[END_REF] pointed to a quasi neglible pressure dependence of T C up to 6 kbar. Our study was aimed to investigate this compound under high pressure for the first time, to establish the phase diagram of URhAl, and to check the existence of a quantum phase transition at a reachable pressure.

Present state of knowledge on URhAl

Anisotropic hybridization

Magnetization density obtained with polarized neutron experiments [START_REF] Paixão | Evidence for Anisotropic Hybridization in Uranium Compounds[END_REF] have shown that a considerable portion (∼ 30%) of the total magnetization is induced on the Rh atoms that lie in the U-Rh plane, but not on the equidistant Rh atoms that lie in the Rh-Al plane. Furthermore, these polarized neutron results give an orbital-to-spin moment ratio µ L /µ S = -1.81 (later confirmed by XMCD studies [START_REF] Grange | 5f-hybridization in URhAl studied by X-ray magnetic circular dichroism[END_REF]) which is reduced compared the free ion value (∼ 3). These observations suggest a large degree of delocalization of the 5f electrons by hybridization with the Rh 4d electrons, which explains the large magnetic 57 anisotropy of URhAl.

Band structure calculations

Ab initio LSDA-based band structure calculations of URhAl were performed by several groups [START_REF] Gasche | Ground-state properties of ternary uranium compounds: I. Hybridization effects[END_REF][START_REF] Oppeneer | Optical and magneto-optical spectroscopy of uranium and plutonium compounds: recent theoretical progress[END_REF][START_REF] Kuneš | Magnetic, magneto-optical, and structural properties of URhAl from first-principles calculations[END_REF][START_REF] Kučera | X-ray magnetic circular dichroism studies of 5f magnetism in UCoAl and UPtAl[END_REF][START_REF] Antonov | Electronic structure and X-ray magnetic circular dichroism in uranium compounds. II. UT Al (T = Co, Rh, and Pt) intermetallics[END_REF]. These studies indicated that the U 5f -Rh 4d hybridization plays a dominant role in the bonding and in the magnetic properties, and successfully accounted for many properties such as lattice volume and compressibility, magnetocrystalline anisotropy, and could reproduce reasonably well the shape of the XMCD and Kerr spectra [START_REF] Grange | 5f-hybridization in URhAl studied by X-ray magnetic circular dichroism[END_REF][START_REF] Kučera | Magneto-optical investigations of URhAl intermetallic compound[END_REF]. The γ-value arising from band structure calculations amounts to ≈ 45 mJ.mol -1 .K -2 , which would mean a many-body effect enhancement of ∼ 1.5 in the experimental value (≈ 67 mJ.mol -1 .K -2 [START_REF] Javorský | Magnetic properties of diluted band ferromagnet URhAl[END_REF]). The main shortcoming of the LSDA-based band calculations in URhAl is the largely underestimated value of the total magnetic moment, even when introducing on-site repulsion (LSDA+U ) and orbital polarization corrections (LSDA+U (OP)).

Localized vs itinerant picture

As for many other U-based compounds, URhAl has the puzzling particularity of behaving like a localized or an itinerant magnetic system depending on which property we look at. Several spectroscopy studies were reported on URhAl that illustrate the localized/itinerant duality:

On the one hand, inelastic neutron scattering experiments showed a peak at 380 meV, interpreted as the signature of an inter-multiplet transition, thus promoting the localized picture. [START_REF] Hiess | Magnetic response function in URhAl[END_REF] On the other hand, the magneto-optical Kerr spectrum measured on URhAl was reported to compare quite well with the calculated spectrum based on the assumption of delocalized 5f electrons. [START_REF] Oppeneer | Optical and magneto-optical spectroscopy of uranium and plutonium compounds: recent theoretical progress[END_REF][START_REF] Kučera | Magneto-optical investigations of URhAl intermetallic compound[END_REF] In order to find a compromise, Oppeneer at al. speculated that the 5f electrons in URhAl might be divided into two groups, based on the strongly anisotropic 5f -ligand hybridization: the possible inter-multiplet transition concerns the rather localized 5f electrons out of the plane, whereas Kerr spectroscopy probes the rather delocalized 5f orbitals in the plane.

Sample characterization

The crystal growth of URhAl was very successful and the ingot was single crystal all through, as attested by Laue photographs, with the c-axis along the pulling direction. Various samples of single crystal URhAl were characterized by resistivity and heat capacity measurements. All samples showed the FM transition at T C ≈ 28 K. The residual resistivity ratio of the different crystals is around 12. Finally, URhAl obeys the Fermi liquid model over a wide temperature range with the T 2 law of resistivity being valid up to 6.2 K. Inelastic terms A J⊥c = 0.27 and A J c = 0.19 µΩ.cm.K -2 are probably largely due to the magnon contribution, as in URhSi (see Chapter 3).

Resistivity

Heat capacity

The heat capacity of URhAl (Fig. 4.3) shows a peak at T C = 27.5 K, in good agreement with resistivity (Fig. 4.1) and magnetization (Fig. 4.6). The T -linear term of C(T ) amounts to γ = 76 mJ.mol -1 .K -2 , which is that of a moderate heavy fermion system. In URhAl, there is no broad hump like in UGe 2 associated to the crossover between FM 1 /FM 2 phases at T ≈ T C 2 , neither in C/T vs T nor in ρ vs T . This suggests a single FM phase for URhAl. The Kadowaki-Woods ratio A/γ 2 = 33 and 47 µΩ.cm.(mol.K/J) 2 for the current and ⊥ to the c-axis, respectively. This is several times larger than the value of 10 µΩ.cm.(mol.K/J) 2 found for «heavier» systems UPt 3 , CeCu 2 Si 2 , CeCu 6 etc. It is probably because the A coefficient of resistivity is largely dominated by the magnon scattering contribution.

Thermal expansion

Thermal expansion measurements were performed on URhAl along the c-and a-axes, using strain gauges (see section 3.3.4 for experimental details). Results are presented in Figure 4.4.

The thermal expansion curves ∆L L vs T were normalized to 0 just above T C (Fig. 4.4-a). On entering the FM phase, there is a dilatation along c which amounts to 10 -4 at 10 K and a contraction perpendicular to c which amounts to -0.5 × 10 -4 . The volume expansion ∆V V = ∆Lc Lc + 2 ∆La La is close to zero, suggesting a flat initial pressure dependence of T C via Ehrenfest relation, in good agreement with the reported value (∂T C /∂P = -0.03 K.GPa -1 ) [START_REF] Cuong | Composition and volume dependence of magnetism in URu (1-x) Rh x Al and URuAl (1-y) Sn[END_REF]. The temperature derivatives of the ∆L L curves, i.e. the thermal expansion coefficients α c and α a , are plotted in T C from the FM phase, there is a little enhancement of α c above 22 K. At the transition, the change of α is very sharp.

Magnetic properties

Magnetic measurements on URhAl were performed using a commercial SQUID magnetometer (MPMS, Quantum Design) down to 2 K and up to 5.5 T.

The Ising-type nature of URhAl is shown in Fig. 4.5 which shows the field dependence of the magnetization at T = 2 K. The saturation magnetization of 0.9 µ B along the caxis (with a hysteresis loop when sweeping between positive and negative field) contrasts with the low, linear magnetic response when H ⊥ c.

The PM-FM transition appears around 28 K on the field-cooling magnetization curves (Fig. 4.6). At low field (H << 0.1 T), there is a sharp increase of M (T ) just below T C saturating at a value which is far below M 0 = 0.9 µ B probably because of the partial cancellation of FM domains. There is an anomaly around 25 K suggesting a two-step transition, probably due to domain walls blocking on crystalline defects. At 0.1 T, the sample seems single domain since the magnetization saturates at M 0 . At higher field (H > 0.1 T), the transition becomes smooth and there is only little increase of M at low temperature.

µ 0 H (T) H || c H ⊥ c T = 2K -1 0 1 -0.2 -0.1 0 0.1 0.2

Critical behaviour

We performed Arrott-Noakes analysis of the PM-FM transition of URhAl by using fieldsweep magnetization measurements around T C . This analysis is based on the equation of state first proposed in 1956 by K. P. Belov and A. N. Goryaga [START_REF] Belov | [END_REF] and soon after by A. Arrott [65]. They started with the Ginzburg-Landau expansion of the free energy:

F (M ) = -HM + a (T -T C ) M 2 + bM 4 + ... (4.1)
and they derived the equation of state in the vicinity of T C :

M 2 = - a b (T -T C ) T C + 1 b H M (4.2)
According to equation 4.2, the so-called Arrott plot consists of M -H isotherms around T C plotted as M 2 vs H M which should form straight and parallel lines. This technique is used to determine T C by the isotherm crossing the origin. However, this description fails in the case of weak itinerant ferromagnets such as pure Ni metal. A. Arrott and J. E. Noakes [START_REF] Arrott | Approximate Equation of State For Nickel Near its Critical Temperature[END_REF] generalized Eq. 4.2 into:

H M 1/γ = T -T C T 1 + M M 1 1/β (4.3)
where M 1 and T 1 are constants dependent on the material, and β,γ are critical exponents. According to equation 4.

M S (T ) = M 0 1 - T T C β for T < T C (4.4a) χ -1 0 (T ) = h 0 M 0 T T C -1 γ for T > T C (4.4b)
In the vicinity of a second-order phase transition, due to the divergence of the correlation length, M S (eq. 4.4a) and χ -1 0 (eq. 4.4b) follow universal scaling laws, and the critical exponents β and γ only depend on the spin and dimensionality of the system (see Table 4.1).

In the modified Arrott plot (Fig. 4.7), the critical exponents under which the M -H curves are straight lines are β = 0.5 and γ = 1.3. Then, from the Y and X intercepts of the modified Arrott plot, we can plot the saturation magnetization M S and the inverse initial susceptibility χ -1 0 as a function of temperature (Fig. 4.8-a). The Curie temperature can be identified with great precision by extrapolating both M S and χ -1 0 to 0, and we obtain T C = 27.9 K The temperature dependence of M S and χ -1 0 were fitted by equations 4.4a and 4.4b in a log-log scale (see Fig. 4.8-b and 4.8-c). The fit to universal scaling laws is successful for M S only in the vicinity of T C , with β = 0.47 (in good agreement with the value obtained by the modified Arrott plot), and an extrapolated value of M 0 = 1.5 µ B , which is far beyond the spontaneous magnetization at T → 0 (0.9 µ B ). This is not surprising since the relations 4.4a and 4.4b are valid only in the critical region. For χ -1 0 , the fit gives γ = 1.03, which is quite different from the Arrot plot value (1.30).

(H/M) 1/γ γ=1.3 23 K 2 5 K 2 6 K 3 0 K 3 1 K b) 0.0 M(µB) H(T) 0.2 0.4 0.6 0.8 0.6 0.4 0.2 0 1 23 K 25 K 26 K 31 K a)

Discussion

The Arrott-Noakes equation of state fails to describe correctly our magnetization data in the vicinity of the PM-FM transition of URhAl:

• the modified Arrott plot shows straight lines which are not parallel, indicating that the parameters T 1 ,M 1 vary accross the transition

• different values of the critical exponent γ are obtained according to the modified Arrott plot (1.3) or the inverse initial susceptibility χ -1 0 (1.03)

• the critical exponent β is found to be ≈ 0.5 in the vicinity of T C , which is that of a mean field system (see Table 4.1), although URhAl is obviously a 3D Ising system.

These observations suggest that the PM-FM transition of URhAl is complicated by the variation of some microscopic parameters, possibly the exchange coupling and/or the quenching of the U orbital moments due to enhanced correlation effects on cooling.

Probing the magnetic order with neutron scattering

URhAl belongs to the large family of UTX compounds cristallizing in the hexagonal ZrNiAl-type structure (space group p 62m). Since the U atoms form a quasi-Kagome lattice, it is naturally expected to have frustrated magnetic moments. As explained in section 4.1, all UTX compounds with the same crystallographic structure as URhAl have FM planes which are ordered ferro-or antiferromagnetically along c, appart from UNiAl which has a low q in-plane modulation in addition to an AF order along c ( q = (0.1, 0.1, 0.5)). In this context, we decided to perform a neutron diffraction experiment on a single crystal of URhAl, to look for possible in-plane or out-of-plane modulations of the magnetization. The neutron diffraction experiment was performed with E. Ressouche at ILL, Grenoble, with the 2-axis diffractometer D23. The sample was a 12 mm 3 single-crystal with roughly cylindrical shape. The structure was refined at 35 K and 2 K using 728 and 594 non-equivalent reflections, respectively. Several q-scans were performed above and below T C (at 35 K and 2 K), so that the reciprocal space was scanned along a dozen of directions (in-plane and out-of-plane), which are listed in Table 4.2. As an example, Figure 4.9 shows the scans (q h ,0,0), 0.3 < q h < 3.3, in the PM and FM phases, along with the substrated spectrum. There is only minor shouldering of the peaks, thus indicating the quasi perfect single crystal sample. Diffraction peaks only appear at integer values of q h , corresponding to lattice reflections. The small temperature-independent peaks around q h = 2.3 and 2.8 are ascribed to the copper sample holder. For all q-scans, the systematic substraction of the PM scans from the FM scans yielded no additional reflections. .9: q-scans of neutron scattering on URhAl along the (q h ,0,0)-direction below (2 K, blue curve) and above (35 K, red curve) the Curie temperature. The difference (green curve) between the two scans is plotted on an expanded scale. q h is expressed in inverse relative lattice units.

As a conclusion, we performed various q-scans spanning the main reciprocal space directions, and found no superlattice reflections. We thus confirm that URhAl has a pure FM ground state.

AC calorimetry under pressure

AC calorimetry measurements have been carried out under pressure, using a Diamond Anvil Cell (DAC), in order to obtain the temperature-pressure phase diagram of URhAl.

Experimental setup

The measurements were carried out using a system that allows to vary the pressure insitu. The details of the apparatus, of the DAC, and the principle of AC calorimetry are presented in Chapter 2.

The sample size was 150 µm diameter by 60 µm thick, and the pressure chamber initial dimensions were 350 µm diameter and 150 µm thick. The thermocouple Au-AuFe(0.07%) was spot-welded onto the sample, using 17 µm-thick Au wire and AuFe foil. The pressure cell was closed with liquid argon as a pressure transmitting medium, setting an initial scan q h q k q l min max min max min max a 0.

-0.5 -0.5 i 0 0 0 0 -0.2 -3.2 j 1 1 0 0 0 -3.2 k 1 1 1 1 0 -3.2
Table 4.2: List of q-scans performed at ILL-D23 on URhAl at 35 K and 2 K, in search for non-FM magnetic structure. Values of q h,k,l are expressed in inverse relative lattice units (r.l.u. -1 ).

pressure of 3 kbar at 300 K. Once cooled to liquid Helium temperature, the residual pressure was 0 kbar. The experiment was performed by increasing pressure step by step, from 0 to 4.8 GPa in a first run, and from 0 to 5.8 GPa in a second run. Between the two runs, the force has been released and the experimental setup has been warmed up to room temperature, but the pressure cell kept the pressure transmitting medium (Ar) inside. These two runs assessed the reproducibility of our measurements.

The AC calorimetry data was obtained by performing temperature scans, starting from the base temperature of the 4 He bath (4.2 K) and increasing the current through a heater attached to the body of the pressure cell. Sometimes the base temperature was decreased by pumping on the bath to start the scan from 2 K.

The same working frequency of 1710 Hz was kept during the entire experiment. The data were smoothed by averaging within 0.1 K windows. The amplitude of the signal is usually presented as S th RT , which we will now refer to as «C/T », keeping in mind that this is only approximatively proportional to the total heat capacity (sample + addenda). The C/T vs T curves were scaled manually to match one another at high temperature. Such corrections (±10 %) compensate the variations of the heating power between two scans. 3). Then T C decreases gradually with pressure. At P = 0, the transition appears as a downward step in C/T vs T , and an upward step in θ vs T . There is a broad hump in C/T vs T centered around 15 K, which is not present in the absolute specific heat measured outside the pressure cell by relaxation method (see Fig. 4.3). This shows that the hump is an artifact. As pressure increases (0 < P < 1.7 GPa), this hump in C/T disappears progressively, and a peak starts to develop just before the step (visible in both C/T and θ). At 1.7 GPa, the peak is sharpest (more spectacular in C/T ), and then gradually decreases with pressure. The broad hump vanishes completely around 2.5 GPa.

Results

AC-calorimetry results for URhAl are presented in

The high pressure curves are better seen in Fig. 4.11, which offers a closer view to

Low temperature behaviour

On cooling below 4.2 K, θ shows a downturn below T max θ ≈ 4 K. This is considered to be an artifact caused by the rapid decrease of the thermocouple response at low temperature. The shoulder-like anomaly merges with the downturn of θ, in such a way that the maximum of θ becomes a broad plateau [4 -7 K] at 5.1 GPa. Above this pressure, no signature of a transition can be seen in θ, but the maximum is shifted to

T max θ ≈ 7 K.

Discussion

At low pressure, the broad hump in C/T vs T around 10 -20 K) indicates that the data below 2.5 GPa have a problem, since there is no such anomaly in the specific heat measured at ambient pressure with the relaxation method (Fig. 4.3). Therefore, we consider the sharp peak in C/T at 1.7 GPa to be an artifact. Above ∼ 2.5 GPa, the C/T vs T curves recover a normal shape.

An anomaly associated to the Curie temperature can be tracked under pressure up to 5.1 GPa in both C/T and in the phase θ. There is an ambiguity at 5.2 GPa (anomaly in C/T only), and no anomaly above. This allows to draw the T -P phase diagram of URhAl (Fig. 4. 13-a). By extrapolating to 0 K where T C collapses, a quantum phase transition is expected at P C = 5.3 ± 0.1 GPa.

We may infer that the ground state of URhAl just above P C is similar to the case of UCoAl at zero pressure. According to the reported lattice parameters measurements under pressure [START_REF] Havela | U ternaries with ZrNiAl structurelattice properties[END_REF], in URhAl the lattice parameter a = 696.5 pm at ambient pressure reduces to ≈ 688 pm at P C , which is 1/3 the way to UCoAl (a = 668.5 pm), while c = 402 pm remains unchanged (in UCoAl: c = 396.5 pm). These lattice considerations are consistent with the picture that the closer in-plane packing increases the 5f -ligand hybridization and wipes out the magnetic moments.

Around P C , there is an increase of C/T at low temperature (see Fig. 4.12 and 4.13-b), and also a shift of T max θ , which are both too sharp to be merely ascribed to a change in the sample environment. Therefore, it is probably due to an increase of the sample heat capacity. This suggests an increase of the Sommerfeld coefficient γ, as it is normally expected in the vicinity of a quantum phase transition.

A tricritical point (TCP) is expected close to P C , where ∂T C /∂P starts to diverge, as for example in the case of UGe 2 : P C -P TCP = 0.8 GPa. But in the case of UCoGe, the FM-PM transition seems already weakly first order at ambient pressure, as suggested by a discontinuous splitting of the 59 Co NMR spectrum at T C [START_REF] Hattori | 59 Co-NMR/Nuclear-Quadrupole-Resonance (NQR) Studies on Ferromagnetic Superconductor UCoGe[END_REF]. A way to identify the TCP would be the observation of metamagnetic «wings» upon application of a magnetic field. Next target will be to probe the possible «wings» and to study the properties of URhAl below 1 K around P C using a resistivity setup in a dilution refrigerator.

