Yannick Klein Co

Keywords: SQUID magnetometry

by solid state reaction . . . . . . . . . .

Résumé

Dans la présente thèse, nous avons étudié la stabilité de la phase isolante de Mott de BaCoS 2 que nous avons choisi comme modèle de systèmes multiorbitaux d'électrons corrélés, un sujet de recherche émergent dans le domaine de la matière condensée et de la physique à plusieurs corps.

Le diagramme de phase du système BaCoS 2 avait été peu étudié jusque là ce qui rendait la synthèse de la phase difficile. Nos résultats nous ont permis de développer et de mettre en oeuvre avec succès une nouvelle technique de self-flux pour la croissance de monocristaux en vue de futures expériences de neutrons inélastiques qui pourraient révéler la nature du couplage entre le magnétisme et l'état isolant dans ce système.

L'étude systématique d'échantillons en poudre et monocristallins utilisant la diffraction des rayons X, la chaleur spécifique et les mesures de susceptibilité magnétique, nous a permis d'élucider la controverse dans la littérature concernant la stabilité de la phase orthorhombique dans la phase antiferromagnetique (AFM) isolante de Mott de BaCoS 2 .

Nos résultats confirment clairement que la phase stable aux conditions ambiantes est orthorhombique Cmma. De plus, nous avons constaté que, remarquablement, la distorsion orthorhombique est concomitante à l'apparition de l'ordre AFM, soulignant ainsi le rôle de l'ordre orbital sur la stabilité de la phase de Mott.

L'étude de la chaleur spécifique a révélé une importante contribution des fluctuations thermiques du paramètre d'ordre à la température T N de transition AFM, suggèrant la proximité d'un état fondamental concurrent caractérisé par un ordre quasi-2D à longue portée.

Nous avons ensuite exploré la possibilité d'induire un état métallique dans BaCoS 2 en utilisant la pression chimique et le dopage électronique comme paramètres de contrôle.

Pour ce faire, nous avons d'abord développé avec succès une technique de synthèse HP/HT pour synthétiser des échantillons de BaCoS 2 polycrystallins monophasés de haute qualité.

Contents

De manière surprenante, nous avons constaté que les propriétés structurelles et physiques des échantillons BaCoS 2 ainsi obtenus diffèrent considérablement de celles des échantillons synthétisés à l'ambiante. Les différences notables sont un rétrécissement important de la maille élémentaire concomitant à une forte réduction de la distorsion orthorhombique et une forte réduction de T N accompagnée d'un étalement du saut de la chaleur spécifique. Ces résultats suggèrent que cette distorsion est nécessaire pour stabiliser l'ordre AFM. Fait intéressant, aucune indication des porteurs mobiles n'est donnée par le comportement à basse température de la chaleur spécifique dans ces échantillons.

Nous avons ensuite appliqué cette technique de synthèse HP/HT à la synthèse d'échantillons Ba 1-x Sr x CoS 2 et Ba 1-x K x CoS 2 où le Ba est partiellement substitué par du Sr ou du K, ce qui devrait favoriser une phase métallique en raison d'un effet de pression chimique ou de dopage de trous respectivement.

Dans le cas de la substitution par du Sr, nous avons réussi à préparer des échantillons de poudre de bonne qualité avec un niveau de substitution allant jusqu'à x = 0.08. Les résultats de nos mesures structurelles confirment que la substitution induit un rétrécissement important de la maille élémentaire, comme prévu en tenant compte de la plus petite taille de l'ion Sr 2+ par rapport au Ba 2+ . Une comparaison avec les données structurelles précédentes d'échantillons étudiés sous pression hydrostatique indique que le niveau de substitution cidessus est équivalent à une pression hydrostatique de 0.21 GPa.

Fait intéressant, les changements structurels induits par la substitution Sr sont radicalement différents de ceux induits par la pression hydrostatique. Plus précisément, la substitution par du Sr (pression chimique) semble être plus efficace pour réduire à la fois les distances dans les plans ionique et apicale Co-S. En effet, la réduction de ces distances est aussi importante que celle obtenue sous une pression hydrostatique de 1.1 GPa qui induit un état métallique paramagnétique. Malgré cette similitude, aucune preuve de propriétés métalliques n'est trouvée dans les échantillons Ba 1-x Sr x CoS 2 dans le comportement à basse température de la chaleur spécifique.

La pression chimique exercée par la substitution Sr semble améliorer les changements structurels produits par la synthèse HP. À savoir, dans les échantillons Ba 1-x Sr x CoS 2 , la distorsion orthorhombique et l'ordre AFM à longue portée sont encore affaiblis lors de l'augmentation de x jusqu'à ce que les deux soient complètement supprimés au niveau de substitution maximum, x = 0.08. Malgré cette suppression, la réponse magnétique conserve encore les principales caractéristiques de la phase non substituée, à savoir une anomalie de type cusp-like à la température caractéristique T N , bien que cette température soit fortement abaissée de 290 K à 240 K par la substitution. Cette observation suggère que de grandes fluctuations de l'AFM à courte portée persistent dans le système.

Notre tentative de synthétiser des échantillons BaCoS 2 substitués par le K sous HP a également été un succès. Similaire au cas de la substitution Sr, nous avons préparé des échantillons de poudre Ba 1-x K x CoS 2 de bonne qualité avec des niveaux de substitution x = 0.07. Contrairement à ce que l'on attendait en comparant les rayons ioniques de Ba 2+ et K 1+ , nous avons trouvé une réduction du volume de cellule unitaire avec x similaire à celle des échantillons Ba 1-x Sr x CoS 2 . Les effets de cette réduction de volume sur les propriétés physiques sont également similaires, à savoir que la distorsion orthorhombique et l'AFM à longue portée sont supprimés par la substitution K tandis que l'anomalie de la susceptibilité magnétique à T N persiste. Cette découverte confirme le scénario de grandes fluctuations de l'AFM à courte portée proposé pour les échantillons Ba 1-x Sr x CoS 2 .

Une différence notable entre les échantillons Ba 1-x Sr x CoS 2 et Ba 1-x K x CoS 2 est que ces derniers échantillons présentent une valeur importante du coefficient de Sommerfeld γ = 5.7 mJ mol -1 K -2 dans le comportement à basse température de la chaleur spécifique. Ce résultat est très intéressant car il suggère que la substitution K induit un état métallique. Compte tenu de la densité relativement faible de trous introduite par la substitution K, la valeur importante ci-dessus peut être expliquée par une masse renormalisée assez grande, cohérente avec un scénario de type mauvais métal.

Introduction

The understanding of the electronic properties of condensed matter beyond the conventional Fermi liquid (FL) regime constitutes a major challenge of modern science. Much of our current understanding of the FL breakdown comes from a simplified one-band Hubbard model, where electronic correlations are approximated by an on-site Coulomb repulsion term U . Until recently, it was generally accepted that the model describes the physics of the Mott metal-insulator transition (MIT) at half-filling controlled by the U/W ratio (W is the bandwidth). In fact, this simple model does not account for the incoherent non-Fermi-liquid (NFL) metallic properties of transition-metal compounds, such as Cu- [START_REF] Basov | Electrodynamics of correlated electron materials[END_REF][START_REF] Hardy | Evidence of Strong Correlations and Coherence-Incoherence Crossover in the Iron Pnictide Superconductor KFe 2 As 2[END_REF] and Fe-based [START_REF] Hardy | Evidence of Strong Correlations and Coherence-Incoherence Crossover in the Iron Pnictide Superconductor KFe 2 As 2[END_REF] high-temperature superconductors (HTSC), vanadates [START_REF] Allen | Resistivity of the high-temperature metallic phase of VO 2[END_REF][START_REF] Laad | Vo2: a two-fluid incoherent metal?[END_REF] and ruthenates [START_REF] Kostic | Non-Fermi-Liquid Behavior of SrRuO 3 : Evidence from Infrared Conductivity[END_REF]. A step towards the description of this anomalous metal was made recently by theoretical studies [START_REF] Costi | Quantum Phase Transition in the Two-Band Hubbard Model[END_REF][START_REF] Werner | Spin Freezing Transition and Non-Fermi-Liquid Self-Energy in a Three-Orbital Model[END_REF] that point at the central role played by the Hund coupling, J, between electrons occupying different d orbitals. Dynamical mean-field theory (DMFT) showed that the interplay between the above coupling and the Hubbard repulsion leads to a crossover from a paramagnetic FL and an incoherent metallic phase with 'frozen' local moments (Hund metal).

Only recently, the availability of exact diagonalisation [START_REF] Perroni | Exact diagonalization dynamical mean-field theory for multiband materials: Effect of Coulomb correlations on the Fermi surface of Na 0.3 CoO 2[END_REF] and Monte Carlo [START_REF] Werner | Continuous-Time Solver for Quantum Impurity Models[END_REF] numerical methods enabled the study of realistic multiorbital correlated systems, such as Fe-based superconductors [10]. These studies have shown the necessity of more powerful theoretical methods to take into account local correlation effects, non-local exchange interactions and relativistic effects. All these points are relevant to transition-metal compounds characterised by hybrid p -d bands, small carrier densities (hence poor screening) and significant spin-orbit splitting of bands [11]. Novel playgrounds are required for experiments as well, for most compounds hitherto studied display a significant electronic-lattice coupling, which hampers a reliable study of the FL-NFL crossover. For example, in the prototype MIT system VO 2 , the MIT is concomitant to a dimerisation of the V ions [12].

In order to address the above issues, in the present thesis we focus on the Mott state in the multiorbital system BaCoS 2 that remains little studied. We believe that this is a model system owing to the following unique characteristics.

Contents

i) Similar to Cu-and Fe-based HTSC's, in BaCoS 2 the Mott MIT is controlled by doping via the Co/Ni substitution in BaCo 1-x Ni x S 2 [13]. However, contrary to the above systems, the lattice plays no role in the MIT and the simple quasi-tetragonal structure of BaCoS 2 is suitable for theoretical studies.

ii) The FL-NFL crossover is sensitive to pressure or Co/Ni substitution level, x. For instance, the NFL phase at x = 0 exhibits a strong enhancement of metallic behaviour under pressure [13] and an unusual linear frequency dependence of the optical conductivity, σ ω , over a broad energy range [14].

A first DMFT study on BaCoS 2 [14] is very encouraging indeed, for it provides a clear indication of a NFL metallic phase arising from a peculiar combination of a strong Hund coupling inherent to the Co 2+ ion and of a vanishing d -p charge-transfer gap. Furthermore, a remarkable orbital-selective Mott properties were found. Namely, the imaginary part of the self-energy of the half-filled x 2 -y 2 and xz/yz orbitals displays pronounced NFL characteristics, while the z 2 orbital has FL behaviour. These results contrast the previous picture of conventional Mott insulator and prompt us to investigate the unusual ground state and transport properties of the BaCo 1-x Ni x S 2 system, which requires suitable theoretical developments, as explained below. iii) The weakly correlated Fermi liquid phase at x = 1 has recently attracted a great deal of interest for its topological properties, namely a hidden spin polarisation due to the Rashba coupling [11] and Dirac cones [15][16][17]. Therefore, the study of the MIT offers a unique opportunity to investigate the effects of electronic correlations on these topological properties, an emerging field of research.

In the present thesis, we attempt to investigate the controversial properties of the Mott state by using the bandwidth W and electronic doping as control parameters. To do so, we first elucidate the relationship between structural properties and an unusual antiferromagnetic (AFM) order concomitant to the Mott phase by means of a systematic study on high-quality BaCoS 2 powders and single crystals. Second, we attempt to tune the Mott phase of BaCoS 2 by employing high-pressure (HP) synthesis, powerful tool to stabilise a metastable structure with altered bond-lengths combined with iso-and hetero-valent chemical substitutions at the Ba site. The former substitution is effective to produce a chemical pressure, while the latter substitution is known to destabilise the insulating state and the AFM order, thus favouring a metallic state. The above substitutions at the Ba site, which acts as charge reservoir for the CoS layer, enables us to avoid introducing disorder in the electronically active Co site.

The manuscript is organised as follows:

In Chapter 1, we recall the main fundamental aspects regarding correlated electron systems and the Mott transition with emphasis on a generalisation of the usual for one-band Hubbard model to the case of multi-orbital systems object of the present work.

Chapter 2 describes the experimental methods used for sample synthesis and characterisation. We first illustrate the conventional solid-state reaction method for the synthesis of polycrystalline powders of BaCoS 2 using and then the growth of single-crystals using conventional and self-flux methods. Second, we illustrate the high-pressure (HP) synthesis technique used for the synthesis of both pristine and Sr-and K-substituted BaCoS 2 . Finally, we briefly illustrate the main characterisation techniques, namely powder and single-crystal xray diffraction (XRD), energy-dispersive spectroscopy (EDS) used to determine sample composition, magnetic magnetisation and specific heat measurements.

Chapter 3 contains the main results regarding the synthesis and a systematic study of the the structural properties of BaCoS 2 . We first report on the promising results obtained using a self-flux method to grow single crystals of BaCoS 2 with high quality with respect to conventional methods. Second, we present a systematic study of the structure-property relationships found in BaCoS 2 that establishes a link between orthorhombic distortion and stability of the AFM order.

In Chapter 4 we report on a successful HP synthesis of Ba 1-x Sr x CoS 2 samples with Sr substitution level up to x = 0.08 and the results of an extensive characterisation of the samples by means of x-ray diffraction, specific heat and magnetisation measurements showing the dramatic effect of the chemical pressure exerted by the Sr substitution on the physical properties of BaCoS 2 .

Chapter 5 reports a similar study as above on the partial substitution of Ba for K. Note that, in this case, the effect of the substitution on the physical properties consists mainly in hole-doping. Finally, in Chapter 6, we draw the conclusions of the thesis and discuss the open questions and perspectives for future work raised by the results obtained.

Chapter 1 Multiorbital Mott insulators and BaCoS 2

In the present thesis, we focus on BaCoS 2 as a model system for studying the stability conditions of the Mott state in the general multiorbital case. In order to illustrate our approach and scientific objectives, in the present Chapter, we first recall the state-of-the art in the field of correlated electron systems and Mott insulators in general. We then discuss to which extent our current understanding of the field is applicable to the BaCoS 2 system, the unusual properties of its insulating state and the open questions. Finally, we outline our strategy to address selected open points.

Beyond conventional band theory

The simple semiclassical theory originally developed by Drude to explain the electrical conductivity of metals has proven to be a rather general framework to account for the electronic and transport properties of a variety of complex materials, like transition metal oxides or chalcogenides, among others. The reason is that the theory is suitable to describe phenomenologically the microscopic mechanism of charge transport within conventional band theory, provided the relevant parameters, namely the density n, lifetime τ and effective mass m * of the carriers are suitably chosen for each electronically active band. These considerations apply also to an interacting electron gas, provided the lifetime of the electronic excitations is sufficiently long, which defines the concept of quasiparticle in Landau's theory of Fermi liquids (FL) [18]. As a result, even systems containing localised orbitals, e.g. d or f (the socalled heavy fermions), can be well described by Drude theory even in the presence of large renormalisations.

The breakdown of Landau's theory occurs when the quasiparticle lifetime is reduced by the interaction until the uncertainty in the quasiparticle energy /τ becomes comparable with the quasiparticle energy itself. This regime is favoured not only by a strong electron-electron interaction but also by a small Fermi energy, E F . Phenomenologically, upon increasing the strength of the interaction, one first observe a crossover from a FL to a non-FL state and then a metal-insulator transition when the Coulomb interaction becomes sufficiently large as compared to the kinetic energy, E F . In 1937, de Boer and Verwey [19] proposed a first approach to the problem to explain the insulating state of NiO, which is expected to be a metal according to band theory because its electronic band is half-filled. According to de Boer and Verwey's model, the electron is not free to move in the solid because it is subject to the Coulomb potential of the hole created by the electron itself in its motion. Later, Mott developed further the model by introducing the notion of critical electron density for the above metal-insulator transition [20], called Mott transition.

For completeness, we recall that an insulating state can still be achieved within conventional band theory as a result of the following phenomena: (i) the filling of a gapped electron band (Bloch-Wilson or band insulator); (ii) the opening of a band gap at E F caused by a spontaneous static lattice deformation (Peierls insulator); (iii) the disorder-induced localisation of the electrons in reduced dimensions, D ≤ 2 (Anderson insulator) [20].

The Hubbard model for the Mott transition

Clearly, a quantitative study of the Mott metal-insulator transition for real systems is an unsolved problem owing to the well-known mathematical difficulties inherent to the many-body problem combined to the complexity of real materials. A popular simplified approach is provided by Hubbard model [21]. In its original formulation, this is a single-band model in the atomic limit that treats the motion of electrons, described by a hopping energy t, in presence of a Coulomb repulsion U for a given electron density or chemical potential. U is defined as the energy required to transfer one electron from site i to site j:

d n i d n j -→ d n-1 i d n +1 j (1.1)
where n=1 or 2 and n =0 or 1 according to Pauli's principle. The model is illustrated schematically in Figure 1.1 and described by the Hamiltonian [START_REF] Imada | Metal-insulator transitions[END_REF]:

H = -t ij (c † iσ c jσ + c † jσ c iσ ) + U i (n i↑ - 1 2 )(n i↓ - 1 2 ) -µ iσ n iσ (1.2)
where, c iσ and c † iσ are electron annihilation and creation operators, respectively, for the i-th site, σ is the spin variable, n iσ is the occupancy operator and µ is the chemical potential. In spite of the radical simplification introduced by the model, no exact solution exists yet, except in 1D, which reflects the aforementioned difficulties of the many-body problem. based on the Hubbard model as a function of the U/t ratio and band filling. BC-and FC-MIT stand for band-and filling-controlled MIT (taken from [START_REF] Imada | Metal-insulator transitions[END_REF]).

Owing to the lack of exact solutions, the model has been the object of numerical studies for decades. The results of these studies (see, for example, [START_REF] Zhang | Mott transition in the d=∞ hubbard model at zero temperature[END_REF] and Figure 1.3) consistently indicate that, at half-filling, i.e. when each site is occupied, on average, by one electron, a metal-insulator transition occurs when the repulsion term U becomes comparable with the kinetic energy term (or bandwidth) t, as expected from general considerations regarding a real gas of electrons. At half-filling, the resulting energy-level scheme consists of two Hubbard bands separated by U , the lower band being completely filled and the 2, 2.7, 3 and 4 (from the top to the bottom), where W is bandwidth, using a Quantum Monte Carlo method combined with a perturbation expansion. Energies ω are measured from the chemical potential (taken from [START_REF] Zhang | Mott transition in the d=∞ hubbard model at zero temperature[END_REF]).

upper band completely empty. Interestingly, in the case of square lattice, the insulating ground state exhibits a Néel-type antiferromagnetic (AFM) order with propagation vector q AF M =(1/2 1/2) as a result of an AFM exchange energy J = t 2 /U [START_REF] Imada | Metal-insulator transitions[END_REF]. As expected intuitively from the above energy-level scheme, further theoretical studies as a function of filling indicate that a metallic state is favoured by hole-or electron-doping the insulating state (the problem clearly possesses an electron-hole symmetry), as schematically shown in figure 1.2. In conclusion, in spite of all its simplifications, the Hubbard model, describes well the phenomenology of the MIT of well-known strongly correlated materials, such as vanadates, manganites or cuprates, as a function of the U/t ratio and of band filling as control parameters. In fact, a generalisation of the model is required to describe real materials, such as transition metal (TM) oxides or sulphides where the electronic bands result from the hybridisation of d and p orbitals, which is the case of BaCoS 2 , as described below.

Mott-Hubbard vs. Charge-Transfer insulators

A first generalisation of the Hubbard model appropriate to treat the electronic structure of TM compounds like BaCoS 2 in a realistic manner is to include the p band into the model. In this case, in addition to the Coulomb repulsion energy, that we shall re-label U dd , required to transfer one d-electron from one site to 1.3. Mott-Hubbard vs. Charge-Transfer insulators another, we consider the energy ∆ required to transfer an electron from the filled p-orbital of the ligand ion to the partially filled d-orbital of the TM ion:

p 6 d n -→ p 5 d n+1
(1.3) This second excitation creates a hole in the ligand ion. In the present case of ionic solids like TM oxides and sulphides, the opposite process is not possible because the p-orbitals of the ligand are assumed to be completely filled.

The charge-transfer energy ∆ is related to U dd and to the energies of the d and p levels, d and p as follows:

∆ = d -p + U dd (1.4)
The resulting energy scheme is given in Figure 1.4. Two different situations leading to an insulating state are found. For ∆ > U , we obtain the Mott-Hubbard insulator described in the previous section. Indeed, in this case, the minimum excitation energy corresponds to the transfer of one electron from one TM site to another. In the opposite situation ∆ < U , the minimum excitation energy corresponds to the transfer of a charge from the ligand ion to the TM ion. This is the case known as 'charge transfer insulator' [START_REF] Zaanen | Band gaps and electronic structure of transition-metal compounds[END_REF][START_REF] Sawatzky | On the electronic structure and related physical properties of 3d transition metal compounds[END_REF].

In this generalised model, one should consider three distinct possibilities of electron hopping from one TM or ligand ionic site to another and therefore the three different energies t dd , t pp and t pd , which considerably enhances the complexity of the model. Some important families of correlated materials like cuprate superconductors and also the BaCoS 2 system object of the present thesis belong to the category of charge-transfer insulators. In the case of TM sulphides like BaCoS 2 , ∆ tends to be comparatively small as compared to the case of TM oxides owing to the larger spatial extension of the 3p orbitals of sulfur, which leads to larger bandwidths and, therefore, to a more pronounced metallic characteristics.

