List of Figures

The -symbol denotes that the oscillation is not present judging from the Fermi surface. The * symbol denotes that the signal could not be observed although the oscillation is expected to be present. 1)The data taken from the paper by Tautz et al. [34] [35] frequencies as a function of magnetic field below and above H m with fields parallel to c-axis in CeRu 2 Si 2 . . . . . . . . . 3.38 TEP (left) and dHvA (right) [35] frequencies as a function of magnetic field of the β branch below and above H m with fields parallel to c-axis in CeRu 2 Si 2 . 3.39 TEP (left) and dHvA (right) effective masses as a function of magnetic field below and above H m with fields parallel to c-axis in CeRu 2 Si 2 . . . . . . . . . The evolution of T N is represented with the scale on the right axis [48]. . . . 4.4 Temperature dependence of the resistivity of Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 with excitation current J along the a and c-axes. The data with J a are from the reference [56] 6 Small and Large Fermi surfaces calculated with the Local density approach (LDA) + spin-orbit coupling method. The sheets are labelled donut D, junglegym J and pillbox P. From an electron point of view, the dark blue side of each sheet is the occupied side and the light yellow side is the unoccupied side. Therefore, D is a hole surface and P is an electron surface. Note that D is a torus in the small Fermi surface case only [72]. . . . . . . . . . . . . . . . . . 5.7 Field dependence of coefficients A (blue triangles, left axis) and γ (red circles, right axis) from FL behaviour in the electrical resistivity and specific heat, respectively for a field in the easy (ab) plane. The inset shows ρ versus T for 8 T and 16 T. Solid lines represent the T

General introduction

The capacity to determine and anticipate the criterion upon which a system becomes magnetic is a crucial issue from both a fundamental and applied perspectives. Heavy Fermion systems, sometimes also referred as Kondo lattice systems, represent probably the most starkly example where these questioning plays a major role. The appellation Heavy Fermion designates usually metallic compounds and alloys containing 4f or 5f ions, forming a matrix of localized magnetic moments, belonging to the rare earth elements family such as Ce or Yb, and also actinide compound like U. Their inner shell conduction electrons generally have effective masses, referred as quasiparticle masses, that could reach a hundred times the ones of the free electrons. In the low temperature regime, many of these compounds display either a magnetic order, or a paramagnetic behaviour, or in some exotic cases a superconducting phase which description goes beyond the traditional BCS theory. As mentioned before, magnetic order in heavy Fermion material has generated a lot of discussions this past few decades. From this debates has emerged, among others, the famous Kondo necklace concept introduced by Doniach [2]. This model somehow resume efficiently the physics of heavy Fermion systems that is principally resulting from the contribution and competition of two major effects. Indeed, in his phase diagram, Doniach considers the competing effects of Kondo compensation, via the singlet formation, of local moments in a metal that produces a non magnetic ground state, and the Ruderman-Kittel-Kasuya-Yosida (RKKY) coupling between these moments that favours long range magnetic order. The screening process of the Kondo effect is continuous and appears when the temperature drops below a characteristic energy scale referred as the Kondo temperature T K . A sudden increase of the resistivity due to the magnetic ions is observed since the quenched magnetic moments act as elastic scattering potentials for electrons. In heavy Fermion material, this ion Kondo effect is replaced by the more general Kondo lattice concept. Within this picture, at every site in the lattice the spin is quenched, but the main difference compared to the isolated Kondo case relies on the development of coherence by the strong scattering process occurring at each lattice site. Finally, the result of this is that at low temperatures the resistivity of the compound suddenly drops below T K . The weak hybridization between the conduction electron and the localized f -electrons leads, in this class of compounds, to a Fermi liquid ground state with narrow bands. Under magnetic fields, the heavy fermion bandwidth can become comparable to the Zeeman energy. A consequence of that, is a potential modification of the Fermi surface as the bands are spin split by the effect of the field and a topological transition can occurs [3]. This class of transition are referred as Lifshitz-type transition. This last few decades, numerous studies have been devoted in the investigation of this typical continuous quantum phase transition [15,16]. Basically, the Lifshitz transition appears in systems of non-interacting Fermion that are characterized by a continuous topological change of the Fermi surface in the Brillouin zone. This picture is highly different of the one obtained in the case of conventional quantum critical points. Magnetic field, chemical doping or pressure are physical parameters that are commonly used to tune heavy Fermion material to a point where their antiferromagnetic ordering temperature is driven continuously to zero resulting in a so-called quantum critical point (QCP) [17].

Changes in the metallic properties, that are usually detected in a wide range of temperature beyond the QCP, are called Non Fermi liquid behaviour and are characteristic of such regions. Recently, a renewed interest from the strongly correlated electron community has emerged in the potential connection between this quantum critical behaviour and the Lifshitztype transition. In this particular context, the two heavy Fermion compounds CeRu 2 Si 2 and YbRh 2 Si 2 have played an important role which was translated into several studies in order to shed light on this issue.

Chapter 1

Theoretical introduction A major issue of condensed matter physics is to understand and predict the behaviour of interacting electrons, when it needs to be described beyond the mean field theory of Landau: this is the field of strongly correlated electron physics, the play ground of the heavy fermion field.

However, even for strongly correlated systems, the Fermi liquid picture remains the reference, either to identify expected behaviours in measured quantities or to reveal deviations from this regime, pointing to a breakdown of Landau's picture.

Fermi-liquid properties at low temperatures. T 2 behaviour of the resistivity

The Drude formula, for non interacting electrons, gives the classical expression of the resistivity:

ρ = m ne 2 τ (1.1)
Where m, e and n are respectively the mass and the charge of the particles, and the density of carriers. The Matthienssens's rule, states that to the total resistivity coming from the contributions of the different scattering mechanisms, is expressed as a sum of these contributions. For example:

ρ(T ) = ρ 0 + ρ el (T ) + ρ ph (T ) + ρ mag (1.2)
Where:

• ρ 0 corresponds to the residual resistivity due to the elastic scattering of the electrons on impurities but also on the lattice defects,

• ρ el corresponds to the contribution coming from electron scattering electron,

• ρ ph that is the contribution of the scattering of electrons by phonons. This contribution is given in the Debye model at low temperatures (T << Θ D ) by: ρ ph ∝ ( T Θ D ) 5 . In the opposite high temperature limit, ρ ph ∼ T . In the high temperature region, for pure metals, ρ ph is predominant, whereas in the low temperature domain ρ ph is usually neglected,

• The last term ρ mag exists in materials exhibiting magnetism. This term is related to the scattering between electrons and the collective long range excitations (magnons), or spin fluctuations.

Another contribution to the resistivity widely studied in the sixties, and very often referred to in the heavy fermion systems, is that of isolated magnetic impurities: in the case of antiferromagnetic exchange coupling between the spin of this impurity, and of the conduction electrons, a collective many body singlet state is progressively formed at low temperatures. This is the celebrated Kondo effect leading to an increase on cooling behaving as ρ K (T ) ∼ -ln(T ) when T is much larger than the so-called Kondo temperature, and a saturation with an initial temperature dependence behaving as ρ

K (T ) = ρ 0 -A(T /T K ) for T << T K [1].
This T 2 behaviour of the resistivity (identified by Nozières to the contribution of a local Fermi liquid ), is a general feature of the Fermi liquid regime, where the energy dependence of the life-time of the electron hole excitations (J(ǫ) ∼ ( 1 (ǫ -ǫ F )

) 2 leads to a T 2 of ρ e-e .

ρ(T ) = ρ + ρ e-e (T ) = ρ 0 + AT 2 (1.3)
For heavy fermion systems, observation of a T 2 of ρ(T ) is usually taken as an indication of the validity of Landau's framework. The coefficient A is proportional to the square of the density if states, and thus, of the effective mass.

Specific heat evolution

The Sommerfeld specific heat coefficient γ is an even more direct measure of the density of states, proportional to the effective mass of the quasiparticles. Indeed, this coefficient is expressed as follows [START_REF] Kittel | Introduction to solid state physics seventh[END_REF]

γ = C el T (1.4)
Where C el is the electronic contribution to the specific heat in an electron gas or in the Fermi liquid. For kT << ǫ F (the Fermi energy): Where:

C el = π 2 3 D(ǫ F )k 2 B T (1.5)
• D(ǫ F ) = 3N 0 2ǫ F
is the density of state at the Fermi level,

• k B is the Boltzmann constant.

The only difference between the free electron gas, and Landau's Fermi liquid is that m (the free electron mass) is replaced by m*, the effective mass renormalized by electronelectron interactions. So, generaly speaking, ǫ F is expressed as ǫ F = 2 k 2 F 2m * And,the Sommerfeld coefficient could be rewritten as:

γ = π 2 k 2 B N m 2 k 2 F (1.6)
Experimentally, in heavy fermion system, the electronic contribution to specific heat does not always present such a linear temperature dependence. This is the a strong indication of a non Fermi-liquid behaviour, often related to the proximity of a quantum critical point (to be discussed later): a good example id the case of YbRh 2 Si 2 , where C/T is clearly not constant down to 40 mK in zero field, but where the usual Fermi liquid regime is restored under field [67]: see figure 1.1.

Specific heat Sommerfeld γ coefficient and A coefficient of the T 2 term of the resistivity are commonly quantitatively compared. The ration A/T 2 , called the Kadowaki-Woods ratio [43], is expected to be roughly constant from one compound to an other, as both A and γ 2 are proportional to the square of the density of states.

Chapter 1

Heavy Fermion systems 1.2.1 Brief introduction

We have already mentioned several times the name heavy fermions: it is time to give a rapid description of this family of strongly correlated electron systems.

Heavy fermions are intermetallics, containing rare-earth (4f-electrons) or actinide (5felectrons): most of the physics, and particularly the strong electronic correlations, are coming from the interplay between f electrons and conduction bands. Depending on the distance between rare-earth or actinide ions, the overlap between f wave-functions might be sufficient or not to create narrow f-bands (Hill criterion [START_REF] Gupta | Selected Topics in magnetism[END_REF]). This will also influence the type of magnetism of these systems, itinerant or localized: however, in many cases, heavy fermion are in between these two extreme limit cases.

The Kondo Model

We have spoken already of the Kondo effect, which may appear in metals with isolated magnetic impurities. But heavy fermion systems are also referred to as Kondo lattices at least in cases where f-electrons have a localized character at high temperature. The Kondo effect of single impurities is characterized by an energy scale, k B T K where T K is the Kondo temperature expressed as:

T K = De -1 N 0 J (1.7)
Where:

• D is the bandwidth of the conduction electrons,

• N 0 is the density of states of the conduction band,

• And J the exchange energy between the local moment and the conduction electrons.

We have already introduced T K as crossover temperature, but k B T K is also the binding energy of the collective singlet ground state. However, in the case where magnetic impurities are not isolated, interaction with the conduction electrons also leads to an indirect coupling between impurities, which is competing with the Kondo effect: the RKKY interactions. These are naturally particularly important in case of a lattice of magnetic impurities (heavy fermion systems). Let us briefly summarizes the main characteristics.

Ruderman-Kittel-Kasuya-Yosida: The RKKY interaction

This interaction was first proposed in 1954 by Rudermann and Kittel [8] as a description of the coupling between nuclear spins through the hyperfine contact interaction and later extended by Kasuya [9] and Yosida [10] to the indirect coupling between localized electronic moments in metals.

H RKKY f = - i,j J RKKY i,j S i S j (1.8)
Where the RKKY-coupling constant is expressed as:

J RKKY i,j = J 2 k 6 F ǫ F 2 V 2 N 2 (2π) 3 F (2k F R ij ) (1.9)
RKKY interaction is clearly a second order effect since J RKKY i,j ≃ J 2 . The function F is expressed as [13]

F (x) = sin(x) -x * cos(x) x 4 (1.10)
It gives an oscillatory dependence to

J RKKY i,j
, which is a function of the distance between the magnetic ions. Therefore, J RKKY i,j will determine whether the long range magnetic interaction are ferromagnetic or antiferromagnetic. The figure 1.3 displays a schematic view of the RKKY interaction. Note that the wave vector of the oscillations is 2k F : like Friedel oscillations, it is the cut-off introduced by the Fermi sea which includes the spatial modulation of J RKKY .

Competition between Kondo effect and RKKY interactions

To understand how the RKKY interaction is competing with the Kondo effect we compare the energy scale of these two. A characteristic energy scale associated to the RKKY effect:

k B T RKKY ∝ J 2 N 0 (1.11)
Remember that the Kondo temperature behaves as:

T K = De -1 N 0 J (1.12)
And thus it can be considered that the Kondo temperature evolves as:

T K ≃ e -1 N 0 J (1.13)
And:

• The Kondo effect quenches the local f-moments via the hybridization of the conduction electrons resulting in the creation of a many-body spin-singlet state,

• The RKKY interaction generates a long-range magnetic coupling between the f-electrons, which could leads to long range order.

Both interactions originate from the same exchange coupling J, but lead to opposite ground states. Doniach addressed this issue, and the so-called Doniach phase diagram schematically represented in the figure 1.4 [11] displays three possible scenarios:

• When the Kondo regime dominates, that is to say T K >> T RKKY , the magnetic moments are quenched and Fermi liquid properties are observed,

• If T K << T RKKY , the RKKY interaction dominates and a magnetic order occurs,

• When T K ≃ T RKKY heavy fermion behaviour can be observed. By varying the coupling constant, the ground state of the system can be tuned between Kondo and the magnetic phase. [11] So far, heavy fermions system is in the vicinity of a zero-temperature magnetic phase transition. Nowadays, this region is referred as a quantum critical region, and the system can usually be tuned through the magnetic to a non-magnetic ground state via the modification of an experimentally accessible external physical parameter such as pressure, doping or magnetic field.

Quantum criticality 1.3.1 Two types of quantum criticality

Introduction

Systems that present a quantum critical point delimiting an antiferromagnetic phase from a paramagnetic one have been described by two different theoretical models [69,75].

The local criticality scenario

Within this model, the antiferromagnetic ground state is induced by local moments, whereas the f-electron would be itinerant in the paramagnetic state. So, the size of the Fermi surface is larger in the paramagnetic phase, which includes a contribution of f-electrons, than in the ordered phase (no f-electron contribution). The quantum critical point marks the reconstruction of the Fermi surface with an abrupt change of its size, and likely also of its topology. It is predicted on approaching the AF phase, the localization of the f-electron induces a divergence of the effective mass. The figure 1.5 displays a schematic illustration of that case. The spin density wave scenario

The second scenario of quantum critical point is the so-called spin density wave type. Hertz Millis and Moriya were the first to develop this model [76,77,78]. Within this scenario, in both the paramagnetic and antiferromagnetic phase, the f -electrons are supposed to remain hybridized with the conduction band. In this situation, the antiferromagnetic can be described as a spin density wave state. In this case the Fermi surface does not change size, but has an instability localized around the (incommensurate) ordering wave vector of the spin density wave state. The regions of the Fermi surface that are connected by these antiferromagnetic ordering wave vectors are called Hot Spots. A divergence of the effective mass at the quantum critical point is then expected only close to the hot spots.

Non-Fermi liquid behaviour

In the vicinity of the critical point, fluctuations will persist down to the lowest temperature. Experimentally, many quantities (like the specific heat, the resistivity, ...) show temperature dependence which are different from those predicted for the Fermi liquid regime; they are called non Fermi liquid behaviours. This could be due to a breakdown of the motion of quasiparticles, but can sometimes be explained by the contribution of magnetic fluctuations as described for example by Moriya's self-consistent renormalization theory (SCR) [76].

For example, in the weak ferromagnet MnSi, the resistivity behaves as ρ ≃ T 5/3 (as predicted by SCR), instead of T 2 [79] close to the QCP (figure 1.6) which is more complex and difficult to reconcile fully with SCR theory, the specific heat diverges as C/T ≃ T lnT at the critical doping in the case of CeCu 5.9 Au 0.1 [80] (figure 1.7).

These are examples of discrepancies that could be observed near a quantum critical point, which reflect, the existence of a non-Fermi liquid behaviour. Usually, the resistivity is com- T indicates a deviation from the Fermi liquid theory [80].

Chapter 1 monly used as a signature to detect such phenomenons. The interest for that technique is the ease to perform measurements also under field and pressure, often used to tune the proximity to the QCP. Quantitative measurements of specific heat or the Pauli susceptibility are experimentally more delicate to realise under pressure. Therefore, are also uses doping (acting as chemical pressure) to tune the system to the QCP. However, the introduces disorder and possibly inhomogeneities, which can also alter the physics of the system.

Chapter 2

Experimental techniques

2.1 Crystal growth techniques

Flux method

In the flux method we used for heavy fermion compounds, the material is dissolved in a metallic flux, which is lowering the melting point of the compound. The single crystal grows from a supersaturated solution. The material used for the crystal growth is placed into a crucible made of non-reactive material tantalum, niobium, or in our case aluminium. The crucible is sealed in a quartz ampoule, under vacuum or Argon atmosphere under 1200 • C. The main advantages of doing so are that the material is in a protective atmosphere during the growth, and that volatile materials will remain contained in the ampoule. Crystal are extracted from the flux by spinning the quartz after a fast enough extraction from the furnace (to avoid solidification of the flux before spinning. This technique has been used for the sample of YbRh 2 Si 2 used in our dHvA measurements. As starting materials, we used stoichiometric quantities of Yb (4N), Si (6N), and Rh (4N), in Indium flux (representing 95 % of the total mass inserted in the ampoule). The flux method to grow this crystal is required by the high vapour pressure of Yb. The figure 2.1 shows a photograph of the quartz ampoule used (a), the flux furnace in which the materials have been melted (b), the spin balance used to centrifuge the ampoule (c) and on of the obtained single crystals of YbRh 2 Si 2 .

Czochralski Technique

The Czochralski method is a popular method of crystal growth that was developed by the Polish chemist Jan Czochralski in 1917 [14]. The material is melted in a crucible, and usually, a seed melt of the target material is used to initiate the crystal growth. The seed is putted into the surface of the melt. Matter starts to solidify at the interface and the seed is slowly pulled to further grow the crystal. In our tetra-arc furnace we use a typical pulling rate of 10 to 15 mm/hr and the seed as well as the crucible rotate to ensure the homogeneity of the melted material. The contact of the melt with the crucible is a potential source of contamination. So, the crucible is made of an inert material and is water cooled. The growth chamber is filled with a protective atmosphere of inert gas such as the Argon. The figure 4.2 shows a photography of the tetra-arc furnace used to growth the samples and schematic view. The photography in the figure 2.3 shows the ingot (left) of U 4 Ru 7 Ge 6 compound obtained by using the tetra-arc furnace. The Czochralski method possesses several advantages:

• The part of the impurities remain in the molten liquid, which improves the crystal quality,

• The compositional change in the solid solution reduced by pulling a small crystal from a large melt,

• The orientation can be controlled by that of the seed,

The single crystal of Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 measured in our pressure study was grown in a tetra-arc furnace using the Czochralski method. The starting materials were prepared in stoichiometric proportions: Ce (purity: 99.9 %-3N), Ru (4N), Rh (4N) and Si (6N) were used in the ratio 1 : 1.84 : 0.16 : 2. These elements were first melted in a high purity Ar atmosphere to form a polycrystalline ingot. In order to obtain a homogeneous phase, the ingot was turned over and melted several times. The single crystal of pure CeRu 2 Si 2 used for the thermopower measurements was grown by Y. Onuki.

Sample preparation 2.2.1 Electro-erosion cutting

A spark-cutter has been used in order to cut the crystal in the desired dimension. This procedure can be decomposed in three stages:

• The sample is mounted on a goniometer. This allows to orientate the sample in the wished direction. Then, the goniometer, is placed in a tank and immerged in oil,

• A thin single-strand copper wire, that is constantly fed from a spool, is placed upon the sample and sustained by two horizontal wire guides. With a 50 µm diameter wire, we can cut slices as thin as 100 µm. Two different electro potentials can be selected to cut the sample. 100 V for a fine cut, 300 V for a faster but rougher cut.

X-ray Laue photograph

X-rays are a powerful tool to acquire information about the crystalline structure of the samples since their wavelengths are approximately the same magnitude as the distances between atoms. We have used extensively Laue diffraction to check the crystals quality and For heavy fermion studies, the temperature plays a major role since most of the interesting physics occurs at low temperatures, below 10K and down to at least 0,1K. This is the range covered by a dilution refrigerator. The central part of the dilution is placed in a vacuum chamber traditionally immersed into a4 He bath. Nowadays, the technique is evolving toward 'cryofree' systems, where the 4 He bath is replaced by pulse tubes. However, even in those new systems, the core of dilution refrigerator remains the circulation of a 3 He/ 4 He mixture, the 'cooling' power coming from the dilution of 3 He into 4 He. This circuit is represented schematically in figure 2.6. Using a system based on a mixture of liquid 3 He and liquid heat exchanger. The circulation of the mixture inside the circuit can be summarized as follow:

• The 3 He is extracted from the mixture by pumping the still. To enhance the efficiency of the pumping the still is heated typically at a temperature around 0.7 K. Mainly evaporated, due to its higher vapour pressure.