Conclusion

We obtained high quality single crystals of URhAl, which have been characterized by resistivity, heat capacity, thermal expansion, and magnetization measurements. The FM order occurs below ∼ 28 K, to an Ising-type ordered state with 0.9 µ B along the c axis. The Sommerfeld coefficient γ = 76 mJ.mol -1 .K -2 is slightly larger than the reported value (67 mJ.mol -1 .K -2 [START_REF] Javorský | Magnetic properties of diluted band ferromagnet URhAl[END_REF]). The resistivity shows a T 2 dependence up to ∼ 6 K, and is largely dominated by magnon scattering contribution.

We performed magnetization measurements along the c-axis around the Curie temperature. The modified Arrott plot analysis fails to describe the critical behaviour correctly, probably because T C is comparable to the onset of the Kondo lattice so the size of the magnetic moment and the degree of delocalization are dynamically changing with temperature.

We checked the magnetic order with neutron diffraction along various reciprocal space directions. No superlattice reflections have been observed at 2 K, thus confirming the simple FM order in URhAl.

Finally, we established the phase diagram of URhAl under pressure by means of AC calorimetry measurements. The Curie temperature decreases and reaches 0 K around 5.3 GPa. At the critical pressure, we observed an increase of C/T which may be related to an effective mass enhancement associated with the quantum phase transition.

Chapter 5 UCoAl

UCoAl belongs to the UTX family of compounds with hexagonal ZrNiAl-type structure, as URhAl (which is presented in Chapter 4). Its ground state is paramagnetic (PM), but it has the very peculiar feature that a ferromagnetic (FM) state is induced when magnetic field is applied along the c-axis. This is a first-order PM-FM transition called metamagnetic transition.

In this chapter, we first review the characteristics of UCoAl and replace it in the general context of FM quantum criticality. Then we present our studies on this compound: pressure studies of Hall effect and resistivity, magnetization, and X-ray magnetic circular dichroism (XMCD).

Introduction

Motivation

Ferromagnetic quantum criticality

The metamagnetism in strongly correlated electron systems with Ising-type ferromagnetic (FM) behaviour is intensively studied because it produces a variety of unconventional effects. In some itinerant ferromagnets, such as UGe 2 [START_REF] Taufour | Tricritical Point and Wing Structure in the Itinerant FerromagnetUGe(2)[END_REF][START_REF] Kotegawa | Evolution toward Quantum Critical End Point in UGe(2)[END_REF] or ZrZn 2 [START_REF] Uhlarz | Quantum phase transitions in the itinerant ferromagnet ZrZn 2[END_REF], one can drive the Curie temperature T C to 0 K by tuning an external control parameter like pressure, and the ground state is found to be paramagnetic (PM) above the quantum phase transition. In theory, it has been suggested -and in some cases experimentally shown -that the second order ferromagnetic transition changes to first order at a tricritical point [START_REF] Dietrich Belitz | Tricritical behavior in itinerant quantum ferromagnets[END_REF]. By applying a magnetic field above the critical point in the paramagnetic phase, such systems eventually recover their ferromagnetic state by undergoing a first-order metamagnetic transition at H m , drawing a wing-shape first order transition plane in the temperature (T ) -pressure (P ) -field (H) phase diagram. The first-order transition terminates at high pressure and high field at T = 0 at the so-called quantum critical endpoint (QCEP). In this critical region, only few experiments were carried out because of the severe experimental conditions of low temperature, high pressure, and high field [START_REF] Uhlarz | Quantum phase transitions in the itinerant ferromagnet ZrZn 2[END_REF][START_REF] Taufour | Tricritical Point and Wing Structure in the Itinerant FerromagnetUGe(2)[END_REF][START_REF] Kotegawa | Evolution toward Quantum Critical End Point in UGe(2)[END_REF][START_REF] Wu | Quantum critical metamagnetism of Sr 3 Ru 2 O 7 under hydrostatic pressure[END_REF].

Due to the recent focus on quantum criticality, the metamagnetism in itinerant ferromagnets has been revisited theoretically (see e.g. Ref. [START_REF] Zacharias | Quantum criticality in itinerant metamagnets[END_REF][START_REF] Bercx | Metamagnetism and Lifshitz transitions in models for heavy fermions[END_REF]. The main debate is whether a Lifshitz-like transition is associated with the occurrence of metamagnetism, as discussed in the case of CeRu 2 Si 2 [START_REF] Daou | Continuous Evolution of the Fermi Surface of CeRu 2 Si 2 across the Metamagnetic Transition[END_REF].

UCoAl : a good candidate for FM criticality studies

A good candidate to investigate itinerant metamagnetism is the heavy fermion compound UCoAl, with ZrNiAl-type hexagonal structure, space group P 62m. At ambient pressure, its ground state is paramagnetic, with strong uniaxial magnetic anisotropy. By applying magnetic field along the easy magnetization axis (c-axis), a sharp first-order metamagnetic transition occurs at low temperature at a critical field H m ∼ 0.7 T [START_REF] Andreev | Metamagnetism of UAlCo[END_REF][START_REF] Mushnikov | Magnetic properties of the 5f itinerant electron metamagnet UCoAl under high pressure[END_REF]. The first-order nature of the metamagnetic transition terminates at a critical end point (CEP) at T 0 ∼ 12 K, and changes into a crossover at higher temperature [START_REF] Aoki | Ferromagnetic Quantum Critical Endpoint in UCoAl[END_REF][START_REF] Karube | Universality and critical behavior at the critical endpoint in the itinerant-electron metamagnet UCoAl[END_REF][START_REF] Palacio-Morales | Metamagnetic Transition in UCoAl Probed by Thermoelectric Measurements[END_REF].

By applying pressure, resistivity and magnetostriction experiments [START_REF] Aoki | Ferromagnetic Quantum Critical Endpoint in UCoAl[END_REF][START_REF] Honda | Electrical resistance and magnetoresistance of UCoAl under high pressure[END_REF] have shown that the critical field increases up to the QCEP, located at P QCEP ∼ 1.6 GPa and H QCEP ∼ 7 T, where an acute enhancement of the effective mass m of the quasiparticles has been detected. Previous magnetization experiments under pressure are in good agreement with an initial pressure increase of H m [START_REF] Mushnikov | Magnetic properties of the 5f itinerant electron metamagnet UCoAl under high pressure[END_REF]. It is worthwhile to remark that just above P QCEP in the paramagnetic ground state in high magnetic field, sharp pseudometamagnetism at H m will replace the sharp first order metamagnetism below P QCEP [START_REF] Zacharias | Quantum criticality in itinerant metamagnets[END_REF]. Further increasing pressure must lead to a broadening of the pseudometamagnetic transition.

State of the art on UCoAl

Magnetic anisotropy

The metamagnetic behaviour of UCoAl was first reported by Andreev et al. in 1985 [74].

It is a highly anisotropic system, with Ising-like behaviour: At low temperature, along the c-axis, there is a high initial susceptibility. A first-order transition occurs at 0.7 T to the FM state with 0.3 µ B , and the magnetization keeps on increasing in the ordered state, with no sign of saturation up to the highest fields (0.6 µ B at 40 T). On the contrary, perpendicular to c, there is a low paramagnetic response, with only 0.1 µ B at 40 T. The magnetic anisotropy is already visible at high temperature (∼ 300 K) with a much stronger magnetic susceptibility along the c-axis. The temperature dependence of the susceptibility exhibits a Curie-Weiss law above 40 K with µ eff ≈ 1.8 µ B -much larger than the ordered moment of 0.3 µ B in the FM state. This indicates the itinerant nature of magnetism in UCoAl.

T max χ

The susceptibility along c shows a broad maximum at ∼ 20 K [START_REF] Aoki | Ferromagnetic Quantum Critical Endpoint in UCoAl[END_REF][START_REF] Havela | 5f -band metamagnetism in UCoAl[END_REF]. This χ max was observed in other itinerant metamagnets : the Co-based pseudo binaries Co(S 1-x Se x ) 2 , YCo 2 , LuCo 2 , etc. [START_REF] Goto | Magnetic properties of the itinerant metamagnetic system Co(S 1-x Se x ) 2 under high magnetic fields and high pressure[END_REF], which are isotropic systems. The singularity of UCoAl, beside the Ising-like anisotropy, is the much lower energy scale (T max χ , T CEP ,H m ), of one or two orders of magnitude compared to the former materials.

Microscopic point of view

Polarized neutron diffraction experiments [START_REF] Wulff | Study of orbital and spin magnetism in UCoAl by polarized neutron diffraction[END_REF][START_REF] Javorský | Magnetization densities in UCoAl studied by polarized neutron diffraction[END_REF] mapped the magnetization density and found that the magnetic moment is exclusively present on the U sites, and that the orbital moment is twice as large and antiparallel to the spin moment. This has been also inferred from band structure calculations and X-ray Magnetic Circular Dichroism experiments [START_REF] Kučera | X-ray magnetic circular dichroism studies of 5f magnetism in UCoAl and UPtAl[END_REF][START_REF] Antonov | Electronic structure and X-ray magnetic circular dichroism in uranium compounds. II. UT Al (T = Co, Rh, and Pt) intermetallics[END_REF].

A recent 59 Co NMR study [START_REF] Nohara | Strong Longitudinal Magnetic Fluctuations Near Critical End Point in UCoAl: A (59)Co-NMR Study[END_REF] has confirmed that the magnetic moment on the Cobalt must be very low (µ Co /µ U <∼ 0.01). They found a linearly M -dependent, positive Knight shift on the Co(1) site (on the same hexagonal plane as U). Because of the strong Co hyperfine coupling constant (A Co ≈ -10 µ B /T), any sizable moment on the Co would have given a contribution to the Knight shift, which would have been non-M -linear since U-5f -Co-d hybridization must be different in the PM and FM states due to lattice change.

UCoAl in the framework of UTX compounds

Systematic studies of UTX compounds have established that UCoAl is a borderline system between non-magnetic and magnetically ordered materials. As a rule, the 5f -ligand hybridization is regarded as the tuning parameter in this family of compounds. Several studies against doping UCo 1-x T x Al showed a high sensitivity of the metamagnetism to small substitutions. But the effect of doping in UCoAl can be quite misleading and counter-intuitive. A typical example is the antagonist effect of Fe/Ni substitution for Co, which are the left-and right-hand side direct neighbours of Co in the periodic table: Fe doping (d-electron depletion) stabilizes a FM ground state with as little as 2% Fe substitution, while Ni doping (d-electron filling) causes the metamagnetic field to increase [START_REF] Andreev | Interplay between doping and pressure effects in magnetism of UCo (1-x) T x Al (T=Fe,Ni) single crystals[END_REF]. As it happens, this is exactly the opposite of what we would naively expect, that the increasing d-filling normally leads to magnetic states (UFeAl is non-magnetic while UNiAl is antiferromagnetic). In UCoAl, the ground state is also affected by the lattice strain induced by substitutions, and thus there is no simple rule.

Magnetoelastic properties

There is a large magnetostriction effect of the order of 10 -4 , which is negative along the c-axis and positive in the perpendicular direction [START_REF] Andreev | Metamagnetism of UAlCo[END_REF], and the total volume expansion is positive. The corollary of this large magnetostriction is the high sensitivity to external pressure: a uniaxial stress of ∼ 0.3 kbar along c tunes the system to a FM ground state with T C ∼ 15 K [START_REF] Ishii | Ferromagnetism induced in UCoAl under uniaxial pressure[END_REF], while hydrostatic pressure increases H m and decreases the magnetization in the polarized state [START_REF] Mushnikov | Magnetic properties of the 5f itinerant electron metamagnet UCoAl under high pressure[END_REF][START_REF] A V Andreev | Effects of external pressure on the 5f-band metamagnetism in UCoAl[END_REF].

Non-Fermi liquid behaviour

At rest, UCoAl is close to a FM instability, and the PM ground state is characterized by strong spin fluctuations and non-Fermi liquid behaviour. Electrical resistivity exhibits T 5/3 or T 3/2 dependence in the PM state, as predicted by the renormalization group theory for 3D Ising FM (T 5/3 ) or AF (T 3/2 ) spin fluctuations [START_REF] Ueda | Contribution of Spin Fluctuations to the Electrical and Thermal Resistivities of Weakly and Nearly Ferromagnetic Metals[END_REF]. UCoAl recovers a usual Fermi liquid T 2 dependence in the FM state [START_REF] Havela | Stability of the non-Fermi liquid state in UCoAl[END_REF][START_REF] Bograd | Non-Fermi liquid behavior in UCoAl[END_REF]. The NFL behaviour in the PM state was also attested with thermoelectric measurements by a diverging S/T at the lowest temperature [START_REF] Palacio-Morales | Metamagnetic Transition in UCoAl Probed by Thermoelectric Measurements[END_REF]. 59 Co and 27 Al NMR studies have further confirmed the 3D spin fluctuations, with longitudinal propagation direction. Furthermore, these NMR studies have evidenced the enhancement of spin fluctuations in the vicinity of the CEP, with critical exponents corresponding to the 3D Ising class of criticality (isomorphic to the gas-liquid transition of water) [START_REF] Nohara | Strong Longitudinal Magnetic Fluctuations Near Critical End Point in UCoAl: A (59)Co-NMR Study[END_REF][START_REF] Karube | Universality and critical behavior at the critical endpoint in the itinerant-electron metamagnet UCoAl[END_REF].

The extension of the NFL behaviour under pressure is unclear.

In 2002, Honda et al. [START_REF] Honda | Electrical resistance and magnetoresistance of UCoAl under high pressure[END_REF] performed the first pressure study of resistivity on UCoAl down to ≈ 4 K, and they concluded to the suppression of NFL behaviour under pressure. This points out to spin fluctuations being responsible for the NFL state in the proximity of the FM instability.

But in 2006, Havela et al. [START_REF] Havela | Non-Fermi liquid behaviour in UCoAl: Pressure variations[END_REF] measured the resistivity in the range [1-18 K] and concluded to a very extended NFL region up to ∼ 5 GPa. This is similar to the case of the helimagnet MnSi above the critical pressure, where NFL behaviour (ρ ∼ T 3/2 ) is observed in a very extended pressure range [START_REF] Doiron-Leyraud | Fermi-liquid breakdown in the paramagnetic phase of a pure metal[END_REF]. Such a robust NFL state in UCoAl may be related to frustrated AF correlations between the U atoms arranged on a quasi-kagome lattice.

Hall effect

The content of this section has been published in September 2013, in the Journal of the Physical Society of Japan [START_REF] Combier | Ferromagnetic Quantum Criticality Studied by Hall Effect Measurements in UCoAl[END_REF]. In this section, we sometimes have to distinguish between the applied magnetic field H and the magnetic induction B. For convenience, we decided to generically use the notation B throughout the entire section on Hall effect, except when it is explicitly stated otherwise.

An interesting tool to unveil the physical properties in the vicinity of a QCEP is the Hall effect as it has two main contributions: the normal Hall effect (NHE), which is linked to the type of carriers and their number, and the anomalous Hall effect (AHE), which is linked to the magnetization M . Changes in the effective mass (m * ) associated (or not) to a Fermi surface instability at the QCEP may have a specific signature.

In this section we present detailed Hall effect measurements performed on a single crystal of UCoAl, at low temperatures down to 150 mK, under hydrostatic pressure up to 2.2 GPa and magnetic field up to 16 T.

Experimental

Single crystals of UCoAl were prepared from depleted U (99.9% -3N), Co (3N) and Al (5N) in stoichiometric proportions. The components were melt in a tetra-arc furnace under Ar gas atmosphere and a single crystal ingot was pulled by Czochralski method with pulling rate of 15 mm/h. The single crystal was checked by X-ray Laue photograph. A flat c-plane rectangular sample of 1.8 × 1.2 × 0.2 mm 3 was cut with a spark-cutter for the Hall effect measurements. Its residual resistivity ratio RRR = 14 is within the highest reported for UCoAl indicating the very good quality of the crystal.

Magnetization measurements at ambient pressure were performed on another sample from the same ingot, with a commercial SQUID magnetometer (Quantum Design) down to 1.8 K and up to 5.5 T. The metamagnetic critical field B m = 0.6 T of that crystal is slightly lower than that of the crystal used for the Hall effect experiment where we found B m = 0.7 T. This shows the high sensitivity of B m to the quality and/or homogeneity of the single crystals.

The Hall effect ρ xy was measured using a four-probe AC lock-in technique. The current was applied perpendicular to the c-axis in the hexagonal planes (typically 1 mA at a frequency f ∼ 17 Hz) and positive and negative fields were applied along the caxis, which is the easy magnetization axis, in order to cancel out contributions of the magnetoresistance R xx due to a small misalignement of the transverse voltage contacts. The symmetric magnetoresistance R xx has been analysed too and good agreement with previous resistivity measurements [START_REF] Aoki | Ferromagnetic Quantum Critical Endpoint in UCoAl[END_REF] has been observed. The sample was set in a hybrid CuBe -NiCrAl piston cylinder cell for the pressure studies with Daphne oil 7373 as pressure transmitting medium. The pressure was determined from the superconducting transition temperature of Pb by AC susceptibility measurements. The Hall effect measurements were performed both in a PPMS (Quantum Design) (2 to 300 K ; 0 to 9 T) and in a homemade dilution refrigerator (0.15 to 10 K ; 0 to 16 T). The metamagnetic transition at B m is of first order, attested by a hysteresis between increasing and decreasing field magnetization curves [START_REF] Andreev | Metamagnetism of UAlCo[END_REF][START_REF] Aoki | Ferromagnetic Quantum Critical Endpoint in UCoAl[END_REF]. The hysteresis reduces and the metamagnetic transition gradually broadens as temperature increases, until it ends at T 0 = 12 K, marking the first-order critical end point (CEP). For T > T 0 a pseudometamagnetic transition fades out into a broad crossover region [START_REF] Palacio-Morales | Metamagnetic Transition in UCoAl Probed by Thermoelectric Measurements[END_REF].

Hall effect at ambient Pressure

The Hall resistivity ρ xy (B), see Fig. 5.1(b), is positive through the entire magnetic field range along the c axis. In difference, a negative Hall resistivity has been reported for B ⊥ c [START_REF] Matsuda | Transport properties of the anisotropic itinerant-electron metamagnet UCoAl[END_REF] indicating the multi-band character. At the lowest temperatures, the metamagnetic transition appears as a step-like increase in ρ xy (B). This feature is a clear illustration of the interplay between ρ xy and the magnetization. With increasing temperatures (4 K < T < T 0 ∼ 12 K), a peak develops just above the critical field B m . Further increasing temperature, the step-like increase at B m fades out and only a broad maximum is observed in ρ xy (B) above T 0 . Similar features were also found in transverse magnetoresistance ρ xx (B) [START_REF] Aoki | Ferromagnetic Quantum Critical Endpoint in UCoAl[END_REF]. This peak in ρ xx was interpreted as an enhancement of the magnetic scattering contribution due to strong magnetic fluctuations around the CEP [START_REF] Nohara | Strong Longitudinal Magnetic Fluctuations Near Critical End Point in UCoAl: A (59)Co-NMR Study[END_REF][START_REF] Karube | Universality and critical behavior at the critical endpoint in the itinerant-electron metamagnet UCoAl[END_REF].