Multiorbital insulators and Hund metals

A further generalisation of the Hubbard model needed to describe the electronic structure of TM compounds in a realistic manner consists in taking into account the multiplicity of the d manifold. The multiplicity of the p manifold is often neglected owing to its wider bands and reduced dependence upon crystal symmetry, except in strongly anisotropic systems. Since the 3d orbital has total angular momentum L = 2 the degeneracy is fivefold:

L z =2,1,0,-1,-2.
In a crystalline solid, the crystal field splits the d manifold into well-separated energy levels d,s . Figure 1.7 shows an example of this splitting for a pyramidal, i.e. C 4v point-group symmetry, relevant to the case of BaCoS 2 . In this case, the highest-energy level is d x 2 -y 2 because the Coulomb repulsion between the lobe of this orbital and the ligand ion is maximum. Opposite consideration holds for the low-lying d xy orbital. The degeneracy of the d xz and d yz orbitals is lifted in presence of an orthorhombic distortion of the pristine tetragonal symmetry and the relevant point group is C 2v .

It appears that the pristine Hubbard model turns into a much more complex multi-band model where the orbital index is treated as a pseudospin. A new ingredient of this multi-band model is the orbital polarisation promoted by lattice distortions, such as those driven by the Jahn-Teller effect. According to a celebrated theory originally proposed by P.W. Anderson [START_REF] Anderson | New approach to the theory of superexchange interactions[END_REF] and further developed by Kanamori and Goodenough [START_REF] Weiss | John b. goodenough: magnetism and the chemical bond. interscience publishers. new york, london 1963. 393 seiten, 89 abbildungen. preis: dm 95 s[END_REF] and later by Kugel and Khomskii [28], orbital polarisation governs the sign and strength of the superexchange interaction between neighboring TM ions. This theory provides a good qualitative understanding of a complex interplay between charge, spin and orbital degrees of freedom in poor conductors, like doped manganites with simple perovskite structure [START_REF] Wollan | Neutron Diffraction Study of the Magnetic Properties of the Series of Perovskite-Type Compounds[(1-x)La, xCa]MnO 3[END_REF], for instance. (from [START_REF] De'medici | Janus-faced influence of hund's rule coupling in strongly correlated materials[END_REF]).

The situation addressed in the present thesis concerns the open issue of describing the regime at the verge of a metal-insulator transition driven by correlations, i.e. when the kinetic energy t is comparable with the repulsion energy U . From a theoretical standpoint, the difficulty lies in dealing with the localised and itinerant characters of electrons on equal footing. In order to address this issue, during the last decade, several authors have focused the attention on the Hund's rule coupling as a fundamental ingredient governing the stability of the metallic phase in multi-orbital correlated TM compounds, called Hund' metals. An emerging scenario relevant to the case of BaCoS 2 is that this stability greatly depends upon the number of electrons filling the active orbitals [START_REF] De'medici | Janus-faced influence of hund's rule coupling in strongly correlated materials[END_REF]. Specifically, ab initio studies suggest that, aside from the well-known cases of a singly or half-filled occupied shell that favour a metallic state or a Mottinsulating state, respectively, the Hund's coupling induces conflicting tendencies and thus causes strong correlations far from the insulating state experimentally observed as a 'bad-metallic' NFL behaviour, as seen in Figure 1.5.

The above picture explains the observed physical properties of a number of TM oxides. In the present thesis, we are interested in investigating this regime in the TM compound BaCoS 2 which exhibits less-pronounced electronic correlations as compared to TM oxides and virtually no interplay between electronic and lattice degrees of freedom, a very favourable characteristic indeed for a reliable theoretical modelling.

1.5 BaCoS 2 , a model system of multi-orbital

Mott insulator

We have seen before that, within a single-band Hubbard model, a Mott MIT is predicted by varying either electronic doping n or the U/t ratio (see 1.2). In fact, this theoretical prediction can not be easily applied to real systems, where the effects of these two control parameters are usually entangled with those of the lattice degrees of freedom, e.g. structural distortion, strain field or defects, which prevents a clear understanding of the microscopic mechanism of the MIT [START_REF] Anderson | Effect of franck-condon displacements on the mobility edge and the energy gap in disordered materials[END_REF][START_REF] Lee | Pa, ramakrishnan tv disordered electronic systems[END_REF][START_REF] Di Sante | Disorder-driven metal-insulator transitions in deformable lattices[END_REF]. A strong interplay between electronic and lattice degrees of freedom is observed even in well-studied prototype systems of the Mott MIT transition, such as V 2 O 3 [START_REF] Mcwhan | Vv. f. brinkman and tm rice[END_REF] and BaVS 3 [35]. In this context, we argue that the quasi-2D BaCo 1-x Ni x S 2 system [38] is appealing for it offers a rather unique opportunity to investigate the Mott MIT in a multi-band system where the electronic degrees of freedom appear to be decoupled from those of the lattice, as originally imagined by Mott.

In BaCo 1-x Ni x S 2 , doping is achieved by partially substituting Co for Ni in a tetragonal or slightly distorted orthorhombic structure, first reported by Grey and Steinfink [36], formed by a peculiar network of puckered edge-sharing (Co/Ni)S 5 pyramids forming a square lattice (see 1.6). The metallic phase, stable for x> 0.22, is paramagnetic (PM), while the insulating one exhibits an antiferromagnetic (AFM) order of the Co/Ni ions in the ab plane below ∼ 305 K [39]. Recently, it was found that the semimetallic endmember (x = 1) displays a unique electronic structure characterised by Dirac nodes arising from the non-symmorphic crystal symmetry [11].

In the present thesis, we focus on the Mott insulating phase, BaCoS 2 . To justify our choice, hereafter we give an overview of the crystal, electronic and magnetic structures of this phase.

Crystal structure

The nuclear structure of the BaCo 1-x Ni x S 2 system, first reported in 1970 by Grey and Steinfink [36] and later by Martinson et al. [40], consists of a nearly square network of edge-sharing puckered CoS 5 or NiS 5 pyramids separated by distorted BaS rock-salt layers (see figure 1.6). The crystal symmetry of the Ni endmember BaNiS 2 is tetragonal P 4/nmm. In this structure, it turns out that the only atomic coordinates not fixed by symmetry are the out-of-plane z-positions of all sites, while the atomic coordinates in the ab-plane of all atoms are fixed by symmetry [38,40,41]. We argue that this symmetry property gives to the crystal structure a great rigidity against lattice deformations, which may explain the absence of coupling between electron and lattice degrees of freedom at the MIT.

The BaCoS 2 structure is slightly distorted as compared to the tetragonal structure of the Ni end member phase BaNiS 2 . Interestingly, the details of these distortions remain to be established. An early structural study by means of xray diffraction on powder samples proposes an orthorhombic distortion of the above tetragonal P 4/nmm symmetry, leading to a Cmma symmetry, as clearly visible as a splitting of the (11l) Bragg reflections [40]. Other single-crystal studies rather suggest the same tetragonal P 4/nmm symmetry reported for BaNiS 2 [42][43][44]. A recent single-crystal study by Baenziger et al. [45] reports a small monoclinic distortion with P 2/n symmetry with β = 90.43(4) • and lattice parameters a = 4.573 [START_REF] Allen | Resistivity of the high-temperature metallic phase of VO 2[END_REF], b = 4.570(2) Å and c = 8.937(2) Å. By refining the structure in the monoclinic P 2/n symmetry, Martinson et al. [40] found a monoclinic angle of 90.43 • C and, consequently, a slight distortion of the square based pyramid with a = b. Both orthorhombic Cmma and monoclinic P 2/n models feature a slight difference between a-and b-axis parameters, a slight reduction of the apical Co-S1 bond length from 2.31 Å to 2.28 Å and a significant increase of the planar Co-S2 from 2.34 Å to 2.44 Å. The structural parameters of the above models are reported in Table 1.1).

Electronic structure: correlations effects

In figure 1.7 [46], we show the crystal field splitting of the 3d manifold expected in the C 4v symmetry corresponding to the square-based pyramidal coordination of the Co or Ni ions. Note that the vertical position of these ions within the pyramid may alter the hierarchy of the orbitals [START_REF] Tripathi | Cobalt(ii) complexes as single-ion magnets[END_REF]. An experimental study of the effect of the z-position of the TM ion on the crystal field has been carried out on BaCrS 2 [START_REF] Fuentes | Synthesis and structural analysis of bacrs2[END_REF] that displays a highly distorted crystal structure similar to that of BaCoS 2 . By moving the Cr ion towards the apical position along the pyramid axis, the d xy orbital pointing between the 3p orbitals of two S ions in the basal plane, lowers its energy. Correspondingly, the S-Cr-S bond angle increases from 90 • to 110 • , thus altering significantly the superexchange path and the strength of the magnetic interaction. The resulting reshuffling of orbitals is schematically shown in Figure 1.7.

A similar hierarchy of orbitals is expected in the BaCo 1-x Ni x S 2 system as well, characterised by a similar out-of-plane shift of the TM ion. This is confirmed experimentally for the weakly correlated semimetal BaNiS 2 , but not for the AFM insulator BaCoS The different hierarchy of orbitals in BaCoS 2 as compared to that in BaNiS 2 points at an orbital reshuffling driven by electronic correlations. This point has been hitherto little studied theoretically. Hase et al. [50] reported ab initio calculations using the full-potential linearised augmented plane wave (FLAPW) method within density functional theory (DFT). By adopting a tetragonal structure for BaCoS 2 as well as for BaNiS 2 phases, they predict a paramagnetic and metallic phase for both compounds, which is clearly in contradiction with the experimental observation of the AFM insulating properties of BaCoS 2 [38,40,42,[START_REF] Fisher | Electronic transport and antiferromagnetism in the layered compound bacos 2[END_REF]. This failure is expected, for a suitable description of the ground state of BaCoS 2 requires ad hoc ab initio studies that properly take into account electronic correlations. This was done later by Zainullina et al. [START_REF] Zainullina | Ground state of bacos 2 as a set of energydegenerate orbital-ordered configurations of co 2+ ions[END_REF] who improved the above DFT scheme in the local density approximation (LDA) by including explicitly a Coulomb repulsion term U that mimics the effect of electronic correlations via a local repulsion potential. As expected, for U = 5 eV and an exchange interaction J = 0.94 eV, this improved scheme yields an AFM insulating ground state. A further theoretical study suggested that the MIT in BaCo 1-x Ni x S 2 is driven by the t dp hopping parameter [START_REF] Krishnakumar | Electronic structure of and covalency driven metal-insulator transition in baco 1-x ni x s 2[END_REF]. However, the theoretical predictions are at odds with the results of an angle-resolved photoemission spectra (ARPES) study carried out at compositions x = 0.18 and x = 0.28 bridging the critical value x c = 0.20 at the MIT [START_REF] Sato | Evolution of metallic states from the hubbard band in the two-dimensional mott system baco 1-x ni x s 2[END_REF]. These results indicate that a tight-binding scheme is not suitable to account for the MIT and confirms the crucial role played by the electronic correlations.

More recently, a theoretical study on BaCoS 2 [START_REF] Santos-Cottin | Linear behavior of the optical conductivity and incoherent charge transport in bacos2[END_REF] carried out within a dynamical mean-field theory (DMFT) has shed some light on the microscopic mechanism leading to the orbital reshuffling that occurs across the MIT and schematically shown in Figure 1.8. Namely, DMFT is well suited to take into account the role of electronic correlations by treating the effect of the Coulomb repulsion dynamically. The crucial role of the dynamic U is apparent by comparing the (i) occupancy and (ii) energy-dependence of the imaginary part of the self-energy for each orbital of the d manifold obtained in the two computational schemes, i.e. conventional DFT+U , where U is treated statically, and DMFT, where U is treated dynamically (see Figure 1.8). As to (i), note that the DMFT calculations yield an almost full occupancy for the z 2 orbital and a half-occupancy for the xz/yz doublet, thus confirming the experimental results of neutron diffraction suggesting a S = 3/2 ground state. As to (ii), it is apparent that both the xz/yz doublet and the x 2 -y 2 state display significant correlation effects, i.e. the imaginary part of the self-energy at ω = 0 is finite for these states, while the remaining z 2 and xy states behave as FL bands. In other words, correlation effects in BaCoS 2 are orbital-selective and the xz/yz doublet and the x 2 -y 2 state appear to be the responsible for the Mott state.

Note that the resulting half-filling of each of the above three states is in full agreement with the experimental observation of a high-spin S = 3/2 state and creates a favourable situation for stabilising a Mott insulator, as discussed above [START_REF] De'medici | Janus-faced influence of hund's rule coupling in strongly correlated materials[END_REF]. The slight orthorhombic distortion observed in BaCoS 2 lifts the degeneracy of the d xz /d yz doublet but the resulting level splitting is expected to be small, so it would not alter significantly the occupancy nor the stability of the Mott state. Though, the splitting is relevant to the anisotropic properties of the magnetic structure, as discussed below. In this thesis, we shall estimate the magnitude of the splitting by analysing the anomalous behaviour of the specific heat observed at low temperature by means of a Schottky model.

Magnetic structure

Previous magnetic susceptibility measurements (χ) on BaCoS 2 powder [38,42,43] as well as single-crystal [START_REF] Mandrus | Magnetism in bacos2[END_REF] samples consistently give evidence of an antiferromagnetic (AFM) ordering at a Néel temperature T N ∼ 300 -310 K. The single-crystal study by Mandrus et al. [55]) further unveils a strong anisotropy of the magnetic properties consistent with a model of magnetic moments lying in the ab-plane, as seen in Figure 1.9. anisotropy (from [START_REF] Mandrus | Magnetism in bacos2[END_REF]).

An early analysis of the data suggested that the Co 2+ ions in BaCoS 2 are in the S = 1/2 spin state [38,40,42,43]. Later, Mandrus et al. found an effective moment µ = 5.26µ B , consistent with the theoretical for Co 2+ ions in the 3 2 spin state (4.3 -5.2µ B ) [START_REF] Mandrus | Magnetism in bacos2[END_REF]. This scenario was confirmed by the aforementioned neutron diffraction results [41]. The magnetic structure proposed for BaCoS 2 on the basis of the only two neutron diffraction experiments hitherto reported [41,[START_REF] Mandrus | Magnetism in bacos2[END_REF] is shown in Figure 1.10). Note that the magnetic structure is formed by AFM chains along the a-axis coupled ferromagnetically (FM) along the b-axis, corresponding to a propagation vector q AF M =(1/2 0 0). Therefore the magnetic structure has a two-fold rotation symmetry. This is remarkable considering that the orthorhombic distortion is rather small, so we would rather expect a conventional AFM structure with q AF M =(1/2 1/2 0) or q AF M =(1/2 1/2 1/2) propagation vector. The refined magnetic moments lie within the ab-plane; their components along the a-and b-axis are 2.56 ± 0.05 µ B and 0.66 ± 0.01 µ B , respectively, as expected for a S = 3/2 spin.

(A) (B) To the best of our knowledge, no theoretical model has been developed yet to study the stability of the above magnetic structure of BaCoS 2 . Our colleagues of the Theory of Quantum Materials group at the IMPMC, M. Casula and B. Lenz, recently initiated a systematic ab initio study combined with a parametrisation of a suitable model described by the Heisenberg Hamiltonian:

H = 1 2 i =j J ij S i • S j (1.5)
where J ij is the magnetic exchange coupling between the i-th and j-th spin site. In this notation, a positive sign for the coupling J indicates AFM coupling. While no definite results of the above study are available yet, we limit ourselves to some qualitative observations that provide an insight into the stability conditions of the unusual stripe-like AFM structure observed in BaCoS 2 . We first recall the possible contributions to the exchange coupling in a solid [START_REF] Blundell | Magnetism in condensed matter[END_REF]:

1. direct exchange interaction arising directly from the Coulomb interaction between the electrons located at two neighbouring ions;

2. superexchange interaction between two magnetic ions bridged by a nonmagnetic ligand ion; As apparent from 1.11, in the case of BaCoS 2 , we expect a direct exchange J 1 between nearest neighbouring Co sites along the ab diagonal (or superexchange via the S atom with 90 • angle) and a superexchange J 2 between next-nearest neighbouring Co sites along the a-or b-axis. The former coupling is ensured by the direct overlap between d xy orbitals and will be FM. The latter coupling is provided by the hybridisation of the 3p x and 3p y orbitals of the S atoms bridging two neighbouring Co ions with the 3d x 2 -y 2 orbitals of the latter ions. This second coupling is AFM, according to the GKA rules [START_REF] Goodenough | Magnetism and the chemical bond, re krieger pub[END_REF] mentioned before 1 . It follows that the system is geometrically frustrated because the two sublattices of Co ions in the upwards and downwards oriented pyramids tend to order AFM due to J 2 , but they are FM coupled through J 1 . This simple observation accounts for the fact that a simple (1/2 1/2) AFM order is not possible in the ab-plane of BaCoS 2 .

In fact, a back-of-the-envelope calculation shows that the magnitude of the FM coupling constant J 1 must be small -although not negligible -because of the comparatively large Co-Co distance of 3.64 Å. On the contrary, the magnitude of J 2 is expected to be fairly large owing to the strong hybridisation of the above 3p and 3d x 2 -y 2 orbitals. We therefore expect two degenerate spin configurations in presence of the C 4 symmetry and thus large fluctuations. Owing to the small J 1 , we put forward the hypothesis that a slight orthorhombic distortion stabilises the experimentally observed AFM structure. Finally, an out-of-plane coupling J 3 is required to ensure the 3D character of the long-range order, in agreement with the well-known Hohenberg-Mermin-Wagner theorem [START_REF] Mermin | Absence of ferromagnetism or antiferromagnetism in one-or two-dimensional isotropic heisenberg models[END_REF][START_REF] Hohenberg | Existence of long-range order in one and two dimensions[END_REF] applicable to Heisenberg FM and AFM structures. The above considerations give a clue to establish a possible relationship between orthorhombic distortion and AFM structure in BaCoS 2 .

In order to verify the above scenario, a quantitative model would be required, which goes beyond to the purpose of the present thesis. Here we limit ourselves to outline the following ideas in view of a future work. Within the frame of the Hubbard model, the exchange coupling J depends upon the t and U parameters according to the expression [START_REF] Sawatzky | On the electronic structure and related physical properties of 3d transition metal compounds[END_REF]:

J = t 2 U (1.6)
or, in the case of charge-transfer insulators suitable for BaCoS 2 :

J = t 2 dd ∆ + 1 2 U pd = t 2 dd U ef f (1.7)
where notations are as before and t dd = t 2 pd /∆ plays the role of an effective d -d hopping parameter. The hopping energy, t, is affected by the interatomic distances, so pressure is expected to be an effective control parameter of the stability of magnetic structures. For instance, in the case of an AFM Mott insulator like BaCoS 2 , pressure is expected to increase t and thus J and T N . This expectation is indeed confirmed in BaCoS 2 as well as in prototype Mottinsulators like NiO, a point to be discussed below, in Chapter 4.

Motivation and open questions

In view of the considerations developed in the preceding sections, we argue that BaCoS 2 is a model system to study the stability of the Mott insulating phase in the general multi-band case, which remains little studied.

In this thesis, we first elucidate the controversy regarding the stability of the BaCoS 2 phase and the relationship between crystal structure and the unusual AFM order experimentally observed. This point is addressed in Chapter 3.

Second, we attempt to answer to the question as to whether it is possible to induce a metallic phase in BaCoS 2 using the usual control parameters known in the single-band case, i.e. U/t ratio and band filling. To realise this objective in the specific case of BaCoS 2 , our strategy is to use iso-or hetero-valent chemical substitutions at the Ba site, e.g. Sr and K. The former substitution is expected to produce a positive chemical pressure, thus enhancing t, while the latter substitution shall introduce holes in the electronic structure. The expected effect of the two substitutions is schematically shown in Figure 1.12. The advantage of these substitutions is that they do not affect the electronically active Co site, which would introduce electronic disorder.

To the best of our knowledge, there exists only one preliminary report on the effects of chemical substitutions at the Ba site, which concerns the K/Ba substitution [60]. Other reports concern substitutions at the Ni site [38,61].

Our choice of Sr-substitution to produce chemical pressure is justified by the smaller ionic radius of Sr 2+ ion in coordination IX (1.31 Å), as compared to that of Ba 2+ in the same coordination (1.47 Å) [62], but still comparable to make the substitution possible. This point is addressed in Chapter 4.

In the case of the K-substitution, the ionic radii of K 1+ and of Ba 2+ in coordination IX are quite different (1.55 Åvs. 1.47 Å, respectively [62]) but, in this case, the radius of the substituent specie is larger, which is expected to produce a negative pressure and thus to stabilise the insulating phase. Therefore, should a metallic phase be observed, the effect would be attributed to the effect of hole doping. This point is addressed in Chapter 5.

K + Sr 2+

Figure 1.12: MIT phase diagram from [START_REF] Imada | Metal-insulator transitions[END_REF] adapted to the current situation of BaCoS 2 , where the band-and filling-controlled (BC, FC) MIT may be achieved by means of the two iso-and hetero-valent substitutions of Sr and K for Ba, respectively.