• After the 3 He has been pumped, it is cleaned through a nitrogen trap, which is outside of the cryostat, before reinsertion. Some refrigerators also have a Helium trap for further cleaning,

• The 3 He is pre-cooled on the 4 He bath, and then the 1 K pot. Usually, a flow impedance is placed before the still, to maintain a sufficiently high pressure,so that 3 He can condense.

• After a series of heat exchangers, the key point to determine the base temperature), the 3 He rich mixture arrives at the top of the mixing chamber. Evaporation of 3 He in the still, and osmotic pressure balance between the still and the mixing chamber, force 3 He to pass from the concentrated to the dilute phase, producing cooling power (like evaporation for a liquid/gaz system), by crossing the phase boundary between the concentrated 3 He phase and the dilute one.

Measurements of resistivity under pressure for Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 and thermopower measurements in CeRu 2 Si 2 were performed in a home-made dilution refrigerators, that could reach temperatures of about 70 mK. It was also equipped with a magnet that generates fields up to 16 T.

High pressures

As mentioned in the introduction, the pressure is commonly used as an external parameter tuning the ground state of the compound. The piston-cylinder cell is used for pressure studies that do not require high pressures, and remain below the limit of roughly 2 or 3 GPa. The figure 2.7 shows a schema of a typical pressure-cylinder cell. A hybrid pressure cell made of copper-beryllium alloy (CuBe) and Nickel-Chrome-Aluminium (NiCrAl) alloy has been used in our experiments. The capsule inside which the sample is placed is filled with pressure transmitting medium, that is selected as a compromise between ease of use and hydrostaticity in the temperature and pressure range required by the measurement. Of course, it should be also chemically non reactive with sample and set-up. The Daphne 7373 oil has been used in this study. A small piece of lead is placed next to the sample inside the Teflon cap (as a pressure sensor, based on its superconducting transition measured by AC susceptibility). A manganin wire, calibrated in pressure at room and low temperatures, is also used to estimate what pressure should be applied initially at 300 K in the pressure cell, to obtain the desired pressure at low temperatures. This step is important, for our study, since we are dealing with small pressures and because we need small steps of pressure. The capsule, made of Teflon, is sealed mechanically before closing the pressure cell. Finally, the inner piston will transmit the pressure to the Teflon cap, applied at room temperature from the top of the cell with a press, and clamped with the locking nut.

Resistivity measurements

To perform resistivity measurements under pressure, four gold wires are spot welded on the sample as indicated in the figure 2.8.

In our measurement, a thin bar-shaped sample of Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 with the dimension of 0.5*0.5*1.5 mm (c-axis long) was cut from a crystal grown by the Czochralski method. The resistivity was measured by a 4-probe AC method at a frequency f ∼ 17 Hz. The current was applied along the c-axis with a value I ∼ 200 µA.

Thermopower measurements

The determination of the Seebeck coefficient relies on the measurement of an electrical potential induced by a thermal gradient. The figure 2.9 displays a schematic representation of the principle. Two dissimilar materials a and b are connected by two junctions in the point A and B. If a temperature gradient dT is applied between A and B, an electrical potential dV will be generated between C and D. When the circuit is open, the Seebeck coefficient of the couple of materials is expressed as:

S ab = dV dT = V C -V D V A -V B (2.2)
The experimental technique of measurement is based on the One heater, two thermometers set-up displayed in the figure 2.10. A thermal gradient is applied to the sample by the heater. Two thermometers will measure the temperature: the nearest to the heater is referred as T Hot whereas the further one is T Cold . Therefore, the Seebeck of the sample is determined via the relation:

S sample = - (V Cold -V Hot ) -(V Cold -V Hot )(P = 0) (T Cold -T Hot ) -(T Cold -T Hot )(P = 0) (2.3)
Usually, gold or silver wires are used for the thermal and voltage (with negligible seebeck coefficient at low temperature compared to that of the sample), and then, superconducting wires up to 4.2 K to measure the voltage. Copper wires, uninterrupted between 4.2K and the DC voltmeter, prolongate the superconducting wires, to minimize parasitic thermo electric effects when measuring V cold -V hot .

Quantum oscillations 2.4.1 Landau quantization

The Fermi surface at zero temperature marks the boundary between the occupied and unoccupied electron states in k-space. When a field is applied electrons undergo a cyclotron motion in the real space. In the k-space, such motion is quantized in the plane perpendicular to the field, and the different states are associated to concentric tubes aligned along the direction of the field and known as the Landau tubes.

Each of this tube correspond to a given energy for the quantized degrees of freedom. The energy of the electrons is given by the following formula:

E n = (n + 1/2) ω c + 2 k 2 z 2m * (2.4)
, and a given tube correspond to a fixed value of the integer n.

ω c is the cyclotron frequency expressed as eH cm * . Increasing the value of the field (and so ω c in 2.4 ) correspond to increasing the diameter of these tubes. When this diameter just matches that of an external section of the Fermi surface (in a plane perpendicular to the field), an anomaly will occur on the density of states, or more generally on the free energy of the metal. This anomaly will reappear at fixed intervals of 1/H corresponding to :

∆(1/H) = 2πe 1 δ ext (ǫ F ) (2.5)
, where δ ext (ǫ F ) is the external area of the fermi surface.

Lifshitz-Kosevish formula

The Lifshitz-Kosevish formula describes quantitatively the oscillatory part of the magnetization and is expressed as follow:

M osc = r i (-1) r r 3/2 A i sin 2πrF i H + β i (2.6)
• The amplitude of the oscillations is given by the term A i ,

• The dHvA frequency is given by F i = 2πe S i . Each extremal cross-section S i of the Fermi surface perpendicular to the magnetic field leads to a specific frequency, As shown by the equation 2.6, the oscillatory part of the magnetization is periodic in 1/H. The amplitude of the oscillations is governed by the term A i whose expression is the following:

A i ∝ H 1/2 | ∂ 2 S i ∂k 2 z | -1/2 R T R D R S (2.7)
The terms R T , R D and R S are reduction factors. The two factors R T and R D will be detailed in the next sections.

R S is called the spin reduction factor and is related to the phase shift between Zeeman split Landeau levels. Its value is given by:

R S = cos πgrm * 2m 0 (2.8)
where g is the g-factor.

The term

| ∂ 2 S i ∂k 2 z
| is called the curvature factor. When the Fermi surface is flat,(perpendicular to the field), this factor is large and therefore the oscillations are reduced. On the other hand, in the case of a two dimensional Fermi surface, this factor is small which induces large oscillations.

The Dingle reduction factor R D

The Dingle reduction factor is related to the Landau level broadening k B T D due to impurity scattering, and is given by:

R D = exp -λrm * T D H (2.9)
where T D is the Dingle temperature and is given by:

T D = 2πk B 1 τ (2.10)
τ is the electron relaxation time due to scattering caused by defects and impurities present in the sample. This Dingle factor is important in the dHvA effect since it introduces a field dependence of the oscillations: dirty samples, or heavy masses, require high fields. The idea is to use the field dependence of the amplitude of the oscillations. Thus, by using the expressions 2.7, 2.9 and 2.14:

A(H) = H 1/2 exp(-λm * T D /H) sinh(λm * T /H) (2.11)
In order to have a simple expression, that could be used experimentally to relate the field dependence of the amplitude A(H) with the Dingle temperature T D , the logarithm of the expression 2.11 is taken. Therefore, the result is given by:

ln A(H)H 1/2 sinh λm * T H = -λm * H T D + cst (2.
12)

The so-called Dingle plot represents ln A(H)H 1/2 sinh λm * T H versus 1/H at a fixed temperature and for a known effective mass. T D is extracted from to the slope of the curve.

Chapter 2

For a Fermi velocity

v F = k F m *
where k F is the Fermi wave-number and m * the known effective mass, the mean free path for a specific orbit is deduced from T D with:

l = v F τ = 2 k F 2πk B m * T D (2.13)
2.4.4 The temperature reduction factor R T

The temperature reduction factor is related to the thermal damping at a finite temperature T and is given, with r the harmonic number, by:

R T = λrm * T /H sinh(λrm * T /H) (2.14)
where λ is given by:

λ = 2π 2 ck B e ≃ 14.69T /K (2.15)
It shows again that heavy masses require low temperatures, and/or high fields. Importantly, it also allows to measure m * for the corresponding extreme orbit, by recording the temperature dependence of the amplitude.

The idea is to rewrite the equation 2.14 in the form:

ln A(T ) T 1 -exp -2λm * T H = - λm * H T + cst (2.16)
The next step is to plot the term ln A(T ) T 1 -exp -2λm * T H versus the temperature for a fixed magnetic field field. Such a figure is referred as a mass plot. The effective mass is then varied to obtain the best fit. The effective mass is then directly extracted from the slope of the curve since it is proportional to it.

Topology of the Fermi surface

As it has been mentioned previously, the dHvA frequency is related to the extremal cross sectional area S of the Fermi surface through the relation:

F (θ) = c 2πe S(θ) (2.17)
Where we have explicitly introduced the angle θ between the applied field and a reference direction in the reciprocal space. Indeed, experimentally a complete dHvA set up includes rotation of the sample, in order to reconstruct completely the Fermi Surface through 2.17, and by comparison with band calculations. Two simple cases can be considered in order to show how this is working.

Spherical Fermi surface

In the case of a spherical Fermi surface, due to the symmetry of the sphere, the extremal cross section is constant and so will be the dHvA frequency. This example is illustrated in the figure 2.12.

Cylindrical Fermi surface

In that case, the situation is different from the previous one. Indeed, if θ is the angle between the cylindrical axis and the direction of the applied magnetic field, thus if one consider the symmetry of the cylinder, one can conclude that extremal cross section will evolve as 1/cos(θ). Therefore, the frequency will follow the same trend as represented in the figure 2.13.

Generally, the Fermi surfaces of compounds are more complicated than this and the band structure calculation is commonly used as a support in the analysis of the dHvA signal.

Field modulation technique Principle of the method

The detection of the dHvA signal is realised by using the so-called field modulation technique. The system, schematically illustrated in the figure 2.14, consists of a pick-up coil, with compensation, and a field modulation coil. The technique consists in adding to the large monotonically increasing field H produced by a superconducting magnet, a small sinusoidally-varying magnetic field h 0 cos(ωt) generated by the modulation coil. Voltage generated in the pick-up coil is amplified and detected by a lock-in amplifier. Moreover, the lock-in is phase-locked to the second harmonic, in order to detect only the second harmonic of the signal. This condition is imposed in order to have a voltage that is independent of the static susceptibility χ 0 and other background effect allowing to obtain a direct measurement of the oscillatory magnetization.

Conditions to observe a dHvA signal

We have already mentioned that the reduction factors R T and R D impose low temperatures, high fields, and high sample purity for sizeable amplitudes. Intuitively, this arises from the disappearance of quantum interference effects when ω c ¡ kbT or ω c τ ¡ 1.

For ω c = eH m , the previous conditions are given by:

H T >> m * k B e τ > 2πm * eH (2.18)
Eventually, the first condition imposes that the measurement has to be performed at high field and low temperatures in order to enhance the ratio H/T . This is especially true if the effective mass is important which is the case of compounds belonging to the heavy fermion family. Regarding the last condition, since τ is related to the quality of sample, its value will decrease if the sample is of low quality. Therefore, here again, according to the second relation of 2.18, high field will be requested to overcome this drawback.

Chapter 3

CeRu 2 Si 2

Introduction

Brief history and presentation

Extensive experimental investigations have been performed on CeRu 2 Si 2 these past few years. CeRu 2 Si 2 is one of the main compounds in the study of the Physics of heavy Fermion systems. Crystallizing into the tetragonal ThCr 2 Si 2 structure, with a = 4.196 Å and c = 9.797 Å [20], this compound presents a large coefficient of the linear term of the specific heat γ = 360 mJ/mole K 2 . [46] One of the remarkable phenomena is the pseudo-metamagnetism that occurs at H m =7.8 T [21] when the field is applied along the easy c-axis, corresponding to a transition from a low field paramagnetic phase (PM) governed by AF correlations to a highly polarized state.

The figure 3.2 shows the field variation of the magnetization and the specific heat coefficient γ. As one can see from the curves, under a magnetic field, a sharp pseudo metamagnetic crossover occurs at H m =7.8 T characterized by a jump in the magnetization and an sudden enhancement of the specific heat coefficient γ [21]. The interesting properties of CeRu 2 Si 2 originate from the hybridization of the f electrons with conduction electrons, but also from the strong correlation among the f electrons of Ce. Physical properties like resistivity, specific heat and others [21,23] have been reported to change drastically through the MT transition indicating that it is intimately related to a modification of the Fermi surface (FS), correlated with the electronic structure change.

Phase diagram

CeRu 2 Si 2 is a Kondo lattice system, with a Kondo temperature T K ∼ 10 -20K [24,22,25], where no superconductivity nor any long range magnetic order has been detected for temperatures even going down to 20 mK [21]. In the low temperature limit, the compound remains in its paramagnetic Fermi liquid state independently of the applied field. The temperature dependence of the specific heat and of the magnetic susceptibility is represented in the figure 4.29. The curve behaves consistently with the PM nature of the ground state.

This compound is known to be strongly anisotropic with a ratio in susceptibility between the c-axis, corresponding to the easy axis, and the basal plane that is superior to 10 at low temperatures. As one can see from the figure 4.7, representing the variation of the susceptibility χ in temperature for a field parallel and perpendicular to the c-axis, the difference in amplitude is well pronounced. While in the perpendicular configuration the signal is weak and increases slightly with decreasing temperature, in the parallel case the susceptibility increases rapidly with decreasing temperature and presents a broad maximum around 10K.

Figure 3.5 represents the phase diagram obtained by thermal expansion and specific heat measurements. The striped low field region corresponds to the PM phase whereas the dashed high fields one is the polarized paramagnetic phase (PPM). This denomination of polarized comes from the large moment induced after H m . Indeed, as we can observe in the figure 3.6, showing the field dependence of the magnetization, at low temperatures, the 4.2 K curve displays a sudden increase of the magnetization at H m = 7.8 T coherent with the metamagnetic nature of the transition. Due to all these considerations about the magnetization, the shapes of the curves and on how they behave under magnetic field, this pseudo metamagnetic-like transition is believed more to be a crossover than a first order transition. This is highlighted by the fact that no hysteresis have been observed [29]. This crossover is concomitant with different atypical phenomenons like a radical modification of the Fermi surface properties or an enhancement of the effective mass that occur in the compound. This discussions are only stressing more the importance of knowing the aspect of the FS, and especially above H m .

Fermi surface

The strongly correlated f electron systems have attracted a lot of interest among the scientific community and this especially due to the low energy phenomena, such as heavy fermion or quantum phase transition, that were some of the main topics. The analysis of the Fermi surface (FS) of these elements is a crucial source of information to understand the Physics sustaining this phenomena. Among the different tools used these past few decades in the investigation of FS, the de Hass van Alphen (dHvA) effect has been the most intensively used method to explore a wide range of f electron systems. In the heavy fermion family, CeRu 2 Si 2 has been the subject of numerous studies to explore its FS. The interest for this compound was motivated by a simple but not yet clear question of whether the f electrons becomes localized or not after the pseudo-metamagnetic transition that occurs at H m = 7.8 T. The volume of the FS was commonly used to ascribe the itinerant or localized nature of the electrons. Nevertheless, this belief has felt into ruins with the discovery that such a modification in FS is not enough sufficient by itself to conclude in the localization of the electrons [30,31]. Moreover, it has been argued that even thought the detected volume of the FS is near to what is usually qualified of small FS, the term localized could not be used in some cases [34]. In the field region under H m , the FS has been well determined from both an experimental and theoretical perspectives [23,35]. CeRu 2 Si 2 is a case where the experimental measured frequencies are extraordinarily consistent with the band calculations [36,37]. The same remark cannot be done on the region above H m where the FS has only been partially discovered. However, based on the existing similarities of the observed frequencies and their angular dependences between CeRu 2 Si 2 and LaRu 2 Si 2 above H m , it has been reported that the La-doped case could be representative of the CeRu 2 Si 2 behaviour in the polarized paramagnetic phase [38]. The results obtained from the band structure calculation, considering that the f electrons belong to the conduction electrons, are represented in the figure 3.7 (a). On the FS are also reported the orbits of the measured dHvA oscillations when a field is applied along the c-axis. Figure 3.7 (b) represents the FS of LaRu 2 Si 2 , that is proposed to be the nearest picture of what could be the FS of CeRu 2 Si 2 above H m [85]. The tables on the figure 3.8 and 3.9 give the corresponding frequencies, and effective masses, of the dHvA orbits below and above H m respectively.

Motivations

Therefore, CeRu 2 Si 2 is the most suitable candidate for a project which aim is double. One of the aims of this study is to investigate by using thermoelectric power (TEP) a compound presenting a well known FS and a well defined MT. Since TEP is a quite sensitive The -symbol denotes that the oscillation is not present judging from the Fermi surface. The * symbol denotes that the signal could not be observed although the oscillation is expected to be present. 1)The data taken from the paper by Tautz et al. [34] Figure 3.9: Frequencies and effective masses above H m [34]. method, this technique represents an indispensable tool to explore phenomenons occurring at FS under magnetic field such as the Lifshitz transition. Thus, CeRu 2 Si 2 appears to be the most relevant compound since dHvA experiments has already been used to analyse the reconstruction of its FS. The fact that this compound has been studied for decades may also provide a strong support in the interpretation of the results obtained with TEP by following the Lifshitz transition. Moreover, CeRu 2 Si 2 is an attractive model since it is a great archetype of the strongly correlated f electron systems. The second aim of this project is to perform a complete study of CeRu 2 Si 2 from the quantum oscillations present in the TEP. Indeed, even if the dHvA method, whose reputation is firmly established in the scientific community, is a powerful technique to examine Fermi surfaces, it presents nevertheless some inconveniences that could prevail us to realize a precise study of the FS. One of the limiting factors is the requested magnetic field to explore some parts of the FS that could become rapidly to high to be experimentally possible. The temperature is also a critical factor since the dHvA signal requires extremely low temperatures to be exploitable. An alternative method, that may represents a convenient complementary tool to follow the FS evolution, is the TEP. Here again, CeRu 2 Si 2 presents an interesting profile that actually makes this approach feasible for different reasons:

• The fact that crystals of great quality of this compound are available, which is a key criterion in the detection of quantum oscillations, and that the thermodynamics of its MMT has been well established through various types of measurements make this compound a solid experimental basis that allows a comparative study of the results acquired via this two different methods.

• As we already have mentioned it before, high magnetic fields are usually necessary to achieve FS study. However, this may cause a problem in the case of the study of strongly correlated f electron compounds where the main interaction is the Kondo effect. Indeed, the effect of high magnetic fields in such a system may result in the modification of the FS properties prevailing us from gaining access to the ground state.

Consequently, the problem arises of how can we probe the FS and have an access to the ground state simultaneously. Once again, CeRu 2 Si 2 made this possible due its strong Ising character that allows us by confronting the FS properties obtained through the two configurations, parallel and perpendicular to the easy c-axis, to study the magnetic field effect on the FS properties.

Finally, the interest will be to appreciate the degree of validity and the relevance of the TEP as a tool to probe the FS of a compound, compare to others methods.

Thermoelectric power investigation of the Lifshitz transition

Introduction

In this section, the obtained results of the TEP measurements realized by following the MMT transition in CeRu 2 Si 2 will be presented. This experiments have been carried out in two configurations:

• H c and ∆T a : the transverse configuration S ⊥ where a transverse thermal gradient is applied in the basal plane. The results are presented in section 3.2.3

• H c and ∆T ⊥ c : the longitudinal configuration S where a thermal gradient is applied along the c-axis. These results are shown in section 3.2.4

The results will then be discussed for the two configurations by comparing them with those obtained in the magnetoresistivity measurements. As we will see in the section 3.2.5, the anomalies that are present in the TEP could be related to the magnetoresistivity response.

Experimental conditions

Single crystals of CeRu 2 Si 2 were grown using the Czochralski method in a tetra-arc furnace by Y. Onuki two decades ago. The single crystal ingots where annealed under ultra high vacuum using the SSE technique. Two samples of high quality were used in this study: one for the transverse configuration with a residual resistance ratio (RRR) of 160 (S ⊥ ), and an other for the longitudinal configuration with a RRR of 100 (S ). A "one heater, two thermometers" set-up has been realized for the TEP measurements in a dilution refrigerator down to 120 mK and up to 16 T. Highly resistive manganin wires have been used in the set-up in order to thermally decouple the thermometers and the heater from the sample holders. The magnetic field was applied along the easy c-axis, and the heat or charge current was applied along [001] for the S and in the basal plane for S ⊥ . The TEP voltage was averaged during several minutes, for the field and temperature dependence, depending of the temperature in the case of a thermal gradient or not.