From the field derivative of ρ xy , the differential Hall constant RH = ∂ρ xy (B)/∂B, see Fig. 5.1(c), we locate the critical field B m at the peak in ∂ρ xy (B)/∂B. The height of the peak is maximal at T 0 = 12 K, while the width does not broaden with increasing temperature until T 0 . Above T 0 , the peak height decreases and the width is strongly increasing. We will use the same criterion to determine the first-order critical end point at (B m , T 0 ) under pressure. Let us note that the differential Hall coefficient RH shows little change across B m (15% decrease), going from 0.18 µΩ • cm/T in the PM region to 0.15 µΩ • cm/T in the FM region. By plotting ∂ρ xy (H, T )/∂B we can draw the (T, H) phase diagram of UCoAl at ambient pressure (see inset in Fig. 5.1(c)) in excellent agreement with previous work [START_REF] Aoki | Ferromagnetic Quantum Critical Endpoint in UCoAl[END_REF]. The crossover region extends at least up to 20 K in the continuity of the metamagnetic transition.

Figure 5.2 shows the temperature dependence of the linear Hall coefficient R H = ρ xy /B at low field (B = 0.2 T). As we can see, R H has a maximum around 30 K, connected with a maximum of the magnetic susceptibility χ at T = 20 K. The inset of Fig. 5.2 shows R H plotted against χ with the temperature as an incipient parameter. R H is proportional to the magnetic susceptibility χ from 300 K down to 70 K. In general, the Hall effect can be expressed as the sum of NHE proportional to the carrier contribution and an anomalous contribution from left-right asymmetric scattering due to the ordered magnetic moments (AHE):

ρ xy (B) = R 0 B + R S M
(5.1) where the second term R S M accounts for the AHE and the R S ∝ ρ xx or ρ 2 xx for dominant skew scattering or side-jump scattering processes, respectively.

The proportionality R H -R 0 ∝ χ is expected in the incoherent high temperature regime with strongly anisotropic crystal field [START_REF] Kontani | Theory of Anomalous Hall Effect in a Heavy Fermion System with a Strong Anisotropic Crystal Field[END_REF]. However, we have to mention that the data can be described similarly by R H = R 0 + Cρχ, where R 0 is the normal Hall effect and C a constant [START_REF] Fert | Theory of the Hall effect in heavy-fermion compounds[END_REF], as ρ is weakly T -dependent in this temperature range. Thus, regardless of the underlying model, the Hall effect is dominated by the AHE in a large temperature range (70 K -300 K). R H is field independent (and so are R 0 and R S ), so it can be fitted using magnetic susceptibility data to obtain R 0 = -0.1 µΩ • cm/T and R S = 97 µΩ • cm/T for the high temperature regime as shown in the inset in Fig. 5.2.

The normal contribution to the Hall effect emerges only at low temperature. Figure 5.3 gives an insight to the relation between Hall effect and magnetization at 2 K, to separately account for the NHE and AHE contributions.

In the ferromagnetic region (B > B m ), at low temperature (2 K), both ρ xy (B) and M (B) show only weak variations. Therefore, R 0 and R S are assumed to be field independent. Indeed, on a plot ρ xy (B)/B vs M (B)/B, see Fig. 5.3(a), the high field data (B > B m ) fall on a straight line which can be fitted by eqn. 5.1 to obtain R 0 = 0.14 µΩ • cm/T and R S = 3.51 µΩ • cm/T in good agreement with Ref. [START_REF] Matsuda | Transport properties of the anisotropic itinerant-electron metamagnet UCoAl[END_REF]. In the PM phase, since both ρ xy and M increase linearly with field, the low-field points (0 T < B < 0.4 T) shrink altogether into a single point. This point falls close to the FM line. The data 0.4 T < B < 0.8 T corresponds to the metamagnetic transition regime. Obviously, there is no drastic change of R 0 between PM and FM phases through the metamagnetic tran- sition, thus there is no direct evidence of a Lifshitz transition at B m .

Figure 5.3(b) shows the field dependence of ρ xy at T = 2 K. On the same scale, we plotted the contributions of R 0 B and R S M as determined in the FM phase shown in Fig. 5.3(a), to compare the relative weight of NHE and AHE. The normal contribution R 0 B is basically linear with H except for a small kink at 0.6 T due to the magnetization density which suddenly amounts to 10% of the internal field (4πM/B ≈ 0.1) when the system becomes ferromagnetic. But since the magnetization only increases very slowly in the FM phase, its consequence on the internal field is negligible, so that B ≈ µ 0 H, the applied field. As a consequence, the slope of the ρ xy (B) curve is mainly determined by the NHE (∂ρ xy (B)/∂B R 0 ), and the step of ρ xy (B) at the transition is due to the AHE, mainly. In a simple approach, the jump in the AHE is due to a step-like increase of the magnetization (∆ρ xy R S ∆M ). However, from the experimental data we cannot exclude that also R S may be affected at the metamagnetic transition, as the resistivity ρ xx -at least in a longitudinal configuration -shows such a step-like increase. Finally, we should mention that the transition appears broader in ρ xy (B) than in M (B). This difference marks the complex interplay between intrinsic and extrinsic (impurity) contributions to the AHE.

Hall Effect under High Pressure

In our pressure study, the Hall effect ρ xy (B) has been measured by performing field scans upwards at various temperatures in the range from 0.15 K up to 20 K. 5.4 represents ρ xy as a function of B at T = 2 K for various pressures up to 2.22 GPa (the curves are vertically spaced by 0.2 µΩcm for clarity, respectively). While at low pressures (P ≤ 1.2 GPa) a clear jump of ρ xy at B m is observed, which is qualitalively similar to the behaviour at ambient pressure, above 1.6 GPa only a rather broad crossover is observed. This is a first indication that the first order metamagnetic transition collapses in the pressure range from 1.2 GPa to 1.6 GPa, thus the pressure of the quantum critical end point P QCEP is located in this pressure window. In agreement with previous measurements [START_REF] Aoki | Ferromagnetic Quantum Critical Endpoint in UCoAl[END_REF], a strong field dependence of ρ xx is also detected at T = 2 K. The magnetoresistance R xx shows a very sharp peak at P = 1.59 GPa while for higher pressures a plateau-like enhancement of R xx is observed. On cooling, it was proved [START_REF] Aoki | Ferromagnetic Quantum Critical Endpoint in UCoAl[END_REF] and verified here that the simultaneous enhancement of the residual resistivity and the enhancement of the A coefficient is correlated with crossing through the QCEP.

Next we concentrate on the behaviour of ρ xy (B) at different temperatures which is shown in Fig. 5.5 for P = 0.9 GPa, 1.59 GPa, and 1.83 GPa. The Hall resistivity ρ xy at different temperatures at 0.9 GPa is rather similar to that at ambient pressure (see Fig. 5.2(b)) with a steplike anomaly at lowest temperatures and a peak which appears with increasing temperature just above B m . At P = 1.59 GPa the behaviour appears likewise, except at lowest temperature (see below). For P = 1.83 GPa no sharp anomaly can be detected, but a broad crossover even at the lowest temperature. Furthermore, below 2 K, the temperature dependence of the Hall resistivity is very small for P < 1.6 GPa which is close to P QCEP .

While low temperature data have been measured only in the pressure regime where the QCEP is expected, we extrapolated ρ xy (B) measured at finite temperature down to T = 0 with a polynomial of second degree for all pressures. As we can see in Fig. 5.5, below 2 K the temperature dependence of the Hall resistivity is very small for P < 1.6 GPa -which is supposed to be close to P QCEP -in such a way that the extrapolated ρ xy (B, T = 0) almost coincides with the one measured at the lowest temperature. Figure 5.6 (upper panel) illustrates the pressure evolution of the Hall resistivity in the limit T → 0 K for selected pressures in absolute values. Interestingly, in the entire pressure range, all the ρ xy (B) curves almost match together in the high field part (B > B m ). This clearly shows that the normal contribution to the Hall effect, which dominates the field dependence above B m , as shown in Fig. 5.3, is pressure independent in the polarized ferromagnetic regime and has a constant slope dρ xy /dB = 0.152 µΩcm/T (indicated by the dashed line). The lower panel of Fig. 5.6 shows the field dependence of the differential Hall coefficient RH = ∂ρ xy /∂B for selected pressures. At low pressure, a very sharp maximum marks the metamagnetic transition. With increasing pressure, the height of the maximum decreases, while the width stays constant. Apparently, dρ xy /dB has changed its sign at 1.59 GPa, while above 1.6 GPa only a broad crossover appears. The dashed lines in the lower panel of Fig. 5.6 indicate a constant slope of dρ xy /dB = 0.152 µΩcm/T which is pressure independent in the FM state above B m .

In contrast to the FM phase, the Hall coefficient in the PM phase below P QCEP varies with field and pressure. Even though the ρ xy (B) curves for P ≤ 0.30 GPa are linear in the low field PM phase, this is not the case at higher pressure, for P → P QCEP .

To summarize the presentation of the experimental data, we plot in Fig. 5.7 the differential Hall coefficient RH in a (B, P ) phase-diagram. It appears clearly that RH is never constant in the paramagnetic state while it varies little in the ferromagnetic state, as discussed above. The width of the step-like transition at B m is delimited by solid lines for P < 1.6 GPa. One can reasonably affirm that the QCEP is located in the pressure window between the characteristic pressures P M = 1.2 GPa and P ∆ = 1.6 GPa. At P M , the differential Hall effect is changing sign, while above P ∆ the broadening of the crossover is clearly visible. The robustness of the FM phase contrasts with the variations in the PM phase, in which a valley of low RH separates two regions of high R H , one at low field and one at high field. It is interesting to observe that there is no drastic difference between the low pressure FM phase and the polarized paramagnetic phase (PPM) which is expected to occur for P > P QCEP through the crossover regime. As a matter of fact, the pressure decrease of the magnetization in the FM region has no effect in the high magnetic field regime.

The pressure dependence of B m , defined by the extrema of RH shown in Fig. 5.7 is in excellent agreement with previous transport measurements [START_REF] Aoki | Ferromagnetic Quantum Critical Endpoint in UCoAl[END_REF]. Magnetization measurements [START_REF] Mushnikov | Magnetic properties of the 5f itinerant electron metamagnet UCoAl under high pressure[END_REF] led to lower pressure dependence of B m , as mentioned above. This discrepancy probably comes from pressure inhomogeneities or differences in the crystal quality. The condition of good hydrostaticity may be crucial in UCoAl since the linear compressibility along the a-axis is known to be about 5 times larger than that along the c-axis [START_REF] Havela | U ternaries with ZrNiAl structurelattice properties[END_REF][START_REF] Mašková | Lattice anisotropy in uranium ternary compounds: UTX[END_REF]. In Figure 5.6, at low field, ρ xy appears P -independent. This observation is in good agreement with pressure magnetization data [START_REF] Mushnikov | Magnetic properties of the 5f itinerant electron metamagnet UCoAl under high pressure[END_REF] and recent magnetostriction experiments [START_REF] Aoki | Ferromagnetic Quantum Critical Endpoint in UCoAl[END_REF] which led to the conclusion that the respective Grüneisen parameter of the magnetization Ω M = 1 M ∂M ∂V V is near 2 for B < B m and 35 for B > B m , while Ω Bm , the Grüneisen parameter of B m , is near 140.

Figure 5.8 shows the jump ∆ρ xy at the metamagnetic transition plotted against the critical field B m . It is compared to the corresponding evolution of the jump of the magnetization ∆M taken from Ref. [START_REF] Mushnikov | Magnetic properties of the 5f itinerant electron metamagnet UCoAl under high pressure[END_REF] which decreases slightly with pressure. In order to correct for differences either in P determination, in hydrostaticity or in crystal quality, the data are plotted as a function of B m . (The upper scale indicates the correspondence with our P determination). In a simple approach ∆ρ xy ∝ ∆(R 0 B) + ∆(R S M ). At least up to P M = 1.2 GPa the main contribution to ∆ρ xy will be the jump in the magnetization ∆M at B m . However, at 1.43 GPa and 1.59 GPa, the corresponding anomaly in ρ xy at B m is no more a finite positive ∆ρ xy , but it appears slightly negative. Thus above 1.2 GPa, the apparent signature of the first order metamagnetic transition is not directly detected in the Hall effect measurement. Clearly, above 1.59 GPa no first order transition has been observed, indicating that the QCEP lies between P M = 1.3 GPa and P ∆ = 1.7 GPa. As already mentioned above, previous resistivity measurements pointed out the strong increase of the inelastic term, namely the A coefficient. Obviously, the associated change of the effective mass has a strong influence on the Hall resistivity. Finally, in Fig. 5.9 we show ∂ρ xy (B, T )/∂B for different pressures as a function of magnetic field and temperature for different pressures in a colour plot. The evolution of RH (T, B) with increasing pressure can be described in simple terms. At ambient pressure, the position of the critical end point is well defined by a sharp maximum in RH . With increasing pressure, the amplitude of RH at the critical end point decreases, T 0 decreases, and the critical field increases. Figure 5.10 shows the pressure evolution of the temperature of the critical end point T 0 (defined by the maximum of RH ) and the width of the metamagnetic transition ∆B m (width of the peak in ∂ρ xy /∂B). Surprisingly, T 0 is not suppressed continuously to zero temperature but rather drops from T 0 = 6 K to zero in the vicinity of P ∆ = 1.7 GPa. This pressure also corresponds precisely to the end of the sharp metamagnetism, unambiguously attested by the sudden increase of the transition width ∆B m on entering a crossover region beyond the QCEP.

Discussion

The Hall effect experiments give detailed insights on the behaviour of the metamagnetic transition in the itinerant heavy fermion system UCoAl. For T T 0 , the step-like feature of the ρ xy (B) curves is connected to the magnetization jump via the anomalous Hall effect. The constant Hall coefficient in the FM phase at low temperature (fieldand pressure-independent) strongly suggests that the coefficients R 0 and R S do not change with pressure and are the same as at ambient pressure, where they are known with precision. On approaching the QCEP, a drastic change occurs in the Hall signal on crossing B m . We associate this to the enhancement of the average effective mass detected via the A coefficient in resistivity measurements.

For Ising-type Uranium ferromagnets, as the jump of the magnetization at B m is often strong (near 0.5 µ B ) and the renormalized Fermi energy is low, Fermi surface reconstruction can be expected at P c and B m as clearly observed in UGe 2 [START_REF] Settai | A change of the Fermi surface in UGe 2 across the critical pressure[END_REF]. In the case of UCoAl, at least at low field (B B m ) and at high field (B > B m ), no major variation of the Hall constant can be pointed out. There is no evident signature of a Fermi surface reconstruction on sweeping from the PM to FM phase for P < P QCEP and from the PM to a polarized paramagnetic phase for P > P QCEP . Recent thermoelectric power experiments at ambient pressure were interpreted only via the decrease of the heaviest hole effective mass through B m [START_REF] Palacio-Morales | Metamagnetic Transition in UCoAl Probed by Thermoelectric Measurements[END_REF]. There was also no signature of a Lifshitz transition. By contrast, in the case of the ferromagnetic superconductor UGe 2 , the crossing from PM to FM is accompanied by a sign change of the Hall coefficient in excellent agreement with a Fermi surface reconstruction detected in quantum oscillations experiments [START_REF] Haga | A change of the Fermi surface across the metamagnetic transition under pressure in UGe 2[END_REF]. However, in these multiband heavy fermion systems, the Hall resistivity response can be complex. A pathological example is the pseudo-metamagnetic transition in CeRu 2 Si 2 where no clear signature of a Fermi surface reconstruction is detected in the Hall constant [START_REF] Kambe | Characteristic of metamagnetic transition in CeRu 2 Si 2 revealed with Hall mobility[END_REF] 1 while there is direct evidence of a Fermi surface change through B m by quantum oscillations experiments [START_REF] Aoki | Transition of f electron nature from itinerant to localized: Metamagnetic transition in CeRu 2 Si 2 studied via the de Haas-van Alphen effect[END_REF] associated with a Lifshitz transition. So the question remains open for UCoAl.

It is interesting to recall the case of the ferromagnetic itinerant system ZrZn 2 (T Curie ∼ 28.5 K, M 0 ∼ 0.17 µ B per Zr atom), which is an isotropic Heisenberg system. As in UGe 2 , evidences are found for two emerging FM phases (FM 2 ,FM 1 ) at low pressure (P C ∼ 1.65 GPa) and one field-induced FM 1 phase when a field is applied from the PM phase above P c . However, at least for FM 1 , it seems that P QCEP almost coincides with P c [START_REF] Uhlarz | Quantum phase transitions in the itinerant ferromagnet ZrZn 2[END_REF]. As a consequence, at 2.1 GPa, the crossover metamagnetic field is so low (0.05 T) that the observation of the Fermi surface in the PM regime is proscribed [START_REF] Kimura | de Haas-van Alphen Effect in ZrZn 2 under Pressure: Crossover between Two Magnetic States[END_REF]. De Haasvan Alphen measurements pointed out the crossover between the FM 1 and FM 2 phases with an invariance of the Fermi surface topology. The change in the exchange splitting of the Fermi surface is the signature of the field crossover FM 1 /FM 2 . Let us point out that in the case of ZrZn 2 the Fermi energy is weakly renormalized, thus the local fluctuations are negligible compared to the case of U compounds where the itinerant-localized duality is a major part of the puzzle.

Finally, a striking point in our results is the appearance of different field regimes in the low field PM response (B < B m ) on approaching P QCEP . It may be connected to the particularity of the quasi-Kagome structure of UCoAl, leading to a complex interplay between frustrated magnetic fluctuations and sole FM interactions. Other evidences of non-conventional FM character of UCoAl is the observation of non-Fermi liquid properties in the PM regime at pressure far below P QCEP [START_REF] Palacio-Morales | Metamagnetic Transition in UCoAl Probed by Thermoelectric Measurements[END_REF]. To solve the UCoAl puzzle, key experiments are now to extend the magnetization measurements through the QCEP and of course to succeed to directly observe the Fermi surface.

Conclusions

Hall effect experiments at ambient and at high pressure have been presented in the itinerant metamagnet UCoAl. At the metamagnetic transition B m , the jump of the Hall effect is dominated by the AHE contribution which scales with the magnetization. The jump at B m disappears at P M ≈ 1.3 GPa, and above P ∆ ≈ 1.7 GPa only a broad pseudo-metamagnetic transition could be detected. While the Hall effect in the polarized ferromagnetic state seems to be pressure independent, the interpretation of the field dependence in the paramagnetic state below B m under pressure on approaching the QCEP appears less straightforward and needs to be revised in the future. Finally, our Hall effect experiments do not allow to conclude on a possible Fermi surface change through the QCEP.

1 although we must say that a kink in the Hall resistivity was ascribed to a Lifshitz transition in Ref. [START_REF] Daou | Continuous Evolution of the Fermi Surface of CeRu 2 Si 2 across the Metamagnetic Transition[END_REF], but in conjunction with other signatures.