Chapter 2 Experimental Methods

This Chapter illustrates the main experimental methods and techniques employed in the thesis. We should mostly provide the technical aspects of those experiments that have originally developed in our laboratory or during the thesis, while we shall limit ourselves to briefly describe those experimental methods and facilities that are well established and routinely employed.

Synthesis of polycrystalline and single-crystal samples

Measurements of structural and physical properties such as electrical and thermal transport or magnetism can be performed directly on polycrystalline samples, such measurements would help researchers to build an approximate image on their systems under study. However, since single crystals has a minimum of impurities and defects, these measurements would be much more accurate if performed on single crystals; that is why basically scientists are interested in single-crystal growth. Due to technical limitations as well as to difficulties in the manual handling of very small crystals, a considerable importance is delivered to the size and the quality of these grown crystals. Millimetric single crystals are grown to perform most of the physical measurements while some experiments like inelastic neutron diffraction requires 1 cm 3 size specimens at minimum. Semiconductor single crystals are widely investigated and also used for industrial purposes. On the other hand, a huge list of different systems, in which the single crystal growth is an essential step of their research, can be build up. Such systems are the perovskites, high temperature superconductors, transition metal oxides/sulfides and many others. Experimentally, the progress in single crystal growth studies of such systems is a promising strategy to understand some old and not well understood physical phenomena in condensed matter physics.

In this work, one of the main challenges was the growth of large and high quality single crystals of BaCoS 2 . Such large crystals are required for the purpose of studying in depth the magnetic structure and magnetic excitations through neutron scattering. Even though the limited time of the thesis did not allow us to complete this study, the efforts put on growing large crystals helped us to establish an efficient and reliable protocol for growing this phase which remained elusive so far. In chapter 3, we will discuss in details our growth self-flux method as well as the main results on the grown BaCoS 2 single crystals.

In this context and starting by the differential thermal analysis (DTA) experiment which is of a strong importance to gather thermal information about materials, I shall highlight in the following subsections my strategy for synthesising both: polycrystalline and single crystal samples of BaCoS 2 .

Phase stability study: differential thermal analysis

In order to choose the appropriate synthesis route for a given target compound, the knowledge of the phase diagram is required. In the case of BaCoS 2 , to the best of our knowledge, no well-established phase diagram is available. We have then first attempted to fill this gap of knowledge by determining precisely the melting temperature of BaCoS 2 . To do so, we have employed a differential thermal analysis (DTA) using the commercial LABSYS evo equipment by Setaram. The experiment was carried out in collaboration with L. Cormier (IMPMC). DTA is a thermoanalytic technique that provides insights onto different phase transformations or reactions that occur during the measurement process due to a temperature sweep. A typical experimental set-up is shown schematically in figure 2.1a. In this technique the material under study and a reference substance1 undergo a constant heating rate and due to their respective heat capacity, a temperature difference between them is recorded [63]. This temperature changing would continue up to a stable state in which a constant temperature difference is achieved, the signal at that point is known as the baseline. Therefor, endothermic or exothermic changes in the sample are determined as related to the reference. When the temperature is rising and a partial melting occurs in the sample, i.e. endothermic process, the sample's temperature stops increasing for a while, so an extra amount of heat is required to flow towards the samples in order to maintain the difference in temperature between the sample and the reference the same. When the melting ends, the temperature reverts rapidly to the baseline. This endothermic process is detected in the DTA curve as a negative peak (see figure 2.1b). On the other hand, in the case of exothermic reaction, the temperature of the sample will increase suddenly and will stabilize at that point for a while, in the same time the heat flow towards the sample will decrease until reaching back the baseline, this is how a positive peak appears in the DTA curve in case of exothermic process (figure 2.1b) [64]. The obtained DTA curve of the polycrystalline BaCoS 2 will be shown and discussed in details in chapter 3.

Synthesis by solid state reaction

In the context of this thesis, polycrystalline (powder) samples of BaCoS 2 as well as some precursors for high pressure synthesis of (Ba,Sr)CoS 2 were prepared by solid state reaction technique. This technique consists of heating two or more solids (starting materials) which react to form the required polyscrystalline compound. In this method, in order to achieve a good diffusion rate necessary for the phase formation, high temperatures are needed to be applied. The higher the temperature, the faster the reaction. However, the choice of temperature should be such that no phase decompositions take place during the growth process. It is also important to keep in mind that it is a 'solid state' reaction, meaning that it occurs in the solid state. There is, therefor, no need to raise the temperature up to the melting point.

In the following, I give the steps leading to a successful precursors preparation needed for poly and single-crystal growths:

• grinding the pure starting materials to obtain a homogeneous mixture.

For the purpose of this thesis, I have used stoichiometric quantities of BaS (99.998%, sigma Aldrich), SrS (99.998%, sigma Aldrich), metallic Co (99.9%, sigma Aldrich) and S (99.99%, sigma Aldrich). The grinding time for all the experiments was 10-15 minutes.

• forming the prepared mixture into pellets using a traditional hydraulic press. This step could help to bring a lot of crystallises together to accelerate the reaction. I have used 1 tonne of force to prepare pellets of diameter of 5 mm.

• the pellets then are placed in a crucible made of a sufficiently inert material with respect to the reactants. This step is to avoid other unwanted reactions with the surrounding. I used graphite crucibles in my synthesis.

• in the case of volatile or air sensitive reactants, the crucible should be sealed in an evacuated silica tube. All the powder and single crystals reactions I have carried out during my Ph.D were prepared in an evacuated (10 -5 mbar) silica tubes (minimum length ∼12 cm) using the glass arc welding station.

• the sealed evacuated tubes are then placed in a resistance heating furnace on which a heating cycle can be chosen. In the following paragraphs I will detail the thermal cycles I have implemented.

Usually, the thermal treatment is repeated twice, separated by an intermediate grinding procedure that turns out to be helpful to bring fresh crystallite surfaces into contact, thus ensuring a faster reaction and an improved phase purity. The heating cycles in powder and single crystals reactions differ mainly in the value of the reaction temperature itself and in the cooling down to the room temperature procedure.

Sulfur is a volatile element, its boiling point being 444 • C at ambient pressure and ∼ 300 • C at P = 10 -5 mbar. In order to avoid high vapour pressure during the chemical reaction, we apply a temperature dwell at 300 • C for 5 hours, which allows the liquid sulfur to react better with the rest of reactants in the mixture [66].

A typical thermal cycle used for powder samples is shown in figure 2.2 where the reactants mixture is heated up to 950 • C, a temperature lower than the melting point (solid state reaction), kept at that temperature for 48-72 hours and then quenched to the room temperature in a bath of liquid nitrogen. The quench has a considerable importance since BaCoS

Self-flux method for single crystals growth

The DTA results discussed in Chapter 3 revealed a non-congruent melting of BaCoS 2 , ruling out the use of standard 'out of melt' methods and making 'flux' methods unavoidable. A successful growth by the melt method requires much more than simply solidification of a melt; that is why different techniques such as: Bridgman, Czochralski, optical floating zone and others, are used for a better control of the nucleation [67,68]. The growth rate of a growth out of melt is normally high (typically 1 mm/h) and this is the main reason of the vast utilization of this method for growing crystals for industry. Since the choice of the suitable growth method is essential and depends strongly on the characteristics of the material and its physical and chemical properties, the growth out of melt is used strictly to grow crystals of materials that melt congruently, i.e. do not undergo any decomposition when cooling down through the liquidus [67].

As shown in section 3.3, this is not the case in BaCoS 2 and a flux method is necessary for single-crystal growth. In flux growth, the components of the material to be crystallized are dissolved at high temperature in a suitable solvent (flux). One can accomplish a flux-growth process folllowing three main steps:

1. realization of supersaturation: which is a thermodynamically non stable state, i.e. more stable states related to a decrease in the free energy can be achieved and resulting in forming crystals.

2. nucleation: due to the supersaturation and before forming a crystal, the randomly moving phase particles come together to form a stable cluster (nucleation). This is always thermodynamically favored since forming such nuclei reduces the total free energy.

3. subsequent growth on the formed nuclei.

However, the nucleation step, most of the times, does not occur homogeneously since the energy is reduced if the clusters form on the crucible walls or on the surface of the flux or even on the surface of an impurity in the mixture. This can be avoided by introducing a seed crystal into the solution such that a subsequent growth starts immediately [69]. After the first successful nucleation, the supersaturation level can be kept low by a regular slow cooling of reaction temperature. This is very important to start the subsequent crystal growth on this formed nucleus. On the other hand, the main challenge in any single crystal growth is to reduce the number of nucleation processes in order to produce small number of large crystals rather than a large number of small crystals.

The main advantages of a flux growth method can be summarized in the following points:

• growing materials below their melting temperature: using fluxes with deep eutectics.

• ability of growing noncongruent melting, multi component systems

• high quality crystals

• No absolute necessity for a seed crystal to start the growth, growth can happen through spontaneous nucleation.

For the single crystal growth, the reaction temperature is raised up to the melting point. In order to achieve the supersaturation state, the temperature has to be increased slightly above the melting temperature and then slowly cooled down to room temperature or to a suitable temperature for the quench. In the case of BaCoS 2 , we made several attempts to optimise the reaction temperature and found it to fall in the 1060-1100 • C range. This is consistent with the DTA analysis (section 3.3) which provided a melting temperature of 1050 • C. The reaction is kept in supersaturation at this temperature for usually 72 hours. Because of this constraint on temperature, the only way to increase the diffusion is to increase the dwell time. Some reactions can last weeks or even months to obtain an efficient growth.

In this work, we concentrated our efforts to optimise reaction temperature, so we did not fully investigated the influence of reaction time. As a final step, the sample is cooled down to 900-950 • C at a cooling rate of 1 • C/hour or even slower. This is an important parameter for the size of the obtained crystals. An example of a thermal cycle for single crystals growth is shown in Figure 2.3.

High pressure/high temperature synthesis

The synthesis under high pressure and high temperature (HP, HT) conditions is a powerful method to synthesise new materials and especially metastable phases. These phases have interesting and unusual properties which give an access to the understanding of some physical phenomena and structural phase transitions. A common limitation on the HP synthesis is the usually limited volume of the recovered sample [70].

In the framework of this thesis, I have used the solid state reaction and flux methods to grow respectively polycrystalline and single crystals samples of BaCoS 2 at ambient pressure. In order to study the Sr and the K-substitutions into the Ba site in our BaCoS 2 system, several series of high pressure synthesis have been undertaken using the belt press at the Néel institute of Grenoble/France and the Paris-Edinburgh press (PE) at IMPMC. In the case of Srsubstitution, due to the smaller ionic radius of Sr 2+ (1.31 Å) with respect to Ba 2+ (1.47 Å) [62], the high-pressure helps to shrink the interatomic distances in the BaCoS 2 lattice and thus favoring the incorporation of Sr ions in the BaS layer. On the contrary, since the K ionic radius (1.55 Å) is comparable to that of Ba [62], high pressure techniques are not needed to stabilize the K-doped phase. However, ambient pressure technique showed a high degree of complexity and found to be inefficient due to high gases pressure of starting materials that can be avoided in a confined high pressure cell.

In the next paragraphs, we describe the characteristics of the belt-type and Paris-Edimbourg (PE) presses used for the HP/HT synthesis of the samples object of the thesis.

Belt press

The belt press is a well-known and widely used apparatus for high-pressure synthesis. It can be used to create pressures ranging from 3 to 8 GPa and temperatures up to 1200 • C [71]. Typically, the sample volume is 20-30 mm 3 . A schematics of the press is shown in Figure 2.4b. The working principle relies on the following features: [START_REF] Chervin | Technologie des hautes pressions[END_REF][START_REF] Wong | Zno varistors as pressure memory and pressure sensor[END_REF]:

• The uniaxial load produced by two opposite anvils is transmitted to the pressure cell through a transmitting medium in the pressure cell (typically pyrophyllite). This uniaxial load produces a three-dimensional stress field in the cell owing to a suitable conical geometry of the anvils.

• Such stress field is converted into a nearly isotropic stress (commonly said hydrostatic pressure) using a ring (or belt) surrounding the cell. The role of this ring is to confine the deformation of the cell along the radial direction.

The anvils and the belt are made of a hard material, typically tungsten carbide.

A compressible teflon gasket is used to provide a lateral support to the pressure cell. A synthesis run can be achieved by following the next steps:

1. The powder is compacted into pellets using 0.2 tonne of applied force.

2. The pellets are loaded into a h-BN capsule of internal diameter 3 mm 3. The capsule fills the space of a hollow-cylinder-shaped graphite furnace; the two pyrophyllite disks are used to cover the bottom and top of the furnace.

4. The resulting assembly is placed between two pyrophyllite cells and loaded in the press. 5. Finally, the teflon gasket is set around the assembly.

Figure 2.5 shows the different parts and a cross section of the belt pressure cell.

A typical synthesis run begins by gradually increasing the pressure from ambient to target with a typical ramp rate of 5 tonne/min. When the setpoint pressure is reached, the cell is heated at a constant rate of 50 W until the setpoint temperature is achieved. For all synthesis runs, we kept the sample at high temperature/pressure conditions for one hour, then we quenched the sample down to room temperature by switching off the power supply and maintaining the water cooling system running. Once the sample approximately reaches room temperature, the pressure is slowly released. Finally, the sample is removed from the assembly and mechanically cleaned.

Paris-Edinburgh (PE) press

Before the 90s, the neutron scattering under high pressure was used for crystal structure refinement up to only 2-3 GPa. However, using neutron scattering for optical, electrical and other measurements require either single crystal samples about 1 mm 3 , or 100 mm 3 powder samples. This constraint makes the use of Diamond anvil cells (DACs) impossible. With this motivation Richard Nelmes, a crystallographer from Edimbourg university doing neutron scattering and interested in the high pressure science, contacted Besson from the "condensed matter physics" laboratory at the Pierre and Mary curie university for the elaboration of a press satisfying this needs. The first PE press (V-type) was constructed in 1990 by Besson,Hamel and Weil,figure 2.6 [74]. This achievement led to the creation of a laboratory dedicated to high pressure studies2 . 1) and ( 2) anvils and anvil seats in tungsten carbide, (s) sample space, (g) toroidal gasket, (i) the incident beam, (d) diffracted beam [START_REF] Klotz | Neutron powder diffraction at pressures beyond 25 gpa[END_REF].

V-type

Afterwards, this PE design was constructed more and more, not only for neuron purposes but also for other laboratories usage. A remarkable step of the development of neutron diffraction under high pressure was achieved in 1995 when a routinely pressure of 25 GPa was reached by S. Klotz, a post-doc at the condensed matter physics laboratory and a very closed collaborator of Besson at that time [START_REF] Klotz | Neutron powder diffraction at pressures beyond 25 gpa[END_REF] Up to this point, high temperature and high pressure were exclusive one from another in neutron facilities. Moreover, around 1994 Besson had already launched the development of high temperature with PE for X-ray diffraction measurements with the LURE synchrotron in Orsay. Around 2000, Y. Le Godec tuned in the high pressure high temperature for neutron scattering up to 7 GPa and 2000 K concomitantly [START_REF] Le Godec | Recent developments using the paris-edinburgh cell for neutron diffraction at high pressure and high temperature and some applications[END_REF]. S. A typical PE assembly preparation starts with compacting a mixed stoichiometric powder into a h-BN crucible which is capsuled afterwards in the graphite heater. Then, this heater is locked in the pyrophyllite gasket with the help of two molybdenum disks. Finally the steel electrodes rings which are filled with ceramic, are placed on the top and bottom of the assembly as can be seen in figure 2.8. The anvil body is made of stainless steel while its core of tungsten carbide.

Once the whole assembly is ready, it is inserted between the two anvils and the pressure can be increased slowly via a computer software.

The ramping rate (at the level of the oil compressor) that I have been using for all my synthesis series with PE press is 0.02 bar/s. When the target pressure is reached, one can start heating the sample. This is achieved by ramping up the current flowing from the lower anvil to the sample through the steel electrodes and the molybdenum disk inducing an increase of the temperature of the graphite heater by Joule dissipation. The current is set to a constant value resulting in a constant power being dissipated into the sample. As the sample heats up, its temperature difference to the environment increases resulting in an increase of the heat loss into the environment. After a while, when the power input and losses are balanced, a steady temperature of the sample is reached. After a complete reaction (two hours last in all the synthesis carried out with the PE press), a quench to ambient temperature is applied by simply switching off the power supply. For the same reasons mentioned in the synthesis at ambient conditions, this temperature quench is necessarily applied to stay away from phase decomposition of BaCoS 2 into Ba 2 CoS 3 (this issue is discussed in details in chapter 3). Moreover, a water cooling system is installed on the anvils to avoid overheating and cracking during the HP/HT stage or even during the decompression. Care should be taken for the calibration of temperature as a function of electrical power since the latter slightly depends on pressure. Moreover, the high pressure environment makes the use of thermocouples for a temperature calibration complicated.

Using the PE press, I have constructed temperature-pressure (T-P) phase diagrams out of ex -situ studies on BaCoS 2 and Ba 0.96 Sr 0.04 CoS 2 . The results are presented in Chapters 3 and 4, respectively. The Ba 1-x K x CoS 2 samples series was carried out also using PE press and the results are shown in Chapter 5.

2.3 Diffraction techniques and methods of structural refinement X-ray diffraction (XRD) is the most common experimental method used to determine the nuclear structure of a material in both powder and single-crystal forms. This technique has been systematically employed to determine quantitatively the phase purity of all the samples synthesised during the course of this thesis as well as to study the structural changes caused by the particular synthesis parameters employed and due to the chemical substitutions. For our routine measurements on powder samples at room temperature, we employed a commercial Panalytical X'pert Pro MPD diffractometer equipped with Co anode (λ = 1.7902 Å) in the Bragg-Brentano geometry. This instrument is one of the facilities operated by the X-ray diffraction platform of the IMPMC (see figure 2.9). Beside powder samples, the single crystals diffraction yields to a full structural information. At IMPMC platform, using Oxford diffractometer equipped with a sealed tube with Mo anode (λ = 0.7107 Å), a CCD detector and a 4-circle goniometer (see Figure 2.10), several series of room temperature XRD were performed on BaCoS 2 single crystals. On top of that, I have used the same diffractometer to orient and test the quality of about 100 BaCoS 2 single crystals which I selected for a magnetic structure study using neutron diffraction at Orphé reactor of the Léon Brillouin Laboratory (LLB). Regarding Ba 1-x Sr x CoS 2 and Ba 1-x K x CoS 2 , we focused on the polycrystalline samples since the available time was not sufficient to develop single crystal growth protocols. Moreover, being achieved using HP/HT techniques the obtained samples are very small in volume (< 20 mm 3 ). Consequently, it is difficult, using a laboratory diffractometer, to proceed a very accurate structural characterisation of the phase using Rietveld structure refinements since it strongly relies on the diffraction intensities. For that reason, a synchrotron X-ray source was needed.

Using the CRISTAL beamline of SOLEIL synchrotron (Gif-sur-Yvette, France) and thanks to B. Baptist, we could obtain room temperature measurements on Ba 1-x Sr x CoS 2 powder, as well as temperature dependence XRD on BaCoS 2 single crystal .

The synchrotron measurements on the polycrystalline samples were performed in transmission geometry on the finely ground powders loaded in borosilicate capillaries (0.1mm diameter). We used the 2-circle diffractometer of the CRISTAL beamline, which is equipped with a MYTHEN2 X 9K detector (DEC-TRIS). The measurements were performed at λ = 0.51490 Å) using 10 min exposure time. At the IMPMC XRD platform, we have carried out some measurements where the samples were also loaded in borosolicate capillaries and the measurements were performed using our Rigaku MM007HF diffractometer equipped with a RAXIS4++ image plate detector and a Mo rotating anode (0.70926 Å-0.713609 Å). The Fit2D program was used for the integration of 2D images into 1D patterns after a calibration with LaB6 standard.

While the single crystal XRD measurements on BaCoS 2 single crystal were performed at different temperatures on the 4-circle diffractometer of the CRISTAL beamline, at λ = 0.6713 Åusing a Cryo Industries of America He gas stream cooler. A 5-10µm crystal was selected in oil and mounted within a cryoloop. The intensities were collected with a 2D CCD detector placed at 80 mm from the sample using a 100x100 µm 2 beam size. For all temperature XRD measurements, data reduction, cell refinement, space group determination, scaling, and empirical or analytical absorption correction [START_REF] Clark | The analytical calculation of absorption in multifaceted crystals[END_REF] were performed using CrysAl-isPro software [START_REF] Rousse | Crystal structures of li6b4o9 and li3b11o18 and application of the dimensional reduction formalism to lithium borates[END_REF]. The structures were solved through the Olex2 [START_REF] Dolomanov | Olex2: a complete structure solution, refinement and analysis program[END_REF] program by direct methods using SHELXS [START_REF] Sheldrick | Shelxt-integrated space-group and crystal-structure determination[END_REF]. The refinement was then carried out with SHELXL [START_REF] Sheldrick | Shelxt-integrated space-group and crystal-structure determination[END_REF] by full-matrix least-squares minimization and difference Fourier methods. All the atoms were refined with anisotropic displacement parameters.

Energy dispersive spectroscopy

Beside the structural analysis by XRD techniques, the chemical analysis of the synthesised materials were performed using the energy dispersive spectroscopy (EDS) technique. This technique was provided through a scanning electron microscope (SEM) as well as a transmitting electron microscope (TEM) both installed at the IMPMC lab complemented with a EDS/TEM study carried out by C. Lepoittevin at Néel institute in Grenoble. Further TEM studies have been carried out in collaboration with D. Taverna (IMPMC).