Transverse configuration

Temperature dependence

We have measured the thermoelectric power (TEP) in the temperature range from 200 mK to 5 K. In the figure 3.10 we show the temperature dependence of the TEP at five different fields. All these curves were obtained by averaging during several minutes the voltage. Two clearly different behaviours could be seen in this figure both corresponding to two regions, one below H m = 7.8 T and one above:

• On heating for the curves below H m , S begins by decreasing, then reaches a broad negative minimum, and then starts to increase and changes sign to become positive at 2.5 K for 0 T and at 4.5 K for 7.5 T. In this region, by increasing the field, the minima of S has been highly broadened as it could be seen from the minima's width at 7.5 T that is almost twice time superior that the one at zero field.

• When the temperature is increased, the behaviour above H m is quite opposite. At 9 T, the curve starts first to increase to reach a maxima at 400 mK, and then decreases to become negative at 700 mK, and finally reaches a broad minimum at 2.5 K before increasing again. The two other curves above H m follows globally the same pattern except that the only both present a maxima at 1.2 K and 1.7 K for 13 T and 16 T respectively. They both then decrease and become negative at 1.6 K for 13 T and 2.8 K for 16 T. However, the 13 T curve seems to show an inflexion point around 3 T meaning that at high temperatures S(T) starts to increase. Concerning the extremum above H m , the same remark than below H m could be done, that is to say that the maximum width at 16 T seems to be broadened almost two times more compared to the 13 T one's.

The figure 3.11 displays the temperature dependence of the TEP divided by the temperature at different magnetic fields for the transverse thermal flow configuration. Basically, the same observations done for S(T) curves apply here. On heating at 0 T, S/T starts first to decrease, becomes negative around 300 mK, reaches a large minimum at around 1 K and then increases to finally becomes positive at 2.7 K. Nevertheless, some remarks can be pointed out:

• The different extrema reported before are more explicit in that figure which allows us to appreciate more clearly the complex temperature dependence characterized by multiple changes of the sign and different extrema,

• Regarding the curve at 9 T, the inflexion point is now evident, and effectively the curve seems to present a minimum.

The results obtained in this transverse configuration will be compared in section 3.2.5, and the differences will be discussed based on the magnetoresistivity results.

Field dependence

The field dependence of the TEP has been measured for the transverse configuration. The figure 3.12 shows S(H) at six different temperatures going from the lowest one 260 mK to the highest 780 mK. The voltage is averaged during several minutes to obtain these curves.

For all the temperatures we can decompose their field dependences in three regions:

• A low field region from 0 T to 6 T where the curves are globally negative. The evolution in field at the different temperatures is the same with a continuous decreases of S(H).

As indicated with a blue arrow, an anomaly is present at 1 T for all the temperatures. This field could be related to the CeRu 2 Si 2 series of doped compound. Indeed, in the La and Ge doped case, neutron elastic scattering experiments has revealed a modification of the magnetic ordering wave vector [39,40]. Therefore, this anomaly could be the consequence of a change in the relative weight of the three detected AF hot spot occurring at this a field around 1 T.

• A region around the MMT that is characterized by a drastic decrease of S(H) resulting in a sharp minimum at H m . This sudden diminution of S(H) is amplified by the increase of the temperature. Indeed, for the lowest temperature 260 mK the diminution is about 40 % while for the highest one's 780 mK the decrease is around 50 %. The inset displays a zoom of this region for the three lowest temperature 260 mK, 330 mK and 420 mK. At 260 mK, as pointed out by the red arrows, the curve of S(H) presents multiples kinks inside the narrow window going from 7 T to 8 T in which the MMT occurs. These kinks, that are well defined at 260 mK, start to become less clear when the temperature is enhanced, and completely disappear above 420 mK. This phenomenon may indicates a cascade of Lifshitz transition happening at the MMT.

• Finally, a last region above 9 T where the curves after having being increased become positive and stabilize at high field. Clear oscillations of S(H) are present in the signal and will be discussed in the section 3.3. In that region, the curves of S(H) are inverted compared to the low fields one's with the signal's amplitude for the low temperatures that is lower than those of the high temperatures. In the six curves presented, an anomaly indicated by a green arrow in the graphic is clearly observed for a field near 13.5 T. A previous ultrasonic measurement has revealed a slight softening around that field that could be related to the observed anomaly in our TEP experiments [41].

Concerning the anomaly detected at the MMT, such a cascade of transitions has already been measured in the TEP in YbRh 2 Si 2 [42,92]. This similar behaviour occurs in this compound at H 0 = 9.5 T which corresponds to its Lifshitz transition. This peculiar comportment of the TEP seems to be a signature of the Lifshitz-type transition that originates from the complexity of the FS's multiband aspect in heavy fermion systems.

Longitudinal configuration

Temperature dependence

The figure 3.13 shows the temperature evolution of the TEP divided by the temperature at different magnetic fields for longitudinal thermal flow configuration. Basically, different behaviours can be observed:

• Below H m , the two curves present globally the same behaviour in temperature. At 0 T on cooling, when the heat current is parallel to the applied field, S(T)/T changes sign from positive to negative at 3 K, has a minimum at 1.4 K, is positive below 750 mK and shows another sharp maximum at T ∼ 260 mK. At 7.5 T, S/T behaves almost samely as at 0 T except that S/T is almost constant above 2 K (up to 5 K),

• Above H m the situation is drastically different. Indeed, on heating, the curve of S/T increases almost linearly with the temperature, reaches a broad maxima around 1.8 K, before decreasing linearly with respect to the temperature.

Here again, the signal displays a complex temperature dependence with different extrema and sign changes. Furthermore, the presence of a peak in S/T for the longitudinal configuration at T = 260 mK reveals the necessity of very low temperatures to estimate correctly S/T for T → 0 K.

Field dependence

The field dependence of the TEP in the longitudinal thermal flow configuration at different low temperatures is represented in the figure 3.14. Three clear domains can be distinguished from the results:

• In the domain below 7 T, two facts can be pointed out. The first one is the similar behaviour of S⊥(H) with the S (H) one's around 1 T. As one can see in the figure, a jump of S(H) occurs at 1 T that as the same origin as for the other configuration. The other point that could be noticed is that near H m the TEP signal begins to slightly increase.

• Close to the MMT field, S(H) is suddenly enhanced and a well pronounced maximum is observed at H m = 7.8 T. Moreover, the amplitude of this maximum is strongly dependent on the temperature and increases with it. Indeed, as it could be observed in the graph, S(H) reaches 0.4 µV K -1 at 270 mK while at 690 mK S(H) = 1.5 µV K -1 which is almost three times more than the lowest temperature. In this configuration, the enhancement of the S(H) absolute value is much more important that the one observed in the transverse configuration at the same field and for comparable temperatures.

The inset shows a zoom of the region surrounding H m . Once again, S(H) displays the multi kinks structure that is associated to the Lifshitz-type transition. This structure disappears above 400 mK in the same way as it was observed for the transverse configuration.

• The last domain corresponds to the high fields above 10 T. In that region S(H) decreases rapidly with increasing field. At 13.5 T a small jump of S(H), comparable to the one seen in the other configuration, is observed for all the temperatures, and is particularly more clear at 390 mK and 690 mK.

The figure 3.15 shows the field dependence of the TEP at temperatures above 1 K and up to 4.9 K. In that temperature region, the comportment of S(H) is completely different of what could be observed in the low temperature region. The first striking point is the absence of the sharp peak at the MMT that was clearly seen in the figure 3.12. The inset shows a zoom of the region near the MMT. The curves can be regrouped in three types:

• At 1.3 K, S(H) begins with a negative value, then displays a broad maximum near 5

T and then starts to increase on approaching H m . At the MMT field, S(H) increases within a quite narrow fields window. Above H m , S(H) shows again a broad maximum around 12 T before decreasing.

• At 2.3 K, S(H) also starts with a negative value, then presents the same broad maximum than at 1.3 K, but then displays a tiny maximum at H m followed by a small minimum at 10 T before increasing at higher field.

• At 3.2 K, 4.2 K and 4.9 K, S(H) begins positively and the behaviour is the same for these three temperatures. At the MMT, S(H) decreases quickly to reach a minimum around 10.5 T, 11.5 T and 12.5 T for 3.2 K, 4.2 K and 4.9 K respectively, and then increases with increasing field. The minimum of the curves is broadened when the temperature is enhanced. Eventually, the emerging picture is a complex one where the behaviour of S(H) is interestingly different of the low temperatures situation that is characterized by the presence of pronounced peak in the TEP signal.

Conclusion

Comparison of the Temperature and Field dependence for the transverse and longitudinal configurations

The first point concerning the differences and similarities between the two measured configurations in our experiments that could be highlighted relies on the complexity of the TEP response. Indeed, as it has been detailed in the previous sections, the transverse and longitudinal configurations present a signal characterised by multiple sign changes and different types of extrema, whether it concerns the temperature dependence or the field one's. Apart from that shared signal's complexity, the two configurations are different in their TEP signal response and this in multiple aspects.

Let us consider first the temperature dependence of S/T. The figure 3.16 shows a comparison of S/T between the two configurations at 0 T, for the case below H m , and at 16 T for the one above:

• Regarding the case at 0 T, the first fact that could be noticed is the presence at 200 mK of a maximum in the longitudinal configuration that is not present in the other one's. By increasing the temperature, the two 0 T curves present roughly the same tendency by starting positively, becoming negative before increasing again. However, these changes in the sign occur at different temperatures with respect to the thermal flow direction.

As an example of that, we see that while S /T becomes negative at 750 mK on heating, for the S ⊥ /T configuration it is already at 300 mK that the TEP becomes negative. Moreover, not only the sign's changes occur at different temperatures, but also the temperature at which an extrema is present. Thus S ⊥ /T is minimum around 1 K while S /T is minimum at 1.4 K.

• Above H m , the curves at 16 T, contrary to what has been said for the 0 T case, does not present the same evolution in temperature. In the temperature region going from the lowest one to 1 K, while S ⊥ /T seems to remain constant around 1 µV K -2 implying that the system is in the Fermi liquid state, S /T decreases almost linearly with the temperature. The Fermi liquid state is never achieved in the longitudinal configuration. S ⊥ /T above 1 K starts to decrease linearly with increasing the temperature while S /T reaches first a broad maximum around 1.5 K before decreasing in the same way as S ⊥ /T.

To continue in our comparative analysis, we consider the field dependence of the TEP. In the figure 3.17, a comparison between the two configurations is reported at the two quasi identical temperatures 660 mK and 690 mK for S ⊥ (H) and S (H) respectively:

• The first remarkable fact is the sharp positive peak of S (H) at the MMT that contrasts with the pronounced minimum of S ⊥ (H). Beyond the fact that the sign is inverted between the both cases,we can however also notice that the amplitude of this drastic modification of S(H) is not the same with respect to the direction. Therefore, the amplitude of S ⊥ (H) is increased in absolute value by a factor two whereas in the case of S (H) it is by a factor three.

• Another aspect that could be stressed on is the sign of S(H). In the longitudinal configuration the sign of the Seebeck coefficient never changes meaning that the nature of the carriers remains the same. On the other hand, for the transverse configuration, the situation is completely different with a signal that changes sign from negative to positive at the MMT. This implies that at H m the type of the carriers is modified through out the transition.

• As an example of similitude, we can nevertheless quote the fact that the small jump of S(H) observed at 1 T and 13.5 T mentioned in the previous sections is present in the both configurations and at the same fields.

This observed anisotropy in the heat current direction has already been detected in a previous study [86]. The origin of such an anisotropy of S/T comes from the nature of the system. Indeed, the fact that CeRu 2 Si 2 has a multiband structure is responsible of the discrepancies between the transverse and longitudinal directions but also of the complexity of the Seebeck response. These strong anomalies observed at H m in the two configurations are indubitably concomitant with a large FS reconstruction happening at the MMT. The sign of the dominant carriers is usually an important question to resolve. In order to answer it, the sign of the thermopower is commonly used as a valuable source of information. Ordinarily, the general rule is that a hole band will participate with a positive contribution to the thermopower whereas an electron band will give a negative contribution. However, this general rule has an exception in the case of heavy fermion systems where the tendency is inverted. In our measurements, in the low temperature regime, the thermopower is positive. This report is consistent with the general observation that underlies that Ce-based heavy fermion compounds have usually a positive TEP in the low temperature limit. In the reality, there is no obvious relation between the TEP sign and the nature of the carriers. Indeed, in a multiband system the sign of the Seebeck can depends on the details of the band structure. Therefore, the fact that the sum of contributions to the TEP is weighted by the band conductivity should be considered in the determination of the carriers-type. Regarding the marked anisotropy between the two configurations in the TEP signal, it has been argued that it should be related to an anisotropy of the hybridization between the f electron and the conduction electron. Moreover, this anisotropy is not founded in the Rhdoped compound where the TEP signal is the same for both current directions [87], and similar to the transverse TEP signal of the pure compound. This is understood as being an effect of the doping process that introduces impurities in the system resulting in an average of the TEP signal of the two directions. Since both directions a and b of the basal plane are equivalent, their TEP signal that is negative will be predominant compared to the positive signal of the c direction.

Comparison with the magnetoresistance measurements

To further appreciate the anisotropy observed between the two thermal flow directions, it would be interesting to have an overview on the magnetoresistivity response. The figure 3.18 displays the magnetic field dependence of the resistivity ρ(H) for the transverse and longitudinal configuration at 30 mK and 1.2 K. Several remarks can be done concerning the evolution in field:

• A general remark that is valuable for the four curves is that at H m an enhancement of ρ(H) is observed relatively to the temperature and the direction considered. The magnetoresistivity response in the transverse case is stronger than in the longitudinal one's as it could have been expected.

• At 1.2 K, ρ ⊥ (H) increases more rapidly with the field than ρ (H). At H m , the peak in ρ ⊥ (H) is much higher than the ρ (H) one's. Finally, above H m , the magnetoresistivity of the two directions diminishes and at 13 T the ρ ⊥ (H) is almost twice time superior to ρ (H),

• At 30 mK, the trend of the field evolution of both ρ (H) and ρ ⊥ (H) is similar to the 1.2 K case. However, by reducing the temperature, the shape of ρ ⊥ (H) changes from a peak aspect at 1.2 K to a step-like increase at 30 mK. Nevertheless, it could be noticed that no such a modification of the magnetoresistivity in the longitudinal direction is observed and thus even down to 30 mK.

Conclusion

Eventually, the magnetoresistivity response is in a good agreement with the evolution of the TEP described in the previous sections. The large enhancement of the magnetoresistivity in the longitudinal direction can be related first to the relatively important jump of the ρ ⊥ (H) at the MMT but also due to large orbital effects. 3.3 Quantum oscillations in the thermoelectric power

Introduction

Report of quantum oscillations in the TEP measurement has been done by C. Papastaikoudis in a single crystal of the compensated metal Gallium [32]. This oscillations were believed to come from a magnetic breakdown. The author has compared is obtained frequencies with those of the magnetoresistance and dHvA measurements. A consistent correspondence has been found except that the amplitude of the TEP oscillations was larger than those of the magnetoresistance. Quantum oscillations in semimetallic system such as graphite have also been reported to occur by J.A. Woollam. Not only measurements of various physical properties like magnetic susceptibility, Hall effect, resistivity were presenting oscillations, but also measurements of TEP has revealed them. Recent experimental investigations in a macroscopic stack of graphene layers [33] has shown that under a strong enough field to push the system in the quantum limit, quantum oscillations of the Nernst effect occurs. As one can see from the figure 3.19, the oscillations are very clear in the Nernst effect, making it a quite sensitive probe.

Oscillations of the Seebeck signal

We first observe TEP oscillations in the measurements of the Seebeck's averaged signal in the study of the Lifshitz transition. Indeed, as we can observe in the figure 3.20, what could have been thought to be a simple effect of the noise in the signal is in fact real oscillations as the recurrence of this effect in temperature attested it. The inset shows a zoom in the high field region in which we can clearly see that the shape of the oscillations' curves at different temperatures remains globally the same. Even if that is not so evident to claim from observing the signal at 260 mK and 510 mK that oscillations are present at fields below H m , they are yet more apparent at 780 mK where we can distinguish undulations of the TEP.

To obtain a better picture of the quantum oscillations in the TEP: • The field has been swept continuously all along the measurements,

• A constant power to the heater was applied during the measurement of the TEP signal in order to have a thermal gradient during the field sweep,

• At the beginning and the end of each sweep, the thermoelectric voltage, when no thermal gradient was applied, was caught.

The field dependence of the TEP obtained from continuous measurements at 510 mK, 790 mK and 1 K is plotted in the figure 3.21. Several aspects can be pointed out:

• In general, the oscillations of the Seebeck effect are clearly visible. The inset shows a zoom of the high field region where we can appreciate the details of the oscillations compared to the averaged signal,

• The oscillations above H m are well and truly present in the signal of the TEP, and remains even at 1 K,

• If the oscillations were not so convincing below H m in the averaged signal, they are this time unambiguously present in the signal, and also remains even up to 1 K,

• The signal behaves differently than it could have been expected, and not only the amplitude is maintained at higher temperature but it becomes also less noisy. All this considerations have an explanation mainly through the nature of the Seebeck effect where the signal is enhanced by the temperature. Ordinarily, the quantum oscillations by TEP method are more preeminent that others. This could be understood if the theoretical expression of the TEP is considered:

S = -π 2 3 k 2 B T e ( ∂ ln(σ(ǫ)) σ(ǫ) ) ǫ F (3.1)
Rather than being related to the density of states such as the others techniques, the Seebeck depend on the derivative of the density of state evaluated at the Fermi level. This peculiar characteristic is at the origin of the remarkable sensitivity that the method has demonstrated particularly through the large amplitude of the observed quantum oscillations.

Oscillations below H m

In order to extract the frequencies of the oscillations from the signal, a Fast Fourier Transform (FFT) is performed after the subtraction of a non-oscillatory polynomial background. Figure 3.22 shows the oscillations of the TEP in the low field region below H m at 270 mK. The data are plotted as function of 1 H . An inverse field window is moved in the data to realize a field-dependent study of the Seebeck signal. The width of the window is selected in such a way that we have a sufficient frequency resolution. To do so, the window should usually at least contain a dozen of period of the TEP oscillations. This criteria is a crucial one, and sometimes just by slightly modifying the width of the FFT window, the obtained results can be drastically different. Indeed, as one can see from the figure 3.23, a small variation of the FFT width results in the situation where the amplitude of the frequency is changed, or occasionally no signal at all is detected after applying an FFT. In this case, the FFT 1 (4.5-7.5 T) is 1 T more than the FFT 2 (4.5-6.5 T), but it is enough to give a radical different Amplitude (arb. units)

1/H (1/T) 1/5.5 1/7.5 H < H m CeRu 2 Si 2
H // c j // a picture with this time a clear pronounced peak. Thus, different windows have been tested in order to follow the evolution in field of the different detected frequencies. Figure 3.24 shows the Fourier transform spectrum of the TEP at 270 mK between 5.5 T and 7.5 T. Strong peaks in the FFT spectrum are observed:

• One at 580 T that is associated at the β branch,

• One at 980 T that corresponds at the γ branch.

In the low field region, the frequencies start to be observed for FFT windows of 3.5-5.5 T. At lower fields, the signal is to noisy to distinguish the peak corresponding to the signal of the branches. H . The high field region just after the pseudo-metamagnetic transition is characterized by oscillations of amplitude much more higher than those below H m . Figure 3.26 displays the FFT spectrum of the TEP at 1.3 K between 11 T and 16 T. Four distinct peaks can be seen: • A peak at 1080 T corresponding at he γ ′ branch. Here too the frequency has increased,

Oscillations above H m

• The peak at 1260 T is associated to the δ branch. The later is only present in the high field region above H m

• A last peak at 27150 T that is attributed to the heavy branch ω . This one also exists only above H m .

The TEP has been measured for 13 different temperatures going from the lowest reached one at 200 mK to the highest one at 4.7 K. For all these temperatures, several FFT have been realized between 0 and 15 T to extract the value of the frequencies. Once in a while, a frequency could be detected at one temperature but not in an another whereas the others factors such as the width of the FFT remains unchanged. This is mainly attributable to the quality of the signal that for some temperatures is not enough strong, or simply too noisy, to be exploitable. The results that will be presented in the next section will be based on the measurements at 270 mK.