Resistivity under pressure

In this section we present a pressure study of the resistivity of UCoAl, down to very low temperature (50 mK), and up to 13 T. Hydrostatic pressure was applied using a piston-cylinder cell. Two samples were mounted in the longitudinal (current J H) and transverse ( J ⊥ H) configurations, respectively, with field applied along the c-axis (easy magnetization axis) in both cases. (Fig. 5.11). A very low noise level is achieved by using cold transfomers thermalized at the temperature of the 1 K pot, which amplify the signal by a factor 100. In the first subsection, we study the Fermi liquid behaviour using temperature scans of the transverse resistivity. In the second subsection, we study both the transverse and longitudinal magnetoresistivity, with a special focus on hysteresis effects at the metamagnetic transition.

Temperature dependence

Temperature scans of the longitudinal and transverse resistivity (ρ zz and ρ xx , respectively) in the range [0.05-0.7 K] were performed at various fields and pressures. Above ∼ 20 K the measurements were perfomed step by step due to the slow thermalization. In the dilution fridge, the T -dependent measurements were performed by continuously recording while decreasing temperature (typical rate -0.2 K/h). The raw data have been processed afterwards, by removing outliers and down-sampling to 2 mK intervals with boxcar smoothing. We started the T -scans at H = 0 and increased the field step by step in one direction. As will be discussed in the next subsection (5.3.2), there are strong hysteresis effect between up and down field sweeps around the metamagnetic transition which also influences the value of the A coefficient.

Figure 5.12 shows the resistivity of both samples between 50 mK and 300 K. The residual resistivity ratio ρ(300 K)/ρ 0 is ≈ 13.5 for both samples. In the inset we plot the low temperature part as (ρ -ρ 0 ). The variation of ρ(T ) from 0 K to 700 mK is about 6 times stronger for current along the a axis than along the c axis. In this section we only show the transverse configuration, since the longitudinal has much weaker A coefficient, thus the analysis is much less precise.

In order to test the Fermi liquid behaviour, ρ xx (T ) was fitted with a power law of the form:

ρ = ρ 0 + AT α (5.2)
On the left hand side of Figure 5.13 (for P < 1 GPa) and Figure 5.14 (for P > 1 GPa), the data are plotted as (ρ xx -ρ 0 ) vs T 2 together with the fit. On the right hand side, the fit parameters ρ 0 , A and α are plotted against the magnetic field. The residual term ρ 0 will be discussed in details in the next subsection (5.3.2). A vertical line indicates the metamagnetic field H m which is determined by the field sweep measurments, also discussed later.

At the lowest pressure P = 0.05 GPa (Fig. 5.13-a) the value of A decreases by a factor ∼ 3.5 between 0 and 1 T. The exponent α ≈ 1.5-1.6 is found at low field, and it changes to α ≈ 1.9 between 0.6 and 0.8 T, which corresponds to the field range of the metamagnetic transition (H m = 0.7 T)2 . This is in rather good agreement with previously reported experiments in which ρ xx was fitted with a T 5 /3 dependence [START_REF] Havela | Stability of the non-Fermi liquid state in UCoAl[END_REF] in the PM phase, and with a T 2 dependence (Fermi liquid) in the FM phase. The residual resistivity ρ 0 first increases between 0-0.4 T, then shows a plateau (0.4-0.8 T), and finally decreases at 1 T.

As P increases (0.28-0.71 GPa) (Fig. 5.13-b&c) the low-field value of α increases to 1.7-1.85 but is still clearly below 2. Surprisingly, α reaches 2 at a field H 0 which is clearly below H m : the Fermi liquid character is recovered within the PM state.

In parallel, the A coefficient shows a two-step decrease : the first decrease occurs around H 0 , the second one just before H m . The residual term ρ 0 shows a similar feature as before : a first increase at low field, followed by a plateau between H 0 and H m , where it decreases. It increases again at higher fields, in a parabolic fashion. This will be explored in the subsequent part.

At 0.90 GPa (Fig. 5.13-d) the FL behaviour is fully established with α = 2 at all fields. The value of A at H = 0 is 3 times smaller compared to the ambient pressure. In the field dependence of A, a first decrease occurs at H 0 = 1-1.5 T in the continuity of previous pressures, and it is connected with the first maximum of ρ 0 . The second shoulder-like anomaly in A grows up, forming a bump just before H m , while there is now a positive jump in ρ 0 at H m . Above H m , A slowly decreases towards a constant value of 0.1 µΩ.cm.K -2 .

At higher pressures (Fig. 5.14) A continues to initially decrease, while ρ 0 symmetrically increases up to H 0 ∼ Hm 2 . A now shows a maximum just before H m , which is sharpest at 1.42 GPa, and then broadens. The amplitude of the positive jump in ρ 0 at H m increases continuously with pressure, even beyond 1.42 GPa, and at least up to 1.80 GPa, which is the maximum pressure at the current stage.

Summary on the non-Fermi liquid behaviour

The Fermi liquid behaviour is fully established for all fields in the pressure range [0.90-1. [START_REF] Nohara | Strong Longitudinal Magnetic Fluctuations Near Critical End Point in UCoAl: A (59)Co-NMR Study[END_REF] GPa], but it breaks down at 1.42 GPa at very low temperature. At this particular pressure, ρ xx (T ) cannot be fitted with a power law in the entire temperature range [0.05-0.7 K]. Below ∼ 400 mK, a power law fit with exponent α ≈ 1.5 is found at low field, and α ≈ 1.7 at high field. Above this temperature, ρ xx (T ) follows a T 2 law at all fields. Due to the limited temperature range of the power law fit, there is a dispersion of the values α, but it seems that the change of α from ≈ 1.5 to ≈ 1.7 occurs at 4 T.

This «re-entrant» NFL behaviour is also noticeable at 1.62 GPa below 2-3 T, and maybe at 1.80 GPa as well, at zero field only. Around H m , because of hysteresis effects

A (µΩ•cm•K -α ) H (T) H m a) b) c) d)
Figure 5.13: Left hand side: T 2 dependence of the transverse resistivity (H c and J a) at various field values and various pressures: P = 0, 0.28, 0.71, 0.90 GPa. Each curve is superimposed with a power law fit (Eq. 5.2), and they are shifted together vertically for clarity. Right hand side: field variation of the exponent α (•), the coefficient A (•) and the residual term ρ 0 ( ) (in µΩ.cm) obtained by fitting ρ xx (T ) with Eq. 5.2. Lines are guides for the eye. (see next), the ρ xx (T ) curves strongly depend on the approach (increasing or decreasing field). Hence, the little variations from the T 2 law observed around H m are due to the crossing of the metamagnetic transition on cooling, and shall not be considered as a NFL effect. The NFL regions are indicated in the field-pressure phase diagram Figure 5.20.

Summary on the A coefficient

Figure 5.15-a summarizes the A coefficient3 versus field at all pressures, obtained from temperature sweeps (increasing field step by step). In the inset we plot the value of A at H = 0 versus pressure, showing a fast initial decrease up to 0.9 GPa where A is only 1 /3 of the intial value, followed by a slow decrease up to the highest pressure. In addition, in Figure 5.15-b we show A determined from field sweep measurements (which will be shown next) at various temperatures between 0.05 K and 0.6 K (typically 5 curves). In the region of H m , there are strong differences between up and down field sweeps, especially above 1.22 GPa when the maximum of A starts to grow. On top of the broad A enhancement, there is a sharp positive peak in decreasing sweeps (at H dec m ) while there is a sharp negative peak in increasing sweep (at H inc m ). It is worthwhile to note that the maximum of A from T scans is broader and occurs before H inc,dec m . As a summary, the A coefficient is enhanced in two distinct field and pressure regions:

• At low-pressure-low-field, A is enhanced due to the proximity of FM instability. It is related to the enhancement of C/T [START_REF] Tatsuma | Specific-Heat Anomaly around Metamagnetic Transition in UCoAl[END_REF] by the Kadowaki-Woods relation A ∼ γ 2 , in the presence of strong spin fluctuations. A decreases abruptly on entering the FM phase, and also decreases with increasing pressure. This is consistent with the picture of strong spin fluctuations associated with the FM instability.

• A is also enhanced at high-pressure-high-field, with a maximum in the broad pressure-field region around 1.4-1.8 GPa -6-9 T. This enhancement is associated to the QCEP. This is very similar to the resistivity data reported in Ref. [START_REF] Aoki | Ferromagnetic Quantum Critical Endpoint in UCoAl[END_REF].

The values of H m and H 0 are plotted in the field-pressure phase diragram of UCoAl Figure 5.20. The change of regime between these two distinct field and pressure regions occurs at ∼ 1.1 GPa. This is probably connected to the end of the Hall effect jump at P M ≈ 1.3 GPa (see Fig. 5.8). The maximum in A coefficient around ≈ 1.6 GPa, is maybe connected to the end of the sharp transition in Hall effect, at P ∆ ≈ 1.7 GPa (see Fig. 5.10). We thus retrieve the characteristic pressures P M and P ∆ established by Hall effect measurements (Section 5.2).

Field dependence -pressure P ≈ 0

Figure 5.16 shows the magnetoresistivity of UCoAl in the transverse (ρ xx ) and longitudinal (ρ zz ) configurations, at P = 0. These measurements were performed in a PPMS down to 5 K, without a pressure cell. At 5 K, when going from the PM to FM state, ρ xx (H) undergoes a step-like decrease and ρ zz (H) a step-like increase. As temperature increases, both ρ xx and ρ zz show a peak at the transition. The peak is maximum at 11 K, marking the end of the first order transition: the critical endpoint (CEP) is located at T 0 = 12 K,H m = 1 T, in agreement with other measurements (thermoelectricity [START_REF] Palacio-Morales | Metamagnetic Transition in UCoAl Probed by Thermoelectric Measurements[END_REF], NMR [START_REF] Nohara | Strong Longitudinal Magnetic Fluctuations Near Critical End Point in UCoAl: A (59)Co-NMR Study[END_REF][START_REF] Karube | Universality and critical behavior at the critical endpoint in the itinerant-electron metamagnet UCoAl[END_REF], etc.).

The increased resistivity around the CEP is ascribed to the scattering on spin fluctuations [START_REF] Karube | Universality and critical behavior at the critical endpoint in the itinerant-electron metamagnet UCoAl[END_REF].

In Figure 5.17, the field dependence of resistivity was measured down to 50 mK in a dilution fridge, with the samples inside the piston-cylinder pressure cell. The pressure cell had been closed with as little force as possible, just enough to seal the pressure chamber with the Daphne oil in it. At low temperature, the pressure was found to be 0.05 GPa. In the longitudinal sample, H m is already shifted to 1 T by the small pressure. Below 1 K the jump in ρ xx collapses, leaving only a small kink at 50 mK. On the contrary, ρ zz still shows a sharp jump at the lowest temperature. There is a hysteretic behaviour in both configurations. In the longitudinal, the hysteresis loop opens up notably in ρ zz (H) on cooling below 4.2 K. In the transverse, the hysteresis consists of a shift of the hump. This is a rather unusual shape at 50 mK, due to the collapse of the jump in ρ xx (H)

To summarize, at (almost) zero pressure, there are 3 important differences between ρ xx and ρ zz :

• i) opposite variation at the metamagnetic transition

• ii) when T → 0 the jump in ρ xx (H) vanishes, while it stays constant in ρ zz (H)

• iii) the A coefficient is much larger in ρ xx
These differences occur at low temperature T << T 0 . At higher temperature, both configurations show a peak in the vicinity of the CEP (T 0 ∼ 11 K) and a broad maximum in the crossover regime (T > T 0 ).

Pressure study -Intermediate Temperatures -Critical EndPoint

Figure 5.18-a shows field-scans ρ zz (H) and ρ xx (H), every 1 K from 50 mK up to 14 K, for 3 different pressures.

In the longitudinal configuration, the pressure evolution does not change the description given at P = 0. There is always a positive step at the metamagnetic transition. On the contrary, the negative step in the transverse resistivity changes into a positive step at high pressure. The temperature T 0 of the critical endpoint is indicated by the green curve in each plot. T 0 corresponds to the sharpest peak, determined by the sharpest variation of ∂ρ/∂H, as shown in Figure 5.18-b. At low pressure, the sharpest peak is rather well defined, so the temperature T 0 of the critical endpoint is well determined. As pressure increases, T 0 decreases, and the peak in resistivity is less pronounced. In ρ zz (H), because of the broadening of the transition, we cannot distinguish the CEP above 1.22 GPa. In ρ xx (H), we can track the CEP down to T 0 ≈ 3.5 ± 1.5 K at 1.62 GPa, but to decide whether there is a sharpest peak is rather subjective, since it is impossible to define T 0 from ∂ρ/∂H. The pressure evolution of T 0 is shown in Figure Finally, the increase of H m with pressure is slightly lower in the longitudinal configuration: at 1.80 GPa, H xx m = 8.8 T while H zz m = 7.4 T. This is probably due to a little pressure inhomogeneity in the sample chamber.

Pressure study -Low temperature

Next we want to discuss the H dependence of ρ xx and ρ xx at low temperature. Previously we observed several anomalies in the H dependence of ρ 0 and the A coefficient. Here we will concentrate on the field range around H m and discuss the observed hysteresis. The field sweep rate was typically 0.03 T/min at the lowest temperature, and we checked that the data were perfectly reproducible for twice faster and twice slower sweep rate, so we ensured that the observed hysteresis effect were not an artifact due to the instrumental time response.

Field scans at lowest temperature (50 mK)

The pressure evolution of resistivity versus field, at the lowest temperature (50 mK), is presented in Figure 5.19 for both configurations.

The longitudinal configuration (Fig. 5.19-a) is the simplest case to start with: there is a clear step-like increase at H m , and the shape of the transition is conserved at all pressures. In the PM state, ρ zz increases with pressure, but it is not affected by pressure in the FM state. In the PM state, ρ zz (H) is basically flat from H = 0 to H 0 ∼ Hm 2 , and shows a broad kink between H 0 ∼ Hm 2 and H m . Above H m , ρ zz (H) has a constant slope of 0.072 µΩ.cm/T, which is pressure-independent. The hysteresis loop at H m reduces with pressure, but it is still visible at 1.93 GPa (see Fig. 5.23). At 1.80 GPa, a bump starts to build up above the transition, and gets more pronounced at 1.93 GPa. From these data it is difficult to locate the critical pressure where the metamagnetism vanishes.

The transverse magnetoresistivity has many particularities (Fig. 5.19-b,c). For all pressures, ρ xx (H) initially increases up to H 0 ∼ Hm 2 from where it forms a broad plateau until the transition occurs (H 0 is related to a feature in A coefficient). In the high field limit, it increases again in a parabolic H 2 dependence. At the transition, the shape of ρ xx (H) is strongly modified by pressure, and notable differences arise between up-sweep (ρ inc xx (H)) and down-sweep ( ρ dec xx (H) ). We can describe the transverse magnetoresistivity at different pressures (Fig. 5.19-b,c) as follows: 4 . P = 1.62,1.80 GPa: the first bump grows up into a peak at H peak , while the second bump at H remains a shoulder-like anomaly and drifts away from the first peak. ρ dec xx (H) starts to present the same features as ρ inc xx (H), in such a way that the difference between up and down sweeps fades out.

P = 0: ρ inc xx (H)
The pressure evolution of H m , H peak , and H , draws the H-T phase diagram of UCoAl in the limit T → 0 (Fig. 5.20). H m (and H peak ) increases almost linearly with pressure. The increase of H m is slower in the longitudinal sample, probably because of a little pressure inhomogeneity. The phase diagram under pressure obtained with transverse resistivity is in good agreement with the two reported experiments [START_REF] Aoki | Ferromagnetic Quantum Critical Endpoint in UCoAl[END_REF][START_REF] Honda | Electrical resistance and magnetoresistance of UCoAl under high pressure[END_REF]. To our knowledge, there is no report on longitudinal resistivity in the literature.

The broad anomaly at H 0 ∼ H m /2 occurs at all pressures, mostly in the transverse resistivity (there is also a corresponding anomaly in ρ zz , which is much weaker, and disappears at 1.93 GPa). It may result from a competition between the scattering near H m and the orbital effect due to the cyclotron motion of the quasiparticles, although we cannot rule out that this kink may also be the signature of a Lifshitz transition within the PM state 5 .

At high field (H >> H m ), a ρ xx ∼ H 2 is observed, with no sign of saturation up to the highest field (13.4 T). This parabolic field dependence is the typical scattering .20: Field-pressure phase diagram of UCoAl (for T → 0 K) from transverse (red) and longitudinal (blue) resistivity, and Hall effect (black). We indicate the non-Fermi liquid (NFL) regions (green), and the unknown phase (orange) between H on and H where up-down difference is observed in transverse resistivity. We also indicate the characteristic pressures P M and P ∆ . contribution due to the cyclotron motion of the quasiparticles in a compensated metal, with closed Fermi surface. It would saturate if it were not a compensated metal. In the other configuration, there is a constant positive magnetoresistance ρ zz ∼ H, which necessarily occurs for non-spherical Fermi surface 6 .

NFL NFL

The plateau of resistivity that appears above 1.22 GPa between H peak and H have already been reported by F. Honda and D. Aoki [START_REF] Aoki | Ferromagnetic Quantum Critical Endpoint in UCoAl[END_REF][START_REF] Honda | Electrical resistance and magnetoresistance of UCoAl under high pressure[END_REF]. This plateau of residual resistivity is observed only in the transverse configuration, whereas the enhancement of A coefficient occurs in both configurations (not shown here). This indicates that the temperature-dependent scattering is rather isotropic, while the residual scattering is highly anisotropic. This is in good agreement with the mechanism of residual resistivity enhancement predicted at a ferromagnetic quantum phase transition [START_REF] Miyake | Enhanced Impurity Scattering due to Quantum Critical Fluctuations: Perturbational Approach[END_REF].

Hysteresis ∆H

At every pressure, we performed up and down field scans at various temperatures, typically every 0.2 K from 0.05 to 2 K, and every 1 K above, in order to get a fine mapping of the phase diagram in the low temperature region. A selection of scans are presented for the longitudinal sample (Fig. 5.21) and the transverse sample (Fig. 5.22), in which we can see the reducing of the hysteresis when temperature increases. In addition to the hysteresis ∆H (which is observed in both samples), there is a difference in absolute value between up and down field sweeps, only in the transverse configuration. This unconventional feature will be referred to as up-down difference, noted ∆ρ xx . First, we focus on the «conventional» hysteresis.

The evolution of the hysteresis ∆H with temperature and pressure is shown in Fig. 5.23) for the two configurations. It is plotted along with the temperature T 0 of the maximum resistivity enhancement around the CEP, determined earlier (see Fig 5.18).

In the longitudinal sample, the hysteresis width is clearly defined. But in the case of the transverse sample, the shape of the hysteresis changes with pressure, in such a way that up and down sweeps are not always super-imposable. We indicate in the insets of Fig. 5.23-a & b how the hysteresis width was determined.