When a solid sample is bombarded with a focused electrons beam, a spectrum of X-ray depending on the contained elements is emitted. EDS makes this spectrum useful for the knowledge on the chemical composition of the investigated sample. In principle, all the elements with atomic number between 4 and 92 can be detected. With EDS, a quantitative analysis aiming at the determination of the stoichiometry of a sample can be carried out by simply comparing the intensities of each element in the sample with the same in a calibrated database. Elemental maps also can be produced.

The error in the analytical result has two contributions: statistical error (precision) and systematic error (accuracy). The statistical error depends both on the apparatus used (typically ±0.1%) and actual physical fluctuations within a given sample or from sample to sample. The systematic error depends on both the apparatus and the way the samples are polished [START_REF] Bob | Energy dispersive spectroscopy on the sem: a primer[END_REF]. For a well polished surface with a standard reference measured on the same microscope, the accuracy is about ±1%. While it could degrade to ±5% without standards. On the other hand, the spatial resolution depends on the sample density. Obviously it is in µm units since the detected x-rays are generated from very deep (some microns) of the interaction volume. In order to decrease the interaction volume, lower accelerating voltages (∼17 KeV) are usually used. Thanks to the large Z elements in BaCoS 2 the spatial resolution with EDS is usually around 1µm 3 .

SQUID magnetometry

Magnetic measurements are essential to investigate the interplay between crystal structure and AFM order in BaCoS 2 , so we have carried out these measurements systematically on our samples using a commercial Quantum Design SQUID magnetometer operated in the MPBT platform of the physics and chemistry departments. With this sensitive magnetometer, the measurements can be done over a temperature ranges from 2 to 400 K and up to 800 K with a furnace installation. It also benefits of the two types of applied magnetic filed: a static magnetic field (DC) ranging from -7 to 7 T and an alternative magnetic field (AC) with an amplitude < 3.5 Oe and a frequency from 0.1 Hz to 1000 Hz.

A SQUID device consists of two superconducting layers separated by a thin insulating barrier to form a Josephson junction. The current density through the thin insulating layers depends on the phase difference of the cooper pairs wave functions on the two sides of the junction. The time derivative of the phase difference is linked somehow with the voltage across the junction. Moreover, in the case of a superconducting ring with two parallel junctions, the phase difference is also influenced by the magnetic flux through the ring. In this way, the magnetic flux can be converted into a voltage signal. This is conceptually what happens in a SQUID magnetometer. When a magnetic sample moves vertically in the axis of the superconducting loops, the resulting magnetic flux is converted to voltage. This voltage is then amplified by electronics and finally read out on the computer screen.

In our case, since the magnetisation M of BaCoS 2 is proportional to H up to reasonably high fields, we can approximate the susceptibility as χ = M/H, justifying the use of the DC option, which is much faster than the AC-one.

Specific heat measurements

Specific heat measurements as a function of temperature, C(T ), and of other control parameters, such as magnetic field or pressure, is a powerful tool to probe accurately phase transition as well as any structural changes that affect the lattice or electronic energies. To perform these measurements, in the present thesis, we systematically employed a commercial physical properties measurement system (PPMS) from Quantum Design. This system is equipped with a thermal-relaxation calorimeter that operates in the temperature range of 2-305 K and a constant pressure of 10 -5 mbar. The measurement method consists of increasing slightly the sample temperature (∆T < 1 % of the initial temperature) and then measuring the time required to return to equilibrium at the initial temperature. During the measurements, a known amount of heat Q is applied at a constant power for a time t until reaching temperature T + ∆T . The heating is followed by a cooling of the same duration and the temperature profile then is directly dependent on the heat capacity of the sample.

Figure 2.11 shows the PPMS sample holder. A heating resistance and a thermometer are located under the sample-support where the sample is deposited. A thin layer of grease is used to ensure a good thermal contact between the sample and the thermometer underneath. A simple difference between the two curves yields the net specific heat signal.

The specific heat measurement is carried out in two steps:

• 1 st : an addenda measurement that consists in calibrating the signal of the deposited thin layer of grease.

• 2 nd : the measurement of the same deposited quantity of grease with the sample (crystal or sintered powder). Note that the sample has to have a sufficient mass (≥ 2 mg). Although this lack of a precise method or recipe for growing single crystals from a flux might strike some as a large hurdle to surmount, it is precisely this imprecision and reliance on either intuition or luck that gives this technique so much of its charm and power.

Canfield et al. [START_REF] Canfield | Growth of single crystals from metallic fluxes[END_REF] Chapter 3

Synthesis and metastable structure of BaCoS 2

In Chapter ??, we emphasised that BaCoS 2 is a model system to study the Mott transition in the multiorbital case. The BaCoS 2 system remains little studied and it turns out that that the optimal synthesis conditions have not been established and some crucial structural properties remain controversial. Elucidating these issues is therefore necessary to grow large single crystals suitable for crucial experiments, e.g. inelastic neutron scattering, and thus to establish clear structure-property relationships that may help us understanding the physics of BaCoS 2 and of the Mott transition in multiorbital systems in general. It is then the purpose of the present chapter to address some important open issues regarding the synthesis and the structural properties of BaCoS 2 . The chapter is instrumental to introduce the presentation of the structure-property relationships object of the next chapters. The chapter is organised as follows:

• In Section 3.1 we present the state of the art in the field and highlight the structural properties of BaCoS 2 object of controversy. To do so, we review the various synthesis methods used so far for BaCoS 2 single crystal growth with emphasis on single-crystal growth techniques.

• Section3.2 shows that by using the available information regarding the phase stability of BaCoS 2 , I could establish a reproducible route to synthesise high-purity BaCoS 2 powder.

• In section 3.3, I will use the outcome of the DTA study that I did during my theses to explain the lack of reproducibility in the previous studies and therefor establish a reliable route for the synthesis of high quality single crystals.

• In 3.4 I will compare between the conventional so far used single crystal growth method and the new self-flux I developed in this work, showing the effect of the new flux on the reproducibility, quality and on the size of the grown single crystals.

• Finally, section 3.5 will present our last experimental results on the stability of the orthorhombic Cmma phase under pressure and temperature. The first goal of the present chapter is to elucidate this controversy by carrying out an detailed temperature-and pressure-dependent study of the crystal structure. We consider that this study is relevant considering the relationship expected between stability conditions of the insulating AFM phase and crystal symmetry. To the best of our knowledge, there exist only two previous reports in the literature. In a first study, Gélabert et al.

[43] investigated the stability of the orthorhombic structure as a function of temperature in the 170 -470 K range using powder samples. The authors find a systematic decrease of the cell parameters with temperature. In a second study, Kanada et al. [84] found that the orthorhombic Cmma structure is stable up to 2 GPa. The results of these studies raise the following open questions:

• In which structure does BaCoS 2 crystallise?

• What is the space group of symmetry that polycrystalline and single crystal phases of BaCoS 2 correspond to?

• Which is the link between orthorhombic distortion and AFM order? Is the AFM to PM transition accompanied by a structural transition? Do the lattice parameters change across the AFM/PM phase transition at T > 310 K?

In order to address these issues, high resolution x-ray diffraction data on high-quality samples is required. At this point, we should discuss on the controversial issue regarding the strategy to follow for the synthesis of the samples and whether a specific synthesis method may affect the structure.

In the next paragraphs, we will discuss in details the controversial results on the BaCoS 2 phase synthesis and the missing information on the phase diagrams of this phase.

Synthesis and metastability of BaCoS 2

Previous reports show that the synthesis of BaCoS 2 is challenging owing to its metastability, the stable phase at standard temperature and pressure conditions being the competing phase Ba 2 CoS 3 [43] (see phase diagram in figure 3.1). In fact, according to classical thermodynamics there is no metastable phase stricto sensu because the system shall irreversibly relax to the ground state after a sufficiently long time. However, in practical terms, due to the multiphases decomposition of BaCoS 2 into Ba 2 CoS 3 , BaS and CoS below 850 • C [43], it has been shown that quenching from high temperature enables to obtain a pure BaCoS 2 phase. These step adds an extra difficulty in controlling the property of the synthesised phase. Moreover, quenching from extreme conditions (high pressure and/or high temperature) leads to the freezing of phases which contain variations of the grains morphology and/or the crystal structure of the synthesised phase.

Synthesis of polycrystalline powders

The standard solid-state method to synthesise single-phase powders of BaCoS 2 consists of grinding a stoichiometric mixture of BaS, Co and S powders of suitable purity, pressing the powders into pellets and sealing the pellets in an evacuated quartz ampoule. The ampoule is then submitted to a heat treatment in a furnace which produces the desired solid state reaction [42,[START_REF] Fisher | Electronic transport and antiferromagnetism in the layered compound bacos 2[END_REF][START_REF] Mandrus | Magnetism in bacos2[END_REF]. The samples are usually kept at typical temperatures in the range of 900-950 • C for 3-4 days and then quenched into a bath of ice-water or liquid nitrogen. Snyder et al. did not observe differences in the powder pattern when using commercial or home-made BaS [42]. On the other hand, Fisher et al. found that only BaCoS z samples which are nominally stoichiometric (z = 2) and with nominal z = 2.1 are single phase BaCoS 2 powder [START_REF] Fisher | Electronic transport and antiferromagnetism in the layered compound bacos 2[END_REF]. Moreover, Gelabert et al. [43] have carried out an intensive synthesis study of BaCoS 2+x (0 ≤ x ≤ 1) in order to study the effect of the sulfur excess on the structure and the purity of the obtained phase. These study has ended up by constructing a temperature-sulfur excess phase diagram shown in figure 3.1. They show that when cooling down BaCoS 2 it decomposes into Ba 2 CoS 3 and CoS. For a stoichiometric BaCoS 2.0 , the decomposition is found to be at temperatures ≤850 • C. The decomposition temperature, presented through the peritectoid line, shifts upwards as x increases meeting the solidus and the liqidus lines at x = 0.4 and 0.6 respectively. It is at the region defined between the peritectoid and liquidus lines where they observed a single phase of both, single crystals and polycrystalline powders of BaCoS 2 .

In spite of the sulfur excess issue, which is still problematic, one of the very first objectives of this work is to synthesis a highly pure and reproducible BaCoS 2 powder. It is at these perspectives that I have concentrated in this work at the region where 0 ≤ x ≤ 0.15 and quenching always at T = 950 • C to avoid the decomposition temperatures. In section 3.2 we show that the above information regarding the stability conditions of BaCoS 2 allow us to establish a successful route for the reproducible synthesis of high-purity BaCoS 2 powders.

Single-crystal growth

A thorough study of BaCoS 2 requires single crystals of sizable size and therefore the knowledge of the phase diagram of the system. Since available data are not available yet for BaCoS 2 , we have carried out a first study using the DTA technique. The determination of the phase diagram of BaCoS 2 was not a direct purpose of this work, nevertheless, it was a very important step to achieve the growth of large single crystals which are needed for studying the magnetic structure of the system through neutron scattering.

In the literature, we find numerous attempts at crystal growth of BaCoS 2 where the authors used empiric methods with a limited knowledge of the phase diagram [42,43,[START_REF] Mandrus | Magnetism in bacos2[END_REF]85]. They obtained single crystals with a size of 0.1 mm 3 which is far too small for experiments such as inelastic neutron scattering.

Since building a phase diagram is a hard task and is beyond the scope of this work, we rather started by testing the reproducibility of the previously reported methods and growth parameters. Afterwards, the strategy was to determine more precisely the melting temperature and the melting type in BaCoS 2 . This informations about the melting temperature and its type are critical at the beginning of any growth process. In the previous single crystal growths [42,43], very thin plate-like single crystals of ≤500 µm were obtained in a solidified matrix1 . However, Mandrus et al. Interestingly, this study was concentrated on the BaCo 1-x Ni x S 2 phase and did not confirm any possibility of growing as large as 1 cm 3 ingot or plate-like single crystals of the BaCoS 2 end member (x = 0). This is, in my opinion, due to the real difficulty in growing the orthorhombic insulating phase of BaCo 1-x Ni x S 2 component.

In this perspective, it would be important to understand the interplay between the structural transformations and the fundamental electronic state of this phase and the impact of the used synthesis method on its properties.

In the light of the previous discussion, the following specific questions can be addressed:

• Using the available information on the single phase stability of BaCoS 2 , can one grow systematically BaCoS 2 single phase polycrystalline powder and/or single crystals? If not, is there a way to optimise the growth conditions to do so?

• Is there any alternative method to grow pure and reproducible single crystals of BaCoS 2 ?

• How can one overcome the lack of information on the BaCoS 2 phase diagram and optimise an efficient self-flux growth using existing binary/ternary phase diagrams?

• Is there any relationship between the adopted growth method and the structural properties of BaCoS 2 phase? If yes, how are these effects reflected on its fundamental state?

Synthesis of high-purity BaCoS 2 powders

We prepared high-quality BaCoS 2 powder samples using a mixture of BaS (99.998%, sigma Aldrich), Co (99.9%, sigma Aldrich) and S(99.99%, sigma Aldrich) powders that we thoroughly grinded and pressed in the form of pellets in a glove box under Ar atmosphere. These pellets were then loaded in a graphite crucible and sealed in an evacuated quartz tube with residual pressure of ∼ 10 -5 mbar. The ampoules were heated up to 950 • C and held at this temperature for 72 hrs. An intermediate heating at 300 • C for 5h is adopted to avoid the strong vapour pressure of sulfur [66]. The final step of this procedure consists of a quench in a bath of liquid nitrogen (see chapter 2 for more details on these steps). The phase composition of each sample has been characterised by x-ray diffraction using a diffractometer equipped with a Co K α source. The measurements sometimes unveil the presence of secondary phases, such as BaS, CoS or Ba 2 CoS 3 . In this case, an intermediate grinding followed by an additional thermal treatment, identical to the one described before, turns out to be effective in improving the phase purity. Figure 3.3 shows the x-ray powder diffractograms after both, the first and the second thermal treatment of a 15 powders containing an excess of sulfur after the 1 st and 2 nd thermal treatments and of BaCoS 2 with no sulfur excess after a single thermal treatment. Note the evident impurity peaks in the pattern of the 1 st thermal treatment (in green). Due to the lack of time, the sulfur concentration was not refined in our study that is why we consider the nominal values.

tice at first sight that the excess of sulfur stabilises secondary phases beside BaCoS 2 . Most of the Rietveld refinements of multiphase powder samples show the BaCoS 2 phase as a major phase (with ∼98 % of the total integrated peaks intensities) with BaS, CoS and Ba 2 CoS 3 representing the remaining ∼ 2% of the mixture. However, repeating the whole synthesis process or starting with a stoichiometric mixture, i.e. without sulfur excess, would improve the purity of the powder. I also have to mention that the quality of the quenching process, which should be done in few seconds, can be responsible for a slight decomposition of the phase. In fact, the difference between the decomposition temperature of 850 • C and the temperature of the quench ∼950 • C is relatively small. The quality of the vacuum in the ampule, and its volume are also hard to be reproduced identically every synthesis, this can play another role on the quality of the powder.

In figure 3.4 the Rietveld refinement of BaCoS 2 powder is presented. The x-ray diffraction data were collected at the SOLEIL synchrotron source using a 2circle diffractometer at the CRISTAL beamline. The structure is confirmed to be in the orthorhombic space group with the refined parameters in agreement with the previous studies (table 3.3). In order to explore the role of the crystal structure on the physical properties, HP/HT synthesis were performed using the PE press described in section 2.2.2) to explore the P-T stability region of the polycrystalline BaCoS 2 . The synthesis were carried out by grinding a mixture of BaS, Co and S powders with molar ratio of 1:1:1. Figure 3.5 shows the constructed P-T phase diagram where the color scale presents the percentage of BaCoS 2 phase found in the multiphases synthesised powder, the other existing phases and the unit cell informations of BaCoS 2 phase are gathered in the table 3.1. From these data, one can say that using PE press the HP/HT conditions have two stability regions for BaCoS 2 : low pressure (≤ 2 GPa) with temperatures ≤ 950 • C and high pressure (4 GPa for instance) with temperatures higher than 1200 • C. Further synthesis can be interesting to confirm the reproducibility of these conditions and to explore more regions in this phase diagram. sample P(GPa) T(

Differential thermal analysis (DTA) study

In the following I present the DTA analysis (2.1.1) that are needed to extrapolate the basic thermal information on the stability and reactions of the BaCoS 2 system. These information are needed in order to determine the best crystal growth conditions.

For this experiment I used 49 mg of highly pure powder of BaCoS 2 , the powder was prepared nominally with a sulfur excess (BaCoS 2.15 ) and then reacted in the furnace through two complete successive thermal cycles to give a pure powder as seen in figure 3 Different phase changes are observed in the DTA curve, figure 3.6b, however other techniques are needed to quantify these observations. In Figure 3.6b-b1, the end-tail exothermic peak at room temperature (or T AF M ) interestingly presents the AFM to PM transition. Furthermore, the small endothermic peak at T 1 falls in the melting temperature interval of sulfur ∼117 The above qualitative DTA data show an obvious decomposition of the BaCoS 2 phase before the complete melt at T m ∼ 1050 • C. This means that an incongruent melting for BaCoS 2 is confirmed and its single crystals can not be grown out of melt but a flux method for its single crystal growth is required.

Flux growth of single crystals

Out of a flux, what favorites a growth of a phase more than another, fundamentally, is their relative stabilities with respect to the flux. The existence of undesired phase which is more stable than the desired one can make the whole growth attempt fail [88,89]. The thermodynamic data to give such information is not generally available especially for ternary component melts. In the absence of such studies on BaCoS 2 , we have performed the growth using different fluxes and different flux concentrations and then the best result was repeated to test its reproducibility. Moreover, when searching for a flux that is not presented as one or more of the constituents of the desired phase, one has to check some points before choosing a suitable one: the reaction temperature, the method that will be used to extract the crystals from the residual flux and in some cases the possibility of unwanted doping or undesired flux inclusions in the crystals [90].

This section will be divided into two parts, in the first one I will present the traditionally used growth method (conventional methods) and the second part will discuss in details the self flux method which I have carried out in these work.

Conventional single-crystal growth method

In the frame of this work, the very first attempts in the growth of BaCoS 2 single crystals were by using the conventional route of heating a stoichiometric mixture of BaCoS 2 above 950 • C and forming a crystallized phase by cooling down the melt at a very small rate 1-5 • C/h. This is the most common method reported in the literature [37,42,43,[START_REF] Mandrus | Magnetism in bacos2[END_REF]. We argue that the term "self-flux" in these studies is not correct since the grown mixture is stoichiometric and no other phase is present in the liquid state. So, we should propose to rather call the method "out of melt" to which we refer in this thesis as "conventional methods". Few of the crystals grown using conventional methods are rather large ( 1 mm) and plate-like, however the number of nucleations is large, which is known to limit the crystal size. This observation may explain why our repeated attempts to grow more large crystals using this method have not been successful in spite of the careful control of growth conditions, e.g. same preparation procedure, same temperature cycles and same furnace. Figure 3.7 shows some examples of crystals grown using this method. At this point, the research for a flux became considerably an important step for the advancement of the growth of high quality and reproducible large single crystals of this phase. Finding a flux which obey all the conditions for growing a pure, large, easily separated crystals without surface contamination is a very difficult task. Here, thinking about self-flux method, where the flux is composed of one or more of the desired phase components, can be an efficient solution. For this purpose, deep eutectic binary phase diagrams of one or more components of BaCoS 2 are needed.

Self-flux method

The relatively deep eutectics in the transition metal-sulfur binaries makes them promising candidates as a flux for growing a wide range of TM-S-X compounds [91]. shown extremely interesting results regarding the size and reproducibility of the grown single crystals. Since at room temperature the flux is solidified, the crystals are trapped in the solid eutectic matrix which contain the flux phase or a mix of different phases. This matrix is called in metallurgy as regulus [92].

Taking the example of HAC1 batch (table 3.2), using the above Co-S binary phase diagram specially the Co 4 S 3 stoichiometry as a flux with 2:1 (BaCoS 2 :Co 4 S 3 ) molar nominal ration, we could grow plate-like single crystals with less number of nucleations and of at least 3 × 3 × 0.5 mm 3 as seen in the SEM image of the sample cross section in figure 3.9a. Mechanically removing the crystals from the regulus is not the most suitable way to do. Even with a lot of attention, the crystals can crack or get destroyed by the large stress applied by the flux matrix and the real size out of the used growth method is not completely precise, it is also time consuming and brings usually contaminated crystal surface that would need to be cleaved afterwards to do a proper measurements. Despite these downsides, it is still a practical option in case of lack of equipment. Some crystals grown by self-flux method, where the Co 4 S 3 used as a flux, are shown in figure 3.9. The crystals were removed mechanically and the size of separated crystals was significantly reduced. It is seen in the SEM image that BaCoS 2 grows in a remarkably interesting way where the flux matrix (dark gray) is sandwiched between the crystals plates (bright plates). It is also noticed that a beginning of a growth occurs nicely in parallel with most of the large platesingle crystals. These spectacular growth is reproduced in a similar way in the batch HAC2. The strategy we have adopted in this growth method was mainly based on the DTA data. In any single crystal growth, in order to grow large crystals one has to reduce the number of nucleation, to do so, we have to melt completely the mixture without heating much further beyond the liquidus temperature2 . Out of our DTA data, the liquidus temperature in the case of BaCoS 2 would be around 1050 • C. We found that heating the mixture at 1100 • C for many days (3-4 days) was a good choice, however, further attempts at lower temperatures can be experimented. The next choice then is that of the quench temperature, at this temperature two conditions have to be fulfill: the used flux has to be liquid and no secondary phases are formed. We deduced from both, the DTA results and Co-S phase diagram that at T = 900-950 • C those two conditions are respected. Table 3 From top to the sample HA19s, conventional method has been used. The other samples have been grown with self-flux method using Co 4 S 3 as a flux. Further analysis on the obtained phases have to be carried out.