Field dependence of the frequencies

The β branch The plot starts at 3.6 T and no coherent signal has been found in our measurements below that field. Indeed, the amplitude of the noise is too large in that region to separate and identify consistently the frequency from the background. As one can observe from the figure, the evolution in field can be decomposed in four steps:

• At low fields, the β frequency remains nearly constant with a value of 558 T up to 7. [START_REF] Kittel | Introduction to solid state physics seventh[END_REF] T,

• As the field approaches H m , β starts to slightly increase in value,

• The situation then drastically changes with a sudden enhancement of the frequency around 7.8 T, and it stabilizes around 8 T. In less than 1 T, the frequency of β has increased by 10 %.

• In the high field region, the frequency becomes almost constant. The scatter of the data points in that region mostly arises from the low quality of the oscillations that make the peak of the β ′ signal in the FFT less pronounced.

This peculiar behaviour of the β frequency indicates that the FS is modified through the MMT.

The γ branch

Here the same procedure than for the β-branch has been applied. The figure 3.28 displays the evolution in field of the γ frequency. In the same way as the β branch, the γ and γ ′ oscillations originate from small hole Fermi surfaces centred at the Z point of the Brillouin zone. The Fourier spectrum starts to present a peak of the γ oscillations at a field around 5 T. The evolution in field of γ can be separated in four stages:

• A low field region (below H m ) between 5 T and 7 T, γ remains at a constant value of 980 T,

• A gapped region between 7 T and 8.5 T where no exploitable signal has been detected. This region corresponds to the MMT that occurs at 7.8 T,

• A region between 8.5 T and 11 T characterized by a monotonous increase of the frequency that goes from 980 to 1080 T, an increase of 10 %, • A last stage corresponding to a plateau where the frequency remains at the constant value of roughly 1080 T.

The field dependence of the γ frequency displays an anomaly at the MMT of the same nature than the one observed for the β branch. The main difference relies on the fact that in this case no sudden enhancement of the frequency is observed, but just a relatively slow augmentation as seen more clearly in the figure 3.29. This difference may be related to the shape of the Fermi surface of the both branches. Indeed, the shift is maybe related to the dimensionality of the Fermi surface of the γ branch that is different from the β one's as seen in the figure 3.7.

The δ branch

Above the MMT, the FS properties change drastically and are no more explainable by the Fermi surface model of the figure 3.7 (a). As we already mentioned, it is commonly accepted, if we assume the localized f electron model above the MMT, that the δ oscillations could be attributed to the electron orbit on the electron Fermi surface presented in the figure 3 • A region between 10 T and 13 T characterized by a continuous decrease of the frequency from 1400 T to 1260 T,

• And a region between 13 T and 15 T where the frequency of δ stays almost constant.

Based on the angular dependence of the δ frequency that can be expressed in 1 cos(θ) [88], the δ oscillations may be ascribed to a FS of cylindrical shape with the long axis parallel to the c-axis.

The ω branch

In the same way as the δ oscillations, the ω oscillations are associated, based on the comparison with the La-doped compound, to the large hole surface centred at the Z point. This large hole surface is usually observed in the isostructural compounds LaRu 2 Ge 2 or CeRu 2 Ge 2 where the f electrons are supposed to be localized or completely absent [63]. The angular dependence of the ω oscillations frequency, and its value are comparable to those of ω in LaRu 2 Si 2 . The figure 3.31 shows the field evolution of the ω frequency below and above H m for a field parallel to the c-axis. The ω oscillations are detectable from 10.5 T to 15 T. Two main domains are present in the graphic:

• Between 10.5 T and 13 T, the frequency slowly and slightly increases from 27100 T to 27135 T. It represents an augmentation of 0.1 %,

• From 13 T to 15 T the value of the frequency stabilizes and remains constant up to 15 T.

Discussion about the frequency changes

The augmentation of the frequency that occurs through out the MMT can be partially explained by a magnetostriction effect. Indeed, the evolution of the volume magnetostriction in field follows globally the one of the frequencies [63]. Thus, near H m , the volume starts to increase and then suddenly jump at MMT to finally continue to increase at higher field. A quantitative correspondence has been founded between the frequency changes of the branches and the magnitude of the volume magnetostriction. Nevertheless, the increase of the frequency is too large to be solely the consequence of a magnetostriction effect. It should be noticed that the observed frequency F obs is not equal to the true frequency F true when F obs is field-depending. The relation between F obs and F true can be written as:

F obs = F true -H dF true dH (3.2)

Field dependence of the effective mass

Temperature dependence of the amplitude

The next target of this study is to extract the effective masses of the different detected frequencies, and their field dependences, based on the TEP measurement. Our measurements have been performed at 13 different temperatures going from 200 mK to 4.7 K. The first step is to obtain the temperature dependence of the oscillations amplitude. To do so, we need to go through three stages of analysis:

• The first step is to perform for each one of the 13 temperatures several FFT covering the totality of the field range used during our measurements, namely from 0 T to 16 T. This represents roughly 20 different FFT windows for each temperature to have an effective field H ef f going from 0 to 16 T.

• Then for each frequency, we gather for each FFT windows the plots obtained for the different temperatures. For some of them, the FFT presents no signal of any frequency meaning that the corresponding oscillations in the TEP are not clear enough to be analysed. The left panel of the figure 3.32 shows the evolution of the peaks' amplitude corresponding to a specific branch at different temperatures for a fixed H ef f .

• Finally, for each one of the FFT windows, at which a specific H ef f is associated, we take the amplitude of the peaks for the different existing temperatures and we plot them versus the temperature. The right panel of the figure 3.32 shows the evolution in temperature of the amplitude for a given frequency.

In order to extract a value of the effective mass for the different frequencies, we have based our analysis on the Lifshitz-Kosevish formula.

The β branch

The figure 3.33 displays the field evolution of the cyclotron effective mass of the β branch. This graph has been obtained following the procedure presented in the previous section. Three domains could be distinguished form the graphic:

• In the low field region, the effective mass is almost constant and has a value between 1.2 m 0 and 1.3 m 0 ,

• Near H m , the effective mass starts to increase slightly, then abruptly increases and decreases before and after H m respectively. At its maximum, the effective mass reaches a value of 2 m 0 ,

• At higher field, the effective mass finally becomes again roughly almost constant with a value around 0.5 m 0 .

The field dependence of the β branch behaves in the same way as its frequency field dependence with a presence of an anomaly at H m surrounded by two regions where the effective mass is constant.

The γ branch

The evolution in field is represented in the figure 3.34. As on can see form the graphic, the γ branch starts to present a signal at 7.5 T whereas in the case of its frequency field dependence a signal was already present at 5 T. This difference could be understood if we consider the fact that for a given effective field H m the signal of the branch is not present in sufficiently enough temperatures to extract a consistent value of the effective mass. Generally, in the data analysis, between three to eight different temperatures are used to obtain the value of the effective mass. Once again, the plot can be decomposed in three parts:

• A low field part where the effective mass remains constant with a value around 1.2 m 0 ,

• A region around H m where the effective mass drastically increases and then decreases after having reached the highest value of 2 m 0 ,

• And finally a high field region where the effective mass stabilizes around a value of 1 m 0 .

The evolution of the γ branch in field seems to be quite similar, and this even at the level of the effective masses' values, to the one of the β branch. The mass enhancement appearing at H m can here again be related to the frequency enhancement observed at the MMT.

The δ branch

In the figure 3.35 is represented the field evolution of δ. A meaningful value of the effective mass has been extracted from 10 T to 16 T. The curve presents two different clear regimes:

• Between 10 T and 13 T, the value of the effective mass diminishes from 5.3 m 0 to 3.5 m 0 with an almost constant rate, • Above 13 T, the effective mass becomes stable at a value around 3.2 m 0 .

The behaviour of the field dependence of the curve of the δ ′ s effective mass is highly similar to the one of its frequency. Above the fact that the tendency of both curves is the same, it is certainly the overlap of the field ranges where the evolutions occur that makes this similarity interesting.

The ω branch

The field evolution is plotted in the figure 3.36. The signal of this heavy branch is observable from 10 T to 15 T. The curve could be decomposed in two regions:

• Between 10 T and 14 T where the effective mass slowly decreases going from almost 10 m 0 to a value around 8 m 0 ,

• And a region above 14 T where the mass seems to stabilize at a value comprised between 8 m 0 and 7.5 m 0 .

As it could be seen from the figure, the data are more scattered than the others. This could be partially explained by the heavy nature of the branch that make it more difficult to be detected. The curve of the ω ′ s effective mass in field behaves, compare to what has been seen up to now for the others branches, in an opposite way to the one of its frequency. Indeed, when its frequency increases in field, its effective mass decreases and vice versa.

Conclusion on the evolution of the effective masses

Near the MMT the effective mass is consequently enhanced for the four branches. This anomaly could be related to the one observed in the field dependence of the frequencies where the same phenomenon occurs nears H m . In a similar way, this mass enhancement has been found to be qualitatively consistent with the effective mass change predicted from the volume expansion. The four branches show globally the same tendency in field:

• Below H m the effective mass of β and γ is nearly constant and then begins to increase in approaching H m ,

• After the enhancement occurring at H m , the effective mass of the four branches decreases with increasing the field and then stabilizes.

The field dependence of the effective masses is quite similar to what has been measured for the specific heat. Indeed, the electronic specific heat coefficient γ is almost constant below H m at low fields, then starts to be enhanced near H m , to finally decrease above H m at high fields. Nevertheless, no such a correspondence is found when it comes to compare the value of the effective masses obtained from the specific heat measurements. Actually, the values above H m measured are much more smaller than those expected from the γ coefficient.

Conclusion: comparison with the dHvA results

Comparison in the field dependence of the frequencies

The figure 3.37 displays the evolution in field of the frequencies for the different branches obtained via two methods:

• In the left panel are grouped the results based on the TEP oscillations presented in the previous sections,

• The right panel shows the dHvA frequencies as a function of the magnetic field below and above H m with fields parallel to the easy c-axis.

The overall field dependences of the frequencies obtained by TEP measurements match quite well with the ones detected by the dHvA effect [35]. Indeed, not only the general pattern of the curves are similar, but also a quantitative correspondence exists. The values of the β, γ (below and above H m ) and δ branches measured in TEP are quasi identical to those of the dHvA effect. In the case of the heaviest branch ω, the difference in value is about 4 % which remains sufficiently low to be consistent.

To estimate the difference of the two techniques we show in the figure 3.38 in more details the comparison between the results obtained via the two methods for the β branch. The left panel presents the field frequency dependence in TEP, and the right one presents the one in the dHvA effect. The thin solid lines are obtained by fitting the experimental data, the thick ones are the calculated true frequencies and the open circles are the experimentally observed frequencies. Several remarks can be made regarding the advantages and disadvantages of the both methods:

• The first remark concerns the general aspect of the data. In the dHvA effect case, the data are scattered due to the spin dependence of the effective mass that provokes a phase change in the oscillations [89]. Even if this effect does not seems to be present in our field dependence measurements, it give us at least a gain of accuracy. Our data presented in the left panel are in good agreement with the solid thick lines of the dHvA data, denoting the true frequencies, which confirm the precision of our obtained results,

• Another aspect , more in favour of the dHvA effect, is the range on which the frequencies are detected. As one can see in the figure, the dHvA oscillations begin to be exploitable already at 2 T for the β branch. In our case, for the same branch, the oscillation starts to be clearly interesting around 4 T. Certainly, below that field the oscillations exist in our data, but the level of the noise does not allow us to distinguish precisely the signal form the background. Nevertheless, this fact is not present at high field where the oscillations' signal is increased by the field. In that case, no real difference exists between both techniques,f

• Finally, the last point is that even if we have been able to determine some heavy branches like the δ branch or more particularly the ω one's, our measurements present no signs of the heaviest branches κ or φ. This is not the case in the dHvA results since this technique is more powerful when it comes to detect heavy branches. Indeed, the effective mass in the TEP oscillations appears to be a more critical factor in the detection of oscillations than it is the case in dHvA effect.

Differences in the effective masses

The evolution in magnetic field of the effective masses extracted via two types of measurements is presented in the figure 3.39

• In the left side are shown the values obtained from the amplitude of the TEP oscillations,

• In the right side are displayed the effective masses as a function of a magnetic field below and above H m with fields parallel to the c-axis.

The overall appearance of the results acquired by this two methods looks the same. Indeed, if we compare the branches one by one, we observe some interesting similarities:

• Let us start with the heaviest branch ω detected in our measurements. The global shape is the same in both TEP and dHvA effects. The highest effective mass value reached by this branch is around 11 m 0 which is slightly more that the one obtained in our measurement that is about 10 m 0 . However, the lowest value is the same in the cases and is around 7 m 0 . The field range where this branch is detected is also the same in the cases except that we were limited to fields up to 16 T in our experiment.

• Concerning the δ branch, in the dHvA measurements its effective mass starts to be acquired at a field around 8 T whereas in our case we have succeeded to obtain a mass around 10 T. This is most likely the consequence of two effects which are the proximity to the MMT that lower the oscillations' amplitude and also the weight of the band that is almost two times heavier at 8 T than at 10 T; and as we already have mentioned it in the previous section the mass is an important restrictive factor in our capacity to detect oscillations. Despite this field limitation, the effective masses of the two methods are quantitatively in excellent agreement. Indeed, as the figure 3.39 demonstrated it, at 10 T the effective mass is around 5.5 m 0 in both cases, and the tendency at higher field is the same. Moreover, the signal of the mass persists up to the highest field that we have reached, probably due to the decrease of the mass, while in the dHvA effect case no mass is plotted beyond 13 T.

• The field range where γ is extracted from the data is comparable for both cases. The same applies for the values of the effective mass that begins around 1.5 m 0 at the lowest field, reaches 2 m 0 around H m and finally decreases up to a value around 1 m 0 . For the dHvA effect, at a field just slightly after H m , the mass begins to decrease, and then increases and to finally stabilize at higher fields. Such a behaviour is not present in our data where the mass simply decreases.

• Regarding the β branch, below H m the evolution is quite the same starting from a value around 1.5 m 0 for the dHvA effect and 1.3 m 0 for the TEP at 5 T and increasing up to a value superior at 2 m 0 near H m . However, an anomaly is present at 6.3 T in the dHvA case that founds its origin in the spin dependence of the masses. Such anomaly does not exist in the field dependence of the β branch. Above H m , the comportment of the mass is quasi identical for the two techniques.

Conclusion

Finally, the main advantage of the dHvA techniques relies on its sensitivity to heavy branches, such as for instance the κ one that we have not been able to consistently detect, and also the possibility to reach experimentally low temperatures which facilitates the acquisition of the masses. Indeed, to obtain a TEP signal, a thermal gradient is applied in the sample that will necessarily increase the lowest experimentally reachable temperature. On the other hand, the TEP oscillations persist even at temperature of the order of the Kelvin which represent from an experimental perspective in the FS investigation a clear advantage. In our experiments, the lowest reached temperature was about 200 mK. It is interesting to notice that we have been able to obtain informations on the FS of CeRu 2 Si 2 that are comparable with those acquired via the dHvA technique's, whereas in this last the usual temperature is much lower (around 30 mK). Therefore, the next target will be to reach lowest temperature in order to try to detect the heaviest branch such as the κ for example. These past few decades, the unusual properties of the heavy fermion compound CeRu 2 Si 2 that partially originates form the competition between the antiferromagnetic (AF) correlations and the Kondo fluctuations have generated interest among the scientific community. The heavy fermion region is usually formed near the AF instability due to the competition of the Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions and the Kondo effect. Although the system remains in the paramagnetic (PM) Fermi liquid down to 20 mK [21], neutron scattering experiments have revealed the presence in the material of AF correlations that emerge below 60 K [22].

The figure 4.1 shows the presence of AF hot spots in the reciprocal space of CeRu 2 Si 2 from inelastic neutron scattering data of the dynamical susceptibility [43]. Three clear peaks, at the wave vectors k 1 = (0.31, 0, 0), k 2 = (0.31, 0.31, 0) and k 3 = (0, 0, 0.35) [43], have been revealed in the dynamical susceptibility measurements in the reciprocal lattice space. The local Kondo fluctuations prevail the intersite interactions maintaining the compound at the verge of an AF state. Each one of these three wave vectors is associated to a specific magnetic correlation. By doping the compound, these AF correlations can be be stabilized and lead to the growth of a long-range magnetic order. Such a phenomenon has already been observed in cases such as the La-doped one where several percentages of Ce atoms have been substituted for La. This structural modifications of the compound induces the emergence of an AF phase with the k 1 vector [44] that reduces the influence of the Kondo effect principally through out a negative pressure effect. The figure 4.2 displays the phase diagram of Ce (1-x) La x Ru 2 Si 2 for different La-concentrations at ambient pressure. The ground state of the non-doped region, which basically corresponds to the pure compound, is characterized by a Fermi liquid (FL) state. As it can be seen from the figure, a tiny percentage of La is sufficient to induce an AF phase in CeRu 2 Si 2 and at 7.5 %. The La doping is mainly a volume effect. Another example would be the Ge doping. Again, with about 5 % of Ge substitution on the Si-site, a long range AF order appears [28]. The first striking point that could be noticed in this phase diagram is the clear decoupling between the two critical fields H c and H m . This remarkable phenomenon will be discussed and detailed later on. In a similar way as the La-doped case, a small amount of Rh in sufficient to involve an AF state that appears around 5 %. This state is the result of the Kondo effect weakening by the substitution process. The evolution of T N in concentration could decomposed in two parts:

• For a concentration of Rh between 5 % and 15 %, T N increases rapidly. At 15 %, the Néel temperature reaches a maximum of about 5.5 K,

• Above 15 % of Rh, the AF state is quickly destroyed with a T N that diminishes and vanishes around 30 %. The critical field H c , delimiting the AF transition, increases linearly with the concentration in the region where an AF phase exists. On the other side, H m starts first to decrease rapidly with the concentration, and then stabilizes at 5 T above 10 % of Rh. Between 20 % and 30 %, the pseudo-metamagnetism disappears.

In our experiment, we have selected the 8 % concentration in Rh indicated by a red circle in the figure 4.3. This choice is justified by multiple reasons:

• The first reason is that this concentration of Rh is in the range of concentration where both an AF state and the pseudo-metamagnetism coexist,

• Another reason is related to decoupling between H c and H m . Indeed, at this value of Rh, the field gap between the two characteristic fields is well pronounced as one can observe from the phase diagram 4.3.

Spin density wave

For the mixed compound Ce(Ru (1-x) Rh x ) 2 Si 2 , the development of a spin density wave (SDW) [49] is reported to appears as a consequence of the substitution of Ru with Rh. This SDW stabilizes for a specific concentration range going from 3 % to 40 % [50,51]. In this Ru-rich region, the ordered magnetic moment is modulated sinusoidally along the easy c-axis with an incommensurate wave vector k 3 [52].The ground state is mainly Fermi liquid although the SDW transition has been inferred to be the consequence of a quasi-particle band nesting [50,51]. Resistivity measurements allow to follow this peculiar phenomenon since a clear anisotropy is present in the data. The figure 4.4 displays the evolution in temperature of the resistivity with the current J parallel to the a and c-axes. Along the a-axis the resistivity behaves as it could be expected for a classical magnetic transition while along the c-axis a clear and pronounced anomaly is present. As we observe in the figure 4.4 an increase of the resistivity of about 15 % is detected below T N resulting in a sort of kink in the measurements. This peculiar behaviour of the resistivity corresponds to the physical manifestation of the opening of a gap in the c-direction that is not present when J is along the a-axis. This point will be detailed in the section 4.3.

Preliminary study of Ce(Ru

0.92 Rh 0.08 ) 2 Si 2

Specific heat measurements

The field dependence of the specific heat divided by the temperature is represented in the figure 4.5 [56]. At the lowest temperature 420 mK, the curves displays a plateau between the two fields 2.8 T and 5.8 T which are respectively corresponding to H c and H m . While increasing temperature, the pseudo-metamagnetic crossover at H m increases in fields on the crossover broadens quickly with temperature. The critical fields to suppress the AF order decreases with increasing temperature, and at H c the specific heat divided by temperature C/T is maximal. The plateau between this two critical fields H m and H c disappears when the temperature is increased. This plateau could be seen more clearly in the figure 4.6 where the field dependence of the Sommerfeld coefficient γ in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 and CeRu 2 Si 2 normalized at H m are represented. The behaviour between the pure and the doped compound are highly different in the region between the two critical fields of Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 . Indeed, while a plateau is observed in the Rh-doped case, a sharp maximum occurs in the pure compound as the field comes close to H m . The strong enhancement of the γ value at H m is due to the suppression of the AF correlations and the appearance of low energy FM fluctuations at H m . On the other hand, strong evidences has been given to support the idea that the γ plateau in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 is the consequence of the competition between field-driven ferromagnetic (FM) instability and AF instability.

Magnetisation measurements

The field dependence of the magnetization obtained from the temperature dependence down to 75 mK at different fields is displayed in the figure 4.7. In this figure, the two critical fields H c and H m are indicated by arrows. These are highlighted by the inset showing the H derivative χ(H) = ∂M/∂H. The two transitions associated to these fields are rather broad in difference to the strong increase of magnetization at H m in the pure system.