∆H decreases with increasing temperature and reaches the instrumental limit, which is the hysteresis of the superconducting magnet (≈ 0.02 T). At low pressure (0.28 GPa), the reducing of ∆H with T looks like an exponential decay, so the temperature T hyst where it collapses is not well defined (T hyst (P = 0) ≈ 7 K). Up to 1.22 GPa, ∆H(T ) is almost pressure-independent, at least up to 4 K (no hysteresis data available at higher temperature for these pressures). Further increasing pressure (P > 1.22 GPa), ∆H(T = 0) reduces and the temperature decay is faster (T hyst decreases). The pressure evolution of ∆H(T = 0) and T hyst are summarized in Figure 5.23-c. By extrapolation, it seems that the hysteresis will collapse at 2 -2.2 GPa, similarly to T 0 . This is probably indicating the termination of the QCEP region. We can try to relate these results to the characteristic pressures of the Hall effect P M ≈ 1.3 GPa and P ∆ ≈ 1.7 GPa. The hysteresis starts to reduce at P ≈ 1.3 GPa ≈ P M , and will collapse at P ≈ 2 GPa which is close to P ∆ .

Yet the hysteresis ∆H seems to vanish at T hyst < T 0 .

Unconventional up-down difference ∆ρ xx

We already mentioned that there is an unconventional difference in ρ xx between up and down sweeps 50 mK (sub kelvin range), around H m , for pressures above 1.22 GPa (Figs. 5.19), and in Figure 5.22 we saw that this up-down difference ∆ρ xx decreases with increasing temperature. In order to study this feature, we must compensate the conventional hysteresis that we have studied in the previous paragraph. We thus define the up-down difference of the transverse resistivity as:

∆ρ xx (H) = ρ inc xx (H + ∆H 2 ) -ρ dec xx (H - ∆H 2 ) (5.3)
where ∆H is the hysteresis that has been determined before (see Fig. 5.23-b). 7∆ρ xx (H) is plotted in Figure 5.24 at various pressures (P = 1.22, 1.42, 1.62, 1.80 GPa) and various temperatures.

∆ρ xx (H) has a triangular shape, with a sharp maximum at H peak . In order to describe ∆ρ xx , we must introduce new characters: it is delimited between fields H on and H (defined earlier), and below the temperature T V .

∆ρ xx appears very small at 1.22 GPa below T V = 0.6 K. It is largest at 1.42 GPa, with the highest temperature limit T V = 1.8 K. At higher pressure, ∆ρ xx decreases and broadens gradually, and the temperature limit decreases: T V = 1.4 K at 1.62 GPa, and 1 K at 1.80 GPa.

The peak of ∆ρ xx (H) occurs at H peak . At 1.42 GPa, there is a shouldering after the peak, and a tail after H . At higher pressure, the anomaly around H is present in both ρ inc xx (H) and ρ dec xx (H), in such a way that it cancels out in the substraction. Consequently, ∆ρ xx (H) is left with a symmetrical peak shape, in contrast with the highly asymmetrical shape of the parent curves ρ xx (H).

Finally, the values of H on , H , and H peak are plotted in the H-P phase diagram of UCoAl (Fig. 5.20). They delimit the region where up-down difference occurs in the transverse magneoresistivity. We remark that this region exactly corresponds to the enhancement of the A coefficient.

T V = 0.6K T V = 1.8K H peak H on H* T V = 1.4K T V = 1K

Discussion

Where is the QCEP?

The transverse resistivity has many variations in temperature, field and pressure, which are different from the longitudinal resistivity. Hence it is not clear where the QCEP is located. The difficulty to localize the QCEP was also noted in the Hall effect measurements (two characteristic pressures P M = 1.3 GPa and P ∆ = 1.7 GPa) (see Section 5.2). In our resistivity measurements, we also identified these characteristic pressures P M and P ∆ :

We observe strong changes in ρ xx around ∼ 1.1 -1.4 GPa: the A coefficient reaches a low value at zero field and rises at H m , the jump in the transverse magnetoresistance ρ xx 0 (H) changes sign and starts to exhibit a peak followed by a plateau, the hysteresis ∆H starts to reduce, and an unconventional up-down difference opens up. This corresponds to the characteristic pressure P M where the jump in Hall effect changes sign. In difference, the longitudinal sample shows no spectacular change at P M : just the reducing of the hysteresis ∆H.

For both configurations, we could follow T 0 under pressure, though with a large uncertainty. It seems to reach 0 K somewhere between 1.6-2.2 GPa (Fig. 5.18). The hysteresis between up and down field sweeps was observed in both configurations up to the highest pressure of the experiment (1.93 GPa for ρ zz (H)). It seems that T hyst and ∆H are going to collapse around 2 GPa, in both longitudinal and transverse configurations (Fig. 5. 23-c). This characteristic pressure of 2 GPa is probably the equivalent of P ∆ = 1.7 GPa where the Hall effect anomaly collapses. The little discrepancy may be due to sample dependence or pressure inhomogeneity.

To summarize, there is a large ambiguity on the location of the QCEP, between 1.2-2.2 GPa and between 6-10 T. This is not an uncertainty due to instrumental limitation. In fact, the QCEP seems to be hidden by the formation of a new phase. To study further the QCEP, thermodynamic probes such as NMR are desired.

Possible scenarios for a new phase around the QCEP

In this paragraph we wish to review some reported examples which could be related to the properties of UCoAl around the QCEP. Quite generally, it was shown that unusual properties arise if a metamagnetic transition is accompanied by a change in the Fermi surface topology (Lifshitz transition), as it is believed to occur at the QCEP in ZrZn 2 , where unconventional universality is expected [START_REF] Yamaji | Quantum Metamagnetic Transitions Induced by Changes in Fermi-Surface Topology: Applications to a Weak Itinerant-Electron Ferromagnet ZrZn 2[END_REF]. First, we discuss a possible analogy with the quantum critical metamagnet Sr 3 Ru 2 O 7 , in which an exotic phase is observed, called nematic phase [START_REF] Grigera | Disorder-Sensitive Phase Formation Linked to Metamagnetic Quantum Criticality[END_REF][START_REF] Borzi | Formation of a Nematic Fluid at High Fields in Sr 3 Ru 2 O 7[END_REF][START_REF] Rost | Thermodynamics of phase formation in the quantum critical metal Sr 3 Ru 2 O 7[END_REF]. In analogy with the nematic state of liquid crystals, which has orientational but no positional order, it has been proposed that the electrons could form nematic liquids in the presence of strong correlations [START_REF] Kivelson | Electronic liquid-crystal phases of a doped Mott insulator[END_REF]. Such an electron nematic state is characterized by a lower rotational symmetry than the lattice symmetry. It was indeed observed in Sr 3 Ru 2 O 7 which has double-layered perovskite structure . In the case of UCoAl, the inversion symmetry is not present in the crystal structure, so the situation is more complicated. The plateau between H m and H may be related to the case of CeRu 2 Si 2 (Rh-doped) where a plateau of A coefficient was observed, between the fields H c and H m . In that case the AF order disappears at H c and the pseudo-metamagnetic transition occurs at H m [START_REF] Sekine | Magnetic Properties and Phase Diagram of Ce(Ru 1-x Rh x ) 2 Si 2 (0 ≤ x < 0.5)[END_REF]. In-between, the system is in a PM state with strong AF fluctuations.This regime occurs only in the doped system, and it was suggested that in the pure compound the two anomalies are actually merged together, with both FM and AF critical fluctuations [START_REF] Sato | Field-induced Ferromagnetic Correlation in the Metamagnetic Crossover in CeRu 2 Si 2 as Studied by Neutron Scattering[END_REF]. This picture may be considered for UCoAl if AF correlations were to be detected at H = 0. The susceptibility for H c-axis at zero pressure shows the characteristic broad peak at T max χ ≈ 20 K. This is typical heavy fermion behaviour with AF correlations. Further studies with microscopic probes are required. One could possibly think that H m and H are two metamagnetic transitions, with a low-moment FM 1 phase in-between, like the FM 1 phase in UGe 2 . The A coefficient also forms a plateau in this region of the phase diagram in UGe 2 . Such a two-step metamagnetic transition will occur if there are two peaks in the DOS due to van Hove singularity [START_REF] Sandeman | Ferromagnetic Superconductivity Driven by Changing Fermi Surface Topology[END_REF][START_REF] Binz | Metamagnetism of itinerant electrons in multi-layer ruthenates[END_REF]. However, the magnetostriction under pressure, which is related to the magnetization (∆V /V ∝ M 2 at first approximation), showed no anomaly at H .

A frustration effect?

We finally propose a last scenario, considering this difference ∆ρ xx between up and down sweep magnetoresistivity. It is a very peculiar phenomenon, observed only in the transverse sample, at low temperature, in the region of the T -P -H phase diagram where the QCEP is expected. We have no explanation for this phenomenon. Yet we may speculate that the vertical hysteresis resembles the memory effect of a spin glass state, which typically occurs in magnetically frustrated systems. It should be noted that the crystal structure of UCoAl forms a quasi-Kagome lattice of U atoms, thus magnetic frustration can potentially exist. When the CEP terminates at T = 0, a new phase due to the frustration -which is hidden by the first-order transition at H m at low pressure -may start to appear. However, this raises the question of how frustration can be realized in an itinerant magnetic system.

Conclusion on the resistivity measurements on UCoAl

We carried out an extensive series of resistivity measurements on UCoAl, under magnetic field, high pressure, and low temperature, with a particularly sensitive instrumentation (cold transformers).

We studied the non-Fermi liquid behaviour along the transverse direction. We found two separate NFL regions: at low-pressure-low-field, due to the presence of spin fluctuations in the vicinity of the FM instability ; and another NFL region at 1.42 GPa. The A coefficient was finely mapped in the field-pressure phase diagram, and it is highly enhanced in the two critical regions: at low-pressure-low-field and around 1.42 GPa, in relation with the NFL behaviour.

The hysteresis between up and down field scans was determined precisely, indicating the collapse of the first-order metamagnetic transition at P ∆ ∼ 2 GPa. We observed striking differences between the longitudinal and transverse configurations. The variation of ρ xx at H m progressively changes sign at P M ≈ 1.2 GPa, where a plateau appears in the transverse residual resistivity, as already reported by F. Honda [START_REF] Honda | Electrical resistance and magnetoresistance of UCoAl under high pressure[END_REF] and D. Aoki [START_REF] Aoki | Ferromagnetic Quantum Critical Endpoint in UCoAl[END_REF]. In addition, an intriguing difference in the transverse resistivity between up and down field sweeps was observed above P M , suggesting an exotic phase emerging in the region of the QCEP. Its location in the phase diagram has been defined precisely, but the nature of this exotic phase is unknown.

Magnetization

Ambient pressure (relaxation effects)

Magnetization measurements are routinely performed as a sample characterization technique, using commercial SQUID extraction magnetometers (MPMS, Quantum Design) available at INAC. In principle, the sharpness of the metamagnetic transition is an aspect of the quality of a sample of UCoAl. Among the numerous samples which we have characterized, one of them showed an extraordinary sharp transition, with the magnetization undergoing a discontinuous jump whatever the field sweep rate. In this section we present magnetic measurements performed at our laboratory with an MPMS down to 2 K, as well as some complementary measurements down to 100 mK, performed with Carley Paulsen (Institut Néel, CNRS -Grenoble) on the same sample, with a home-made instrument (a dilution fridge mounted on an extraction SQUID magnetometer).

The sample was cone-shaped, cleaved out near the tail of the ingot. Single crystal and orientation was checked by X-ray Laue photograph. In the MPMS, the sample was glued with GE varnish onto a plastic foil inserted inside a plastic straw. For the experiment in dilution fridge, the sample was glued with GE varnish onto a copper plate, and thermal contact was provided by copper foil wrapped around the sample and Apiezon N-grease. The field was applied along the c-axis.

Magnetization versus field

Figure 5.25 shows M -H measurements at various temperatures from 0.1 K to 13 K. Points are taken every 5 Oe in the transition region, with a delay of 1 30 between two measurements. Below 10 K, we can see discontinuities at the transition, and a clear hysteresis between up and down sweeps. From these data we obtain the T -H phase diagram of UCoAl at zero pressure (Fig. 5.26). The hysteresis expands rapidly on cooling below ∼ 2 K. The temperature of the critical endpoint is found at T 0 = 10 K.

Metamagnetic transition: relaxation effects

Actually, in the vicinity of H m before the transition, the system is unstable, and it eventually transits spontaneously after some time. This transient regime could be observed in both directions : from PM to FM (relaxation «up») and from FM to PM (relaxation «down»). From now on, we distinguish the critical field for up-sweep (H inc m ) and for down-sweep (H dec m ). Figure 5.27 shows the details of the hysteresis at 2 K and indicates the unstable regions where the relaxation phenomena are observed.

In order to measure relaxation processes, the field was rapidly increased from 0 to H H inc m in the case of up-sweeps, or rapidly decreased from ∼ 2 T to H H dec m in the case of down-sweeps. The measurements were started after cooling the shunt in order to have the magnet in persistent mode (∼ 15 s delay), or as soon as the field H was established (for short-time relaxation measurements). In any case the field was steady during the whole measurements. The results are independent of the magnet state (driven/persistent mode) and of the measuring rate (typical extraction time is 4 s, and the delay between points can be shrinked to arbitrary low value). Figure 5.28 shows relaxation curves (up and down) at 2 K for different steady fields H. Time is plotted on a log scale. Relaxation curves M (t) could be measured in a field range of about 50 Oe before H inc m and H dec m , where the time scale of the relaxation ranges from a few seconds to a few hours.

The relaxation curves are not typical exponential decays. The magnetization relaxes step by step. We observe two rapid jumps of ≈ 0.08 µ B each, where the slope dM/dt diverges. The last jump leads to the stable FM state with M = 0.3 µ B . Starting from different initial values, all M (t) curves have the same steps at the same values of M , and they all finish in the same stable final state. Similar relaxation curves are observed up to 5 K, but over a gradually reduced field window.

During the experiment performed at CNRS with the dilution fridge, we measured relaxation curves at 100 mK, 500 mK, 1.2 K and 2 K. The data are similar to those obtained at INAC, but not identical. At 100 mK, the relaxation appears to be too slow to measure over reasonable time scales and we do not show the data here. The relaxation at 500 mK are shown in Figure 5.29. It is interesting to note that the points of inflection in the relaxation occur at a given value of magnetization which is temperature-and field-independent. We note that there are some strange problems with reproducibility. We observed step-like features in the relaxation at given values of magnetization in all applied fields. However, these features disappeared after the extractor was replaced (see Fig. 5.29-b) and were not observed during the time that we took the 500 mK data shown in Fig. 5.29-a, despite many different measuring protocols being attempted so that to get the multiple step features back again. Eventually, after a few days of M vs T , the features returned again when measuring at 1.2 K etc. This indicates that the shape of the relaxation curves is highly sensitive to the sample environment, possibly through mechanical stress induced by the glue and sample holder.

Effect of field and pressure on the relaxation time

M H (t) curves taken at a given temperature and different fields are superimposable in a logarithmic time scale, which means that they only differ by a time scaling. Hence there are two aspects to consider:

• the shape of a relaxation curve, with plateaux and jumps

• the effect of field and temperature on the characteristic time of the relaxation curves where ∆H > 0 (∆H < 0) for relaxation up (resp. down). The constant h has the dimension of a field, and it is the characteristic width of the unstable region, such that: 8 . In a simple picture, the field window within which relaxation curves are measureable in a reasonable time is ∆H obs = 4h. This field window reduces with temperature: from ∆H obs ∼ 50 Oe at 500 mK to only ∼ 10 Oe at 5 K.

τ (H inc m -0h) = 1 s ; τ (H inc m -4h) ≈ 1 min ; τ (H inc m -8h) ≈ 1 hour
We already noted that relaxation curves M (t) at a given temperature only differ by their time scale. The farther from H m , the slower the relaxation. We can therefore write the relation between two relaxation curves taken at H and H = H -∆H:

M H-∆H (t) = M H te -∆H/h (5.6)
and in a log time scale, they are just shifted horizontally:

M H-∆H (ln t) = M H ln t - ∆H h (5.7)
The external field actually tunes the speed of the relaxation.

Interpretation of the relaxation effects

These magnetic relaxation effects are very unusual. To our knowledge, such relaxation curves have never been observed in materials more or less related to our topic 9 . So far, we found no explanation for these phenomena. We put forward 3 possible mechanisms that could be relevant in the context of UCoAl: i) A percolation mechanism. Clusters of FM phase nucleate throughout the sample, and grow up independently. When a critical density is reached, the multitude of independent clusters merge into a single entity. This critical point is a singularity, in the sense that the delocalized spin-polarized electrons suddenly escape from microscopic confinement (cluster-size) to the infinite lattice, producing a sudden Fermi surface reconstruction. This picture could explain the diverging ∂M/∂t that occurs in the relaxation curves.

ii) Lattice strain relaxation : this invokes the magnetostriction as a source of irreversibility responsible for the simultaneous onset of ferromagnetism within the basal plane and between the planes. There is a strong magnetostriction accompanying the metamagnetic transition in UCoAl. As a consequence, when a part of the sample becomes ferromagnetic, a strain field forms at the boundaries, which stands as a potential barrier separating the two states of the system. In phase space, the system has to overcome this potential in order to go from one state to the other. This is the direct manifestation of the first-order nature of the transition, giving rise to a hysteresis. In real space, when the field is close to but below H m , pockets of FM phase nucleate by thermal activation over the potential barrier ∆ with a probability τ -1 = exp ∆-µB k B T . This is consistent with the exponential increase of the hysteresis with decreasing temperature observed in magnetization (Fig 5 .26) and in resistivity (Fig. 5.23). At low temperature, the relaxation becomes extremely fast (we observe a discontinuous jump at 100 mK), as if the band splitting between spin-up / spin-down was occuring simultaneously in the whole sample.

iii) A spin glass state: The existence of AF interactions -if any -would give rise to frustration on the quasi-Kagome lattice of U atoms, and may thus produce relaxation effects. This possibility was also put forward in Section 5.3.5 to explain the peculiar difference between up and down field sweep in the transverse magnetoresistance.

Magnetization under pressure

We performed magnetization measurements of UCoAl under pressure with a non-magnetic pressure cell (Cu-Ti alloy) which fits in a commercial SQUID magnetometer (MPMS, Quantum Design). This pressure cell is based on the model developped by Kobayashi et al. [START_REF] Kobayashi | Nonmagnetic indentertype high-pressure cell for magnetic measurements[END_REF] and Tateiwa et al. [START_REF] Tateiwa | Miniature ceramicanvil high-pressure cell for magnetic measurements in a commercial superconducting quantum interference device magnetometer[END_REF], and modifed by Yusei Shimizu in 2009 at Hokkaido University. We used Daphne oil 7373 as a pressure transmitting medium. The sample was a single crystal oriented along the c-axis, and cut into hexagonal shape in order to optimize the sample space. Pressure was determined from the superconducting transition of Pb. The background -mainly due to the NiCrAl gasket -was measured independently and substracted from the data. In Fig. 5.31-a we present M -H magnetization curves of UCoAl along the c-axis, at 2K, for various pressures up to 1.8 GPa. Pressure shifts the transition to higher fields, as expected. We can track the transition up to 5.5 T, which is the limit of our magnetometer. The magnetization in the FM state slowly decreases with pressure. But pressure has no effect on the slope dM/dH, neither in the PM or FM state. The transition anormaly broadens with pressure: from 0.25 T at zero pressure to 0.7 T at 0.7 GPa and 1 T at 1.3 GPa. At 1.8 GPa, the end of the transition is beyond the limit of the magnet. The pressure dependence of H m (taken as the middle of the transition) is plotted in Fig. 5.31-b along with the values obtained by other measurements. Our magnetization measurements are not in agreement with the other probes : there is a lower pressure dependence, which is comparable to the magnetization data reported by Mushnikov et al.. There is definitely a problem with magnetization measurements under pressure. The probable reason is the very high sensitivity of UCoAl to non-hydrostaticity. We tried to improve the pressure conditions by reducing the size of the sample, by using different gaskets, without notable improvement. Attemps to improve the hydrostaticity by using liquid Argon as a pressure transmitting medium failed, as we did not succeed to obtain any pressure (Ar was probably not successfully trapped in the pressure chamber). Resistivity data was reported in [START_REF] Aoki | Ferromagnetic Quantum Critical Endpoint in UCoAl[END_REF] and magnetization data is taken from [START_REF] Mushnikov | Magnetic properties of the 5f itinerant electron metamagnet UCoAl under high pressure[END_REF].