Conventional vs. self-flux single-crystal growth

In order to assess the improved crystal quality obtained using the self-flux method, we compare the structural refinements obtained on representative single crystals grown using the conventional or self-flux methods. We first studied a representative single crystal grown using the former method by high-resolution X-ray diffraction at the CRISTAL beamline of the SOLEIL synchrotron. We then studied a representative single crystal grown using the latter method using a laboratory single crystal x-ray diffractometer equipped with a Mo K α source.

For the structural refinements, we integrated the intensity of 369 and 316 independent reflections, respectively, using the Olex2 software. The results of the two structural refinements are reported in Table 3.3. The refinements on both crystals show very similar results as to the structural parameters, but the refinement on the crystal grown by the self-flux method shows better R-factors in spite of the fact that the data were obtained using a laboratory XRD diffractometer. The improved R-factors is a good indicator of the improved crystalline quality of the crystal. In Figure 3.10, we compare the x-ray diffraction Laue images of the three principal sections for representative crystals grown using the two techniques to show visually the improved quality of the crystals grown using the self-flux method. Note, for instance, that the Bragg's spots of these crystals are better defined, the contrast is higher and the shape of the diffraction spots is more rounded. The grown crystals found to be twinned even at small scale (< 100 µm), this is probably due to the quench from high temperature where the stabilised phase has a tetragonal structure.

Mechanical separation of the crystals from the flux

In the cas of the flux method, the mechanical separation of the crystals from the residual solution is one of the critical problems, particularly in the case of chemically unstable phases for which the choice of suitable solvents to leach the flux is limited. One can overcome this difficulty by using a high temperature solution that allows the flux to be poured off. However, this decanting process has the following drawbacks:

• The hot crucible has to be removed very quickly from the furnace to allow the flux the pour out.

• The crystals can suffer a thermal shock. 3 The orthorhombic distortion is defined as: δ = a-b a+b 2

. The error bar is given by the relation

σ(δ) = √ σ 2 a +σ 2 b a+b 2 self-flux method
Conventional method (100) (010) (001) (001) (010) (100) Figure 3.10: X-ray diffraction Laue diffractograms of BaCoS 2 single crystals grown using the conventional and flux methods. Images are taken along the three principal directions. Note that in the self-flux crystals the Bragg's spots are better defined, the contrast is higher and the shape is more rounded, which indicates an improve in the crystals quality using this growth method.

• Residual droplets of the solution may remain on the crystal surface.

In the case of BaCoS 2 , a first tentative of pouring off the flux was carried out. We have used a decanting disc with holes diameter of ∼ 0.7 mm. This attempt ended up by ampule explosion while handling it at high temperature. No other efforts have been put to improve this process since it was quite dangerous.

Therefore, a smooth flux separation technique is needed. Centrifuging with a frit-disc crucible can be a good and clean choice [93]. Even if it is an expensive process that needs a careful handling and optimisation of the experimental conditions, the method is attractive for it allows not only to separate crystals with clean surfaces but also to recover the flux for further analysis.

In this thesis we limited ourselves to the mechanical separation method because of the limited available time and also the fact that the lab is not equipped for other methods but this, indeed, opens an interesting perspective for future work.

-9 ≤ h ≤ 9 -8 ≤ k ≤ 9 -6 ≤ l ≤ 8 -7 ≤ h ≤ 5 -4 ≤ k ≤ 6 - 4

Stability of the orthorhombic phase

Considering the controversial studies regarding the symmetry of the stable phase and the possible relationship between orthorhombic distortion, onset of antiferromagnetic phase and the structural changes concomitant to the Mott transition, we studied the pressure and temperature dependence of the nuclear structure.

Pressure dependence

It is well known that the BaCoS 2 system becomes metallic and paramagnetic under pressure, but there is only one study which has been reported on the pressure dependence of the lattice parameters and the structural changes in the system [84]. In order to improve the resolution of these data, we have planned a new single-crystal in situ diffraction study at the SOLEIL synchrotron. Kanada et al. have shown the pressure dependence of the lattice parameters a, b and c at room temperature, they have observed that at thermodynamic equilibrium i.e. without decompression, the value of a and b decrease when increasing the pressure. Moreover, a rapid decrease is observed at MIT pressure (P c = 1.1 GPa). On the other hand, the c-parameter does not show the same behavior at P c . The abrupt change around P c was explained by a redistribution of the electronic coordination in the 3d-orbitals, in other words, the electronic occupation in (x 2 -y 2 ) decreases and that of z 2 and xz, yz increases with increasing the pressure through P c . Interestingly, the out-of-equilibrium synthesis do not exhibit a similar behavior, but instead, an on/off behavior is observed at P ≥ 1 GPa. This means that quenching from HP/HT freezes the lattice parameters with a shrinked unit cell and with a negligible orthorhombic distortion as shown in figure 3.11. 3.11: Comparison between the effect of hydrostatic pressure [84] and that of synthesis pressure on the lattice parameters of BaCoS 2 . The notation ex situ for the samples synthesised under high pressure indicates that the lattice parameters have been measured on the recovered samples. Note that synthesis pressure stabilises a tetragonal phase above a threshold value of ∼1 GPa.

Temperature dependence

The evolution of the cell parameter, a, distortion as well as the magnetic susceptibility and the specific heat of BaCoS 2 as a function of temperature are shown in figure 3.12. The single crystal refinements of the lattice parameters are also presented in table 3.4.

Structure and suppression of the orthorhombic distortion

Our analysis of the x-ray diffraction indicates that the symmetry remains Cmma in the whole temperature range studied (20-360 K). Decreasing the temperature decreases the in-plane parameters (a and b) while the perpendicular to the plane (c) does not show any particular tendency. Nevertheless, the orthorhombic distortion interestingly shows a Gaussian dependency around T N which is defined with the gray vertical band shown in figure 3.12. There might be some discrepancy of the T N value here and in previous reports, but this could be due to the definition of T N in the susceptibility. In our study we have referred to

T N with the well defined transition peak in the specific heat curve.

Magnetic susceptibility and specific heat

We remind here that the BaCoS 2 phase shows a unique magnetic structure due to its nematicity as proposed by the two previous neutron studies [41,[START_REF] Mandrus | Magnetism in bacos2[END_REF]. In both studies, the nematicity of the magnetic order in ab-plane consists in an AFM stripes in a direction and FM ones in the other direction at T N = 310 K. Furthermore, The magnetic susceptibility curve of BaCoS 2 shows a large fluctuation region around T N . This large region can categorize the observed fluctuation into a short-range order type. We will see in next chapters that this fluctuations can explain the disappearing of the characteristic AFM transition peak in the specific heat curves of both Sr and K-substituted phases.

As mentioned before, we identified the T N value through the very pronounced transition peak observed in the specific heat curve which found to be around 290 K. Nevertheless, there are some unusual aspects in this transition: in addition to the typical behaviour in the susceptibility of an AFM system, we do observe a clear Curie-Weiss like upturn at low temperature which may explain the non zero extrapolation of the susceptibility down to room temperature.

These observations indicate the presence of some paramagnetic impurities that superpose with the AFM order. Moreover, at temperatures above T N and in the paramagnetic phase, the orthorhombic distortion is suppressed down to almost zero at 365 K.

It can be seen that the orthorhombic distortion coincides with the AFM order. We also observe that the orthorhombic distortion has a fluctuation zone which is spread over almost 60 K. This is consistent with the previous DTA data and shows a fluctuation region with the same temperature width (inset of the specific heat curve, figure 3.12).

Surprisingly, our temperature-dependence experiment of the lattice parameters on a single crystal of BaCoS 2 does not exhibit any structural transition around 223 K as reported previously on powders [43]. This could be due to changes in the sulfur content between powder and single-crystalline samples. Further electron diffraction studies can elucidate this controversy and confirm or not the occurrence of superstructure around 200 K.

Note that the three set of data show a clear anomaly at T N : the Gaussianlike behaviour of the orthorhombic distortion, the typical cusp-like behaviour of the susceptibility and the discontinuity of the specific heat. This shows a strong link between the stability of the antiferromagnetism and the stability of the orthorhombic structure. The synthesis of BaCoS 2 at HP/HT confirms this hypothesis where, as we have seen previously, the HP/HT synthesis stabilize a structure with strongly reduced orthorhombic distortion. Moreover, in figure 3.13 one can remark that BaCoS 2 under pressure shifts the T N to a smaller value (∼ 260 K) with an enlarged transition peak in the specific heat curve.

A support of this picture is provided by ab-initio calculations carried out by B. Lenz and M. Casula of the TQM team at IMPMC, who calculated the energy difference between the PM and the AFM states within the framework of GGA + U . Taking into account the orthorhombic distortion in BaCoS 2 , this calculations propose that at T < T N the two possible stripe orders, AFSa with ordering vector Q=(π,0) and AFSb with Q=(0,π), also proposed by neutron studies [41,[START_REF] Mandrus | Magnetism in bacos2[END_REF], are not at the same energy and only one is stabilized at low temperature by the orthorhombic distortion in the system. Zainullina et al. proposed that the BaCoS 2 ground state can be considered as a set of energydegenerate orbital-ordered configurations of Co 2+ ions in the high-spin state.

Although qualitatively agreeing with this picture, the calculations of Lenz et al. suggest that for an order of a certain magnetic stripe (AFSa or AFSb) only specific orbital configurations would explain the observed metal-insulator transitions as a function of temperature and pressure in a coherent way. The fact that the energy differences of the different configurations are of the order of 20 meV would also suggest that the transition happens from a certain AFS state (in a-or b-direction) to a mixture of the two AFS states, whose energy difference is within GGA+U far smaller than that to the PM state. This interpretation of fluctuations between different spin and orbital configurations could also explain the broad transition peak around T N seen in figure 3.12.

Low temperature specific heat

As seen in figure 3.12, the C-T curve of BaCoS 2 shows a λ shaped sharp anomaly at 290 K corresponding to the AFM transition seen also as a cusp-like behaviour in the magnetic susceptibility curve. This λ anomaly around the AFM transition was also observed in the Bi-doped BaCoS 2 [61]. However, in a typical AFM system, e.g. the prototype NiO or the monoxide CoO, the transition will be presented in the specific heat curve as a clear jump as seen in [94].

In figure 3.14a we plot the C/T vs. T 2 curve of BaCoS 2 (synthesised at AP and HP) at low temperature. Within a conventional Debye model, the curve is expected to follow a linear behavior C/T ∼ γ + βT 2 , where the Sommerfeld constant γ is proportional to the carrier density and found to have a non zero value in metallic phases. Thus, one could expect that the specific heat curve at low temperature in BaCoS 2 would follow a linear behaviour extrapolated to zero. However, we found that in BaCoS 2 , this linear behaviour is deviated at T < 40 K.

The anomaly then can be explained by a Schottky term which is expected in a two level system. Due to the othorhombic distortion in BaCoS 2 , the degeneracy due to the tetragonal symmetry is weakly lifted and the system can be considered as a two level system. We remind here that in a two level system with ∆ energy difference, at enough low temperature (T ∆/k) the upper level will scarcely be populated, whereas at T ∆/k both levels will be almost equally populated. But at temperatures comparable to ∆/k, transitions from one level to another would take place. This change in the internal energy corresponds to a large specific heat that becomes zero at both low and high temperatures. Thus in such systems, there is this eccentric possibility of a hump in the specific heat curve which is in general superimposed on the lattice and other contributions. This Schottky anomaly recognizable as a peak in the specific heat curve appears at very low temperatures as seen in figure 3.14 where the remarkable turn up of the curve is nothing but the tail of this peak. Hence, the low temperature C/T Vs.T 2 of BaCoS 2 can be described as:

C T = βx + 4 R z 2 1 √ x e z (1 + e z ) 2 (3.1)
where

β = (12π 4 R/5θ 3 D ), x = T 2 , z = ∆/ √ x and R = N A k B = 8.3145 Jmol -1 K -1
. These observations yield to γ ∼ 0 within the experimental resolution, as expected for an insulator. The Debye temperatures are estimated from the fitted parameter β and found to be θ D = 142 K and 145 K for AP and HP samples respectively . Moreover, the two levels separation energy ∆ sch found to be in the order of 0.6 meV for AP sample and ∼0.4 meV for the HP sample. This result is consistent with the reduced orthorhombic distortion when quenching from high pressure.

Conclusions

In conclusion, in this Chapter we first elucidated the phase diagram of the BaCoS 2 system that has been little studied. Our results enabled us to develop and implement successfully a novel self-flux technique for the growth of single crystals. The as-grown crystals grown are significantly larger (more than 4 × 2 mm) and of better crystalline quality than to those grown by conventional crystal growth methods reported previously. This result is very promising indeed in view of future inelastic neutron experiments that may unveil the nature of the coupling between magnetism and the insulating state in the system. In order to use the crystals for future experiments, we envisage to develop a suitable method to separate the crystals from the flux.

Second, by means of a systematic study of powder and single crystalline samples, we elucidated the controversy regarding the stability of the orthorhombic phase in the AFM Mott insulating phase of BaCoS 2 . Our results clearly confirm that the stable phase at ambient conditions is orthorhombic Cmma and that, remarkably, the orthorhombic distortion is concomitant to the onset of AFM order. This results suggests a scenario of orbital order for the Mott phase.

In order to estimate the splitting of the d xz /d yz doublet caused by the orthorhombic distortion, we used a simple Schottky model to analyse quantitatively the upturn of the specific heat at low temperature associated with the contribution of the doublet to the entropy. The above specific heat study further unveiled a large contribution of the thermal fluctuations of the order parameter at the AFM ordering temperature T N , which suggests the vicinity of the AFM phase to a competing ground state or the quasi-2D character of the long-range order. Further inelastic neutron scattering studies on single crystals are required to probe the spectrum of these fluctuations.

Finally, in order to explore the possibility of inducing a metallic state in BaCoS 2 using chemical pressure or electronic doping, we developed successfully a HP/HT synthesis technique to synthesise high-quality single-phase BaCoS 2 powder samples under high pressure. We found that the structural and physical properties of the recovered BaCoS 2 samples differ substantially from those of the samples synthesised under ambient conditions. Notable differences are a sizeable shrinking of the unit cell concomitant to a strong reduction of the orthorhombic distortion and large reduction of T N accompanied by a smearing of the specific heat jump. These results suggests that this distortion is required to stabilise the AFM order. Interestingly, no indication of mobile carriers is given by the low-temperature behaviour of the specific heat in these samples. This Chapter is devoted to our attempt to probe the Mott insulating phase of BaCoS 2 by using the first control parameter of the Mott MIT, i.e. the U/t ratio. To do so, we adopt the strategy of exerting a chemical pressure on the crystal structure, which is expected by partially substituting Ba for the smaller Sr. Indeed, chemical pressure is expected to increase the bandwidth without introducing any disorder in the electronically active Co site.

In this Chapter, we first report on the HP/HT synthesis of Ba 1-x Sr x CoS 2 powders and then on a study of their structural and thermodynamic properties carried out by means of x-ray diffraction, magnetic susceptibility and specific heat measurements.

Synthesis and phase stability of Ba

1-x Sr x CoS 2
We synthesised Ba 1-x Sr x CoS 2 at HP/HT using the belt press in a collaboration with P. Toulemonde at the Néel institute in Grenoble. The synthesis route followed two steps: we first prepared precursor powders by thoroughly mixing stoichiometric quantities of BaS (99.998%, sigma Aldrich), SrS (99.998%, sigma Aldrich), Co (99.9%, sigma Aldrich) and S(99.99%, sigma Aldrich) powders of nominal compositions x = 0.02, 0.04 and 0.07 and prepared them in the form of pellets. The pellets were thermally treated in evacuated quartz ampoules using the conventional solid-state synthesis method described in section 2.1.2. The prereacted powders were then loaded into the high-pressure cell for high-pressure synthesis at ∼ 3.5 GPa and 950 • C during 1 hour. The as-prepared samples were then quenched from high temperature to avoid phase decomposition, as discussed in the previous chapter. Representative x-ray diffractograms of the samples are shown in figure 4.1. Our samples show a purity of more than 96 %. In order to explore the HP/HT stability region of the phase we used our local PE press (see 2.2.2). For this purpose, a powder mixture (without passing through the solid-state reaction step1 ) of Ba 0.96 Sr 0.04 CoS 2 was used to establish the pressure-temperature (P -T ) phase diagram shown in figure 4.2. The synthesis of the same phase carried out at ambient pressure (AP) using a solidstate reaction with two thermal cycles is also added to the diagram as well as that obtained from the belt press.

The secondary phases (Ba 2 CoS 3 , CoS and BaS) found in the as-prepared powders, details on the synthesis conditions as well as the refined cell parameters of the (Ba,Sr)CoS 2 phase are summarised in table 4.1. The trend emerging from these data is that the synthesis of pure Ba 0.96 Sr 0.04 CoS 2 phase using the PE press requires higher temperature conditions (T > 1100 • C, probably increasing with pressure).

The highest degree of purity for the Ba 0.96 Sr 0.04 CoS 2 composition was obtained for synthesis at 0.5 and 1 GPa and temperatures of 1065 • C and 1040 • C using the PE press. Remarkably, the double thermal cycle solid-state reaction synthesis provided an even higher degree of purity, which makes this technique promising even if the solubility limit is 4% only. More synthesis runs are required to complete this diagram and to verify the reproducibility of the synthesis conditions. Careful chemical composition and stoichiometric analysis are also required to confirm the Sr-substitution level of the AP phases.

In the next sections, we shall mainly discuss the measurements performed on the Sr-substituted polycrystalline samples series synthesised using the belt press. The representative major phase of this series is isostructural with BaCoS 2 and was refined in the orthorhombic Cmma space group. Sample name We carried out a systematic XRD room temperature study on the series of Ba 1-x Sr x CoS 2 samples with nominal composition 0.02 ≤ x ≤ 0.07 synthesised under high pressure using the belt press at the 2-circle difractometer CRISTAL beamline of the SOLEIL synchrotron. We were delighted to see that the data indicate a phase purity better than 96 %. Impurity phases are CoS (≤4%) or/and BaS (≤1%). The high quality of the data enabled us to obtain a highquality structural refinement of the pure and the Sr-substituted BaCoS In the next paragraphs we discuss the solubility limit of Sr-substitution on the Ba site and the effect of this substitution on the lattice parameters and on the orthorhombic distortion.

P [GPa] power [W] T [ • C] phase content [%] a [Å] b [Å] c [Å] V [Å 3 ] remarks

Solubility limit of Sr in Ba

1-x Sr x CoS 2
In order to determine the Sr-substitution level x in our samples, chemical analysis using EDS were carried out by C. Lepoittevin of the Néel Institute in Grenoble. The measurements give x values of 0.04, 0.07 and 0.08 for the samples with nominal composition of x = 0.02, 0.04 and 0.07. Since the amount of sample obtained with a single HP/HT synthesis is relatively small, high resolution XRD data were collected at 300 K using SOLEIL synchrotron source.

The refined parameters and the evolution of the cell parameters of all samples synthesised using the belt press as a function of the measured x are given in table Note that all cell parameters decrease when increasing x and a deviation of Vegard's law begins at 8 % which gives an indication of the solubility limit.

crystalline samples with a Sr solubility limit of 8 % at the HP/HT conditions of 3.5 GPa and 950 • C. The unit cell volume of BaCoS 2 shows a progressive reduction with x. It was found that the cell volume at a substitution of 8 % is reduced down to ∼ 1 % of its volume at ambient pressure. However, one has to keep in mind that a partial reduction of the unit cell volume is due to the quenching from high temperature and under pressure (ex-situ synthesis).