Preliminary pressure study

The figure 4.9 shows the field dependence of the A coefficient of the resistivity ρ = ρ 0 + AT 2 at three different temperatures [56]. The aim of this study was to have an overview on the pressure response of the compound. Thus, evolution of H c and H m in pressure as been roughly estimated via the A coefficient determination. At ambient pressure the plateau is here again present. By increasing the pressure, the plateau is enlarged and disappear above 1.5 kbar. Concerning H c , it also vanishes at pressure superior to 1.5 kbar meaning the ordered AF phase is no more existing.

Motivations

Our attention will be first focused on the evolution of the phase diagram with pressure, and more specially on the two critical fields. One of the peculiar properties of Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 relies on the noteworthy split between the critical fields H c and H m . This intriguing feature is commonly observed when the pure compound is mixed with Rh. The ground state is also characterized by an AF state. However, the ordered wave vector is k 3 in this case. As a matter of fact, the longitudinal nature of k 3 could be linked to the detected decoupling in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 in opposition to the transverse aspect of k 1 and k 2 in the La-doped compound where H c and H m coincide. Therefore, our first target will be to follow precisely in pressure the evolution of the two critical fields. The previous preliminary pressure study has revealed the necessity of a fine pressure tuning in order to follow precisely how H c and H m evolves and to determine at which critical pressure P c the AF state vanishes. Moreover, the preliminary pressure was not sufficient to conclude if the critical field H c terminates at P c as a critical end point as in the La-doped case [18]. The second aim of this project is to study the evolution in pressure of the nesting phenomenon that occurs when the current is applied along the c-axis. Comparable behaviours have been previously measured by resistivity in the cubic heavy fermion compound YbBiPt [54] and on the BCC Cr metal [55]. The resistivity presents a sharp increase below the Néel temperature T N only in the c-direction. The anisotropic comportment of the resistivity has been interpreted in terms of partial gaping of the Fermi surface at T N . We have investigated the field and pressure evolution of the resistivity in order to follow how the gap evolves when those two external physical parameters are changed. Eventually, an estimation of the gapped and ungapped percentage of the Fermi surface with respect to the field and the pressure will be extracted from the experimental data.

Phase diagram under pressure: Evolution of H m

and H c

Introduction

The results acquired from the resistivity measurements in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 will be presented in this section. In order to study and characterize the sample, different types of measurements have been realized:

• For each pressure, a temperature scan at constant field has been performed in the low temperature region going from 150 mK to 1 K as well as in the high one's from 1 K to 5 or 6 K. The A coefficient of the T 2 term will be extracted from this data in order to follow its evolution in field. The results will be presented in the section 4.2.3.

• The field dependence of the resistivity at different low and high temperatures has been measured up to 16 T. These scans allowed to construct the phase diagram by the determination of the two critical fields. The results are shown in the section 4.2.4.

• Finally in the section 4.2.5 we show the evolution in pressure of the phase diagram constructed based on the results of the field and temperature dependence of the resistivity. Furthermore, the evolution in pressure of the characteristic fields H c , H m and temperature T N will also be presented and discussed.

Experimental conditions

Single crystals of Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 were grown using the Czochralski method in a tetra-arc furnace. The single crystal ingots were orientated by the X-ray Laue photograph, and then were cut by using a spark cutter. A thin bar-shaped sample with the dimension of 0.5*0.5*1.5 mm (c-axis long) was prepared. The sample was put in a hybrid (NiCrAl-CuBe)-type piston cylinder pressure cell applying the pressure up to 2 kbar. The resistivity was measured by a 4-probe AC method with the frequency f ∼ 17 Hz. The current was applied along the c-axis with a value I ∼ 200 µA. The temperature was decreased down to to 100 mK in a home-made dilution refrigerator. A high field up to 16 T was applied parallel to the easy c-axis. The Néel temperature T N , easily identified here through the marked kink that occurs in the resistivity, decreases with H as expected for a classical AF system. Its value is extracted from the derivative dρ/dT of the resistivity with the temperature. The ordered phase is no longer present at 3 T and the compound turns to a paramagnetic (PM) state. The curves present different behaviours depending on the field that we consider. Basically, their comportments can be regrouped in three categories :

• At 0 T and 1 T, the resistivity initially starts to decrease down to T N . For both cases, T N is almost the same and is equal to 4.4 K and 4.3 K at 0 T and 1 T, respectively. At this ordering temperature, the curves suddenly increase and the resistivity is rapidly enhanced by about 15 %. After having reached this maximum, both curves begin to slightly decrease on cooling the temperature to finally stabilize below 1 K.

• At 2 T, the curve initially presents the same trend than the two previous one except that T N has been decreased by about 20 % compare to its value at 1 T. Thus, it seems that in approaching H c , the Néel temperature rapidly decreases. Below T N , the resistivity suddenly increases here again by about 15%, but however, at low temperatures, no sharp diminution of the resistivity is observed and the curve tends to be stable.

• Finally, at higher fields, the behaviour is radically different. At 3 T, no kink of the resistivity is present meaning that we already are above H c and therefore that the systems now in its PM phase. From 3 T to 5 T, at the lowest temperature, the curves become closer and closer when the field is increased. This is justified by the proximity to H m = 5.8 T. At 6 T, the system is now in the PPM phase and the curve of the resistivity is clearly below the one at 5 T. Then from 6 T to 9 T, the curves are the same with a continuous diminution of the resistivity with respect to the temperature.

Basically, the field acts on the gap by first broadening the kink present in the resistivity from 0 T to 2 T, then by rapidly weeping it out when it is close to H c = 2.8 T. Above this critical field, the resistivity behaves as expected for a PM compound.

Temperature dependence of the resistivity under pressure

The resistivity has been measured in a pressure range going from 0 kbar to 1.9 kbar. A fine tuning of the pressure was necessary to follow the evolution of Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 since the pressure response of the compound is quite sensitive. Therefore, we have measured at 0.5, 1, 1.6 and 1.9 kbar the temperature evolution from 150 mK to 5 K at different constant fields of Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 .

The figure 4.12 displays the temperature dependence of the resistivity at 0.5 kbar (left) and 1 kbar (right). Let us compare first the results at this two pressures with those obtained at ambient pressure and then between themselves:

• The temperature dependence of ρ at fixed field at 0.5 kbar is very similar to that at 0 kbar. T N is slightly lower. The only difference that could be highlighted relies in the quality of the signal that is more noisy at 0.5 kbar. This could be seen as a direct consequence of the pressure effect on the signal's quality. • The situation at 1 kbar is drastically different than before as it could be seen in the right panel of the figure 4.12. The first point to notice is the shape of the kink of the resistivity at low temperatures that after having reached its maximum value do not decreases any more and becomes stable. The evolution in temperature of the resistivity at this pressure is quasi identical at the one at 2 T at 0 kbar. Another fact to notice is the resistivity's augmentation that is here more around 12 % at H = 0 T, which less than the 15 % previously observed. The temperature evolution from 0 T to 3 T, that we have more detailed at this pressure, reveal that the kink is here more rapidly suppressed and has almost been vanished at 2 T. This could be opposed at the ambient pressure situation where the kink is still present and well pronounced.

The figure 4.13 shows the temperature dependence of the resistivity at 1.6 kbar (left) and 1.9 kbar (right). At his two pressures the temperature response of the resistivity is very different of what could have been seen up to now:

• At P = 1.6 kbar, no increase of ρ(T ) appears at T N . To follow the evolution of the Néel temperature in field, the derivative of the resistivity is used. As it could be observed from the figure, at 1 T the system is already in the PM state. This picture is quite far from the situation at 1 kbar since at 0 T and 1.6 kbar the system is already quite close to be in the PM region, and thus we are near to the critical pressure P c . Once again, at high field, despite the discrepancies stated just before, the curves behave similarly to what has been observed in the previous pressures and this even for the value of the resistivity.

• The right panel of the figure 4.13 shows the results of the highest measured pressure. The striking point is that at 0 T the resistivity decreases almost continuously with respect to the temperature and globally adopts the same behaviour that could be expected in PM region and this even at the lowest temperature reached. Therefore, this observation tends to indicate that the system has reached the critical pressure, in other words that the AF phase has been wept out. The second point to remark relies on the augmentation of resistivity when the field is increased from 0 T up to 9 T. At this pressure H m is around 10 T, and thus when the system come close to that critical field, the magnetic fluctuations, such as the AF or FM correlations, become more predominant which will induce an augmentation of the resistivity around that region. Far from that region, at 16 T the resistivity has decreased again, and the same remark that has been done in the previous pressures applies here to, that is to say that the resistivity at high pressure is similar in shape and value at the one at ambient pressure.

Conclusion

The impact of a hydrostatic pressure in the temperature dependence of the resistivity with a current applied along the c-axis and without field at different pressures is shown in the figure 4.14. In the region above T N , the slope of the resistivity remains almost constant for the different measured pressures. As can be seen from the figure, just by slightly pressurizing the sample, T N promptly starts to diminish drastically from 0.5 kbar to completely vanishes at the critical pressure P c = 1.9 kbar. Therefore, the AF order is rapidly suppressed and the system becomes paramagnetic at P c . On the other hand, the kink of the resistivity that was determined at ambient pressure still persists and broaden at 1 kbar before entirely disappear at P c . This high sensitivity of the gap towards the pressurization as an element that could be consistently related to the Fermi surface instability. The observed diminution of the resistivity after the curve has reached its maximum at 0 kbar and 0.5 kbar could be understood has a diminution of the weight of the gap in its contribution in the total resistivity. The figure 4.15 displays the evolution of the Néel temperature T N with respect to the pressure. The dashed lines are guide for the eyes. T N first remains almost constant when the pressure is increased at 0.5 kbar. Then, when the pressure is increased at 1 kbar, T N diminishes rapidly by about 20 %. Finally, T N drops radically with the pressure by about 40 % between 1 kbar and 1.6 kbar, and by about 50 % between 1.6 kbar and P c = 1.9 kbar.

Field dependence of the resistivity

Field dependence of the resistivity at ambient pressure The resistivity of Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 has been measured in the field range going from 0 T to 9 T, and in the temperature range from 200 mK and up to 4 K. The field dependence of the resistivity at different temperatures at ambient pressure is represented in the figure 4. 16. The values of the critical fields are obtained from the derivative of the resistivity with the field dρ/dH. Globally, for all the measured temperatures, the different curves can be decomposed in three distinctive field regions:

• In the low fields region from 0 T to H c (T ), the resistivity drops drastically. As an example of this important diminution, the 200 mK curve can be considered since in less than 2 T its resistivity experience a diminution of about 70 %. The augmentation of the temperature induces a reduction of this diminution. Thus, at 4 K the resistivity is reduced of about 30 % through the AF transition. However, this jump remains non negligible and is clearly observed in the graphic independently on the temperature,

• The second field region is comprised between the two critical fields H c and H m . This region is characterized by the formation of a plateau of the resistivity at low temperature,

• Finally, the last region corresponds to the one above H m defined by a continuous and slow decrease of the resistivity. The curves of the different temperatures merge at higher field.

From the figure 4.16, two critical fields corresponding at two transitions can be extracted. The first one occurs at the extrapolated value H c (0) = 2.8 T and refers to the AF transition while the second one, H m (0) = 5.8 T relates to the pseudo-metamagnetic crossover. The AF transition is marked by a well defined drop of the resistivity whereas H m is enlarged with increasing temperature. H c slightly decreases with the temperature at the beginning and then decreases rapidly with increasing temperature. On the other hand, the pseudo-metamagnetic transition displays solely a weak temperature dependence but remains perceptible even at 4 K, despite the fact that it is markedly broadened above 1 K.

Field dependence of the resistivity on pressure

The left panel of the figure 4.17 shows the field dependence of the resistivity at 0.5 kbar for different constant temperatures. As in the case for the temperature dependence, no significant difference exists between this pressure and the 0 kbar case. H c and H m deduced from the field dependence coincide with the values from the temperature dependence. The plateau of the resistivity is also present here between H c and H m , and follows the same evolution in temperature as the 0 kbar situation. The picture is radically different if the results obtained at 1 kbar are regarded. The evolution in field at different temperatures of the resistivity at 1 kbar is displayed in the right panel of the figure 4.17. Several remarks can here be done:

• The first striking point relies on the presence of an anomaly referred as H * around 5.5 T that is present in the two low temperatures curves. This anomaly was not present at the previous pressure, and was not seen too in the temperature-dependent measurements at the same pressure,

• The plateau of the resistivity that was observed between the critical fields at ambient pressure is absent here, and this observation could be done even at the lowest temperature 200 mK. Instead of that, the resistivity presents a sharp increase just above H c , and then reaches a maximum at H m before decreasing in the same way as the 0 kbar and 0.5 kbar cases,

• The AF transition vanishes around 3.5 K as it could be seen in the figure 4.17 where H c disappears in this temperature region.

• The last remark concern the quality of the signal that is more noisy than before due to the pressurization.

The evolution in field of the resistivity at 1.6 kbar and 1.9 kbar for different temperatures is presented in the figure 4.18. The emerging picture from this two graphics is drastically different of what could have been seen up to know:

• Let us first consider the graphic from the left panel of the figure 4.18 treating the 1.6 kbar situation. It could be noticed that the anomaly that has emerged at 1 kbar at H * is here present again. Moreover, this anomaly appears to be well more defined now that the pressure has been increased. However, a slight augmentation of the temperature seems to have a radical effect on the anomaly since it already disappears at 1 K, in the same way as it was the case at 1 kbar. Concerning the AF transition, the critical field H c has almost vanished at this pressure. However, below 2 K H c is still perceptible in the curves of resistivity. Furthermore, below H c , no sudden jump of the curves is observed as it was detected previously. Instead of that, the resistivity slightly increases of about 15 % for temperatures below 1 K. The situation regarding the MMT is different since the pronounced peak that was present at H m at low temperatures is progressively replaced by a broad maxima more characteristic of a the crossover nature of the transition. • Finally, the results obtained at 1.9 kbar are shown in the right panel of the figure 4.18.

Once again, the anomaly present at H * is sill there, as clearly pronounced as it was the case at 1.6 kbar. However, whereas in the previous pressures the anomaly was vanishing rapidly with the temperature, here it is still possible to detect it at temperature around 2 K. The magnetoresistance is monotonously increasing up to H m . This shows that the AF order is suppressed at 1.9 kbar. The widening of the MMT observed at H m with the temperature appears to be delayed by the pressure effect. Indeed, the broadening observed at 5 K at 1.6 kbar is comparable at the one reached at 1.9 kbar but at 6 K.

Eventually, the pressure effect in the field dependence of the resistivity pointed out once more the high sensitivity of the system toward the pressure. Indeed, as it has been remarked previously, a simple increase of the pressure of about 0.3 kbar, which represents a low pressurization, is enough to provoke important modifications of the signal with the emergence or the suppression of anomalies.

Evolution of the A-coefficient

The A coefficient of the T 2 term of the low temperature dependence of the resistivity has been extracted from the analysis of the results obtained at the different measured pressures in the temperature range from 200 mK to around 1 K. The figure 4. 19 (a) shows the field dependence of the A coefficient up to 1.9 kbar for H c-axis in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 . It could be noticed that for the pressure range from 0 kbar to 1 kbar, the figure exhibits a missing part for the fields below H c . This fact should directly be related to the nesting phenomenon that occurs in this compound. Indeed, as it has been detailed in the sections dealing with the temperature and the field dependence of the resistivity, the observed gap is still predominant in this pressure range prevailing us, with the sudden enhancement of the resistivity below T N , from extracting a meaningful value of the A coefficient from the low temperatures resistivity measurements. As it can be seen in the figure 4.19 (a), A(H) is maximal at H c and above H m the A coefficient is strongly suppressed. The field dependence of A is very similar to that observed in the transversal configuration. With increasing pressure, A(H c ) and A(H m ) is decreasing. As mentioned, it is not possible to plot A(H=0) as function of the pressure. However, from the data the current is parallel to the a-axis, it gets obvious that A(H=0) is maximal near P c . The anomaly occurring at H * is clearly present in the graphic. Moreover, while it was not possible to detect it at 0 kbar and 0.5 kbar in the resistivity data, the curves of the A coefficient at this respective pressures display clearly the anomaly. Figure 4.19 (b) represents the normalized value of the A coefficient at H m . A noticeable effect of the augmentation of the pressure is linked to the shift of H c to the lower field that could be understood as the progressive destruction of the AF by increasing pressure until it completely vanished at P c = 1.9 kbar leading to a PM order. On the other hand, the pseudo-metamagnetic crossover H m is shifted to the higher field with increasing the pressure. The decrease of A-coefficient above H m can be well scaled for different pressures, indicating that the spin fluctuations are suppressed with increasing field, as observed in the pure system CeRu 2 Si 2 under pressure

The figure 4.20 displays the evolution in field of the residual resistivity ρ 0 at different pressures extracted from the low temperature dependence data. Basically the same comments done for the A coefficient applies here too. The critical fields H c and H m present in the curves coincide well with the values reported before. However, contrary to the field dependence of the A coefficient, no evident anomaly is detected at H * . As seen in this figure, three main domains exists delimited by two transitions. H c slightly decreases with the temperature and then suddenly drops above 3 K. The phase boundary steeply decreases and becomes zero at T N = 4.2 K. On the other hand, H m remains roughly unchanged below 1 K and then begins to increase up to 4 K where finally it appears to saturate for the higher temperatures. The PM and AF character has been confirmed by previous magnetization and neutron scattering experiments [56].

Evolution of the phase diagram under pressure

The figure 4.22 shows the evolution of the (H, T ) phase diagram at the four different pressures applied on the sample. The evolution can be decomposed in four steps, with each step corresponding to a pressure: different. Indeed, the decoupling between H c and H m is more important, and the PM region covers a bigger area in the phase diagram. In parallel, the AF phase area has been reduced. This modification is seen in the evolution of H c that remains almost invariant up to 2 K, and then drops drastically before vanishing in less that 1 K. On the other side, H m has the same behaviour that previously at this exception that its value has been increased.

• In the
• On the figure 4.22 (c) is shown the 1.6 kbar case where the situation rapidly evolves compare to the pressure before. Indeed, the AF state only represents a small pocket in the phase diagram. Moreover, the evolution of H c is more radical since it stays almost constant with increasing and then diminishes to disappear in less than 0.5 kbar. This observation highlights the proximity of the system to the critical pressure P c . The situation is also different for the evolution of H m with an augmentation of its value that already starts at low temperatures.

• At 1.9 kbar 4.22 (d) no AF phase has been observed while the temperature dependence of H m is almost unchanged compared to AF regime at lower pressures.

Conclusion

When the current is longitudinal to the field, a sudden enhancement of the resistivity occurs. This phenomenon is anisotropic since such augmentation is not detected in the transversal configuration. This fact could be related to the nature of the wave vector responsible of the ordering. In the Rh-doped case, the k 3 longitudinal wave vector is associated to the AF phase. Therefore, when a current is applied along this k 3 wave vector, the resistivity will presents a kink in its curve below T N that materializes the opening of gap in that direction. Such a phenomenon is not observed when the current is applied along the a-axis since in that case the current is transversal to the k 3 wave vector. A simple way to understand the pressure response of the system will be to consider the figure 4.24 displaying the evolution of both the Kondo temperature T K and the Néel temperature T N with respect to the pressure. Within the Doniach's framework (see chapter 1.2.4), when no hydrostatic pressure is applied, the system is chemically pressurized due to the presence of the Rh-atoms in the structure. As seen before, a direct consequence of that is the emergence of an AF order. Therefore, the pressure induced in the system by the presence of the Rh-atoms was sufficiently low to predominantly favour the RKKY interactions. Nevertheless, if the pressure in increased mechanically in the system (meaning in the Doniach's diagram that the parameter J increases), the Kondo compensation of the local moments should dominate the RKKY interactions and then suppress the development of long-rang magnetic order. This affirmation, which follows simply by considering that T N depends geometrically on J whereas T K evolves exponentially, is corroborated by the figure 4.24. Indeed, even if T N and T K are quite stable up to 0.5 kbar, it could be seen that if the pressure is increased further, T K increases rapidly with the pressure while one the other hand T N drops.

Fermi surface nesting under magnetic field and pressure 4.3.1 Introduction

As it has been reported in the previous sections, the resistivity of Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 presents a sudden enhancement that occurs below T N when a current is applied along c-axis that is collinear to k 3 AF wave vector. The target was to investigate the evolution of this partial gaping of the Fermi surface both in field and in pressure. Therefore, the idea was to obtain the evolution in percentage of the gapped and ungapped part of the FS. In order to achieve that objective, further analysis on the field and pressure dependence of the resistivity can be performed in order to investigate more specifically the evolution of the gap. Those kinds of studies are similar to those performed by Movshovich et al. on YbBiPt [54], or by McWhan and Rice on Chromium [55]. By considering that the total Fermi surface could be divided into two separate parts, an ungapped and gapped region, one can assumes that the total conductivity corresponds to the sum of this two distinct parts. Therefore the conductivity is written as σ = σ 1 + σ 2 where

σ p -σ g σ p = ρ g -ρ p ρ g = σ 2p σ p (1 - σ 2g σ 2p ) (4.1)
The indices g and p correspond to the case where respectively the region 2 is gapped or made paramagnetic by field or pressure.