X-ray Magnetic Circular Dichroism

The degree of delocalization of the 5f electrons governs the magnetic properties of Uranium intermetallics. The metamagnetism in UCoAl is usually considered in a itinerant picture (band metamagnetism). The strong spin-orbit coupling in the 5f states induces a sizeable orbital moment which is comparable and antiparallel to the spin moment. The ratio of orbital-to-spin moments provides information on the degree of delocalization of the 5f electrons [START_REF] Lander | Magnetic and structural study of La 1.8 Sr 0.2 NiO 4[END_REF]. In many U compounds (URhAl, UIrAl, UPtAl, etc.) this ratio is reduced (≈ -2) compared to the free ion values (-3.42 and -2.57 for 5f 2 and 5f 3 configurations, respectively). Furthermore, the individual values of µ L and µ S of the 5f orbitals are key parameters to understand the metamagnetic behaviour of UCoAl. These microscopic properties can be adressed by X-ray magnetic circular dichroism (XMCD) measurements. We performed an XMCD experiment at the European Synchrotron Radiation Facility (beamline ID12), in collaboration with Alexandra Palacio Morales, Andrei Rogalev and Fabrice Wilhelm. The motivation was to measure the orbital and spin part of the magnetization in order to compare with band structure calculations and polarized neutron diffraction experiments, and to see whether there is a change of the orbital-to-spin moment ratio between the PM and FM state of UCoAl on the U site. In addition, Coand U-specific magnetization measurements may address the question of whether the Co holds an intrinsic or an induced moment. In the past, XMCD experiments had been performed at the M 4,5 absorption edges of Uranium on a polished crystal, at ∼ 10 K, between 0.7-7 T. The authors could not conclude about the question of a change in µ L /µ S . We repeated the measurements at the U M 4,5 -edges, with improved experimental conditions: lower temperature T = 2 K, more extended field range H = 0.5, 1, 17 T, higher surface quality (cleaved sample), and a high quality sample of UCoAl (sharper metamagnetic transition). We also measured the XMCD signal of UCoAl at the Cobalt K-edge, which, to our knowledge, have never been reported before. In this section, we introduce the principle of the XMCD technique, and how we can retrieve the orbital and spin moments by applying sum rules. We then present our results and compare with reported XMCD experiments. We also relate our results with neutron and NMR experiments reported in the literature [START_REF] Nohara | Strong Longitudinal Magnetic Fluctuations Near Critical End Point in UCoAl: A (59)Co-NMR Study[END_REF][START_REF] Karube | Universality and critical behavior at the critical endpoint in the itinerant-electron metamagnet UCoAl[END_REF][START_REF] Wulff | Study of orbital and spin magnetism in UCoAl by polarized neutron diffraction[END_REF].

Principle

X-ray magnetic circular dichroism (XMCD) is a spectroscopic technique that utilizes circularly polarized X-rays from a synchrotron source to obtain information on the microscopic magnetic properties of a material. It is sensitive to the spin and orbital magnetic moments of an absorbing atom. By definition, XMCD is the difference of absorption spectrum (dichroism) between left and right circularly polarized photons. It is essentially an element-specific and orbital-selective magnetometry tool, which is particularly adapted for the study of 3d, 4d, 5d transition metals, rare earths and actinides.

X-ray absorption is proportional to the density of unoccupied states (DOS above the Fermi level), with different absorption cross-sections (α + , α -) according to the helicity of the photons -parallel or antiparallel to the projection of the spin or orbital angular momentum µ s , µ l of the electrons (which are decided by the field direction). The XMCD signal is written as:

∆α = α + -α - (5.8)
At a given absorption edge, XMCD is sensitive to the difference of the spin ↑/spin↓ DOS above the Fermi level. The uranium M 4,5 absorption edges correspond to excitations of 3d electrons to the 5f states10 :

UM 4 : 3d 3/2 → 5f 5/2 UM 5 : 3d 5/2 → 5f 7/2,(5/2)
Left or right handed photons transfer their angular momentum to the excited photoelectrons, and the destination empty states act as a detector for this angular momentum. Since the initial state is known, and the transfered angular momentum as well, the XMCD at the U M 4,5 is sensitive to the difference between the 5f spin ↑/spin↓ DOS above the Fermi level.

Sum rules

From the absorption and XMCD spectra of a pair of spin-orbit split core levels, we can extract microscopic information on the ground state of an atom. For this purpose, Thole, Carra et co-workers established a set of sum rules [START_REF] Thole | X-ray circular dichroism as a probe of orbital magnetization[END_REF][START_REF] Carra | X-ray circular dichroism and local magnetic fields[END_REF]. They relate the integrated intensities of the isotropic absorption spectra (referred to as XANES: X-ray absorption near-edge spectroscopy) (I M 4,5 ) and XMCD (∆I M 4,5 ), to:

• the angular part of the spin-orbit splitting of the 5f valence shell

• the orbital moment

• the spin moment These sum rules were derived within an atomic model. When applied to solids, some input parameters are required (the number of valence electrons and the spin-orbit coupling scheme) thus adding some uncertainty to the results.

First sum rule: angular part of the spin-orbit coupling

The first sum rule is used to get the number of spin-orbit split 5f 5/2 and 5f 7/2 empty states. The simple idea is that the ratio of integrated intensities of the XANES (I M 4,5 ), called branching ratio, reflects the ratio of 5f 5/2 and 5f 7/2 empty states.

Formally, the branching ratio is related to the average value of the spin-orbit interaction in the 5f states:

l • s = - 15 4 n 5f h I M 5 I M 5 + I M 4 - 3 5 + 3 2 n 5f h • ∆ (5.9)
which, in turn, is related to the occupation number of 5f 5/2 and 5f 7/2 states:

l • s = 3 2 n 7/2 -2n 5/2 (5.10) 14 -n 5f h = n 7/2 + n 5/2
(5.11) (5.12)

The second term in eq. 5.9 is a correction which takes into account the overlap of XANES spectra. When l • s is large, this term is vanishing. From the system of equations {5.10,5.11} we can obtain n 7/2 and n 7/2 if we know the number of valence electrons. • U-M 4 : Perfect matching is achieved at all fields with ∆α 0.5 T /∆α 17 T = 45 and ∆α 1 T /∆α 17 T = 1.725.

• U-M 5 : the XMCD spectra at 1 T and 17 T can be superimposed with a ratio ∆α 1 T /∆α 17 T = 1.42 which is different than at U-M 4 . But the 0.5 T spectrum cannot be satisfactorily superimposed to the others, despite the tolerance allowed by the relatively large standard deviation of the data points. Eventually, the negative part can be superimposed to the other spectra with a ratio ∆α 0.5 T /∆α 17 T = 45 as at U-M 4 , but then the positive part is ∼ 2 times larger.

Van der Laan et al. have predicted that the shape of the XMCD spectra at the U N 5 and M 5 edges is sensitive to the number of valence electrons in the U atom [START_REF] Van Der Laan | X-ray-absorption sum rules in jj-coupled operators and ground-state moments of actinide ions[END_REF]. However, because of the weak signal, the spectral shape of the U-M 5 XMCD at 0.5 T should be considered with precautions, especially since different spectral shapes were observed in the reported experiments:

• At the same U M 5 -edge, the XMCD signal was found to have no positive part at low field (0.7 T) [START_REF] Kučera | X-ray magnetic circular dichroism studies of 5f magnetism in UCoAl and UPtAl[END_REF].

• In the soft X-rays, all XMCD spectra at U N 5 -edge could be superimposable [START_REF] Takeda | Separation of magnetic properties at uranium and cobalt sites in UCoAl using soft X-ray magnetic circular dichroism[END_REF] (no change of spectral shape).

As a conclusion, we can hardly confirm or infirm a possible valence change at the metamagnetic transition in UCoAl, owing to the important variability of the XMCD spectral shape at low field among the various experiments.

From the second and third sum rules (eqs. 5.13 and 5.15), we have ∆I M 4 ∝ L z -2 S eff z and ∆I M 5 ∝ L z + 4 S eff z /3. At U-M 4 , the large negative symmetrical XMCD means that µ L and µ eff S contributions add up. At U-M 5 , the small S-shape XMCD indicates that their contributions almost cancel out.

From these simple observations, it already appears that µ L and µ eff S are antiparallel, with µ L /µ eff S ∼ -2. Precise determination of these values will be discussed in Section 5.5.5.

In Figure 5.34 we plot the field dependence of the maxima of XMCD signal at U M 4,5 -edges, and we compare with the data of Ref. [START_REF] Kučera | X-ray magnetic circular dichroism studies of 5f magnetism in UCoAl and UPtAl[END_REF]. The amplitudes of the U-M 4,5 XANES peaks (corrected for self-absorption and normalized to edge jump) are similar, but in our experiment the XMCD signal is quite larger : ∼ 1.5 times larger at U-M 4 , and 2-3 times larger at U M 5 -edge.

Co K-edge

The Cobalt K-edge corresponds to the transitions 1s → 4p. Since there is no spinorbit splitting of the starting levels, XMCD only measures the orbital moment of the 4p electrons. XANES and XMCD at the Co K-edge are presented in Fig. 5.35. Measurements were performed at 0.5, 1, and 17 T. Only isotropic XANES are shown since left and right polarizations would not be distinguished on the graph. The XMCD signal is indeed very small (XMCD/XANES ∼ 10 -3 at 17 T). Since 4p states are in the conduction electrons continuum, there is no white line peak on top of the Co K XANES edge jump, and it is very difficult to apply sum rules. The structures above ∼ 7730 eV are resonances ascribed to the local environment of the Co atoms, but the study of this so-called EXAFS (Extended X-ray Absorption Fine Structure) outreaches the scope of our experiment. Around 7752 eV and 7760 eV, there are non-reproducible features that are due to diffraction peaks.

In the XMCD spectra, the sharp positive and negative peaks around 7710-7715 eV are dominant, and do not exist in Co metal. They are known to originate from a large spinorbit coupling with U orbitals 11 , so it indicates that the Co moment is mostly induced by the U. These induced peaks can be used to detect a change of the hybridization between U-5f and Co-4d bands, by following the evolution of the peak signal with field.

Element-specific magnetization

Co-specific magnetization curve M -H was obtained by measuring XMCD at a fixed photon energy hν = 7710.9 eV, corresponding to the maximum of the positive peak which originates from Co(4p)-U(5f ) hybridization. The field was increased step by step along a loop pathway H = 0 → +17 T → -17 T → 0, and the signal was recorded for the two different beam polarizations at every field value. The dichroism was separated from the residual contribution by averaging over positive and negative fields. Similarly, U-specific M -H curve was obtained at fixed energy hν = 3725.8 eV corresponding to the maximum of XMCD signal at U M 4 -edge. Results are presented in Figure 5.36, in which the Co-and U-specific magnetizations are compared with the bulk magnetization of UCoAl measured with a SQUID magnetometer. 12 The residual contributions are also plotted, showing no field dependence. Both U and Co magnetizations show a sharp step-like increase at H m . The U curve almost perfectly scales with the bulk magnetization, following the same negative curvature above H m , while the Co magnetization has no curvature. The difference is small, thus the proportion of U moment induced on the Co is quite constant. A possible interpretation for the absence of curvature in the Co magnetization is that the Pauli paramagnetism is non-negligible compared to the U-induced polarization, thus adding a significant linear contribution. That the Pauli paramagnetism is visible would imply a very small Co moment, which is indeed consistent with NMR and neutron diffraction results [START_REF] Nohara | Strong Longitudinal Magnetic Fluctuations Near Critical End Point in UCoAl: A (59)Co-NMR Study[END_REF][START_REF] Karube | Universality and critical behavior at the critical endpoint in the itinerant-electron metamagnet UCoAl[END_REF][START_REF] Wulff | Study of orbital and spin magnetism in UCoAl by polarized neutron diffraction[END_REF]. In the reported soft-XMCD experiment [START_REF] Takeda | Separation of magnetic properties at uranium and cobalt sites in UCoAl using soft X-ray magnetic circular dichroism[END_REF], the authors compared the field dependence of the XMCD signal at the U N 5 and Co L 2 edges. They found a different magnetization profile on U and Co, but their result might be biased by the overlap of Co L 2 with U N 4 edge. In the inset of Fig. 5.36, we see the details of the metamagnetic transition with up and down sweeps, compared with bulk magnetization at 2 K. Nominal temperature of the XMCD experiment was 2 K. The transition is as sharp as the bulk magnetization in up-sweep curves (< 150 Oe), but broader in down-sweeps (∼ 300 Oe). The shape of the transition may be affected by the sample environment: for the XMCD experiment, the sample was strongly maintained by GE varnish inside a cavity in the Cu plate, and may thus experience stress.

The up-sweep critical field is similar for both XMCD signals and almost matches (∼ 100 Oe higher) with the bulk magnetization. On the contrary, down-sweep critical fields are quite different, with a larger hysteresis of the U M 4 XMCD signal and a smaller hysteresis at Co K-edge as compared to the bulk magnetization.

The ∼ 2 times smaller hysteresis in Co than in U could be ascribed to a discrepancy between the nominal temperature of the XMCD experiment (2 K) and the temperature of the sample under X-ray irradiation. Indeed, the transition in the Co signal resembles the bulk magnetization at ∼ 3 K (see Fig. 5.25). The higher luminosity of the beam at the Co K-edge could explain a higher sample temperature. But in any case the sample heating is very limited and does not impact the results at higher field. Beside a possible experimental bias, these hysteresis curves would otherwise mean that Co demagnetizes first. This would be again another aspect of the peculiar properties of UCoAl observed in magnetic relaxation effects and resistivity measurements.

Analysis

In this section we present the sum rule analysis of the U M 4,5 spectra. All sum rules require the number of 5f holes as an input, which is unknown. We thus consider the two atomic cases for the U electronic configuration: 5f 3 (U 3+ ) and 5f 2 (U 4+ ), and we get n 5f h = 14 -n 5f e -= 11 and 12, respectively. First sum rule involves the branching ratio BR =

I M 5 I M 5 +I M 4
, which is determined experimentally, and is field-independent: BR = 0.698 Solving eqs. {5.10,5.11} for the two electronic configurations leads: with values of the correction ∆ taken from the literature [START_REF] Wilhelm | X-ray Magnetic Circular Dichroism studies of Magnetism in Actinides[END_REF]. The 5f 2 configuration leads to non-physical values of n 7/2 and n 5/2 . Nevertheless, our results confirm that Second and third sum rules (Eqs. 5.13 and 5.15) yield the orbital moment µ L and the effective spin moment µ eff S , which we determine for the two electronic configurations 5f 2 and 5f 3 . The spin moment µ S can be determined from the effective spin moment µ eff S using theoretical values of the dipolar magnetic term µ DM , as shown in eq. 5.16. Table 5 Alternatively, the spin moment µ S can be determined independently of µ eff S , by considering that the magnetic moment is entirely carried by the U 5f orbitals:

µ bulk = µ L + µ S ( 5.22)
Results are summarized in Table 5.2, at H = 0.5, 1, 17 T, for the 2 electronic configurations 5f 2 and 5f 3 .

In the first part of the table we show µ L , µ eff S , and µ L /µ eff S obtained from sum rules. The ratio µ L /µ eff S changes from -0.55 in the PM state to ≈-0.75 in the FM state, and it is independent of the number of 5f electrons since it only results from the XMCD spectra and does not involve any external parameter. In the second part, we used the bulk magnetization µ bulk as an input to derive µ S , µ L /µ S , and µ DM /µ S . In the PM state, bulk measurements performed on the same sample with two different magnetometers yielded µ bulk (0.5 T) = 0.12 µ B and 0.16 µ B . This deviation is comparable to |µ S | and thus undermines the ratios µ L /µ S and µ DM /µ S at 0.5 T. In the FM state, the error in the bulk magnetization is negligible. There is only the uncertainty on the number of valence electrons, which propagates to the ratio µ L /µ S , but the differences between 5f 2 and 5f 3 remain quite small: the ratio µ L /µ S is found between -2.40 --2.15 at 1 T and between -2.15 --1.95 at 17 T. These values are notably reduced compared to the free ion values of -3.42 (5f 2 ) and -2.57 (5f 3 ). The values of µ DM /µ S inferred from our results are close to 2, which corresponds to the theoretical value ( 1.85) for the 5f 3 configuration in the intermediate coupling scheme. In the last part of the table, we determined µ S , µ L /µ S , and the total U moment phase. A small additional linear contribution in the Co magnetization is ascribed to a sizeable Pauli paramagnetic effect. This suggests that the Co moment is very small, which is consistent with NMR and polarized neutron diffraction results [START_REF] Nohara | Strong Longitudinal Magnetic Fluctuations Near Critical End Point in UCoAl: A (59)Co-NMR Study[END_REF][START_REF] Karube | Universality and critical behavior at the critical endpoint in the itinerant-electron metamagnet UCoAl[END_REF][START_REF] Wulff | Study of orbital and spin magnetism in UCoAl by polarized neutron diffraction[END_REF]. Looking at the hysteresis at the metamagnetic transition, we observed different critical fields for U and Co on decreasing field sweep, which could suggest that the Co site demagnetizes before the U site. This peculiar result should be considered with precaution as long as it is not confirmed by another microscopic probe such as NMR. Individual values of the orbital and spin moments of Uranium were obtained by applying sum rules. They are in good agreement with other XMCD [START_REF] Kučera | X-ray magnetic circular dichroism studies of 5f magnetism in UCoAl and UPtAl[END_REF] and polarized neutron diffraction results [START_REF] Wulff | Study of orbital and spin magnetism in UCoAl by polarized neutron diffraction[END_REF]. Our results confirm that the magnetism in UCoAl is dominated by the orbital component, which is larger and antiparallel to the spin moment. The ratio of orbital-to-spin moment µ L /µ S is close to -2 -which is reduced from the free ion value -in good agreement with ab initio calculations [START_REF] Antonov | Electronic structure and X-ray magnetic circular dichroism in uranium compounds. II. UT Al (T = Co, Rh, and Pt) intermetallics[END_REF].