Chemical pressure induced by Sr-substitution

As shown in chapter 3, the high-pressure synthesis route requires a quenching of the sample that tends to stabilize a shrinked unit cell. This shrinking is observed for synthesis pressures above 0.5 GPa. Interestingly, the out-of-equilibrium shrinked cell is almost tetragonal, consistent with the fact that the quenching tends to 'freeze' the high-temperature tetragonal phase. In order to decouple the effect of the synthesis pressure from the effect of the Sr-substitution on the Ba sites (chemical pressure) and to estimate the effective pressure applied on the unit cell through the chemical substitution of Sr, the effect of the Srsubstitution on the unit cell is compared to that of the hydrostatic pressure on BaCoS 2 which was reported by Kanada et al. [84]. These comparison allowed us to estimate the chemical pressure induced by the Sr-substitution to be ∼ 0.21 GPa for a substitution level of 8 % (see figure 4.5). In order to evaluate this correspondence we forced the substitution-to-pressure conversion coefficient to be such that the slope obtained for our substituted samples equals that given by Kanada et al.. In Figure 4.6, we compare the effect of chemical pressure on the interatomic distances in the Ba 1-x Sr x CoS 2 system with that of hydrostatic pressure in the pristine BaCoS 2 phase previously reported. Note that the ambient pressure data of Kanada et al. are not consistent either with those of Snyder et al. [42] or of those of the present work. We attribute this discrepancy is attributed to a pre-pressurisation of the sample during the loading of the latter in the diamondanvil cell used in the experiment, which is inevitable during sample preparation for a in situ XRD study. Typically, this pre-pressurisation amounts to ∼ 0.5 GPa. So we corrected this point to P=0.3 GPa (the minimum value of the prepressure loaded to the DAC cell) instead of P=0 to be compatible with the rest of measurements. In spite of this caveat that may partly affects our quantitative analysis, we note a remarkable reduction of the planar Co-S1 bond length with

x (i.e. with chemical pressure) and that the reduction of Co-S2 obtained with x = 0.08 is larger than that obtained under an hydrostatic pressure of 0.6 GPa. Namely, the two distances are 2.42 Å and 2.43 Å, respectively. Moreover, the Co-S2 bond length tends to increase at P > P c [84]. Further in situ structural measurements are required to confirm the statistical relevance of the difference between our apical distances, Co-S1, and that of Kanada et al.

Suppression of the orthorhombic distortion

A progressive decrease in the orthorhombic distortion is observed when increasing x, figure 4.7. One has to keep in mind that the the term 'chemical pressure' 400) and (040). From top to bottom, the indicated Sr-substitution levels corresponds to samples with nominal values, x = 0.07, 0.04 and 0 synthesised at HP/HT using the belt press and x = 0 for a sample synthesised at ambient pressure. It is noticed that the orthorhombic distortion is strongly suppressed using HP synthesis and upon Sr substitution.

means that the general effect of the chemical substitution results in a reduction of the unit cell volume as that seen by applying a hydostatic pressure. However, the reduction due to chemical substitution is not necessarily isotropic in the whole cell directions. In figure 4.8 the normalised cell parameters a and c are shown. It is obviously seen that when increasing the Sr content, i.e. chemical pressure, the in-plane cell parameter, a, is reduced much faster than the out of plane parameter, c. These results show that the chemical pressure suppresses in advantage the orthorhombic distortion and anticipates the reduction in the a-parameter agreeing with the strong reduction in the Co-S1 distances. The effect of Sr-substitution on the structure is similar to that of pressure, however the effect is larger on the a-axis than on the c-axis. samples in the Cmma symmetry. The samples were synthesised under HP using a belt press (see text). The refined crystal structure of a BaCoS 2 sample synthesised at ambient pressure (AP) is also presented for comparison. The data of the x = 0.02 sample were collected using a laboratory diffractometer equipped with a Co K α source, while the data of all the other samples were collected at the CRISTAL diffractometer of the SOLEL synchrotron.

Suppression of AFM order by chemical pressure

Contrary to the previously reported case of hydrostatic pressure, we find that chemical pressure significantly reduces T N down to ∼ 240 K (figures 4.9) for a substitution level of 8 at% only, corresponding to an effective pressure of 0.21 GPa, previously estimated (figure 4.5). This reduction of the T N which we observe is surprising considering the previous reports on AFM systems such as NiO and even on pure BaCoS 2 phase under pressure as shown in figure 4.10. Taking the case of the prototype NiO studied by V.A. Sidorov [START_REF] Sidorov | Differential thermal analysis of magnetic transitions at high pressure: néel temperature of nio up to 8 gpa[END_REF] where we observe an increase of 2 % of the T N value by increasing the external hydrostatic pressure of 1 GPa. Usually, when increasing pressure the hopping energy control parameter of Hubbard model, t, increases as well since the interatomic distances are decreased with pressure. In fact, in the study of NiO by Sidorov the data are interpreted within a Bloch model. Bloch proposed an empirical law for the volume dependence of superexchange interactions in insulating magnetic materials, such as: In the inset is the T N behaviour as a function of hydrostatic pressure of the prototype AFM system NiO [START_REF] Sidorov | Differential thermal analysis of magnetic transitions at high pressure: néel temperature of nio up to 8 gpa[END_REF]. Note the strong decrease in T N with chemical pressure.

- ∂ ln J ∂ ln V ≈ 10
Since the temperature of magnetic ordering is directly proportional to J in all theories of magnetism, the pressure dependence of the Neél temperature is a direct measure of the strength of superexchange interactions. This model explains the T N behaviour under hydrostatic pressure of multiple transitions metal monoxides: MnO, CoO and FeO. It seems also to be well adopted to explain the T N behaviour of BaCoS 2 under hydrostatic pressure up to p c = 1.1 GPa [95](see figure 4.10). Surprisingly, we observe exactly the opposite effect when introducing chemical pressure, even, as we mentioned previously, the chemical pressure decreases also the interatomic distances as hydrostatic pressure does (see figure 4.6). However, we have seen in 3 that the AFM order in the pristine BaCoS 2 coincides with the orthorhombic distodtion, thus, one can relate the reduction of T N in the chemical pressure case to the strong reduction in the orthorhombic distortion (figure 4.9b) as deduced from the structural analysis (figure 4.4a)

4.4 Specific heat of Ba 1-x Sr x CoS 2

The temperature dependence of the specific heat of a sintered polycrystalline 7% Sr-substituted BaCoS 2 was measured using the PPMS device in the temperature range of 2-305 K in zero field. The full scan curve is presented in figure 4.11. Note that the characteristic transition peak seen in the pure phase of BaCoS 2 is replaced by a slope discontinuity (i.e. very smeared peak) at the transition temperature T N ∼ 240 K. In the following paragraphs, we shall discuss the high and the low temperature behaviour of this phase and compare it with the pristine phase synthesised at ambient and high pressure.

High temperature specific heat

For the pristine phase BaCoS 2 synthesised at AP, we have seen that the C-T curve shows a distinguishable peak at 290 K corresponding to the antiferomagnetic transition seen in the magnetic susceptibility curve. This peak found to be enlarged and shifted to smaller T N The transition is shifted to even smaller temperatures by chemical pressure (Sr-substitution) as seen in figure 4.12. Remarkably, this shift is concomitant to a gradual smearing of the transition peak which can be explained by the exchanging of the long range AFM order in BaCoS 2 by thermodynamic fluctuations around T N when substituting with Sr. Below T N , no anomaly than at T N is observed and the value of C decreases with decreasing temperature indicating that no additional transitions exist in this doped system. in Debye model (C/T = βT 2 ). As seen previously in chapter 3 the C/T vs T 2 curve at low temperature of the pure phase BaCoS 2 is not completely linear and make a hump-shape corresponds to a Schottky anomaly at very low temperatures. .13: Specific heat of Ba 0.93 Sr 0.07 CoS 2 at low temperature where it is seen that the fit is extrapolated to zero at very low temperatures and no conduction electrons contributing to the total specific heat curve. The curve deviation due to Shottcky effect is as obvious as in our BaCoS 2 . In inset is a log scale showing the very good agreement between the fit and the experimental points.

Low temperature specific heat

In the Sr-substituted BaCoS 2 (figure 4.13) it is noticed that the curve is well fitted to the same equation used for the pristine BaCoS 2 and no other terms are needed to describe the present behaviour. Thereby, no electronic contribution is observed in the total specific heat of the system (γ = 0). Thus, the low temperature specific heat of Ba 0.93 Sr 0.07 CoS 2 behaves similarly as that of BaCoS 2 and reflects an insulating state. Since the specific heat is a direct measure of the conduction electrons in the system, this observation leads us to say that the Sr-substitution is in the right track to modify the U/t ratio without disturbing the electronically active CoS layer.

Conclusions

With the purpose of studying the effects of chemical pressure on the stability of the Mott phase in BaCoS 2 , we succeeded in preparing good-quality powder BaCoS 2 samples with substitution level up to x = 0.08. Our structural study confirms that the substitution does induce a sizeable shrinking of the unit cell, as expected by taking into account the smaller size of the Sr 2+ ion as compared to the Ba 2+ ion. A comparison with previous structural data of samples studied under hydrostatic pressure indicates that the above substitution level is equivalent to an hydrostatic pressure of 0.21 GPa. Interestingly, the structural changes induced by the Sr-substitution are radically different from those induced by hydrostatic pressure. Namely, the substitution appears to be much more effective in reducing both the ion-plane and apical Co-S distances. Specifically, the reduction of these distances is as large as that obtained under an hydrostatic pressure of 1.1 GPa that induces a paramagnetic metallic state. In spite of this similarity, no evidence of metallic properties is found in any of the Ba 1-x Sr x CoS 2 samples from the low-temperature behaviour of the specific heat.

The chemical pressure exerted by the Sr-substitution appears to enhance the structural changes produced by HP synthesis. Namely, in the Ba 1-x Sr x CoS 2 samples, the orthorhombic distortion and the long-range AFM order are further weakened upon increasing x until both are completely suppressed at the maximum substitution level, x = 0.08. In spite of this suppression, the magnetic response still preserves the main features of the unsubstituted phase, namely a cusp-like anomaly at the characteristic temperature T N , although this temperature is strongly lowered down to 240 K by the substitution. This observation suggests that large short-range AFM fluctuations persist in the system.

Chapter 5

Tuning the MIT of BaCoS 2 by electronic doping

As said in the introduction, within a simple Hubbard model, the Mott state can be typically controlled using either the U/t ratio or the filling parameter n. In the previous chapter, we discussed the possibility of modifying U/t by applying the chemical pressure exerted by the small Sr 2+ ion, which is expected to increase t. In this Chapter, we attempt to use the second control parameter n by means of the heterovalent substitution of Ba 2+ for K 1+ . For a substitution level x in the chemical formula Ba 1-x K x CoS 2 , we thus expect to introduce x holes in the 3d 7 electronic configuration of the Co 2+ ion.

The structure of the present Chapter is identical to that of the preceding Chapter. Namely, we first report on the HP/HT synthesis of Ba 1-x K x CoS 2 powders and then on a study of their structural and thermodynamic properties carried out by means of x-ray diffraction, magnetic susceptibility and specific heat measurements.

Synthesis, phase stability and solubility limit

In order to synthesise Ba 1-x K x CoS 2 powders, we used the very same HP/HT technique employed for the synthesis of the Ba 1-x Sr x CoS 2 . The only difference is that we exclusively employed the PE press. An early attempt to synthesise Ba 1-x K x CoS 2 was reported by Gélabert et al. by using a conventional solid state synthesis method [60]. In order to avoid the high vapour pressure of K in the commonly used K 2 S phase, the authors used K 2 CoS 2 as alternative reagent. As first attempts we followed their protocol even though it was complicated and relied on a complex setup. However we failed to reproduce their synthesis because of some problems we faced including tubes explosions or sample sticking on the crucible walls. We then came up with the idea of using HP/HT techniques where the synthesis occurs within a HP cell which can afford the high vapour pressure of potassium in K 2 S phase. We then prepared stoichiometric powders of Ba 1-x K x CoS 2 with nominal x = 0.04, 0.07 and 0.10 using BaS (99.98%), K 2 S (99.5%), Co and S as starting materials. With the PE press we successfully stabilised 7-10% K-doped BaCoS 2 phase at a pressure of 4 GPa and T ∼ 1500 • C. In the previous chapter, we have tried a wide range of synthesis conditions and this exploratory work has been useful to find in a short time the good conditions also for the synthesis of the K-substitution phase. In fact we were expecting that the conditions to synthesis pure samples of K-substituted phase would be very similar to that of Sr-substitution. For these reason, it was not necessary at this stage to explore a wide region of conditions. Although the as-prepared powders were not single phase as revealed by X-ray diffraction, the structural phase analyses ensured that, in the case of x = 0.04, the secondary phase found to be CoS. At higher doping level, x = 0.07 and 0.10, beside BaCoS 2 and CoS, the BaS starts to stabilise with ∼5 % of the total peaks integral intensities (figure 5.1). These observations are interpreted as a saturation of the K-doping (confirmed by the cell parameters saturation as shown in section 5.2) and establish a solubility limit at x = 0.07. . The X-ray powder diffraction was carried out using a diffractometer equipped with a Co K α source. A solubility limit at a nominal 7 to 10 % doping is achieved and presented in the powder diffraction as emergence of BaS secondary phase beside CoS. The circle, triangle and star symbols indicate the secondary phases: CoS, BaS and BN respectively. Note that while extracting the sample from the rest of HP/HT assembly, some BN from the crucible can be stuck and mixed up with the sample powders and this is the only source of BN in our X-ray diffraction. In the figure underneath is the Rietveld powder refinement on the Ba 0.96 K 0.04 CoS 2 .

Structural analysis

In the samples with nominal K composition 0.04 ≤ x ≤ 0.1, the purity of the desired phase was better than 90 %. Secondary phases are CoS (≤6.5%) or/and BaS (5%). Thus, it was possible to carry out a high-quality structural refinement of the BaCoS 2 phase in the orthorhombic Cmma space group. The results of the refinements are reported in table 5. level is shown in figure 5.3. Interestingly, the K-substitution mainly reduces the a-axis parameter while the second planar dimension, b, and the out-ofplane lattice parameter, c, do not exhibit significant changes. A reduction of cell volume with x is also observed as seen in figure 5.3b. We have shown previously in chapter 3 that the reduction of unit cell volume is partially due to the quenching from high pressure and is presented as an on/off effect at a pressure larger than the threshold value 1 GPa. A rapid suppression of the monoclinic distortion with the substitution of K for Ba is also observed and shown in figure 5.4a. Interestingly, these K-doping shows a similarity to the stabilisation of the tetragonal phase when doping the Co-sites by Cu atoms [START_REF] Schweitzer | Structural, electrical, and magnetic properties of ba co 1-x cu x s 2[END_REF].

The amount of impurity phases increases with increasing x. The main impurity phase turns out to be CoS. Our analysis of the structural data unveils this impurity for concentrations of 2% and 6.5% for x = 0.04 and 0.1, respectively. There are correlations with the starting sulphides materials and the reaction temperatures used in the synthesis. So, such multiple phases product can be purified using other HP/HT conditions. Moreover, different synthesis conditions can help to exceed the solubility limit presented as a saturation in the cell volume at the nominal K-content 0.07 ≤ x ≤ 0.1. Finally, it is obvious that the net effect of the K-doping at the Ba site results in a slight reduction in the unit cell volume and an anisotropic changing in the lattice parameter (reduction in the a-direction only as seen in figure 5.4b).

Magnetic susceptibility measurements

We have shown in chapter 1 that the magnetic susceptibility (χ) at high temperature confirms a high spin state of 3 2 in BaCoS 2 . Thus, the electronic configuration of the 3d-orbitals has to be coherent with a 3 2 spin state with a dominant Hund coupling. It follows that the (x 2 -y 2 ) and xz, yz are half-filled (see figure 1.8). Then, the energy separation of these levels depends strongly on the geometry of the CoS 5 pyramid. Doping with K + will modify the half-filling state of BaCoS 2 , therefor, a changing in the electronic occupancy which would affect the AFM ordering in the system is expected as we have discussed in chapter 3. Figure 5.5a shows the temperature dependence of the magnetic susceptibility of Ba 1-x K x CoS 2 . The measurements were performed on powder samples with x = 0.04, 0.07 and 0.1 using SQUID magnetometer in 10 kOe and in temperature range 2-350 K. We have seen in chapter 3 that for the undoped sample (x=0), the Néel temperature (T N ) corresponds to a characteristic cusp-like anomaly with the antiferromagnetic to paramagnetic transition in these layered compounds at about 290 K. In the Ba 1-x K x CoS 2 series, with increasing K content this characteristic cusp-like peak shifts rapidly down to 240 K at only 7% of K-doping. These measurements confirm the coincidence between the the AFM order suppression and the orthorhombic distortion as seen in figure 5.5b which was also observed in both unsubstituted and Sr-substituted samples synthesised under high pressure (see Chapters 3 and 4).

Specific heat study

The temperature dependence of the heat capacity of a polycrystalline Ba 0.93 K 0.07 CoS 2 sample was measured in the temperature range of 2-305 K in zero field (see figure 5.6). Hereafter, we shall discuss separately the high and low temperature behaviour of this sample as compared with that of the pristine phase synthesised at ambient and high pressure. The specific heat curves of two BaCoS 2 synthesised at ambient pressure and high pressure and Ba 0.93 K 0.07 CoS 2 , measured between 2 and 305 K under zero magnetic field. The coloured vertical lines indicates the T N of each sample. Note that the cusp-like anomaly at the AFM transition is gradually broadened and disappears completely for 7% of K substitution.

High temperature specific heat

A close-up view of the data in the high-temperature region near T N is shown in figure 5.7. Remarkably, the K-substitution suppresses almost completely the clear anomaly seen at T N ∼ 290 K in the unsubstituted phase BaCoS 2 prepared at ambient pressure conditions. Instead, one notes a slope discontinuity at much lower temperature T N ∼ 240 K, similarly to what previously observed in the Sr-substituted sample. Typically, the smearing of the anomaly at second-order phase transition is due to an enhancement of short-range fluctuations of the order temperature. This picture is plausible in our case as well, considering that the suppression of the orthorhombic distortion enhances the two-dimensional character of the electronic state and thus the magnitude of the short-range AFM correlations. A further factor that may lead to stronger fluctuations is the disorder unavoidable in substituted samples. Below T N , no further anomalies are observed and the C decreases smoothly in a conventional manner. ). As seen previously in chapter 3 the C/T vs T 2 curve at low temperature of the pure phase BaCoS 2 is not completely linear and make a hump-shape at very low temperature. This hump was explained by taking in considerations that BaCoS 2 is an orthorhombic AFM phase where it is considered then as a two level system where a Schottky term is required to explain the low temperature heat capacity of the system. However, in the K-doped BaCoS 2 , figure 5.8, it is noticed that a linear fit extrapolated to zero does not fit the experimental points but is parallel with a small down shift. Since BaCoS 2 is an insulating phase, obviously no conduction electrons contribute to the total specific heat of the system, on the other hand, modifying the filling, n, in such a Mott system is expected to result in a metallisation of the phase. Thus, in the case of Ba 0.93 K 0.07 CoS 2 , the low temperature behaviour of the specific heat is well described by the expression: C = γT + βT 3 + C Sch , where γ is the Sommerfeld coefficient arising from conduction electrons. Our data analysis yields γ = 5.7 mJ/mol K 2 , about two times larger than the value previously reported for metallic BaNiS 2 , γ BN S = 3 mJ/mol K 2 [39, 98]. This interesting results reflects the need for further experiments including transport measurements and aiming to the understanding of the electronic nature of this doped compound. Even if the specific heat measurements of the Ksubstituted phase gives a non zero γ value, suggesting a metallic state. Indeed, the K-substitution is just a modification of the insulating state of BaCoS 2 , thus, we expect that the corresponding electronic state of this compound would be different than that of BaNiS 2 .

Low temperature specific heat

In fact, at this level, I can not say quantitatively how the crystal field and the electronic occupancy are changed with K-substitution. Indeed, ab-initio calculations are needed. But, I can confirm that significant changing happened at the energy separation of the orbitals.

Conclusions

In conclusion, we succeeded in the synthesis of a series of Ba 1-x K x CoS 2 samples with x up to 0.07 using the HP/HT method developed in Chapter 3. The results of our structural refinement of XRD data indicate a sample purity better than 90 %. As in the previous case of Sr-substituted samples, also the Ba 1-x K x CoS 2 samples exhibit a progressive reduction of the orthorhombic distortion with x.

Similar is also the behaviour of the magnetic susceptibility, namely we observed a progressive reduction of T N with x. In fact, the pronounced smearing of the anomaly in the specific heat curve rules out a AFM long-range order and instead suggests large short-range AFM fluctuations, similar to the case of the BaCoS 2 and Ba 1-x Sr x CoS 2 samples. In Table 5.2 we summarise the results of the analysis of the specific heat data for selected samples synthesised at ambient and high pressure. In summary, the data support the scenario that the stability of the AFM order requires a sizeable orthorhombic distortion. As a further tool to estimate the energy separation of the d xz /d yz doublet produced by the distortion, we applied a simple Schottky model that accounts quantitatively for an anomalous upturn of the specific heat at low temperatures. In the Ba 0.93 K 0.07 CoS 2 sample, our data analysis yields a very small value of this splitting, ∆ Sch = 8 × 10 -5 meV, which confirms the presence of an almost degenerate d xz /d xz doublet, as indicated by the tetragonal symmetry of the crystal structure.

The finding of a sizeable value of the Sommerfeld coefficient γ is important, for it indicates that our objective of electronically doping the insulating BaCoS 2 phase has been reached. This interesting finding raises the question as to which 3d-orbital is affected by the doping and whether the induced metallic state would exhibit FL or non-FL (or bad metallic) properties. According to [START_REF] Santos-Cottin | Linear behavior of the optical conductivity and incoherent charge transport in bacos2[END_REF] and Figure 1.8, we expect FL properties for a doping of the z 2 orbital, while adding a hole to the x 2 -y 2 or xz, yz orbitals is expected to introduce a non-coherent metal (non-Fermi liquid). In order to elucidate this point, further ab initio calculations would be required.

From the experimental side, the present results prompt us to probe the transport properties by means of optical conductivity or electrical resistivity measurements that could measure directly the presence or the absence of a Drude peak. To do so, we envisage that sizeable single crystals of the K-doped phase of 200-300 µm size may be grown under HP using the self-flux method described in Chapter 3.