Field effect on the pressure dependence of the gap

Fermi surface gapped at ambient pressure We first consider the situation where the system is at ambient pressure, and the PM state is established by applying a field of 3 T, superior to the critical field H c . We define the following ratio:

ρ(B) -ρ(B = 3T ) ρ(B) (4.2)
The relative resistivity ratio at different fields at ambient pressure is represented in the figure 4.25. The curves can be decomposed in different parts:

• A temperature independent region for temperature above T N ascribed to a sort of paramagnetic background effect. This part is neglected by subtracting it, • The last part of the three domains corresponds to the low temperature region, roughly estimated between 0 K and 3 K, where a visible increase of the resistivity is observed. Indeed, as it could be seen in the figure 4.25, just after the part where the curves coincide well with the fit, a noteworthy augmentation of the resistivity occurs. This enhancement could be understood in the light of the figure 4.11. Indeed, just after the marked kink in the curves at low fields, a clear decrease of the resistivity is observed in a temperature range that matches with the one in the figure 4.25 where the curves are no more fitted. In that region, the contribution of the gap to the total resistivity is no more predominant involving a diminution of it.

The equation used for the fit to a Bardeen-Cooper-Schrieffer (BCS) function for the temperature dependence of the gap [59,60]. The term σ 2p σp in the equation (1) refers to the percentage of the Fermi surface that is gapped. This term could be extrapolated from the figure by taking the limit when T → 0 K of the fit. In that case, the other term σ 2g σ 2p referring to the contribution of the gapped magnetic part is null [61,62]. Therefore, we obtain if we consider the relative resistivity ratio at 0 T σ 2p σp = 0.43, meaning that about 43 % of the Fermi surface is gapped at ambient pressure when no field is applied.

Evolution of the gap under pressure

The figure 4.26 displays the evolution under pressure of the relative resistivity ratio. Basically, the same procedure as the ambient pressure situation has been applied by each time considering the new field at which the PM is established in the determination of the ratio. Each one of the three graphics, related to three different pressures, shows interesting • Once again, at 0.5 kbar, the situation is not that much different than at ambient pressure as it can be seen in the figure 4.26 (a). Indeed, it is difficult to observe graphically a difference, and it is only when the term σ 2p σp , representing the percentage of the Fermi surface, is extracted by taking the limit when T → 0 K of the fit, that we detect a small difference.

• The situation is a bit different when the figure 4.26 (b) showing the 1 kbar case. The first fact that could be pointed out is that this time the field of reference is 2 T. Indeed, as it has been seen previously, at that pressure H c = 1.9 T. Thus, the closest field of that critical field is no more 3 T but 2 T. Another striking point is that the curves of ratio fit in the full temperature range the curve of the ratio. No augmentation of the ratio is observed in the low temperature region as it was the case at lower pressure. This could be explained if the evolution in pressure of the Néel temperature is considered (figure 4.15). Indeed, at this pressure, T N is around 3.4 K. If one consider the fact that the gap opens below T N meaning that this time the gap will remains predominant at low temperatures, one can understand that no increase of the ratio is observed in this temperature range.

• Finally, at 1.6 kbar the situation has evolved rapidly with the pressure as the figure 4.26 (c) shows it. The first noticeable fact is the field at which the PM state is established. Indeed, in that case 1 T is the field of reference in the determination of the ratio since at this pressure H c is around 0.8 T. Moreover, in a comparable way and for the same reasons at the previous pressure, no augmentation of the ratio, beyond the fit, is detected at low temperatures. The main difference relies here on the extrapolated value of the term σ 2p σp that has been drastically reduced whereas this was not the case in the passage from 0.5 kbar to 1 kbar.

Conclusion

The figure 4.27 presents the results acquired based on the treatment by using the equation 4.2. Two distinctive regions are present on this graphic:

• At low pressure, typically below 1 kbar, the percentage of the Fermi surface ∆ slightly evolves by diminishing with increasing pressure but remains globally around 40 %,

• Beyond 1 kbar, ∆ drops rapidly with respect to the pressure, and vanishes at 1.9 kbar. At P c , the gap has been completely suppressed, and the system is totally ungapped.

Eventually, the pressure dependence the gapped and ungapped part of the Fermi surface acts has the one of the Néel temperature. Indeed, if the evolution in pressure of T N , obtained by the temperature dependence of the resistivity, displayed in the figure 4.15 is compared at the one of ∆, it could be noticed that they both globally shows the same trend in pressure. This appears to be normal since the formation of the gap is intrinsically linked to the AF order which evolution is delimited by T N .

General conclusion: comparison with the La-doped case

Eventually, the pressure response of the system regarding the evolution of the transitions almost acts as if applying the pressure leads to come back to a situation near to the pure situation case. Indeed, the doping process could be perceived as a chemical pressure that is involved in the emergence of an AF order. Therefore, when the critical pressure P c = 1.9 kbar is achieved, the system is no more magnetically ordered, and the MMT critical field H m is quite near in value of the one of the pure compound. To see more clearly this effect, let us consider the figure 4.28. Thus, the system starts in pure region, then is shifted to the left in the AF region due to the Rh-doping, and finally by applying a pressure on the compound, the system seems to be pushed back in the pure region of the phase diagram. Thus, the application of a hydrostatic pressure on the compound will somehow cancel the effect of the doping, even if the system keeps the physical properties induced by the doping. The figure 4.29 shows the schematic (H, T ) phase diagram at different pressures for two different cases. In the pure or in the La-doped system, H m is merged into H c at low temperature and for low pressures. At higher temperature, both H c and H m are observed. When the temperature is increased, T N decreases while H c remains almost constant. This is explained by the strong Ising properties of CeRu 2 Si 2 as it has been seen in the previous chapter. Further increase of the pressure will result in the progressive suppression of the AF state marked by the continuous decrease of T N in the phase digram. However, the first order transition changes into H m which is the pseudo-metamagnetic crossover. The change from H c to H m is so-called quantum critical end point (QCEP) since the first order transition terminates there at T = 0 K. The situation is radically different in the Rh-doped case with H c and H m that are already decoupled in the initial state. Thus, both H c and T N decrease and the AF phase is shrunk under pressure. Finally, when the critical pressure is reached, only H m remains. In this case, there is no QCEP. The quite unusual physical properties near the quantum critical point (QCP) of the heavy fermion YbRh 2 Si 2 compound have attracted a lot interest. The Kondo temperature of this system is T K = 25 K and the Sommerfeld coefficient, obtained by extrapolation of the specific heat curve C/T → 0, is γ = 1.6 J/molK 2 . At ambient pressure, the system is already close to a QCOand strong non Fermi liquid behaviour has been observed. The properties at this QCP cannot b explained by the standard Spin-fluctuation theory. One of the main reasons for such an interest in that compound relies on the experimental possibility to tune it through a field-induced QCP. Following this theoretical concept, the observed step in in the Hall effect at H c has been interpreted as a signature of a Fermi surface reconstruction wich is driven by the Kondo breakdown.

This ternary compound has a ThCr 2 Si 2 -type crystal structure as displayed in the figure 5.1. The figure 5.2 shows the magnetic susceptibility along the a and c-axes in YbRh 2 Si 2 down to 20 mK. As shown in this graphic, a strong magnetocrystalline anisotropy is present between the two directions. The (ab) basal plane corresponds to the easy magnetization direction whereas the c-axis is the hard magnetization direction. The susceptibility along a is 20 times larger at 2K than along the c-axis. YbRh 2 Si 2 orders antiferromagnetically a Néel temperature T N = 0.07 K. A small magnetic field H c is sufficient to destroy the magnetic order continuously with increasing the field through H c . The figure 5.3 shows the evolution in temperature of the electronic specific heat divided by the temperature C el /T in the basal plane for small fields below 0.06 T [67]. When H = 0 T, the curve exhibit a sharp peak at T N traducing the AF order. A slight increase of the magnetic field results in the progressive destruction of the magnetic order characterized by a diminution of the peak's height until it vanishes at H c . At H c , the specific heat increases down to the lowest temperature. The figure 5.4 presents an field-temperature ambient pressure phase digram of YbRh 2 Si 2 for an in-plane magnetic field [73]. From this phase diagram, different regions and transitions can be highlighted:

• The weak AF order characterize by a T N of about 70 mK that can be driven to zero by a small critical field H c of about 60 mT. For a field applied along the c-axis, the requested field to wipe out the AF phase is about 0.6 T, which is 10 times larger than in the basal plane. This important difference could be related the strong anisotropy that presents this system,

• Above H c a field-induced QCP is reported to occur. In that region, strong deviations from the Landau Fermi liquid (FL) behaviour in thermodynamic and transport properties referred as a non Fermi-liquid (NFL) behaviour have been observed to appear. As an example of such discrepancies, the temperature dependence of C el /T is presented in the figure 5.5 [67]. For fields above H c , after having reached a maxima, the C el /T curves decrease by adopting a behaviour in C ( el)/T ∝ -ln(T ) characteristic of a NFL behaviour, instead of having a C/T = Const. FL comportment. Other quantities such as the resistivity shows a deviation form the FL behaviour with ρ ∝ T instead of ρ ∝ T 2 trend near the QCP [68],

• At field superior to H c , and in the low temperatures regime, the FL behaviour is recovered. This could be particularly seen in the figure 5.5 where a saturation of C el /T is present when the temperature is decreased and for fields above H c , involving a Sommerfeld coefficient typical of the FL state. The electronic configuration of YbRh 2 Si 2 in the trivalent configuration of Yb presents one electron missing in the 4f shell. The hole of this 4f shell in YbRh 2 Si 2 has been the origin of several intriguing and interesting phenomena that have generated a lot of interest related to that discussion in a similar way as the electron in the 4f shell of the Ce based heavy fermion compound. Therefore, the Kondo effect gives rise near the Fermi energy to the formation of quasiparticles made by the combination of the 4f hole with the conduction electrons that possess highly renormalized properties such as the enhancement of their effective masses. This 4f 13 configuration has a magnetic moment that gives usually rise to long range magnetic order. However, in some compounds the Yb-ion stabilizes in a 4f 14 configuration that is Y b 2+ . Small deviations from the 4f 13 configuration give rise to the heavy fermion behaviour. YbRh 2 Si 2 has mainly attracted the attention of the scientific community due to the quantum critical behaviour at low field. The drop of Hall effect in this field region has been interpreted as a signature of the reconstruction of the Fermi surface [64]. In addition, the question of the violation of the Wiedemann-Franz law at the critical field is strongly debated [92]. In the high magnetic field region, and for a direction of the field in the easy basal plane, the physical quantities depending on the density of states such as the Sommerfeld coefficient of the electronic specific heat γ or the A coefficient of the residual resistivity demonstrate a unusual behaviour around 10 T. This could be clearly seen in the figure 5.7 representing the field dependence of the resistivity's A coefficient and the γ specific heat term. Indeed, both display a pronounced decrease just above 10 T that traduces a drastic reduction of the heavy fermion character of the quasiparticles.

Therefore, this study will be focused in the high field region around 10 T. The quantum oscillations, and more specifically the dHvA effect will be used as a probe in order to investigate this phase diagram domain. The idea will be to find out elements that can shed some light on the nature of the phenomena occurring near H 0 . Indeed, this transition is clearly not associated with a metamagnetic transition as the figure 5.8 shows it. With increasing the field, the magnetization first starts to increases rapidly. On approaching H 0 , the curve progressively bends and beyond H 0 the magnetization saturates at a value around 1.3 µ B /Yb. This comportment persists even at extremely high field whereas in the case of the metamagnetism it is generally expected that, due to the polarization of the electrons, the magnetization does not saturates. In fact, this transition is associated to a Lifshitz-type transition that is observable through the magnetoresistivity and thermopower measurements as shown in the figure 5.9 displaying the thermoelectric power S and magnetoresistance ρ(H) of YbRh 2 Si 2 for H [110] and J [110] in the basal plane as a function of magnetic field [42]. ρ(H) shows several anomalies crossing through H 0 and the fields of these anomalies (extrema and inflection points) in ρ(H) coincide perfectly with the sharp anomalies in the thermopower. Moreover, the TEP is an efficient probe to detect topological transitions of the Fermi surface. These multiple anomalies observed in the TEP signal have been interpreted as topological Lifshitz transition [42]. It is this specific Lifshitz transition occurring at H 0 ≃ 10 T that has attracted several dHvA investigations principally performed within the easy basal-plane [71]. The main results of these studies were confirming the Lifshitz nature of the transition at H 0 [91].

5.2 Angular dependence of YbRh 2 Si 2

Introduction

In this section, the obtained results from the de Hass van Alphen (dHvA) measurements performed across the anomaly at 10 T are presented. The angular dependence of the YbRh 2 Si 2 compound has been realized through three sets of experiment:

• In the section 5.2.3 angular dependence has been done from the easy a-axis [100] to the hard c-axis [001],

• The section 5.2.4 presents the results of the angular dependence from the a-axis [100] to the (ab) plane [110],

• Finally, the section 5.2.5 shows the results of the angular dependence from the basal plane [110] to the c-axis [001].

Experimental conditions

The dHvA oscillations measurements were realized in YbRh 2 Si 2 across the field region around H 0 in order to probe the evolution of the Fermi surface. Using the classical field modulation technique, the field was swept from 6 T up to 15 T. The samples, which are platelet-type grown from indium flux, have been mounted on a dilution refrigerator with a base temperature of about 20 mK. The magnetic field was rotated across a range of 90 • for all the explored directions.

Angular dependence from [100] to [001] direction

The angular dependence from the a-axis to the c-axis has been measured in two steps. The first step was to measure the dHvA signal from [100] to [001]. In that case, the signal remains up to 54 • . Then, the angular dependence has been completed by measuring the signal in the other direction, that is to say from [001] to [100]. In that configuration, the signal is rapidly lost when the field is aligned 9 • from the [001]. The figure 5.10 (a) shows the dHvA oscillations obtained when the field is applied along the a-axis at a temperature of ∼ 25 mK. The amplitude of the oscillations increases quickly with the field as it could be expected from a dHvA signal. The arrow in the graphic denotes the position of H 0 . The evolution of the oscillations' amplitude with the temperature is shown in the figure 5.12. Here also, the temperature has a drastic consequences in the amplitude, and just a slight augmentation of it is sufficient to reduce strongly the oscillations. When the field is exactly along the a-axis, five peaks are clearly seen in the Fast Fourier spectrum. The same procedure as before has been applied at different angles and the results are plotted Amplitude (arb. units) • A high frequency region with also three different signals. A first one with a frequency of 5640 T (f 4) that vanishes rapidly at 13.5 • , a second one at 6050 T (f 5) that disappear at 9 • , and a frequency of 6490 T (f 6) that persists up to 54 • and that is completed at 90 • by the rotation from the c-axis to the a-axis.

f 1 = 2945T f 2 = 3200T f 4 = 5640T f 5 = 6050T f 6 = 6490T m * f 1 = 14.4m 0 m * f 2 = 8.6m 0 m * f 4 =
Before pursuing further in the analysis of the results, it is important in the identification of the frequencies to verify if one of them is not the harmonic of another one. This checking procedure could be done in two steps. Basically, the first one is to check directly by comparing the value of the frequencies if their not related to each other by a multiplying factor. To do so, in the figure 5.11, the two frequencies f 1 and f 2 have been multiplied by 2 and reported in the graphic as a blue solid line and a red solid line. Regarding the blue line, even if it is close to the f 5 frequency there is not a clear matching between them. Thus, it can not be concluded from that that f 5 is the second harmonic of f 1. On the other, if the red line is considered, it can be seen that it perfectly matches with the f 6 frequency. Moreover, not only this correspondence is found in the [100] direction, but also in the all angular range where f 2 is detected. Therefore, this observation gives a strong support to the affirmation that f 6 is the second harmonic of f 2.

The second step of this checking procedure is based the effective mass of the quasiparticles. This renormalized mass has been extracted by using the Lifshitz-Kosevish formula on the temperature dependence of the amplitude of the oscillations. The obtained masses are reported in the table 5.1 on which also appears the results obtained by Sutton et al. in their study [71]. Let us applied the same procedure as for the frequency by firstly considering f 1. The reported value of the masses in the table indicate clearly that f 5 in is not the second harmonic of f 1 since its mass is inferior at the f 1 one's. Moreover, even if the f 4 frequency would have been though to be the second harmonic of f 1, despite the fact that its value is furthest from f 1 than f 6 is, its effective mass is here again inferior to the f 1 one's, excluding again this possibility. Concerning the f 2 frequency, the effective mass of f 6 is exactly the double of the f 2 one's. This relation between the effective mass of this two branches, associated with the correspondence of their frequencies in value and in angular dependence confirm that f 6 is the second harmonic of f 2.

The figure 5.13 displays the evolution of the dHvA signal in field at different angles and the figure 5.14 the angular dependence of H 0 . From both figures, two important remarks can be done:

• The dHvA oscillations presented in the figure 5.13 are well defined up to 9 • and then suddenly drop. This comportment could be understood in the light of the figure 5.14. Indeed, the position of H 0 remains stable at low angles and then starts to increase 10 • . In view of the fact that the oscillations are principally observable above H 0 , when the sample is rotated, the oscillations are shifted to higher fields in the same way as H 0 does, making the dHvA signal not detectable in the field range experimentally reachable in our measurements,

• When the field is aligned along the c-axis, if one follows the black solid line of the figure 5.14, one assumes that H 0 at that position (90 • ) is around 25-30 T. This means that the detected Fermi surface near the c-axis reported in the figure 5.11 corresponds to a state where the Fermi surface has not yet been affected by the transition at H 0 since its value is far way from the FFT windows (10-15 T) used in the analysis.

Angular dependence in the basal plane from [100] to [110] direction

In this section is presented the first part of the results obtained from the angular dependence from the a-axis to the c-axis through the basal plane. The signal remains after a rotation of about 58 • from the a-axis and is lost above. The dHvA oscillations after a rotation of 36 • from the [100] are displayed in the figure 5.15 (a) and the corresponding FFT spectrum in the (b). The signal of a high frequency is clearly detected around 14 kT and its nature and origin will be discussed later in this section. The angular dependence from [100] to [110] is plotted in the figure 5.16. We observed excellent agreement with previous data [72,71] in the basal plane. In comparison with the previous direction, the figure is more rich in branches and can be basically decomposed in three domains:

• A low frequency region presenting four branches: one at 2980 T that stars at 0 • a1, another at 2635 T a2 that begins at 4.5 • , another that stars 34 • at 2590 T a3, and the last one at 3384 T a4 beginning at 36 • ,

• In the intermediate frequency region five branches can be seen: two that starts at 0 • with a frequency of 5735 T a5 and 6063 T a6, one at 5108 T a7 that begins at 7 • , another on at 6978 T a8 starting at 25 • , and the last one at 5330 T a9 that begins at 36 • .

• Finally, in the high frequency region, there is only one branch at 13.8 kT a10 starting at 31.5 • and that remains detectable up to 52 • . The similarities in shape that presents the branch a1, a2 and a3 with respectively a6, a7 and a9 suggest that these last ones are the second harmonics of the first ones. To clarify the situation, the same checking procedure as the previous angular dependence has been applied. The frequencies of the second harmonics of a1, a2 and a3 has been reported in the figure 5.16 with a red, green and blue solid line respectively. The red line does not fit perfectly a6, but is sill yet really close to this branch. Thus, the value of the frequency alone is not sufficient to identify or not a6 as the second harmonic of a1. However, the reported value of the different effective masses presented in the tables 5.2 and 5.3 indicates that a1 and a6 are not related since their effective masses are to close in value. The picture is quite different if the green curve is considered. Indeed, it matches perfectly with the a7 meaning that they could be related one to each other. The effective mass can not be used as a support for this correspondence since their values have not been measured for this range of angle. However, the fact that the angular dependence of a2 and a7 are identical, and that the factor two between the two branches is respected in the full angular range where they are detected is a strong evidence for the belief that a7 is the second harmonic of a2. Finally, regarding the blue line, it could be noticed that it is unlikely that a9 is the second harmonic of a3 since nether the angular dependence nor the value of the frequencies are comparable between this two branches. However, we have to keep in mind that the angular dependence of the detected branches is not complete and a2 and a3 could be the same Fermi surface branch. A particular attention should be paid to the high frequency branch since it has been the object of different discussions in previous paper [72,71,74]. The object of the debate was whether the 13.8 kT (a10) branch is the second harmonic of the 7 kT (a6) one or not. In their paper [71], Sutton etal concluded that it is not the case based on the discrepancies in the angular dependence of this two branches since the values of the frequency and the effective mass were inconclusive due to the presence of a factor roughly equal to two. In this study, not only the discrepancies in angular dependence is observed, but also in the effective mass since m * a10 is far away to be the double of m * a6 as it could be seen in the table 5.3. Therefore, in the light of this new measurements it appears that this high frequency branch is a real one, and not simply the expression of the harmonic of a lower branch.