Moreover, we observed a change of µ L /µ S from -1.6 to -2.1 accross the metamagnetic transition. This change occurs while the ratio of induced Co to U moment is conserved, hence the U-5f -Co-4d hybridization is basically unchanged. We recall that the orbitalto-spin moment ratio is normally considered as a measure of the 5f -ligand hybridization [START_REF] Lander | Magnetic and structural study of La 1.8 Sr 0.2 NiO 4[END_REF]. Therefore in this case this is not completely true. It means that µ L /µ S is affected by the splitting into spin-up-spin-down bands at the metamagnetic transition, possibly via a change in the spin-orbit coupling of the 5f electrons, or maybe the quenching of the orbital moment by the crystal field is partially released when the large magnetostriction occurs.

General conclusion

The aim of this thesis was to study new materials in order to explore the physics of ferromagnetic quantum criticality. This follows the discovery of the ferromagnetic superconductors in uranium-based intermetallic compounds, and the need to find ideal systems to study the mechanisms which allow the coexistence of ferromagnetism and superconductivity.

In this thesis we have investigated three uranium-based ternary intermetallic compounds: URhSi, URhAl, and UCoAl. These are strongly correlated electron systems, with moderate heavy fermion behaviour (γ ∼ 100 mJ.mol -1 .K -2 ). URhSi and URhAl are itinerant ferromagnets, while UCoAl is a paramagnet being close to a ferromagnetic instability. The quantum critical behaviour in these compounds have been studied under high pressure and magnetic field. Single crystals of these materials have been grown and characterized by resistivity, magnetic, and specific heat measurements.

In Chapter 3 we discussed URhSi. High quality single crystals are difficult to obtain. We have succeeded to grow single crystals of URhSi, however the quality is very low (RRR 4). With thermal expansion measurements we observed that the Curie temperature T C = 10.5 K reduces upon application of a magnetic field perpendicular to the easy magnetization axis. By extrapolating T C to 0 K, a quantum phase transition is expected at ≈ 40 T, in relation with a resistivity anomaly. Given the strong similarities with the ferromagnetic superconductor URhGe, URhSi is expected to attract a lot of attention, now that its phase diagram is being unveiled. However, the main limitation for now is the low crystal quality.

The pure FM ground state in URhAl has been confirmed by neutron diffraction. The critical behaviour of the PM-FM transition at 27 K has been studied with Arrott-Noakes analysis of the magnetization. Although URhAl is certainly a 3D Ising system, the critical behaviour does not correspond to any universal class of criticality, probably because at these temperatures close to the Kondo coherence the system is in the frontier region between the Curie-Weiss picture of localized moments and the itinerant picture of band magnetism. URhAl has been studied under high pressure, focusing on the Curie temperature with AC calorimetry measurements. T C decreases with pressure and reaches 0 K at ≈ 5.3 GPa. We thus established the pressure phase diagram of URhAl for the first time, thereby opening the way to further studies. Preliminary results of resistivity 139 recently obtained by our team are indicating that the A coefficient becomes very large in the quantum critical region (∼ 5 GPa), where non-Fermi liquid behaviour is observed.

The chapter on UCoAl is the main part of this thesis. The microscopic properties of the magnetization have been investigated with X-ray magnetic circular dichroism (XMCD), by probing the U 5f and Co 4p density of states. The U moment is dominated by the orbital part, which is antiparallel to the spin part. The orbital-to-spin moment ratio changes from -1.6 in the PM phase to -2.1 in the FM phase. We confirmed that the magnetism of UCoAl is almost entirely carried by the U 5f orbitals, and that the small Co moment is not intrinsic, but is induced by hybridization with 5f orbitals. The ratio of Co and U magnetization is conserved on either side of the metamagnetic transition, thus the hybridization seems unchanged.

The metamagnetic transition has been studied by bulk magnetization measurements. Very unusual relaxation phenomena have been observed, showing a succession of steps. The field tunes the speed of the relaxation but does not affect the steps. We proposed several possible explanations for these phenomena.

The transport properties have been studied under pressure by Hall effect and resistivity measurements. We have drawn the wing-shape temperature-field-pressure phase diagram of UCoAl, and we explored the region of the quantum critical endpoint. The Hall coefficient does not change significantly between the PM and FM phase. In the meantime, thermoelectric power measurements performed in our laboratory showed that S/T changes drastically. Since Hall effect and thermoelectricity are sensitive to light and heavy bands, respectively, we thus believe that they are probing different bands, so the question of a Fermi surface change is not so clear. The positive jump in Hall effect at the metamagnetic transition disappears at the pressure P M ≈ 1.3 GPa, while a positive jump appears in the transverse resistivity at the same pressure. The anisotropy of the resistivity under high pressure has been investigated in detail. In the longitudinal resistivity, nothing happens at P M . The metamagnetic transition then terminates at a pressure P ∆ ∼ 2 T. In the region between P M and P ∆ , a plateau appears in the transverse resistivity, with a large difference between increasing and decreasing field sweep. This up-down difference is a very peculiar phenomenon, which should be further investigated in the future. This is suggesting that a new exotic phase appears in the region of the quantum critical endpoint.

Eventually, UCoAl is not a simple system, probably because of frustration on the quasi-Kagome lattice formed by the uranium atoms. Instead, UCoAl is a promising candidate for several unexpected physical phenomena, which may lead physicists to discover new aspects of solid state and quantum matter.

Figure 1 . 1 :

 11 Figure 1.1: Doniach phase diagram of a Kondo lattice system. There is an opposite pressure response for Ce and Yb because of the opposite sign of the charge carriers. Credits: E. Bauer (TU Wien, Austria).

Figure 1 . 2 :

 12 Figure 1.2: (taken from [19]) Free energy F as a function of the order parameter M (magnetization), in a FM system with second-order transition (a,b) and first-order transition (c,d). The panels illustrate the role of temperature at zero field (a,c), and the role of field H M close to the FM transition (b,d). Dots indicate the stable states. Insets show the field dependence of M associated with panels b & d.

Figure 1 . 3 :

 13 Figure 1.3: Schematic Temperature(T )-Pressure(P )-Field(H) phase diagram of an itinerant ferromagnet (FM), where the Curie temperature T C is suppressed by pressure.The ground state is paramagnetic (PM) above the critical pressure P c . The transition becomes first-order at the tricritical point (TCP). Above the critical pressure P c , there is a metamagnetic «wing» at temperatures below the line of critical endpoints (CEP), which terminates at a quantum critical endpoint (QCEP).

Figure 1 . 5 :

 15 Figure1.5: (taken from[START_REF] Aoki | Ferromagnetism and Superconductivity in Uranium Compounds[END_REF]) Crystal structure of URhSi (TiNiSi-type). The arrows on the U sites denote the direction of the magnetic moments.

Figure 1 . 6 :

 16 Figure1.6: (taken and adapted from[START_REF] Sechovský | Chapter 1 Magnetism of ternary intermetallic compounds of Uranium[END_REF]) Schematic plot illustrating evolution of some electronic properties (type of magnetic ground state, magnetic ordering temperature, γvalue (linear coefficient of the specific heat), spontaneous magnetic moment (black part of a column), induced magnetic moment in field of 35 T (white top parts of a column) of UTSi and UTGe compounds with the orthorhombic TiNiSi-type structure or the closelyrelated CeCu 2 -type structure. PM/SF stands for paramagnetic with spin fluctuations.

Figure 1 . 7 :

 17 Figure1.7: (taken from[START_REF] Aoki | Ferromagnetism and Superconductivity in Uranium Compounds[END_REF]) Magnetization curves and temperature-field phase diagram for H b-axis in URhGe. SC, RSC and FM denote superconductivity, re-entrant superconductivity and ferromagnetism, respectively. The inset shows the field dependence of resistivity at low temperatures (≈ 80 mK). It is noted that the field range of RSC is very sensitive against a small mis-orientation to the c-axis.

Figure 2 . 1 :

 21 Figure 2.1: Picture of the radio-frequency furnace. We can see the coil and, inside the quartz tube, the material melting in the crucible.

Figure 2 . 2 :

 22 Figure 2.2: Crystal growth in a tetra-arc furnace by the Czochralski technique

Figure 2 . 3 :

 23 Figure 2.3: X-ray Laue schematic and photograph. Here in the case of UCoAl.

  .4 has been used for resistivity, Hall effect, magnetization with Hall

Figure 2 . 4 :

 24 Figure 2.4: Details of a piston-cylinder pressure cell, with pictures of a setup (here, for Hall effect measurements).

Figure 2 . 5 :

 25 Figure 2.5: Pictures of the indenter-type pressure cell for magnetic measurements.

Figure 2 . 6 :

 26 Figure 2.6: Sketch of the indenter-type cell for magnetic measurements. Credits: Y. Shimizu

Figure 2 . 7 :Figure 2 . 8 :

 2728 Figure 2.7: Magnetization of the empty cell (background), and with a sample of UCoAl inside. Here, P = 0.

Figure 2 . 9 :

 29 Figure 2.9: Diamond anvil cell pictures: a) sample of URhAl with thermocouple Au-AuFe for AC calorimetry; b) stainless steel gasket showing the footprint of the anvil and a hole of ø350 µm for the sample space; c) pressurized cell.

Figure 2 . 11 :

 211 Figure 2.11: Thermocouple response used in the analysis of AC calorimetry data. This model was obtained by fitting an average over several different thermocouple calibrations.

Figure 2 . 12 :Figure 2 . 13 :

 212213 Figure 2.12: Electrical circuit with two strain gauges mounted in a Wheatstone bridge.

Figure 3 . 1 :

 31 Figure 3.1: (courtesy of D. Aoki) Magnetization vs field in URhSi along the main crystallographic directions.

Figure 3 . 2 :

 32 Figure 3.2: Resistivity vs temperature of URhSi, from 300 K to 2 K (logarithmic scale) measured on a bar-shape sample of RRR = 3.5, with current along a-axis. The arrow indicate the Curie temperature. Inset shows the data plotted against T 2 , and a solid line shows a T 2 fit of the lowest temperature.

Figure 3 . 3 :

 33 Figure 3.3: C/T vs T 2 of URhSi. The value of T C = 10.5 K is obtained by equal entropy partition (thin solid path). Fits to equation 3.1 are represented by the solid line (low temperature) and dashed lines (T > T C ).

Figure 3 . 4 :

 34 Figure 3.4: Thermal expansion of URhSi, measured with strain gauges along the crystallographic directions a, b and c.

Figure 3 . 5 :

 35 Figure 3.5: Thermal expansion along the b-axis, with a field of 9 T applied along the b-axis. The sample was mounted on a 1-axis rotator with angle varying from c-to b-axis by ±0.25 deg steps around the H ⊥ c position. Curves are shifted vertically for clarity.

  3.7).Further studies were performed at high fields by W. Knafo (LNCMI-Toulouse) by measuring magnetoresistivity and magnetization of URhSi with H b, but without rotator. His results (in proceedings of SCES2013) show a maximum in magnetoresistance around H max ρ = 42 T at ∼ 2 K (Fig.3.8). It is associated to an anomaly in magnetization, such as in URhGe or UCoGe( [50]). As temperature increases, this anomaly shifts to lower field, until it collapses at 10 K in URhSi.

Figure 3 . 9 -Figure 3 . 6 :

 3936 Figure 3.6: Field dependence of the thermal expansion coefficients along the b-and caxes. Field was applied along b, and the sample was mounted on the rotator to ensure the H ⊥ c condition. Curves are shifted vertically for clarity.

Figure 3 . 7 :

 37 Figure 3.7: Field-temperature phase diagram of URhSi obtained from α b and α c curves of Fig. 3.6, for field applied along b, with the condition H ⊥ c respected. Lines are fits using the relation (T C -T C0 ) ∝ (-H 2 ).

Figure 3 . 8 :Figure 3 . 9 :

 3839 Figure 3.8: Pulsed-field magnetoresistance of URhSi with H b, for various temperatures (1.5 to 50 K). Courtesy of W. Knafo (LNCMI-Toulouse) (unpublished).

Figure 4 . 1 :

 41 Figure 4.1: Resistivity of URhAl with current along the reciprocal a direction (red symbols) and c direction (blue symbols). Inset focuses on the low temerature part. Arrows indicate the Curie temperature.

Figure 4 .

 4 Figure 4.1 shows the temperature dependence of the resistivity along two different crystallographic directions. In a wide temperature range (300 K -30 K), ρ is almost temperature independent. There is a sharp change around 28 K which corresponds to the Curie temperature. ρ drops dramatically on cooling below T C , like in the ferromagnet URhSi (see 3.3). A small hump at T C could indicate a drop of the carrier number, suggesting the opening of a gap in the Fermi surface.Finally, URhAl obeys the Fermi liquid model over a wide temperature range with the T 2 law of resistivity being valid up to 6.2 K. Inelastic terms A J⊥c = 0.27 and A J c = 0.19 µΩ.cm.K -2 are probably largely due to the magnon contribution, as in URhSi (see Chapter 3).

2 Figure 4 . 2 :

 242 Figure 4.2: Resistivity of URhAl vs T 2 for current along a * (red symbols) and c-axis (blue symbols), and their respective low-temperature fit (straight lines). The T 2 fit is valid up to T ≈ 6.2 K for both directions, with elastic scattering coefficients A J⊥c = 0.27 and A J c = 0.19 µΩ.cm.K -2 .

Figure 4 Figure 4 . 3 :

 443 Figure 4.3: Heat capacity of URhAl as C/T vs T . A specific heat jump occurs at the Curie temperature T C = 27.5 K, and the Sommerfeld coefficient γ amounts to 76 mJ.mol -1 K -2 .

Figure 4

 4 Figure 4.4: a) Thermal expansion of URhAl along c-and a-axes, and volume expansion (green). b) Thermal expansion coefficients along c-and a-axes.

Figure 4 . 5 :Figure 4 . 6 :

 4546 Figure 4.5: Magnetization of URhAl versus magnetic field along c (red symbols) and a-axis (blue symbols), at 2 K. Inset shows the hysteresis loop of the magnetization reversal.

  3, the modified Arrott plot now consists of M -H isotherms plotted as M 1/β vs H M 1/γ . This technique was applied in Fig. 4.7 to determine the critical exponents β and γ. Straight lines have been obtained with β = 0.5 and γ = 1.3. From equation 4.3, we can derive the saturation magnetization M S = lim H→0 M (H) and the inverse initial susceptibility χ -1 0 = lim M →0 H M :

Figure 4 .

 4 Figure 4.7: a) M -H isotherms of URhAl around T C ; b) modified Arrott plot of URhAl obtained with critical exponents β = 0.5 and γ = 1.3. Labels indicate the temperature (every 0.5 K between 26 K and 30 K). Lines show linear fits. T C is located between 27.5 and 28 K, when M 1/β ∝ (H/M ) 1/γ .

Figure 4

 4 Figure 4.8: a) Temperature variation of M S (0, T ) obtained from Y intercepts and χ -1 0 (T) from X intercepts in Figure 4.7, b) ln(M S ) vs ln(1-T /T C ) and c) ln(χ -1 0 ) vs ln(T /T C -1), along with the fits (solid lines) to Eqs. 4.4a and 4.4b, respectively.

Figure 4

 4 Figure 4.9: q-scans of neutron scattering on URhAl along the (q h ,0,0)-direction below (2 K, blue curve) and above (35 K, red curve) the Curie temperature. The difference (green curve) between the two scans is plotted on an expanded scale. q h is expressed in inverse relative lattice units.

Figure 4 .

 4 10.

Figure 4 .

 4 10-a shows S th RT ∝ C T vs T , and Figure 4.10-b offers an expanded view of the former plot by substracting

Figure 4 . 10 :

 410 Figure 4.10: Results of AC calorimetry of URhAl under pressure: temperature dependence of a) C T , b) C T -C T 5.8 GPa and c) phase shift θ. Legend indicates the pressure values. Curves in (a) and (b) are shifted vertically for clarity.

Figure 4 . 11 :

 411 Figure 4.11: Focus on the high pressure curves of Figs. 4.10-b (left) and 4.10-c (right). Full arrows indicate T C , and empty arrows indicate the lower and upper limits of the transition. These arrows are located at the same coordinates in both graphs. Curves are shifted vertically for clarity.

Figure 4 . 12 :Figure 4 .CT 5 . 8

 412458 Figure 4.12: Focus on the low temperature part of selected curves from Fig. 4.10-a.

C T -C T 5 . 8

 58 GPa and θ vs T . On every curve, the transition region is delimited by two cursors, and an arrow indicate the probable transition point. Around 4.3 GPa, the sharp step in θ turns into a broad shoulder-like anomaly, and the peak structure in C/T smears out.Further increasing pressure, an anomaly associated to the transition continues to appear. It consists of a change of the slope in C T -C T 5.8 GPa (visible up to 5.2 GPa). In parallel, the transition appears at the same temperature values in θ vs T as a shoulderlike anomaly, clearly visible up to 5.1 GPa. The transition temperature is plotted versus pressure in Figure4.13-a, drawing the phase diagram of URhAl. Above 5.2 GPa, no transition can be detected, at least down to 2 K.

Figure 4 .

 4 Figure 4.12 shows C/T vs T in the region of pressure and temperature where T C collapses. We observe an increase of C/T with pressure. The pressure evolution of C/T taken at 3 K and of T max θ

Figure 5 .Figure 5 . 1 :

 551 Figure 5.1(a) shows the field dependence of the magnetization at ambient pressure for B c and B ⊥ c-axis at various temperatures. At low temperatures (T = 2 K) a sharp

Figure 5 . 2 :

 52 Figure 5.2: Temperature dependence of the Hall coefficient R H for B = 0.2 T c-axis(full circles). The dotted line is a high temperature fit using magnetic susceptibility data (dotted line). The inset shows R H plotted against the magnetic susceptibility. The linear dependence indicates that the high temperature regime is dominated by the AHE. We obtain R 0 = -0.1 µΩ • cm/T and R S = 97 µΩ • cm/T.

Figure 5 . 3 :

 53 Figure 5.3: a) Determination of the coefficients R 0 and R S in the FM phase: ρ xy /B vs 4πM/B plot at 2 K (P = 0) up to 5 T, where M is the magnetization density expressed in 10 -4 emu • cm -3 and B = H + 4πM is the internal field in Tesla. The high-field part (0.8 T < B < 5 T) is fitted by eqn. 5.1 (dotted line) and we obtain R 0 = 0.14 µΩ • cm/T and R S = 3.51 µΩ • cm/T. The low-field points (0 < B < 0.4 T) shrink altogether into a single point. The data 0.4 < B < 0.7 T is not shown here since it corresponds to a transition regime which has no physical interest (different samples for ρ xy and M measurements). b) ρ xy and its constituent contributions R 0 B (NHE) and R S M (AHE) vs H (external field), in the same experimental conditions and using the values of R 0 and R S obtained in (a).

Figure 5 . 4 :

 54 Figure 5.4: Pressure evolution of ρ xy vs B at 2 K for different pressures (labels indicate the pressure in GPa). The curves are vertically spaced by 0.2 µΩ • cm for clarity. In addition, we plot the symmetric part of the measured signal, the magnetoresistance R xx , at T = 2 K as a function of B for different pressures (1.59, 1.83, 1.97, and 2.2 GPa, right scale).