Chapter 6

Conclusions and perspectives

General Conclusions. In the present thesis, we investigated the stability of the Mott insulating phase of BaCoS 2 that we have chosen as a model system of multi-orbital correlated electron system, an emerging subject of research in condensed matter and many-body physics.

To do so, we first elucidated the phase diagram of the BaCoS 2 system that has been little studied. Our results enabled us to develop and implement successfully a novel self-flux technique for the growth of single crystals in view of future inelastic neutron experiments that may unveil the nature of the coupling between magnetism and the insulating state in this system.

Second, by means of a systematic study of both powder and single crystalline samples using x-ray diffraction, specific heat and susceptibility measurements, we elucidated the controversy in the literature regarding the stability of the orthorhombic phase in the AFM Mott insulating phase of BaCoS 2 . Our results clearly confirm that the stable phase at ambient conditions is orthorhombic Cmma. In addition, we found that, remarkably, the orthorhombic distortion is concomitant to the onset of AFM order, thus pointing at the role of orbital ordering on the stability of the Mott phase.

The above specific heat study further unveiled a large contribution of the thermal fluctuations of the order parameter at the AFM ordering temperature T N , which suggests the vicinity of the AFM phase to a competing ground state or the quasi-2D character of the long-range order.

Next, we explored the possibility of inducing a metallic state in BaCoS 2 using chemical pressure and electronic doping as control parameters.

To do so, we first developed successfully a HP/HT synthesis technique to synthesise high-quality single-phase BaCoS 2 powder samples under high pressure. Surprisingly, we found that the structural and physical properties of the recovered BaCoS 2 samples differ substantially from those of the samples synthesised under ambient conditions. Notable differences are a sizeable shrinking of the unit cell concomitant to a strong reduction of the orthorhombic distortion and large reduction of T N accompanied by a smearing of the specific heat jump. These results suggests that this distortion is required to stabilise the AFM order. Interestingly, no indication of mobile carriers is given by the low-temperature behaviour of the specific heat in these samples.

We then applied the above HP/HT synthesis technique to the synthesis of Ba 1-x Sr x CoS 2 and Ba 1-x K x CoS 2 samples where Ba is partially substituted for Sr or K, which is expected to favour a metallic phase owing to an effect of chemical pressure or hole-doping, respectively.

In the case of Sr-substitution, we succeeded in preparing good-quality powder samples with substitution level up to x = 0.08. The results of our structural measurements confirm that the substitution does induce a sizeable shrinking of the unit cell, as expected by taking into account the smaller size of the Sr 2+ ion as compared to the Ba 2+ ion. A comparison with previous structural data of samples studied under hydrostatic pressure indicates that the above substitution level is equivalent to an hydrostatic pressure of 0.21 GPa.

Interestingly, the structural changes induced by the Sr-substitution are radically different from those induced by hydrostatic pressure. Namely, the substitution appears to be much more effective in reducing both the ion-plane and apical Co-S distances. Specifically, the reduction of these distances is as large as that obtained under an hydrostatic pressure of 1.1 GPa that induces a paramagnetic metallic state. In spite of this similarity, no evidence of metallic properties is found in any of the Ba 1-x Sr x CoS 2 samples from the low-temperature behaviour of the specific heat.

The chemical pressure exerted by the Sr-substitution appears to enhance the structural changes produced by HP synthesis. Namely, in the Ba 1-x Sr x CoS 2 samples, the orthorhombic distortion and the long-range AFM order are further weakened upon increasing x until both are completely suppressed at the maximum substitution level, x = 0.08. In spite of this suppression, the magnetic response still preserves the main features of the unsubstituted phase, namely a cusp-like anomaly at the characteristic temperature T N , although this temperature is strongly lowered down to 240 K by the substitution. This observation suggests that large short-range AFM fluctuations persist in the system.

Our attempt to synthesise K-substituted BaCoS 2 samples under HP was also successful. Similar to the case of Sr-substitution, we prepared good-quality Ba 1-x K x CoS 2 powder samples with substitution levels x = 0.07. Contrary to what expected by comparing the ionic radii of Ba 2+ and K 1+ , we found a reduction of the unit cell volume with x similar to that of the Ba 1-x Sr x CoS 2 samples. The effects of this volume reduction on the physical properties are also similar, namely both the orthorhombic distortion and the long-range AFM are suppressed by the K-substitution while the anomaly of the magnetic susceptibility at T N persists. This finding confirms the scenario of large short-range AFM fluctuations proposed for the Ba 1-x Sr x CoS 2 samples.

A notable difference between Ba 1-x Sr x CoS 2 and Ba 1-x K x CoS 2 samples is that the latter samples exhibit a sizeable value of the Sommerfeld coefficient γ = 5.7 mJ mol -1 K -2 . This result is very interesting for it suggests that the K-substitution does induce a metallic state. Considering the comparatively small density of holes introduced by the K-substitution, the above sizeable value may be explained by a rather large renormalised mass, consistent with a picture of bad metal.

Perspectives. We believe that the above results raise a number of open questions that deserve to be addressed in a future work. We should propose the following points.

The self-flux technique successfully developed in this work is promising to obtain sizeable single crystals that would be suitable for elastic and inelastic neutron scattering measurements.

The latter measurements would be very important indeed to study the spectrum of the AFM fluctuations that appear to play a crucial role on the interplay between magnetism and Mott state in the BaCoS 2 system. For instance, these measurements may enable to determine the J parameters of the effective Heisenberg Hamiltonian and the role of the out-of-plane coupling parameter J 3 to stabilise the long range order.

The strikingly different effects of hydrostatic vs. chemical pressure on the crystal structure and on the physical properties of the BaCoS 2 system, e.g. the AFM order, suggest further in situ studies under high pressure, such as optical conductivity measurements, in order to establish whether the Mott state can be stable in the absence of long-range AFM order.

The above studies would require the synthesis of Ba 1-x Sr x CoS 2 single crystals under high pressure. To achieve this, it would be interesting to apply the above self-flux single-crystal growth method to the high-pressure techniques successfully employed to synthesise powder Ba 1-x Sr x CoS 2 samples.

The indication of metallic behaviour in the Ba 1-x K x CoS 2 samples is interesting as it suggests that electronic doping by means of heterovalent substitutions is possible without introducing disorder in the electronically active Co site. To verify the scenario of metallic state, the synthesis of Ba 1-x K x CoS 2 single crystals under high pressure would be important to investigate the intrinsic transport properties of the samples by means of optical conductivity or electrical resistivity and Hall effect measurements. These measurements may enable to establish whether the metallic properties are described by a FL or NFL scenario.

The comparatively large ionic radius of the Ba 2+ ion severely limits the choice of chemical substitutions of this ion. As a matter of fact, K 1+ appears to be the only possibility of heterovalent chemical substitution, which leads to hole doping. In order to study the effect of electron doping, one may consider the alternative strategy of employing an ionic gating technique on freshly cleaved BaCoS 2 crystals. This would enable us to explore the stability of the Mott insulating and AFM phase using both types of doping and thus to investigate the electron-hole symmetry of the electronic phase diagram.

This last consideration and our results on Ba 1-x K x CoS 2 raise the question of the analogy between the electronic phase diagram of BaCoS 2 and that of unconventional superconductors, such as cuprates or pnictides. Indeed, all these systems are TM compounds featuring a quasi-2D lattice of the TM ions that exhibits a long-range AFM order in the pristine undoped phase. This order is rapidly suppressed upon electronic doping or pressure. Upon further doping, in both cuprates and pnictides this suppression is followed by the appearance of a superconducting dome-shaped region at low temperature and by a region of bad metallic properties at high temperature. Our results on Ba 1-x K x CoS 2 unveils a close similarity between the BaCoS 2 system and these unconventional superconductors. The question is whether a superconducting instability may also exist in BaCoS 2 . In order to answer this question, one should require to explore the electronic phase diagram at higher hole-doping levels or to achieve electron doping, as discussed above. In any case, we expect that high-doping levels should progressively lead to a FL state.

Last but not least, the observation of topological Dirac states in the Ni endmember BaNiS 2 [11] raises the question of the fate of these topological states across the MIT transition. Indeed, while the topological properties of materials have been extensively studied in the case of weak electronic correlations, the role of correlations on the stability of the topological properties constitutes an almost unexplored research line, especially on the experimental side [START_REF] Rachel | Interacting topological insulators: a review[END_REF]. This appears to be an exciting perspective considering that the most fascinating effects in condensed matter physics, such as unconventional superconductivity, the Kondo effect and the Mott-Hubbard transition, are driven by electronic correlations. Along this line, we believe that the results obtained in the present thesis are promising to drive the Mott state of BaCoS 2 across the MIT, towards the topological Dirac states of BaNiS 2 . To the best of our knowledge, this tunability has not been observed before experimentally in the solid state; previous reports are either theoretical [100,101] or experimental in optical lattices [102].

Figure 1 . 1 :

 11 Figure 1.1: A cartoon of a 2D lattice described by the single-band Hubbard model (see eq. 1.2). Red and green arrows indicate up and down electronic spins. U and t are the Coulomb repulsion and the hopping energy, respectively.

Figure 1 . 2 :

 12 Figure 1.2: Electronic phase diagram of the Mott metal-insulator transition (MIT) based on the Hubbard model as a function of the U/t ratio and band filling. BC-andFC-MIT stand for band-and filling-controlled MIT (taken from[START_REF] Imada | Metal-insulator transitions[END_REF]).

Figure 1 . 3 :

 13 Figure 1.3: Density of states (DOS) of the single-band Hubbard model calculated for different ratios U/W =1, 2, 2.7, 3 and 4 (from the top to the bottom), where W is bandwidth, using a Quantum Monte Carlo method combined with a perturbation expansion. Energies ω are measured from the chemical potential (taken from[START_REF] Zhang | Mott transition in the d=∞ hubbard model at zero temperature[END_REF]).

Figure 1 . 4 :

 14 Figure 1.4: Schematic energy-level diagrams of (a) Mott-Hubbard and (b) charge-transfer insulators (from[START_REF] Imada | Metal-insulator transitions[END_REF]).

Figure 1 . 5 :

 15 Figure 1.5: Electronic phase diagram showing the FL vs. NFL stability as a function of the Coulomb repulsion U over bandwidth D ratio and the number of d electrons-from empty (0) to full (6). A measured of this tendency is the value of the quasiparticle weight Z shown in false colours, i.e. darker regions correspond to good metals and lighter regions to bad metals. Black bars signal the Mott-insulating phases(from[START_REF] De'medici | Janus-faced influence of hund's rule coupling in strongly correlated materials[END_REF]).

Figure 1 . 6 :

 16 Figure 1.6: Left: crystal structure of BaCo 1-x Ni x S 2 [36]. Ba, Co/Ni and S atoms are indicated as green, blue and yellow spheres. The unit cell is indicated by the sold black line. Right: schematic electronic phase diagram as a function of the Ni substitution level, x. AFI, PI and M denote the antiferromagnetic insulating, paramagnetic insulating and metallic phases (from [37]).

Figure 1 . 7 :

 17 Figure 1.7: The d-orbital splitting of a square-based pyramid with a metal (in blue) and the ligand (in yellow): (A) the metal in the basal plane and (B) the metal is outof the basal plane. The graph is redrawn from[START_REF] Tripathi | Cobalt(ii) complexes as single-ion magnets[END_REF].

Figure 1 . 8 :

 18 Figure 1.8: (A) Pyramidal coordination of the Co ion in BaCoS 2 . Note the location of the ion out of the basal plane of the pyramid. (B) Occupancy of the 3d orbitals calculated using two different ab intio schemes: (i) LDA supplemented by a static Coulomb repulsion term U and a LDA+DMFT+U (ω), where this term is treated dynamically [49]. (C) Imaginary part of the electron self energy as a function of Matsubara frequencies obtained from the above two schemes [49]. (D) Predicted hierarchy of the orbital levels according to the two schemes. Note an orbital reshuffling induced by the use of a dynamical U (ω).

Figure 1 . 9 :

 19 Figure 1.9: Magnetic susceptibility of single crystal of BaCoS 2 . Note the strong anisotropy (from [55]).

Figure 1 .

 1 Figure 1.10: (A) Structural model proposed by Kodama et al. for the magnetic structure of BaCoS 2 [41]. (B) The same as in (A) for the study by Mandrus et al. The numbers next to the symbols of the Co sites indicate the fractional z-coordinates of these sites [55].

3 .Figure 1 .

 31 Figure 1.11: Nearest-and next-nearest neighbour exchange couplings, J 1 and J 2 , respectively, in the ab-plane of BaCoS 2 . The black solid line indicates the unit cell.

Figure 2 . 1 :

 21 Figure 2.1: (A) A schema of a differential thermal analysis apparatus [63]. (B) DTA responses to melting and exothermic processes of a pure material under ideal conditions where a is an onset temperatures (taken here as equal to melting point, TM), b being the peak signals and c is a peak temperature [65].

2 Figure 2 . 2 :

 222 Figure 2.2: Typical used thermal cycle for powder samples

Figure 2 . 3 :

 23 Figure 2.3: An example of thermal cycle for the growth of BaCoS 2 single crystals.

Figure 2 . 4 :Figure 2 . 5 :

 2425 Figure 2.4: (A) The belt press installed at the Néel Instute of Grenoble enabling synthesis up to 7 GPa and 1200 • C. (B) A schematic drawing of the belt apparatus.

Figure 2 . 6 :

 26 Figure 2.6: The V-type PE press. Typically 10-50 kg mass and achieves up to 25 GPa with 10-100 mm 3 produced sample. (1) and (2) anvils and anvil seats in tungsten carbide, (s) sample space, (g) toroidal gasket, (i) the incident beam, (d) diffracted beam[START_REF] Klotz | Neutron powder diffraction at pressures beyond 25 gpa[END_REF].

Figure 2 . 7 :

 27 Figure 2.7: (a) The VX-type PE press. Typically 60 kg mass and achieves up to 10 or 25 GPa with sample volume 100 or 30 mm 3 respectively. (b) Cross section of V-type PE press shows the different elements of PE body. (c) A side view of the press shows the azimuthal window of ±30 • [77].

Figure 2 . 8 :

 28 Figure 2.8: A schematic drawing of the PE sample assembly.

Figure 2 . 9 :

 29 Figure 2.9: Sample holder and θ -2θ goniometer of the Panalytical X'pert Pro MPD powder diffractometer.

Figure 2 .

 2 Figure 2.10: The IMPMC Oxford diffractometer used for single crystals XRD measurements.

Figure 2 .

 2 Figure 2.11: A representative photo for the typical sample holder used in the specific heat option of the PPMS.

3. 1

 1 State of the art 3.1.1 Controversial structure of the Mott insulator BaCoS 2

Figure 3 . 1 :

 31 Figure 3.1: Phase diagram of BaCoS 2+x proposed by Gelabert et al.[43]

  [START_REF] Mandrus | Magnetism in bacos2[END_REF] affirmed the formation of 3 × 3 × 0.1 mm 3 single crystals out of a molten stoichiometric mixture of BaCoS 2 . On the other hand,Shamoto et al. [37] have reported a 1 cm 3 ingot-like crystal of BaCo 1-x Ni x S 2 phase. Even they attempted a pseudobinary phase diagram of BaNiS 2 -Ni 3 S 2+γ and a ternary phase diagram of the components BaM S 2 , where M is Co 1-x Ni x (figure3.2) they confirmed the need for a precise description of these phase diagrams specially at x value different than 0.3.

Figure 3 . 2 :

 32 Figure 3.2: A tentative ternary BaCo 1-x Ni x S 2 and pseudobinary BaNiS 2 -Ni 3 S 2+γ (inset). The shaded area presents the area where BaCo 1-x Ni x S 2 crystals grow [37].

Figure 3 . 3 :

 33 Figure 3.3: (A) XRD patterns of BaCoS 2.15 powders containing an excess of sulfur after the 1 st and 2 nd thermal treatments and of BaCoS 2 with no sulfur excess after a single thermal treatment. Note the evident impurity peaks in the pattern of the 1 st thermal treatment (in green). Due to the lack of time, the sulfur concentration was not refined in our study that is why we consider the nominal values.

Figure 3 . 4 :

 34 Figure 3.4: Rietveld refinement of the structural model for BaCoS 2 sample. The red dots in the figure are the experimental data points and the black colored line is Rietveld refined XRD patterns. The Bragg's reflection positions are indicated in light green color symbols and presents the Bragg's reflection positions for BaCoS 2 , CoS and BaS respectively. The differences between obtained XRD data and the Rietveld refined pattern is shown in blue colour. The measured powder sample was synthesised at ambient pressure

Figure 3 . 5 :

 35 Figure 3.5: A P-T phase diagram of BaCoS 2 . Synthesis are done using PE press, one synthesis out of Belt press and another at ambient pressure are also indicated. The white region presents non explored points of pressure and temperature. Note the difference pressures used at belt and PE presses to obtain the same phase purity.

. 4 .Figure 3 . 6 :

 436 Figure 3.6: (a) An overall DTA curve of BaCoS 2 powder from room temperature to 1200 • C. (b) Several enlargements of zones containing endothermic (T 1 , T 3 , T 4 and T 6 ) and exothermic (T AF M , T 2 and T 5 ) peaks.

phase. T 5 (

 5 figure 3.6b-b4) suggests the BaCoS 2 phase formation which was indicated in figure3.1 around T ∼ 1025 • C. Moreover, the T 6 ∼ 1040 • C at which a quit deep endothermic effect occur present a partial melting of the mixture. In this experiment the thermogravimetry (TG) curve was not recorded but most probably the abrupt fall in the heat flow observed after 1050 • C indicates a continuous decomposition and a complete melting (T m ).

Figure 3 . 7 :

 37 Figure 3.7: Examples of single crystals grown out of a stoichiometric melt (HA05). Photos were taken by optical microscope. In photo (B) crystals of the same batch were aligned in (110) direction on an Al support for a purpose of neutron diffraction study.

Figure 3 .

 3 8 shows a binary phase diagram of Co-S on which our self-flux method relied. Working around the eutectic point of Co-S phase diagram has

Figure 3 . 8 :

 38 Figure 3.8: Binary phase diagram of Co-S [91] where the compound Co 4 S 3 around the eutectic point was used as a flux to grow BaCoS 2 single crystals.

Figure 3 . 9 :

 39 Figure 3.9: (A) a SEM photo shows a polished surface of the regulus where one can see the trapped crystals with their real size. (B) an example of single crystals batch of HAC1 where crystals are grown by self-flux method using a molar ratio of 2 BaCoS 2 : 1 Co 4 S 3 . Photos were taken by a polarised light microscope.

Figure

  Figure 3.11: Comparison between the effect of hydrostatic pressure[84] and that of synthesis pressure on the lattice parameters of BaCoS 2 . The notation ex situ for the samples synthesised under high pressure indicates that the lattice parameters have been measured on the recovered samples. Note that synthesis pressure stabilises a tetragonal phase above a threshold value of ∼1 GPa.

Figure 3 . 12 :

 312 Figure 3.12: Temperature dependence of (from top to bottom): orthorhombic distortion, ZFC magnetic susceptibility and specific heat of BaCoS 2 phase where the vertical gray bar indicates the Néel temperature. Remarque the coincidence between the different measurements specially the reduction in the orthorhombic distortion in the PM phase at T > T N . The heat capacity curve shows the characteristic peak of the AFM transition at T N = 290 K. In the inset, the continuity between two calorimetric measurements, heat capacity and DTA, is also shown.

Figure 3 . 13 :

 313 Figure 3.13: Temperature dependence of ZFC magnetic susceptibility and specific heat of BaCoS 2 phase synthesised at both ambient anf high pressure. The vertical bars indicate the T N . Remarque the enlarged transition peak of the HC curve of the HP/HT synthesised sample which coincides with the reduction of the orthorhombic distortion in this sample.

Figure 3 .

 3 Figure 3.14: (A) Specific heat of BaCoS 2 at low temperature where it is seen that the fit goes to zero at 0 K and no conduction electrons contributing to the total specific heat curve . In inset is a log scale. (B)

Chapter 4 Effect

 4 of chemical pressure on the Mott phase of BaCoS 2

Figure 4 . 1 :

 41 Figure 4.1: X-ray diffractogram of a polycrystalline BaCoS 2 sample synthesised at ambient pressure compared with that of two Ba 1-x Sr x CoS 2 samples synthesised under HP with x = 0 and x = 7% Sr-substituted. Circle and triangles indicate the Bragg peaks of CoS and BaS impurity phases, respectively.

Figure 4 . 2 :

 42 Figure 4.2: P-T phase diagram of Ba 0.96 Sr 0.04 CoS 2 . The white area in the diagram is the non-explored P-T conditions. The color scale presents the percentage of the (Ba,Sr)CoS 2 phase content. This percentage is calculated out of the total integrated intensities of the XRD peaks which were refined using a Rietveld refinement method.

Figure 4 . 3 :

 43 Figure 4.3: Experimental and Rietveld refined diffractogram of a Ba 0.93 Sr 0.07 CoS 2 sample synthesised using the belt press at 3.5 GPa and 950 • C. The green vertical ticks indicate the position of the Bragg reflections for the following phases included in the refinement (from top to bottom): BaCoS 2 , CoS and BaS. The difference between observed and calculated profile is shown in blue.