Angular dependence in the basal plane from [110] to [001] direction

In this section is shown the second part of the angular dependence from the [100] to the [001] direction via the basal plane. For this configuration, the measurements have started in the [110] direction and the signal remains perceptible after a rotation of 40 • as it could be seen in the figure 5.17. Basically, once again, the figure can be separated in three areas:

• In the low frequency region, four branches are detected: b1 with a frequency of 2600 T at 0 • , b2 at 3280 T, b3 at 3650 T, and b4 at 4150 T,

• The intermediate frequency region consists in two branches: one at 5630 T b5 and another one at 6610 T b6 in the [110] direction. • Finally, the high frequency branch at 13.8 kT b7 detected previously is still well present and remains in an angular window of 15 • .

In a similar way as the previous section, the high frequency b7 could be suspected to be the second harmonic of b6. To check that, this last has been plotted in the figure 5.17 as a blue solid line. Even if a resemblance could effectively be observed between b7 and the blue line in their angular evolution, it is not enough to conclude. However, if the effective masses reported in the table 5.4 and the values of the frequencies are considered, it could be inferred that b7 is a real branch and not a second harmonic. Indeed, not only m * b7 is not equal to twice the value of m * b6 , but moreover a divergence going from 5 % to 10 % is observed between the values of the two frequencies. From bandstructure calculations [97] this large orbit is expected and should have a heavy mass in agreement with the observation in the present experiment. It can be noticed that all these frequencies change around H 0 ∼ 10 T; some of them more than others:

• The two lows frequencies f 1 and f 2 are the most affected since they experience a decrease of about 30 % and an increase of about 50 % in the field window going from 7 T to 10 T respectively,

• The intermediate frequencies on the other hand are less affected by their passage through H 0 since globally the change of their frequencies is of the order of about 5 %,

• It is interesting to notice the lowest frequency f 1 is the only which value increases in approaching H 0 .

In the figure 5.20 are displayed the evolution in field of the effective masses of the frequencies reported in the precedent section when the field is aligned along the a-axis. Basically, the same observations done for the field dependence of the frequencies can be done on their mass dependences. Indeed, globally the effective masses are suddenly enhanced near H ∼ 10 T and stabilizes at much higher fields. This behaviour of the frequencies through H 0 is concomitant with the transition detected at a magnetic field of 10 T in thermodynamic and transport measurements [92,72]. The remaining question is to know whether this transition is associated or not to a reconstruction of the Fermi-surface, in other words that above H 0 the Fermi-surface is small or not. Some elements could conclude in this way as the figure 5.21 representing a comparison between the characteristic spin fluctuation temperature T 0 (left axis) with that of the characteristic field B * (in our case referred as H 0 ) under hydrostatic pressure. T 0 is estimated by fitting the specific-heat increment with ∆C(T )/T α ln(T 0 /T ) [65]. It has been assumed that the pressure dependence of T 0 represents that of the Kondo temperature T K (in accordance with the exponential decrease of T K with pressure) since at ambient pressure T 0 matches with the single-ion Kondo temperature T K . Therefore, one might associate H 0 with the polarisation of the Kondo singlet states and a suppression of the Kondo effect based on the fact that T K and H 0 present comparable energy scales and demonstrate scalable pressure dependencies. This elements have been argued to yield for a small Fermi-surface model beyond H 0 [65]. Nevertheless, the evolution in field of the frequencies highlighted previously are in opposition with this model based on the suppression of the local Kondo effect. Instead of that, the continuous evolution of the quantum oscillations frequencies through H 0 also pointed out by our dHvA measurements is more consistent with the progressive and complete depopulation of one Fermi-surface branch. Eventually, the observation of this peculiar frequency's behaviour provides a strong support to the belief that, rather having a transition toward a small Fermisurface, this indicates large Fermi-surface character since this comportment is no expected for a system with localized electrons. It is more believable to consider this transition in terms of a Lifshitz-type transition of spin-splitted bands where while a hole Fermi-surface decreases, an electron-type one increases through H 0 . Local density approximation (LDA) band structure calculations based assuming the 4f -electrons fully itinerant do not exactly reproduce our results [90] once compared to the dHvA frequencies' angular dependences. However, recent theoretical developments on the role of the Zeeman energy to drive a Lifshitz transition have been reported [91,96]. A specific study on the YbRh 2 Si 2 case has been realized by looking to the magnetic field induced changes of the heavy quasiparticle described in realistic renormalized band model [97]. The anomalies at H 0 result from a van Hove singularity in the density of state (DOS) which appears below the Fermi level at H = 0 and that can be revealed in magnetic field. It is this combined effect of local many body effects and coherence given by the periodicity of the lattice which produces this van Hove singularity [97]. However the present theoretical work in magnetic field is restricted to the analysis of the quasiparticle density of states, a further step will be to indicate the Fermi surface change in order to allows a comparison with the experimental results.

Conclusion

In this thesis, we have devoted efforts in three axes of research based on experiments performed on Ce-based and Yb-based heavy Fermion compounds.

The first part oh this thesis was dedicated to the study of the pure CeRu 2 Si 2 compound. The aim of this part was to present a complete study of the TEP in a system where a FS reconstruction under magnetic field is already well established by de Haas-van Alphen (dHvA) experiments. In multiband systems, like heavy Fermion compounds, the TEP response is complex. The thermoelectric response is the sum of the contribution of every subband weighted by its relative conductivity. We choose for this study the Ising-type heavy fermion compound CeRu 2 Si 2 . We have presented a detailed study of the TEP of CeRu 2 Si 2 for heat current applied parallel or transverse to the magnetic field. These longitudinal and transverse measurements, with respect to the applied magnetic field along the c-axis, reveal quite contrasting responses. Strong anomalies are detected for both configurations at the pseudo-metamagnetic transition H m . It is clearly associated with large Fermi surface reconstruction which occurs at H m . The anomalies inside the transition at H m show that the FS evolution may occur in field window as recently detected for YbRh 2 Si 2 [92,42]. Furthermore, additional anomalies have been detected at fields around 1 T and 13.5 T. A crude analysis on the magnetoresistivity response was made to understand qualitatively S(H). In conclusion, we showed that the thermopower is particularly suitable due its high sensitivity for revealing field induced changes in the FS of a correlated metal, hardly detectable by any other probe. The combination of improvements in the crystal quality and in reducing the signal to noise level of the measurement lead to observe directly significant quantum oscillations in the TEP signal. An important achievement of this thesis was to extract valuable informations on the evolution of the Fermi surface through TEP quantum oscillations. We have tried to demonstrate that this type of oscillations may represent a convenient complementary tool to follow the Fermi surface evolution. The main advantage of the dHvA technique relies on its sensitivity to heavy branches and also the possibility to reach experimentally low temperatures which facilitates the acquisition of the quasiparticles masses. On the other hand, the TEP oscillations persist even at temperature of the order of the Kelvin which represent from an experimental perspective in the FS investigation a clear advantage. For CeRu 2 Si 2 the FS in the low field paramagnetic phase (H < H m ) is excellently known. As clear direct evidence is given for a FS reconstruction through the critical pseudo-metamagnetic field H m and a broad data basis of various measurements, including the spin dynamics exists, the observed effects in the TEP can serve as reference for future experimental and theoretical studies in heavy-fermion systems and in particular in the detection of FS instabilities.

Another part of this work focused on the investigation of a potential quantum critical behaviour of the CeRu 2 Si 2 compound doped with Rh. The starting point of that study was to try to understand how it is that if we consider the CeRu 2 Si 2 compounds doped with La, we observe in its phase diagram a quantum critical end point when this system is pressurized at the critical pressure P c , and that this last is not observed when CeRu 2 Si 2 is doped with Rh. Moreover, another specificity of the Rh-doped compound that has attracted our interest during this study, relies on the clear decoupling detected in its phase diagram between the critical fields H c of the AF transition and H m of the pseudo-metamagnetic crossover whereas in the pure or the La-doped system H m is merged into H c . To investigate this points, the pressure has been used in order to follow how, by tuning the system with field and temperature, the phase diagram, and most particularly the ordered magnetic phase, was going to evolve. We have finally succeed to show that under pressure, whereas in the case of La only T N reduces while H c remains almost constant, for the Rh-doped system they both are reduced resulting in the progressive suppression of the AF order without any quantum critical end point. During this study, we also have pointed out the existence of an anisotropic opening of gap since this last manifests only when the current is applied along the c-axis. This anisotropy is justified by the longitudinal character of the AF wave vector k 3 c that is the same as the requested direction of the current j to observe the opening of a gap.

The last part of this work was dealing with the study of another Lifshitz-type transition on YbRh 2 Si 2 by using dHvA quantum oscillations as a probe to follow and investigate the evolution of the Fermi surface at the vicinity of the critical field H 0 ≃ 10 T. Recently, the effect of a magnetic field on heavy Fermions systems has become an important issue. Most of the informations on the quasiparticles characteristic of this system originates from the dHvA experiments done at high field. Nevertheless, band structure calculations, necessary to analyse these experimental results, exist almost exclusively at zero field [93,72]. Specific heat, susceptibility, and magnetostriction measurements revealed anomalies at the critical field H 0 [65,94]. As we have seen it, it was first thought to be the manifestation of the breakdown of the Kondo screening. However, this conclusion has changed in the light of dHvA experiments that were interpreted in terms of Lifshitz transition were a spin-split band disappears [72,95]. The results of our dHvA measurements presented in this study go in that stream. We have seen that rather than having an abrupt modification of the FS at H 0 , it is most likely that this last is progressively changed through H 0 as our results of the field dependence of the frequencies and the effective masses attested it. Finally, our dHvA investigation is consistent with a Lifshitz-type transition of spin-splitted bands where while a hole Fermi-surface decreases, an electron-type one increases through H 0 .

1. 1 T 5 3

 15 Electronic specific heat of YbRh 2 Si 2 divided by the temperature from 0.06 T and up to 1.5 T[67]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Function F (x) that determines the evolution of the oscillatory behaviour of RKKY exchange integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Schematic view of the RKKY interaction . . . . . . . . . . . . . . . . . . . . 1.4 Schematic Doniach phase diagram. By varying the coupling constant, the ground state of the system can be tuned between Kondo and the magnetic phase. [11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Schematic representation of the evolution of the Fermi surface through the QCP within the local criticality picture. The localization of the f -electrons induces a strong reconstruction of the Fermi surface on entering the AF state.[12] 1.6 (a) Phase diagram of MnSi showing that the quantum critical point is located at the C,D points where the pressure is equal 14.8 kbar. (b) The closer the system approach the quantum critical point, the more the resistivity behaves as a [79]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Specific heat divided by the temperature of CeCu 5.9 Au 0.1 as a function of the temperature. The T lnT form of C T indicates a deviation from the Fermi liquid theory[80]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Photograph of: (a) the quartz ampoule, (b) the flux furnace, (c) the spin balance and (d) the obtained sample of YbRh 2 Si 2 . . . . . . . . . . . . . . . 2.2 (left) Photograph of the tetra-arc furnace used to growth the samples and (right) schematic view of the Czochralski crystal growth technique. . . . . . 2.3 Photograph of the U 4 Ru 7 Ge 6 sample grown by using the Czochralski method in a tetra-arc furnace. In the photography are both present the grown ingot (left) and the rest of the starting material. . . . . . . . . . . . . . . . . . . . 2.4 Photograph of the Laue device based on a CCD camera. . . . . . . . . . . . 2.5 Schema illustrating the Bragg's relation that links the desired crystal structure parameter to the wavelength of the beam. . . . . . . . . . . . . . . . . . . . 2.6 Schema illustrating the principle of a 3 He/ 4 He dilution refrigerator. . . . . . 2.7 Schema detailing the different components of the piston-cylinder pressure cell. 2.8 (left)Schema illustrating the 4-points contact AC method usually used to measure the resistivity and (right) the corresponding photograph. . . . . . . . . 2.9 Schema of the thermoelectric principle. . . . . . . . . . . . . . . . . . . . . . Chapter 2.10 Schema illustrating the experimental set-up based on the One heater, two thermometers technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Fermi surface for a free electron gas: (a) no magnetic field, (b) under magnetic field showing the appearence of the Landau tubes. . . . . . . . . . . . . . . . 2.12 dHvA frequency in the case of a spherical Fermi surface. . . . . . . . . . . . 2.13 dHvA frequency in the case of a cylindrical Fermi surface. . . . . . . . . . . 2.14 Schema of the pick coil used in the field modulation technique. . . . . . . . . 3.1 Tetragonal ThCr 2 Si 2 structure of CeRu 2 Si 2 . . . . . . . . . . . . . . . . . . 3.2 Field variation of the magnetization and the differential susceptibility χ m (left). Field variation of the specific heat coefficient γ (right)[21]. . . . . . . . 3.3 Temperature dependence of the specific heat (left)[46] and of the magnetic susceptibility (right)[48]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Temperature dependence of the magnetic susceptibility χ and χ ⊥ in CeRu 2 Si 2 [21]. 43 3.5 The crossover phase pseudo-diagram T α (H) derived from the thermal expansion measurements (♦). The high-field data ( ) are the temperature of the C/T maxima [27]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Field dependence of the magnetization at different temperatures in CeRu 2 Si 2 [21]. 3.7 (a) Fermi surface of CeRu 2 Si 2 [84].(b) Fermi surface of LaRu 2 Si 2 [85]. The orbits which are responsible for the observed dHvA oscillations are also shown on the Fermi surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Frequencies and effective masses of the dHvA oscillations observed below H m .

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Frequencies and effective masses above H m [34]. . . . . . . . . . . . . . . . . 3.10 Temperature dependence of the TEP at different magnetic fields for the transverse thermal flow configuration. The insert is a zoom of the region below 1 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Temperature dependence of the TEP divided by the temperature at different magnetic fields for the transverse thermal flow configuration. . . . . . . . . . 3.12 Field dependence of the TEP at different temperatures for the transverse thermal flow configuration. The insert is a zoom of the MMT region. . . . . . . . 3.13 Temperature dependence of the TEP divided by the temperature at different magnetic field for longitudinal thermal flow configuration. The presence of a peak in S/T at 200 mK underlines the difficulty of extrapolating correctly S/T when T → 0 K (black line). . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14 Field dependence of the TEP at different low temperatures for the longitudinal thermal flow configuration. The insert is a zoom of the MMT region. . . . . 3.15 Field dependence of the TEP at different temperatures from 1.3 to 4.9 K for the longitudinal thermal flow configuration. The insert is a zoom of the MMT region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.16 Temperature dependence of the TEP divided by the temperature at 0 T and 16 T for the: (a) transverse and (b) longitudinal thermal flow configurations. 3.17 Comparison of the field dependence of the Seebeck signal S(H) in the transverse and longitudinal configurations. . . . . . . . . . . . . . . . . . . . . . . 3.18 Magnetic field dependence of the variation of the magnetoresistivity ρ(H) for transverse (blue) and longitudinal (black) configuration at 30 mK (solid lines) and 1.2 K (dashed lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.19 The field dependence of the Nernst signal in Highly Oriented Pyrolytic Graphite (HOPG) sample at different temperatures[33]. . . . . . . . . . . . . . . . . . 3.20 Magnetic field dependence of the TEP at different temperatures. The inset shows a zoom in the high field region. . . . . . . . . . . . . . . . . . . . . . . 3.21 Magnetic field dependence of the TEP at different temperatures with the field swept continuously. The inset shows a zoom in the high field region. . . . . . 3.22 TEP oscillations observed at 270 mK for a field parallel to the c-axis below H m . 3.23 FFT spectra of the oscillations obtained from the TEP for two FFT windows at 270 mK for a field parallel to the c-axis below H m . . . . . . . . . . . . . . 3.24 FFT spectrum of the oscillations obtained from the TEP from 5.5 T to 7.5 T at 270 mK for a field parallel to the c-axis below H m . . . . . . . . . . . . . . 3.25 TEP oscillations observed at 1.3 K for a field parallel to the c-axis above H m . 3.26 FFT spectrum of the oscillations obtained from the TEP at 1.3 K for a field parallel to the c-axis from 11 T to 16 T (above H m ). . . . . . . . . . . . . . 3.27 Field dependence of the β frequency below and above H m for a field parallel to the c-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.28 Field dependence of the γ frequency below and above H m for a field parallel to the c-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.29 Field dependence of the β and γ frequencies below and above H m for a field parallel to the c-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.30 Field dependence of the δ frequency below and above H m for a field parallel to the c-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.31 Field dependence of the ω frequency below and above H m for a field parallel to the c-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.32 Evolution of the amplitude of the signal of the branches for a field parallel to the c-axis. In the left panel are represented the mass plot of each branches. . 3.33 Field dependence of the effective of the β frequency below and above H m for a field parallel to the c-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.34 Field dependence of the effective of the γ frequency below and above H m for a field parallel to the c-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.35 Field dependence of the effective of the δ frequency above H m for a field parallel to the c-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.36 Field dependence of the effective of the ω frequency above H m for a field parallel to the c-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.37 TEP (left) and dHvA (right)

Chapter 4 . 1

 41 Appearance of three AF hot spots in the reciprocal space of CeRu 2 Si 2 from inelastic neutron scattering data of the dynamical susceptibility[43]. . . . . . 4.2 Phase diagram of Ce (1-x) La x Ru 2 Si 2 for different concentrations of La at p = 0 [19]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Magnetic phase diagram of Ce(Ru (1-x) Rh x ) 2 Si 2 for different concentrations.

93 4 . 5 . 1 ThCr 2

 4512 11 Temperature dependence of the resistivity for the current along the c-axis at different constant fields in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 . . . . . . . . . . . . . . . . . 4.12 Temperature dependence of the resistivity for the current along the c-axis at different constant fields in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 at 0.5 kbar (left) and 1 kbar (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13 Temperature dependence of the resistivity for the current along the c-axis at different constant fields in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 at 1.6 kbar (left) and 1.9 kbar (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.14 Temperature dependence of the resistivity along the c-axis for different pressures at H = 0 T in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 . . . . . . . . . . . . . . . . . . . . 4.15 Evolution of the Néel Temperature T N with the pressure in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 . The dashed lines are guide for the eyes. . . . . . . . . . . . . . . . . . . . . . 4.16 Field dependence of the resistivity for the current along the c-axis at different constant temperatures in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 . . . . . . . . . . . . . . . . . . 4.17 Field dependence of the resistivity for the current along the c-axis at different constant temperatures in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 at 0.5 kbar (left) and 1 kbar (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.18 Field dependence of the resistivity for the current along the c-axis at different constant temperatures in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 at 1.6 kbar (left) and 1.9 kbar (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.19 (a) Field dependence of the A coefficient of resistivity at different pressures for H c-axis in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 and (b) the corresponding normalized values at H m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.20 (a) Field dependence of the ρ 0 coefficient of resistivity at different pressures for H c-axis in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 and (b) the corresponding normalized values at H m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.21 (H, T ) phase diagram of Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 determined by resistivity measurements at ambient pressure. . . . . . . . . . . . . . . . . . . . . . . . . . 4.22 (H, T ) phase diagram of Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 determined by resistivity measurements at (a) 0.5 kbar, (b) 1 kbar, (c) 1.6 kbar and (d) 1.9 kbar. . . . . . 4.23 (H, p) phase diagram of Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 representing the evolution of H c and H m with the pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.24 (T, p) phase diagram of Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 representing the evolution of T K and T N with the pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.25 Relative resistivity ratio with the current along the c-axis for different fields in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 at ambient pressure (the dotted line corresponds is a fit based on a BCS function). . . . . . . . . . . . . . . . . . . . . . . . . . . 4.26 Relative resistivity ratio with the current along the c-axis for different fields in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 at (a) 0.5 kbar, (b) 1 kbar and (c) 1.6 kbar. . . . . . 4.27 Evolution of the percentage of the gapped and ungapped part of the Fermi surface in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 4.28 (H, p, T ) schematic phase diagram of CeRu 2 Si 2 and the CeRu 2Si 2 -La doped. The dashed area correspond to the domain where the FM component might play a role in the enhancement of γ [19]. . . . . . . . . . . . . . . . . . . . . 4.29 Schematic (H, T ) phase diagrams in (a) CeRu 2 Si 2 and CeRu 2 Si 2 -La doped cases and (b) CeRu 2 Si 2 -Rh doped case. The dotted line corresponds to the ambient pressure situation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Si 2 -type crystal structure of YbRh 2 Si 2 . The lattice parameters are a = 4.010 A and c = 9.841 A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Magnetic susceptibility for both crystallographic directions (a and c) in YbRh 2 Si 2 . The inset shows the sharp AF transition occurring at 70 mK [66]. . . . . . . 5.3 Electronic specific heat of YbRh 2 Si 2 divided by the temperature at zero and small fields below 0.06 T[67]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Ambient pressure phase diagram of YbRh 2 Si 2 for a field in the (ab) plane. The quantum critical point (QCP) lies between the antiferromagnetic (AF) and Landau Fermi liquid (FL) regimes. Regions of small and larg Fermi surface (FS) are labelled, with dashed arrows indicating supposed continuous Fermi surface crossovers. T 0 is identified with the Kondo temperature, T K = 25 K, and H 0 = 10 T is the field at which Kondo fluctuations supposedly are suppressed. The T * line indicates a crossover in the Hall effect and other quantities [73]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Electronic specific heat of YbRh 2 Si 2 divided by the temperature from 0.06 T and up to 1.5 T[67]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 5.