Figure

Figure 5 . 5 :

 55 Figure 5.5: ρ xy as a function of field at various temperatures (0.15 to 11 K) under pressure (0.90/1.59/1.83 GPa). For clarity, curves at 1.59 and 1.83 GPa are shifted vertically by -0.4 and -1 µΩ • cm, respectively.

Figure 5 . 6 :

 56 Figure 5.6: (upper panel) ρ xy vs B for various pressures extrapolated to T = 0 (labels indicate the pressure in GPa). The dotted line indicates the pressure independent linear increase of ρ xy in the ferromagnetic state above B m which has a linear slope of 0.152 µΩcm/T. (lower panel) Field dependence of the differential Hall coefficient ∂ρ xy /∂B for different pressures.

Figure 5 . 7 :

 57 Figure

Figure 5 . 8 :

 58 Figure 5.8: Comparison of the jump in ρ xy (B) and M (B) taken from Ref. 75. Data are plotted versus B m to correct for discrepancy in pressure determination. Top axis indicates our corresponding pressure scale. Lines are guides for the eye. At P M the positive jump δρ xy vanishes, and above P ∆ only a crossover from the PM to the field induced FM state occurs.

Figure 5 . 9 :Figure 5 . 10 :

 59510 Figure 5.9: (B, T ) phase diagram of UCoAl and its evolution under pressure as obtained by Hall effect measurements. The colour scale corresponds to ∂ρ xy /∂B. Each image was obtained from a mesh of field scans at P = 0, 0.75 and 1.59 GPa.

Figure 5 . 11 :

 511 Figure 5.11: Picture of the experimental setup for resistivity measurements of UCoAl under pressure. There are two samples (for the transverse and longitudinal configurations) with H c.

Figure 5 . 12 :

 512 Figure 5.12: Resistivity versus temperature up to 300 K at zero field and P = 0.28 GPa, for the transverse (ρ xx ) and longitudinal (ρ zz ) configurations. Inset shows the low temperature dependence of (ρ -ρ 0 ).

Figure 5 . 14 :

 514 Figure 5.14: Continuation of Fig. 5.13 for the pressures P = 1.22, 1.42, 1.62, 1.80 GPa. At P = 1.42 GPa, the fit is successful only up to 0.4 K.

Figure 5 .

 5 Figure 5.15: a) A coefficient vs field, at various pressures, obtained from T 2 law fit of ρ xx in the range [0.05-0.6 K], from temperature scans (dots). b) these values of A are shifted vertically, and plotted along with A determined from up and down field scans (continuous lines). The fits were obtained over the same temperature range, involving ∼ 300 points in the T scans, and only ∼ 5 points in the H scans. Labels indicate the pressure in GPa. Arrows indicate the field sweep direction, and markers indicate H inc,dec m . Inset of panel (a) shows the value of A at H = 0 versus pressure. The line is a guide for the eye.

Figure 5 . 16 :

 516 Figure 5.16: Magnetoresistance of UCoAl at zero pressure for H c in the transverse (J a, left panel) and longitudinal (J c, right panel) configurations. The temperature T 0 = 11 K corresponds to the sharpest peak (bold line).

Figure 5 . 17 :

 517 Figure 5.17: Magnetoresistance of UCoAl at T = 4.2 K and 50 mK (P = 0.05 GPa), in the transverse ( J ⊥ H, left panel) and longitudinal ( J H, right panel) configurations, showing up and down sweeps. The curves at 50 mK for the transverse configuration are also shown on an expanded scale.

Figure 5 .

 5 Figure 5.18: a) Longitudinal (ρ zz ) and transverse magnetoresistivity (ρ xx ) versus field at various temperatures (50 mK,1 K,... every 1 K up to 14 K) for 3 different pressures. Green curves indicate the temperature T 0 of the CEP, which is determined as the sharpest peak (sharpest change in ∂ρ/∂H, panel b). Panel c shows the pressure evolution of T 0 from both resistivity configurations, and from Hall effect.

Fig 5 .

 5 18-c.

Figure 5 . 19 :

 519 Figure 5.19: Magnetoresistivity at 50 mK for various pressures, a) in the longitudinal and b) transverse configurations. In [b], the curves are shifted vertically for clarity. Panel [c] shows the low-pressure transverse resistivity on an expanded scale. Labels indicate the pressure values in GPa. Colour arrows indicate the field sweep direction (up or down). Black arrows indicate the metamagnetic field H inc,dec m

Figure 5

 5 Figure 5.20: Field-pressure phase diagram of UCoAl (for T → 0 K) from transverse (red) and longitudinal (blue) resistivity, and Hall effect (black). We indicate the non-Fermi liquid (NFL) regions (green), and the unknown phase (orange) between H on and H where up-down difference is observed in transverse resistivity. We also indicate the characteristic pressures P M and P ∆ .

Figure 5 . 21 :

 521 Figure 5.21: Longitudinal magnetoresistivity ρ zz vs H -details of the hysteresis for P = 0.28 and 1.93 GPa. Labels indicate the temperature.

Figure 5 . 22 :

 522 Figure 5.22: Transverse magnetoresistivity ρ xx vs H -details of the hysteresis for various pressures (top labels). Colour labels indicate the temperature.

Figure 5 .

 5 Figure 5.23: a) Longitudinal and, b) transverse hysteresis width ∆H vs temperature, at various pressures. The dashed line indicates approximatively the residual hysteresis of the superconducting magnet (200 Oe). Insets show examples of hysteresis loops. c) Pressure dependence of ∆H at 50 mK (right scale) and of T hyst and T 0 (left scale). T hyst is the temperature at which the hysteresis collapses, and T 0 is the sharpest peak in ρ(H) (see Fig. 5.18) for longitudinal (blue) and transverse (red) samples. Lines are guides for the eye.

Figure 5 . 25 :Figure 5 . 26 :Figure 5 . 27 :

 525526527 Figure 5.25: Magnetization as a function of applied field, with up and down sweeps. Colour labels indicate the temperature.

Figure 5 . 28 :

 528 Figure 5.28: Magnetization as a function of time (log scale) at the various steady fields at 2 K. The field was rapidly increased (decreased) from 0 (2 T) to the steady value, for relaxation «up» and «down», respectively.

Figure 5 .

 5 Figure 5.29: a) Magnetization as a function of time (relaxation up) at various steady fields at 500 mK. The sample was warmed to 1 K before regulating at 500 mK. b) Before and after the extractor has been replaced.

Figure 5 .

 5 Figure 5.31: a) Magnetization of UCoAl at 2 K at various pressures, measured in a SQUID magnetometer. b) Pressure dependence of H m measured with various probes.Resistivity data was reported in[START_REF] Aoki | Ferromagnetic Quantum Critical Endpoint in UCoAl[END_REF] and magnetization data is taken from[START_REF] Mushnikov | Magnetic properties of the 5f itinerant electron metamagnet UCoAl under high pressure[END_REF].

Figure 5

 5 Figure 5.32:XANES and XMCD spectra at the U M 4,5 edges, at various fields (0.5, 1, 17 T). The shaded areas represent the integral of the XANES peaks. Absorption spectra are shown at 17 T only, for both polarizations. XMCD and XANES are given in the same arbitrary units.

4 Figure 5 . 33 :

 4533 Figure 5.33: XMCD spectra at the U M 4,5 edges, at various fields (0.5, 1, 17 T), scaled to match the 17 T spectra. The 1 T and 17 T spectra can be hardly distinguished in the M 4 edge. In addition, we show a smoothed curve of the U-M 5 spectrum at 0.5 T for better reading.

4 U-M 5 Figure 5 . 34 :

 45534 Figure 5.34: Field evolution of the XMCD extrema at the U M 4,5 edges, and comparison with values of ref. [59]. XMCD spectra at 17 T are shown on the right hand side as a reference.

Figure 5 . 35 :

 535 Figure 5.35: XANES and XMCD spectra at the Co K-edge, at H = 0.5, 1, 17 T.

Figure 5 . 36 :

 536 Figure 5.36: Magnetization of Cobalt at K-edge (left axis) and of Uranium at M 4 -edge (right axis) ; residual contributions ; and bulk magnetization of UCoAl (free axis). Inset shows the details of the hysteresis between up and down sweeps at the metamagnetic transition, with bulk magnetization of the same sample at 2 K.

Table 4 .

 4

	1: Critical exponents predicted by mean field theory (MF) and 3D Ising model
	[67]
	Model	β		γ
	MF theory 0.5	1.0
	3D Ising	0.325 1.24
			0.4
		β=0.5	0.3
		M 1/β	0.2
			0.1
			0
			0	0.4	0.8	1.2	1.6

  and ρ dec xx (H) show an anomaly with opposite sign P = 0.28 GPa: the plateau appears between [ Hm 2 ,H m ], and the hysteresis consists in a little shoulder on ρ dec xx (H) P = 0.71 GPa: ρ dec xx (H) starts to show a little peak at the transition P = 0.90,1.09 GPa: ρ xx now has a positive step on entering the FM phase. Up and down sweeps are similar with a hysteresis ∆H P = 1.22 GPa: in addition to the hysteresis ∆H, there is now a positive hump in ρ inc xx (H) P = 1.42 GPa: up and down sweeps are now very different in the transition region. A two-bump structure builds up at H m (which is now referred to as H peak ) and at H , in ρ inc xx (H) only, while ρ dec xx (H) is smooth. Thus we cannot infer the hysteresis ∆H at the lowest temperature

  Figure 5.24: Up-down difference ∆ρ xx (H) = ρ inc xx (H + ∆H 2 ) -ρ dec xx (H -∆H 2 ), at various temperatures (colour scale), at P = 1.22, 1.42, 1.62, 1.80 GPa. We indicate the fields H on , H peak , H and temperature T V (see in text). Horizontal axis is broken.

							1.42				1.62			1.80 GPa
		0.25
										H peak
		0.20
	xx (µΩ.cm) ∆ρ	0.15 0.10 0.05	H m ,H peak ?	H*?				H on			H*	H on		H peak	H*
		0
			5.0	6.0	7.0	6.0	7.0	8.0	6.0	7.0	8.0	9.0	7.0	8.0	9.0	10.0

Table 5 .

 5 .1 lists the theoretical values of µ DM /µ S in the intermediate coupling scheme, with 5f 2 and 5f 3 configurations: 1: Theoretical values of µ DM /µ S in the intermediate coupling scheme (IC) 5f 2 5f 3 µ DM /µ S 3.48 1.85

we used a film until

2011![START_REF] Löhneysen | Fermi-liquid instabilities at magnetic quantum phase transitions[END_REF] The notation with 4 indices (hkjl) with h + k + j = 0 is used for hexagonal structures, in order to lift the ambiguity between direct and reciprocal space: 0001 = c, 10 10 = a (direct space), 1 100 = a (reciprocal space).

do not confuse: a strain (synonym of deformation) is a relative variation of length (unitless), whereas a stress is a force per unit surface (dimension of a pressure).

the criterion to define H m is discussed in the field-scans of resistivity.

The A cofficients obtained from power law fits with different α exponent have different units, so in principle they should not be plotted on the same scale. But when the exponent is close to 2, the value of A is only weakly affected when forcing the fit with α = 2. For this reason, in the summary of A vs H for all pressures (Fig.5.15), we show the values obtained by fitting ρ xx (T ) with a T 2 law.

but we can infer ∆H at higher temperature, see next.

M. Brando (private discussions)

H. Harima (private discussions)

at 1.42 GPa, we could not determine ∆H below 1.2 K because ρ dec xx (H) is smooth, so we extrapolated ∆H at low T using an exponential law.

This is for relaxation-up ; for relaxation-down, one should consider (H dec m + 4h), (H dec m + 8h) . . .

The steps in the relaxation curves may look like the case of some single molecule magnets[START_REF] Gatteschi | Quantum Tunneling of Magnetization and Related Phenomena in Molecular Materials[END_REF][START_REF] Mougel | Uranium and manganese assembled in a wheelshaped nanoscale single-molecule magnet with high spin-reversal barrier[END_REF], but these are completely different systems

There is a little branching 3d 5/2 → 5f 5/2 at the U M 5 edge, which cannot be neglected.

A. Rogalev and F. Wilhelm (private discussions)

Some precision about the bulk magnetization data in Fig.5.36: Since we do not have magnetization data at high field for this sample, the data shown in the main graph was measured at 90 mK on a different sample. Nevertheless, we checked that both samples have exactly the same magnetization profile above H m , and we can see in Fig.5.25 that there is no difference in the magnetization between 0.1 K and 2 K in the FM state. Differences only appear at the metamagnetic transition, where the critical field and hysteresis width are sample-and temperature-dependent. This is why the bulk magnetization in the inset of Fig.5.36 was measured at 2 K, on the same sample as the XMCD data.

Acknowledgements

For the latter, we propose a simple phenomenological description.

Since the relaxation curves measured in the dilution fridge cannot be fit to an exponential function, to infer a relaxation time we estimated the time it takes the magnetization to reach (1 -1/e)M max . For the curves measured in the 4 He fridge (MPMS), we defined the characteristic time τ as the duration of the plateau between the two characteristic steps. In order to find a relation between τ and H, in Figure 5.30 we plot ln τ against (H -H inc m ) for up-relaxation, and (H dec m -H) for down-relaxation, so that we have -|H -H m | as a common horizontal scale for up and down relaxation curves at various temperatures (2, 3, 4, 5 K). This plot yields the empirical relations:

where τ is expressed in seconds. The constant h is the same for up and down relaxation, but decreases with increasing temperature. It therefore defines H inc,dec m as the field at which the characteristic relaxation time τ = 1 second is too short to observe relaxation effects experimentally.

From equations 5.4a and 5.4b, we can derive the scaling relation between two time constants τ taken at different fields H and H = H -∆H:

Second sum rule: orbital moment

From the integrated intensities of the XANES (I M 4,5 ) and XMCD (∆I M 4,5 ) spectra, it is possible to extract the orbital part of the magnetic moment µ L = -L z µ B using the relation:

Third sum rule: spin moment

The determination of the spin part of the magnetic moment µ S = -2 S z µ B is less straightforward, because we have access to an effective spin moment of the 5f orbitals:

which is obtained from combined analysis of XMCD and XANES spectra:

µ S is hidden by the so-called dipolar magnetic term:

where T z is the intra-atomic magnetic dipole operator, which measures the anisotropy of the spin distribution (due to the anisotropy of charge) in a shell. It depends on the spin-orbit coupling and on the crystal field. The difficulty is that there is no way to experimentally determine T z unambiguously. There are, however, some theoretical calculations of T z for three spin-orbit coupling schemes [START_REF] Stöhr | Determination of Spin-and Orbital-Moment Anisotropies in Transition Metals by Angle-Dependent X-Ray Magnetic Circular Dichroism[END_REF]: LS (Russel-Saunders), JJ, and intermediate coupling (IC). Uranium corresponds to the IC scheme.

Experimental details

The sample was a single-crystal with a cleaved surface perpendicular to the c-direction, and it was glued onto a Cu plate mounted on the cold finger. The temperature of the sample was 2 K during the whole experiment.

XMCD measurements have been performed at the U M 4 (3.728 keV) and M 5 (3.552 keV) edges, and at the Co K edge (7.709 keV) (values given in the tables [START_REF] Thompson | X-Ray Data Booklet[END_REF]), at 3 different fields (0.5, 1, 17 T). Energy scans were performed with left and right handed polarizations, and for positive and negative field in order to remove the residual contribution, thus eliminating the possible experimental bias coming from linear dichroism. We shall thus refer to parallel (antiparallel) scan when the photons polarization is parallel (antiparallel) to the applied field. XANES and XMCD are therefore obtained from the parallel ( ) and antiparallel (∦) scans averaged over positve and negative field, as follows: XANES = 1 2 scan pos + scan neg (5.17)

(5.21)

The measurements were typically repeated 10 times (20 times at 0.5 T) at every edge and every field value.

Results

U M 4,5 -edges

The XANES were corrected for self-absorption and the edge jumps were normalized by the degeneracy of initial states: between [0, 1] for U M 5 , and between [1, 5/3] for U-M 4 .

Figure 5.32 shows XANES spectra (left-and right-hand polarization, and isotropic) and XMCD spectra at the U M 4,5 edges. XMCD is shown for H = 0.5, 1, 17 T, while XANES is shown only at 17 T since the isotropic XANES has negligible field dependence. The data at 0.5 T correspond to the paramagnetic state, and 1 T, 17 T to the FM state. The XANES of different helicities cannot be distinguished at U-M 5 , but are clearly visible at the U-M 4 edge.

The XANES at U-M 4,5 edges are typical absorption spectra, consisting of a resonance peak (called white line) on top of a step-like edge jump. In order to integrate the peak intensities (shaded area on the graph), the edge jump was accounted for by an arctan function.

At U-M 4 , there is a rather large negative XMCD peak, with symmetrical shape, whereas the XMCD at U-M 5 is comparatively much smaller, and it has an asymmetrical S-shape with a negative part and a smaller positive part. At first look, these spectral shapes agree with the reported XMCD experiments at U-M 4,5 [START_REF] Kučera | X-ray magnetic circular dichroism studies of 5f magnetism in UCoAl and UPtAl[END_REF] and U-N 4,5 [START_REF] Takeda | Separation of magnetic properties at uranium and cobalt sites in UCoAl using soft X-ray magnetic circular dichroism[END_REF] on UCoAl.

However, the low-field XMCD signals are small, and require a closer look. In Figure 5.33 the XMCD spectra are scaled and superimposed to one another. it is clear that the 5f 2 configuration can be dismissed, while 5f 3 yields realistic values, close to the bulk magnetization. In this configuration, the ratio µ L /µ S changes from -1.6 to ≈-2.1 between the PM and FM states. In reality, the number of 5f electrons is probably intermediate, but there are theoretical values of the dipolar magnetic term for integer valence numbers only.

Conclusions

We measured the absorption and XMCD of UCoAl, at the M 4,5 edges of Uranium and at the K-edge of Cobalt, at 2 K, in magnetic fields up to 17 T. This synchrotron experiment was carried out under presumably improved experimental conditions, compared to the first XMCD experiment on UCoAl reported in 2002 [START_REF] Kučera | X-ray magnetic circular dichroism studies of 5f magnetism in UCoAl and UPtAl[END_REF] (lower temperature, high quality sample with sharp transition, cleaved surface). We compared our results with that of Kučera et al. and Takeda et al. [START_REF] Takeda | Separation of magnetic properties at uranium and cobalt sites in UCoAl using soft X-ray magnetic circular dichroism[END_REF]. We could reproduce the XMCD spectra in the FM state, but we obtained a slightly different U M 5 XMCD spectral shape in the PM state, probably because of low signal-to-noise ratio. The XMCD spectrum at the Co K edge exhibits two main peaks which are , not present in the Co metal, and which are ascribed to a polarization induced by the U orbitals via spin-orbit coupling. The Co moment is therefore mostly induced by the U orbitals (absence of intrinsic Co moment). By comparing the U-and Co-specific magnetizations versus field, we found that their proportion is basically constant, hence the hybridization between U 5f and Co 4p orbitals is not significantly changed between the PM and FM