Figure 4 . 4 :

 44 Figure 4.4: The effect of Sr-substitution on the unit cell parameters a, b, c and the cell volume, V . The experimental points are plotted as a function of the measures x values and the error bars are smaller than symbol size. The XRD data were collected using synchrotron radiation except the point of nominal x = 0.02 (measured x = 0.04). Note that all cell parameters decrease when increasing x and a deviation of Vegard's law begins at 8 % which gives an indication of the solubility limit.

Figure 4 . 5 :

 45 Figure 4.5: Effect of the Sr-substitution on the unit cell as compared to that of hydrostatic pressure (fromKanada et al. [84]) This comparison allows us to estimate the chemical pressure induced by the substitution to be ∼0.21 GPa for a substution level of 8%.

Figure 4 . 6 :

 46 Figure 4.6: The interatomic distances of the CoS 5 pyramids layer of BaCoS 2 as a function of pressure. Co-S1 is the distance between the Co and the apical S atoms, Co-S2 is the distance between the Co and the planer S atoms and d represents the distance between the Co atom and the square base of the CoS 5 pyramid. The small and big diamond symbols are the pure BaCoS 2 re-drawn from [84] and [42] respectively, note that we have corrected the point at P=0 to P=0.3 which is the expected minimum loading pressure in a DAC cell. The open stars symbols are the Ba 1-x Sr x CoS 2 phases with measured x = 0.04, 0.07 and 0.08 and the filled star is BaCoS 2 synthesised with the conventional solid-state reaction at ambient pressure.

Figure 4 . 7 :

 47 Figure 4.7: BaCoS 2 diffraction peaks at the positions (hkl): (400) and (040). From top to bottom, the indicated Sr-substitution levels corresponds to samples with nominal values, x = 0.07, 0.04 and 0 synthesised at HP/HT using the belt press and x = 0 for a sample synthesised at ambient pressure. It is noticed that the orthorhombic distortion is strongly suppressed using HP synthesis and upon Sr substitution.

Figure 4 . 8 :

 48 Figure 4.8: The effect of Sr-substitution on the structure is similar to that of pressure, however the effect is larger on the a-axis than on the c-axis.

Figure 4 . 9 :

 49 Figure 4.9: (A) Temperature dependence of the magnetic susceptibility (χ) of a series of Ba 1-x Sr x CoS 2 samples of nominal composition x = 0, 0.02, 0.04 and 0.07 synthesised at HP/HT conditions. The coloured vertical lines indicate the T N for each sample. The AFM transition is shifted to smaller T N values when increasing the Sr content and reduced down to ∼ 240 K for a nominal substitution level x = 0.07. Notice that the Sr-substituted phases have also the same overall behaviour of the magnetic susceptibility as BaCoS 2 where the transition peak is broad and at T < 50 K the curves reflect paramagnetic impurities. (B) A plot of T N as a function of the orthorhombic distortion (δ) where the coincidence between AFM order and orthorhombicity is observed.

Figure 4 .

 4 Figure 4.10: The pressure dependence of T N of BaCoS 2 (filled squares) [95] compared to that of the Sr-substituted BaCoS 2 (open diamonds).In the inset is the T N behaviour as a function of hydrostatic pressure of the prototype AFM system NiO[START_REF] Sidorov | Differential thermal analysis of magnetic transitions at high pressure: néel temperature of nio up to 8 gpa[END_REF]. Note the strong decrease in T N with chemical pressure.

Figure 4 .

 4 Figure 4.13 shows the C/T vs T 2 plot for the Ba 0.93 Sr 0.07 CoS 2 component where the dashed line corresponds to a linear fit. This linear behaviour of the C/T versus T 2 at low temperature originally describes the lattice vibrations as phonons

  Figure 4.13: Specific heat of Ba 0.93 Sr 0.07 CoS 2 at low temperature where it is seen that the fit is extrapolated to zero at very low temperatures and no conduction electrons contributing to the total specific heat curve. The curve deviation due to Shottcky effect is as obvious as in our BaCoS 2 . In inset is a log scale showing the very good agreement between the fit and the experimental points.

  Figure 5.1 shows the different X-ray diffractograms of Ba 1-x K x CoS 2 powders.

Figure 5 . 1 :

 51 Figure5.1: X-ray diffraction of the polycrystalline 4 %, 7 % and 10 % K-doped BaCoS 2 . The X-ray powder diffraction was carried out using a diffractometer equipped with a Co K α source. A solubility limit at a nominal 7 to 10 % doping is achieved and presented in the powder diffraction as emergence of BaS secondary phase beside CoS. The circle, triangle and star symbols indicate the secondary phases: CoS, BaS and BN respectively. Note that while extracting the sample from the rest of HP/HT assembly, some BN from the crucible can be stuck and mixed up with the sample powders and this is the only source of BN in our X-ray diffraction. In the figure underneath is the Rietveld powder refinement on the Ba 0.96 K 0.04 CoS 2 .

Figure 5 . 2 :

 52 Figure 5.2: X-ray powder diffraction of Ba-0.96K 0.04 CoS-2 carried out using a diffractometer equipped with a Co K α source. The red dots in the figure are the experimental data points and the black colored line is Rietveld refined XRD pattern. The Bragg's reflection positions of BaCoS 2 and CoS are indicated in light green color symbols and the differences between obtained XRD data and the Rietveld refined pattern is shown in blue colored line. The Rietveld refinement shows a good agreement between the experimental and calculated diffraction profile. Differences in the intensities can be due to large grains powder but, preferred orientation and site occupancy also highly affect the intensities.

Figure 5 . 3 :

 53 Figure 5.3: Dependence of the cell parameters and cell volume of Ba 1-x K x CoS 2 on the K-substitution level, x. Note a deviation of Vegard's law at 7 %, which gives an indication of the solubility limit.

Figure 5 . 4 :

 54 Figure 5.4: Dependence of the orthorhombic distortion δ and of the normalized variations of the a-and c-axis parameters with x. Note that the latter parameter is unchanged with x.

Figure 5 . 5 :

 55 Figure 5.5: Left: FC magnetic susceptibility of a series of Ba 1-x K x CoS 2 powder samples measured in a field of 10 kOe for x = 0.04, 0.07 and 0.1. Broken vertical bars denote the cusp-like anomaly signature of an antiferromagnetic ordering at T N .Right: correlation between T N and the in-plane lattice parameter a.

  Figure 5.6: The specific heat curves of two BaCoS 2 synthesised at ambient pressure and high pressure and Ba 0.93 K 0.07 CoS 2 , measured between 2 and 305 K under zero magnetic field. The coloured vertical lines indicates the T N of each sample. Note that the cusp-like anomaly at the AFM transition is gradually broadened and disappears completely for 7% of K substitution.

Figure 5 .

 5 Figure 5.8 shows the C/T vs T 2 plot for the Ba 0.93 K 0.07 CoS 2 component, where the dashed line corresponds to a linear fit. This linear behaviour of the C/T versus T 2 at low temperature originally describes the lattice vibrations, as phonons, in Debye model (C/T = βT 2 ). As seen previously in chapter 3 the C/T vs T 2 curve at low temperature of the pure phase BaCoS 2 is not completely linear and make a hump-shape at very low temperature. This hump was explained by taking in considerations that BaCoS 2 is an orthorhombic AFM phase where it is considered then as a two level system where a Schottky term is required to explain the low temperature heat capacity of the system. However, in the K-doped BaCoS 2 , figure5.8, it is noticed that a linear fit extrapolated to zero does not fit the experimental points but is parallel with a small down shift. Since BaCoS 2 is an insulating phase, obviously no conduction electrons contribute to the total specific heat of the system, on the other hand, modifying the filling, n, in such a Mott system is expected to result in a metallisation of the phase. Thus, in the case of Ba 0.93 K 0.07 CoS 2 , the low temperature behaviour of the specific heat is well described by the expression: C = γT + βT 3 + C Sch , where γ is the Sommerfeld coefficient arising from conduction electrons. Our data analysis yields γ = 5.7 mJ/mol K 2 , about two times larger than the value previously reported for metallic BaNiS 2 , γ BN S = 3 mJ/mol K2 [39, 98].

Figure 5 . 8 :

 58 Figure 5.8: Specific heat of Ba 0.93 K 0.07 CoS 2 at low temperature where a remarkable shift between the experimental curve and the linear fit (extrapolated to zero in green dashed line) indicates a non-zero value for γ. Meaning that a conduction electrons contribute to the total specific heat curve. The curve deviation due to Shottcky effect is again as obvious as in the pure BaCoS 2 . In inset is a log scale showing the very good agreement between the fit and the experimental points.

Table 1 . 1 :

 11 Structural models previously reported for BaCoS 2 .

	Reference	Baenziger et al. [45]	Snyder et al. [42]	Gelabert et al. [43]
	Crystal type	single crystal	polycrystalline	polycrystalline single crystal
	Crystal system	monoclinic	orthorhombic	orthorhombic	tetragonal
	Space group	P2/n	Cmma	Cmma	P4/nmm
	a (Å)	4.573(3)	6.4413(3)	6.4390(3)	4.568(1)
	b (Å)	8.937(2)	6.4926(3)	6.4909(3)	4.568(1)
	c (Å)	4.570(2)	8.9406(4)	8.9379(4)	8.942(2)
	comments	β = 90.43(4)	lattice parameters of single sulfur deficient sulfur deficient
			crystals not considered reliable	= 0.2	= 0

  BaCoS 2 sample containing sulfur excess (BaCoS 2.15 ), as well as for a BaCoS 2 sample with no sulfur excess after a single thermal treatment. One can no-

		1.4 1.6		2 nd thermal cycle_BaCoS 2.15 1 st thermal cycle_BaCoS 2	BN	* Ba 2 CoS 3
	Intensity [a.u.]	0.2 0.4 0.6 0.8 1.0 1.2	*	1 st thermal cycle_BaCoS 2.15 calculated intensity BaCoS 2 * *		*
		0.0
		20.0		22.5	25.0	27.5	30.0	32.5	35.0	37.5	40.0
							2��[�]

Table 3 .

 3

1: Explored HP/HT conditions on the BaCoS 2 phase stability. % Cont being the percentage of the total integrated intensity of each phase

  • C [86] , that can be interpreted as a non-stoichiometry of the compound and reveals a sulfur excess. Since BaCoS 2 is a metastable phase as shown in the phase diagram in figure3.1, we expect a transformation towards Ba 2 CoS 3 and thus the exothermic effect observed at T 2 ∼ 660

• C (figure 3.6b-b2) reveals the Ba 2 CoS 3 phase formation [87]. On the other hand, the endothermic peaks observed at T 3 , T 4 (figure 3.6b-b3) correspond to the intermediate decomposition of Ba 2 CoS 3

Table 3 .

 3 .2 presents the single crystal growths we have tried out with different temperatures and different BaCoS 2 :Co 4 S 3 ratios. 2: examples on single crystal growth of BaCoS 2 carried out in this work.

	name nominal composition	temperature dwell ramp quench	remarks
			( • C) ( • C)/h ( • C)
	HA05	BaCoS 2 (powder)	1150 (48h)	1	950	many (300 -700) µm crystals and several 1 mm size
	HA16s	BaCoS 2	1020 1030	0.5 0.5	1000 1010	ampoule was exploded very small crystals
			1040	0.5	1010	small crystals (< 400µm)
	HA17s	BaCoS 2.15	1020 1030	0.5 0.5	1000 few small crystals (50 -100) µm 1010 almost no crystals
			1040	0.5	1010	good crystals (300 -700) µm
	HA18s	BaCo 0.9 Ni 0.1 S 2	1030 1040	0.5 0.5	1010 1000	No crystals very small crystals (< 300µm)
	HA19s	BaCo 0.9 Ni 0.1 S 2.15	1030 1040	0.5 0.5	1010 1000 few big crystals (500 -700)µm No crystals
	HAC1	BaCo 3 S 3.5	1100 (72h)	1	900	highly pure, large plate-like crystals
		2 BaCoS 2 : 1 Co 4 S 3				of many mm size
	HAC2	BaCo 3 S 3.5	1100 (72h)	1	900	similar to HAC1
		2 BaCoS 2 : 1 Co 4 S 3
	HAC4	BaCo 9 S 8	1100 (72h)	1	900	crystals < 100µm and few metallic pearls
		1 BaCoS 2 : 2 Co 4 S 3				of 100-300 µm diameter
	HAC5	BaCo 5 S 5	1100 (72h)	1	900	needle-shape crystals
		1 BaCoS 2 : 1 Co 4 S 3				with another cell parameters!
	HAC6	Ba 3 Co 7 S 12	1100 (72h)	1	900	no crystals but a very
		3 BaCoS 2 : 1 Co 4 S 3				homogeneous regulus
	HAC7	BaCo 13 S 11	1100 (72h)	1	900	no crystals found
		1 BaCoS 2 : 3 Co 4 S 3
	HAC8	BaCo 9 S 8	1060 (72h)	1	950	few and tiny crystals
		1 BaCoS 2 : 2 Co 4 S 3
	HAC9	BaCo 11 S 10.5	1060 (72h)	1	950	no crystals found
		1 BaCoS 2 : 2.5 Co 4 S 3
	HAC10	BaCo 7 S 7.5	1060 (72h)	1	950	few and tiny crystals
		1 BaCoS 2 : 1.5 Co 4 S 3

Table 3 . 3 :

 33 Refined structure in the Cmma space group of powder and single crystalline BaCoS 2 samples synthesised at ambient and high pressure (AP,HP).

Table 3 . 4 :

 34 Lattice parameters refinement of a single crystal of BaCoS 2 . The XRD measurements were carried out at the beamline of CRISTAL of SOLEIL synchrotron at a temperature range of 16-365 K with a step of 25 K.

	Empirical formula			BaCoS 2
	Formula weight			260.39
	Z			4
	Crystal system			orthorhombic
	Space group			Cmme
	Crystal size/mm 3		0.015 × 0.01 × 0.01
	Radiation		synchrotron (λ = 0.6713)
	Temperature	a[Å]	b[Å]	c[Å]	V[ 3 ]	ρ calc [g/cm 3 ]
	365	6.4819(6) 6.4820(5) 8.9619(15) 376.54(8)	4.593
	340	6.4685(15) 6.4715(15) 8.950(8)	374.6(3)	4.617
	315	6.4703(6) 6.4756(6)	8.936(2) 374.39(10)	4.620
	295	6.4640(15) 6.4592(15) 8.958(8)	374.0(3)	4.624
	265	6.4554(5) 6.4522(5) 8.9465(13) 372.64(7)	4.641
	240	6.4443(14) 6.4507(14) 8.929(7)	371.2(3)	4.660
	215	6.4441(5) 6.4408(5) 8.9385(18) 370.99(9)	4.662
	190	6.4364(16) 6.4344(16) 8.932(7)	369.9(3)	4.675
	165	6.4312(5) 6.4366(5) 8.9282(13) 369.58(7)	4.680
	140	6.4265(14) 6.4279(14) 8.942(6)	369.4(3)	4.682
	115	6.4211(4) 6.4329(5) 8.9329(16) 368.98(8)	4.687
	90	6.4248(15) 6.4238(16) 8.923(7)	368.3(3)	4.696
	65	6.4241(5) 6.4256(5) 8.9255(11) 368.43(6)	4.694
	16	6.4247(7) 6.4146(6)	8.927(2) 367.91(10)	4.701

Table 4 . 1 :

 41 HP/HT synthesis conditions of Ba Structural properties of Ba 1-x Sr x CoS 2

	BaCoS Ba CoS BaS CoS	350 970 65 11 23 1 6.47999 6.40763 8.95169 371.686 traces of another phase	375 1040 90 8 0.6 1.4 6.49254 6.44484 8.93374 373.818 -	380 1052 72 16 11 1 6.48735 6.44949 8.93373 373.788 traces of another phase	385 1065 96 -2 2 6.48319 6.44364 8.92618 372.899 clean sample	300 832 52 36 3 9 6.4459 6.44224 8.91592 370.243 -	350 970 75 -17 8 6.46123 6.44455 8.91304 371.137 a very small peak around 20 • C	370 1024 67 21 3 9 6.46107 6.43794 8.91802 370.954 -	375 1040 95 -2 3 6.48298 6.44128 8.92518 372.704 have unknown peaks	300 830 -70 20 10 ----a very small peak around 20 • C	375 1040 50 29 11 10 6.45153 6.44189 8.91025 370.31 very small peak around 18 • C	375 1119 48 11 26 15 6.45377 6.44017 8.90369 370.067 -	375 1119 33 49 10 8 6.45476 6.43595 8.91611 370.397 -	300 897 -57 36 7 -----	325 971 -34 58 8 ----unknown phase with large peaks	350 1045 41 11 29 18 6.43775 6.41967 8.8839 367.155 unknown phase with large peaks	-375 1119 63 -22 15 6.44344 6.43208 8.89305 368.57	1-x Sr x CoS 2 samples and structure	refinement details, lattice parameters and cell volumes calculated using Rietveld re-	finement method. The XRD data were collected using a Co-source diffractometer on	powder samples over 2θ range 10-75 • .
		0.5	0.5	0.5	0.5	1	1	1	1	2	2	3.5	3.5	4	4	4	4
		PE_HA56	PE_HA56	PE_HA57R	PE_HA58R	PE_Sr1	PE_HA38	PE_HA49	PE_HA44	PE_Sr2	PE_HA47	PE_HA46	PE_HA50	(HA03 precursor)	PE_Sr3	PE_HA41	PE_HA39	PE_HA45

  2 phases in the orthorhombic Cmma space group. We summarise in table 4.2 the results of the refinements. A representative diffractogram showing the good agreement between experimental and calculated profile is shown in figure 4.3.

	Bragg positions
	BaCoS 2
	CoS
	BaS

2θ [°] Ba 0.93 Sr 0.07 CoS 2 @ 3.5 GPa / 950°C Y data Y calc. Y data -Y calc.

  4.2 and figure 4.4. We have succeeded in synthesising (Ba,Sr)CoS 2 poly-

	a, b parameters [Å]	6.44 6.45 6.46 6.47 6.48 6.49	Ba 1 x Sr x CoS 2	a_ ex situ HP/HT_Belt a_ambient pressure a_ ex situ HP/HT_Belt b_ambient pressure	c parameter [Å]	8.940 8.915 8.920 8.925 8.930 8.935 8.910	Ba 1 x Sr x CoS 2 c_ ex situ HP/HT_Belt c_ ambient pressure	369.5
			0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Sr content, x			0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Sr content, x
			(a)

Table 4 . 2 :

 42 Rietveld refined crystal structures of a series of powder Ba 1-x Sr x CoS 2

  The total specific heat curves of: BaCoS 2 synthesised at ambient pressure, HP/HT synthesised BaCoS 2 and Ba 0.93 Sr 0.07 CoS 2 , measured between 2 and 305 K under zero magnetic field. The colourful vertical lines indicate the T N of each sample. It is noticed that the characteristic peak of the AFM transition is gradually broadened and disappearing completely at 7 % of Sr-substitution.

		120	BaCoS 2 _AP
		100	BaCoS 2 _BELT Ba 0.93 Sr 0.07 CoS 2 _BELT
	]
	K 1	80
	C [J mol 1	40 60
		20
		0
		0	50	100	150	200	250	300
					T [K]
	Figure 4.11:

  Figure 4.12: A zoomed curve of the specific heat of: BaCoS 2 synthesised at ambient pressure, BaCoS 2 synthesised at HP using a belt press and Ba 0.93 Sr 0.07 CoS 2 synthesised under pressure using also a belt press.

	BaCoS 2 _AP BaCoS 2 _BELT Ba 0.93 Sr 0.07 CoS 2 _BELT

Table 5 .

 5

1: Refined crystal structures of (Ba,K)CoS 2 samples in the Cmma symmetry. AP and HP denote ambient-and high-pressure synthesis. For the latter synthesis, we used a belt-or Paris-Edimbourg-(PE) type of press.

  Figure 5.7: A zoomed curve of the specific heat of: BaCoS 2 synthesised at ambient pressure, BaCoS 2 synthesised at HP using a belt press and Ba 0.93 K 0.07 CoS 2 synthesised under pressure using a PE press.

	K 1 ]	110 120	BaCoS 2 _AP BaCoS 2 _BELT Ba 0.93 K 0.07 CoS 2 _PE
	C [J mol 1	90 100
		80
		70	T [K] 125 150 175 200 225 250 275 300

Table 5 . 2 :

 52 Relationship between orthorhombic distortion δ, T N and the specific heat parameters ∆ Sch , γ and the Debye temperature θ D .

	sample	δ	T N [K]	θ D [K]	∆ Sch [mev]	γ [mJ mol -1 K -2 ]
	BCS_AP	0.00764 290 142	0.620	0
	BCS_PE	0.000789	-	230	0.674	0
	BCS_BELT	0.00434 260 145	0.432	0
	7% Sr_BELT 0.000396 230 137	0.608	0
	7% K_PE	0.001263 230 139 8 × 10 -5	5.7

In case of superexchange interaction J 1 , the coupling would be also FM according to GKA rules.

In our experiment, we used an empty Al

O

crucible as reference.

In 2005, this laboratory was merged with the laboratory of "crystallography and mineralogy" to give birth to our IMPMC laboratory

V stands for 'vérin' and X for 'single crystal'

Mechanical separation was adopted in all previous growths, even though it limits so far the size of the separated crystals.

the temperature above which the material is completely liquid.

One sample of this series was synthesised using the same pre-reacted precursors used in the belt press, however, the obtained powder contained ∼70% secondary phases.