  2 behaviour [65]. . . . . . . . . . . . 5.8 Magnetization M of YbRh 2 Si 2 measured in pulsed fields at 1.6 K on oriented powder samples (B ⊥ c). [65]. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Thermoelectric power S and magnetoresistance ρ(H) of YbRh 2 Si 2 for H [110] and J [110] in the basal plane as a function of magnetic field measured at 120 mK and 30 mK respectively. All the different anomalies (extrema and inflection points) are indicated by double-vertical arrows underlying the excellent agreement between thermopower and resistivity measurements. The inset shows the ρ(H) at various temperatures in the field range around H 0 in an enlarged view. [42]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 (a) dHvA oscillations for a field applied in the [100] direction at 25 mK in YbRh 2 Si 2 and (b) the corresponding FFT spectrum in the field range from 9.5 T to 15 T. The inset of (a) shows a zoom of the region between 1/15 T and 1/13 T. The arrows denote the detected frequencies. . . . . . . . . . . . . . . 5.11 Angular dependence from the a-axis [100] to the c-axis [001] in YbRh 2 Si 2 . . 5.12 Field dependence of the dHvA signal at different constant temperatures. The signal is rapidly reduced when the temperature is increased. . . . . . . . . . 5.13 Evolution of the dHvA signal in field at different angles starting from a magnetic field aligned to the a-axis and rotating toward the c-axis. . . . . . . . . 5.14 Angular dependence of the position of the transition occurring at H 0 . The black solid line is a guide to the eyes. . . . . . . . . . . . . . . . . . . . . . . 5.15 (a) dHvA oscillations for a field applied at 36 • from the [100] direction at 25 mK in YbRh 2 Si 2 and (b) the corresponding FFT spectrum in the field range from 8 T to 15 T. The arrows denotes the detected frequencies. . . . . . . . 5.16 Angular dependence from the a-axis [100] to [110] in YbRh 2 Si 2 . . . . . . . . 5.17 Angular dependence from the a-axis [110] to [001] in YbRh 2 Si 2 . . . . . . . . 5.18 Angular dependence of the frequencies in YbRh 2 Si 2 . . . . . . . . . . . . . . 5.19 (a) Field dependence of the dHvA frequencies in the [100] direction (b) and the corresponding normalized values at each orbit's frequency at 13.5 T. . . 5.20 Evolution in field of the effective masses of the frequencies detected when a magnetic field is applied along the [100] direction in YbRh 2 Si 2 . . . . . . . . 5.21 Comparison of the pressure dependence of the characteristic spin fluctuation temperature T 0 (left axis) with that of the characteristic field B * (right axis) for YbRh 2 Si 2 in a semi-logarithmic representation [65]. . . . . . . . . . . . . List of Tables 5.1 Effective masses of the frequencies detected when the field is along the [100] direction. The last line corresponds to value obtain by Suton et al. in their dHvA measurements [71]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Effective masses of the frequencies detected when the field is along the [100] direction (Rotation in the basal plane). . . . . . . . . . . . . . . . . . . . . . 5.3 Effective masses of the frequencies detected when the field is along the [110] direction. The last line corresponds to value obtain by Suton et al. in their dHvA measurements [71]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Effective masses of the frequencies detected when the field is along the [110] direction (Rotation towards the [001] direction). . . . . . . . . . . . . . . . .

Figure 1 . 1 :

 11 Figure 1.1: Electronic specific heat of YbRh 2 Si 2 divided by the temperature from 0.06 T and up to 1.5 T[67].

Figure 1 . 2 :

 12 Figure 1.2: Function F (x) that determines the evolution of the oscillatory behaviour of RKKY exchange integral.

Chapter 1 Figure 1 . 3 :

 113 Figure 1.3: Schematic view of the RKKY interaction

Figure 1 . 4 :

 14 Figure 1.4: Schematic Doniach phase diagram.By varying the coupling constant, the ground state of the system can be tuned between Kondo and the magnetic phase.[11]

Figure 1 . 5 :

 15 Figure 1.5: Schematic representation of the evolution of the Fermi surface through the QCP within the local criticality picture. The localization of the f -electrons induces a strong reconstruction of the Fermi surface on entering the AF state.[12]

Figure 1 . 6 : 5 3

 165 Figure 1.6: (a) Phase diagram of MnSi showing that the quantum critical point is located at the C,D points where the pressure is equal 14.8 kbar. (b) The closer the system approach the quantum critical point, the more the resistivity behaves as a T 5 3 [79].

Figure 1 . 7 :

 17 Figure 1.7: Specific heat divided by the temperature of CeCu 5.9 Au 0.1 as a function of the temperature. The T lnT form of C T indicates a deviation from the Fermi liquid theory[80].

Figure 2 . 1 :

 21 Figure 2.1: Photograph of: (a) the quartz ampoule, (b) the flux furnace, (c) the spin balance and (d) the obtained sample of YbRh 2 Si 2 .

Figure 2 . 2 :

 22 Figure 2.2: (left) Photograph of the tetra-arc furnace used to growth the samples and (right) schematic view of the Czochralski crystal growth technique.

Figure 2 . 3 :

 23 Figure 2.3: Photograph of the U 4 Ru 7 Ge 6 sample grown by using the Czochralski method in a tetra-arc furnace. In the photography are both present the grown ingot (left) and the rest of the starting material.

Figure 2 . 4 :

 24 Figure 2.4: Photograph of the Laue device based on a CCD camera.

Figure 2 . 5 :

 25 Figure 2.5: Schema illustrating the Bragg's relation that links the desired crystal structure parameter to the wavelength of the beam.

Figure 2 . 6 :

 26 Figure 2.6: Schema illustrating the principle of a 3 He/ 4 He dilution refrigerator.

Figure 2 . 8 :

 28 Figure 2.8: (left)Schema illustrating the 4-points contact AC method usually used to measure the resistivity and (right) the corresponding photograph.

Figure 2 . 9 :

 29 Figure 2.9: Schema of the thermoelectric principle.

Figure 2 . 10 :

 210 Figure 2.10: Schema illustrating the experimental set-up based on the One heater, two thermometers technique.

Figure 2 . 11 :

 211 Figure 2.11: Fermi surface for a free electron gas: (a) no magnetic field, (b) under magnetic field showing the appearence of the Landau tubes.

Chapter 2 Figure 2 .

 22 Figure 2.12: dHvA frequency in the case of a spherical Fermi surface.

Figure 2 .

 2 Figure 2.13: dHvA frequency in the case of a cylindrical Fermi surface.

Figure 2 . 14 :

 214 Figure 2.14: Schema of the pick coil used in the field modulation technique.

Figure 3 . 1 :

 31 Figure 3.1: Tetragonal ThCr 2 Si 2 structure of CeRu 2 Si 2

Figure 3 . 2 :

 32 Figure 3.2: Field variation of the magnetization and the differential susceptibility χ m (left). Field variation of the specific heat coefficient γ (right)[21].

Figure 3 . 3 :

 33 Figure 3.3: Temperature dependence of the specific heat (left)[46] and of the magnetic susceptibility (right)[48].

Figure 3 . 4 :

 34 Figure 3.4: Temperature dependence of the magnetic susceptibility χ and χ ⊥ in CeRu 2 Si 2 [21].

Figure 3 . 5 :

 35 Figure 3.5: The crossover phase pseudo-diagram T α (H) derived from the thermal expansion measurements (♦). The high-field data ( ) are the temperature of the C/T maxima [27].

Figure 3 . 6 :

 36 Figure 3.6: Field dependence of the magnetization at different temperatures in CeRu 2 Si 2 [21].

Figure 3 . 7 :

 37 Figure 3.7: (a) Fermi surface of CeRu 2 Si 2 [84].(b) Fermi surface of LaRu 2 Si 2 [85]. The orbits which are responsible for the observed dHvA oscillations are also shown on the Fermi surface.

Figure 3 . 8 :

 38 Figure 3.8: Frequencies and effective masses of the dHvA oscillations observed below H m .The -symbol denotes that the oscillation is not present judging from the Fermi surface. The * symbol denotes that the signal could not be observed although the oscillation is expected to be present. 1)The data taken from the paper by Tautz et al.[34]

Figure 3 . 10 :

 310 Figure 3.10: Temperature dependence of the TEP at different magnetic fields for the transverse thermal flow configuration. The insert is a zoom of the region below 1 K.

Figure 3 . 11 :

 311 Figure 3.11: Temperature dependence of the TEP divided by the temperature at different magnetic fields for the transverse thermal flow configuration.

Figure 3 . 12 :

 312 Figure 3.12: Field dependence of the TEP at different temperatures for the transverse thermal flow configuration. The insert is a zoom of the MMT region.

Figure 3 . 13 :

 313 Figure 3.13: Temperature dependence of the TEP divided by the temperature at different magnetic field for longitudinal thermal flow configuration. The presence of a peak in S/T at 200 mK underlines the difficulty of extrapolating correctly S/T when T → 0 K (black line).

Figure 3 . 14 :

 314 Figure 3.14: Field dependence of the TEP at different low temperatures for the longitudinal thermal flow configuration. The insert is a zoom of the MMT region.

Figure 3 . 15 :

 315 Figure 3.15: Field dependence of the TEP at different temperatures from 1.3 to 4.9 K for the longitudinal thermal flow configuration. The insert is a zoom of the MMT region.

Figure 3 . 16 : 3 - 2 Figure 3 . 17 :

 31632317 Figure 3.16: Temperature dependence of the TEP divided by the temperature at 0 T and T for the: (a) transverse and (b) longitudinal thermal flow configurations.

Figure 3 . 18 :

 318 Figure 3.18: Magnetic field dependence of the variation of the magnetoresistivity ρ(H) for transverse (blue) and longitudinal (black) configuration at 30 mK (solid lines) and 1.2 K (dashed lines).

Figure 3 . 19 :

 319 Figure 3.19: The field dependence of the Nernst signal in Highly Oriented Pyrolytic Graphite (HOPG) sample at different temperatures[33].

Figure 3 . 20 :

 320 Figure 3.20: Magnetic field dependence of the TEP at different temperatures. The inset shows a zoom in the high field region.

Figure 3 . 21 :

 321 Figure 3.21: Magnetic field dependence of the TEP at different temperatures with the field swept continuously. The inset shows a zoom in the high field region.

Figure 3 . 22 :

 322 Figure 3.22: TEP oscillations observed at 270 mK for a field parallel to the c-axis below H m .

Figure 3 .

 3 Figure 3.25 shows the oscillations of the TEP in the region above H m at 1.3 K. Here again the data are plotted as function of 1H . The high field region just after the pseudo-metamagnetic transition is characterized by oscillations of amplitude much more higher than those below H m . Figure3.26 displays the FFT spectrum of the TEP at 1.3 K between 11 T and 16 T. Four distinct peaks can be seen:

•Figure 3 . 23 :Figure 3 . 24 :Figure 3 . 25 :

 323324325 Figure 3.23: FFT spectra of the oscillations obtained from the TEP for two FFT windows at 270 mK for a field parallel to the c-axis below H m .

Figure 3 .Figure 3 . 26 :FrequencyFigure 3 . 27 :

 3326327 Figure 3.27 shows the frequencies of the TEP oscillations as a function of the magnetic field applied along the c-axis. They are detected both below and above the MMT. Each point

Figure 3 . 28 :

 328 Figure 3.28: Field dependence of the γ frequency below and above H m for a field parallel to the c-axis.

Figure 3 . 29 :Figure 3 . 30 :Figure 3 . 31 :

 329330331 Figure 3.29: Field dependence of the β and γ frequencies below and above H m for a field parallel to the c-axis.

Figure 3 . 32 :

 332 Figure 3.32: Evolution of the amplitude of the signal of the branches for a field parallel to the c-axis. In the left panel are represented the mass plot of each branches.

Figure 3 . 33 :

 333 Figure 3.33: Field dependence of the effective of the β frequency below and above H m for a field parallel to the c-axis.

Figure 3 . 34 :

 334 Figure 3.34: Field dependence of the effective of the γ frequency below and above H m for a field parallel to the c-axis.

Figure 3 . 35 :

 335 Figure 3.35: Field dependence of the effective of the δ frequency above H m for a field parallel to the c-axis.

Figure 3 . 36 :

 336 Figure 3.36: Field dependence of the effective of the ω frequency above H m for a field parallel to the c-axis.

10 FrequencyFigure 3 .Figure 3 .

 1033 Figure 3.37: TEP (left) and dHvA (right)[35] frequencies as a function of magnetic field below and above H m with fields parallel to c-axis in CeRu 2 Si 2 .

Figure 3 . 39 :

 339 Figure 3.39: TEP (left) and dHvA (right) effective masses as a function of magnetic field below and above H m with fields parallel to c-axis in CeRu 2 Si 2 .

Figure 4 . 1 :

 41 Figure 4.1: Appearance of three AF hot spots in the reciprocal space of CeRu 2 Si 2 from inelastic neutron scattering data of the dynamical susceptibility[43].

Figure 4 . 2 :

 42 Figure 4.2: Phase diagram of Ce (1-x) La x Ru 2 Si 2 for different concentrations of La at p = [19].

Figure 4 . 3 :

 43 Figure 4.3: Magnetic phase diagram of Ce(Ru (1-x) Rh x ) 2 Si 2 for different concentrations. The evolution of T N is represented with the scale on the right axis[48].

4. 1 . 2

 12 Presentation of Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 : Decoupling between H m and H c Another type of doping that presents interesting perspectives is the one based on Rh. Substitution of Ru by Rh provokes in the pure CeRu 2 Si 2 compound intriguing features. The isostructural CeRh 2 Si 2 is a well localized AF compound with a Néel temperature T N = 36 K [45]. Due to the similarities in the structure, the hybridization between the 4f and conduction electrons is expected to change without any negative impact on the 4f sites periodicity. Doping with Rh stabilizes the k 3 wave vector inducing a magnetic phase transition in the pure CeRu 2 Si 2 compound at the critical field H c . The phase diagram of Ce(Ru (1-x) Rh x ) 2 Si 2 is shown in the figure 4.3 for different concentrations of Rh in the Ru rich region. The closed circles correspond to the ordering temperature T N obtained by specific heat measurements and the open circles to those acquired via magnetic susceptibility measurements. The critical fields H c and H m are respectively associated to the AF transition and the pseudometamagnetic crossover (MMT) both present in the Rh-doped compound.

Figure 4 . 4 :

 44 Figure 4.4: Temperature dependence of the resistivity of Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 with excitation current J along the a and c-axes.The data with J a are from the reference[56]

Figure 4 . 5 :

 45 Figure 4.5: Field dependence of the specific heat in the form of C/T vs H at different temperatures in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 . H c (T ) and H m (T ) are indicated by small arrows [56].

Figure 4 . 6 :

 46 Figure 4.6: Field dependence of the Sommerfeld coefficient γ in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 and CeRu 2 Si 2 normalized at H m and the value of γ at H m for the field and γ respectively.[56].

Figure 4 . 7 :

 47 Figure 4.7: Magnetization curve extrapolated to 0 K obtained by the temperature dependence at different fields in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 . The inset shows the field derivative of the magnetization curve.[56].

Figure 4 . 8 :

 48 Figure 4.8: (H, T ) phase diagram obtained by magnetostriction measurements in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 .[56].

Figure 4 . 9 :

 49 Figure 4.9: Field dependence of the A coefficient of resistivity at different pressures for H c-axis in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 .[56].

4. 1 . 4

 14 Phase diagram at ambient pressure In the figure 4.8 is shown the phase diagram of Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 constructed from the data of the field dependence of the magnetostriction in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 [56]. The figure displays a clear decoupling between the two critical fields H c and H m that marks the AF transition and the pseudo-metamagnetic crossover respectively both occurring in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 . This is different from the La or Ge doped case of CeRu 2 Si 2 where the critical fields H c and H m merge.

Figure 4 . 10 :

 410 Figure 4.10: Schematic magnetic phase diagrams of (a) Ce 0 .8La 0 .2Ru 2 Si 2 and (b) Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 .[56].

4. 2 . 3 Figure 4 . 11 :

 23411 Figure 4.11: Temperature dependence of the resistivity for the current along the c-axis at different constant fields in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 .

Figure 4 . 12 :

 412 Figure 4.12: Temperature dependence of the resistivity for the current along the c-axis at different constant fields in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 at 0.5 kbar (left) and 1 kbar (right).

Figure 4 . 13 :

 413 Figure 4.13: Temperature dependence of the resistivity for the current along the c-axis at different constant fields in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 at 1.6 kbar (left) and 1.9 kbar (right).

Figure 4 . 14 :Figure 4 . 15 :

 414415 Figure 4.14: Temperature dependence of the resistivity along the c-axis for different pressures at H = 0 T in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 .

Figure 4 . 16 :

 416 Figure 4.16: Field dependence of the resistivity for the current along the c-axis at different constant temperatures in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 .

Figure 4 . 17 :

 417 Figure 4.17: Field dependence of the resistivity for the current along the c-axis at different constant temperatures in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 at 0.5 kbar (left) and 1 kbar (right).

Figure 4 . 18 :

 418 Figure 4.18: Field dependence of the resistivity for the current along the c-axis at different constant temperatures in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 at 1.6 kbar (left) and 1.9 kbar (right).

Figure 4 .

 4 Figure 4.19: (a) Field dependence of the A coefficient of resistivity at different pressures for H c-axis in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 and (b) the corresponding normalized values at H m .

Figure 4 .Figure 4 .

 44 Figure 4.20: (a) Field dependence of the ρ 0 coefficient of resistivity at different pressures for H c-axis in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 and (b) the corresponding normalized values at H m .

Figure 4 .

 4 Figure 4.21 represents the (H, T ) phase diagram constructed from the field and temperature dependence of the resistivity with a current applied along the c-axis at ambient pressure.As seen in this figure, three main domains exists delimited by two transitions. H c slightly decreases with the temperature and then suddenly drops above 3 K. The phase boundary steeply decreases and becomes zero at T N = 4.2 K. On the other hand, H m remains roughly unchanged below 1 K and then begins to increase up to 4 K where finally it appears to saturate for the higher temperatures. The PM and AF character has been confirmed by previous magnetization and neutron scattering experiments[56].The figure4.22 shows the evolution of the (H, T ) phase diagram at the four different pressures applied on the sample. The evolution can be decomposed in four steps, with each step corresponding to a pressure:

Figure 4 .

 4 Figure 4.22: (H, T ) phase diagram of Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 determined by resistivity measurements at (a) 0.5 kbar, (b) 1 kbar, (c) 1.6 kbar and (d) 1.9 kbar.

Figure 4 .

 4 Figure 4.23: (H, p) phase diagram of Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 representing the evolution of H c and H m with the pressure.

Figure 4 .

 4 Figure 4.24: (T, p) phase diagram of Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 representing the evolution of T K and T N with the pressure.

Figure 4 . 25 :

 425 Figure 4.25: Relative resistivity ratio with the current along the c-axis for different fields in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 at ambient pressure (the dotted line corresponds is a fit based on a BCS function).

Figure 4 . 26 :

 426 Figure 4.26: Relative resistivity ratio with the current along the c-axis for different fields in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 at (a) 0.5 kbar, (b) 1 kbar and (c) 1.6 kbar.

Figure 4 . 27 :

 427 Figure 4.27: Evolution of the percentage of the gapped and ungapped part of the Fermi surface in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 .

Figure 4 .

 4 Figure 4.28: (H, p, T ) schematic phase diagram of CeRu 2 Si 2 and the CeRu 2 Si 2 -La doped. The dashed area correspond to the domain where the FM component might play a role in the enhancement of γ [19].

Figure 4 . 5 YbRh 2 2 5. 1

 45221 Figure 4.29: Schematic (H, T ) phase diagrams in (a) CeRu 2 Si 2 and CeRu 2 Si 2 -La doped cases and (b) CeRu 2 Si 2 -Rh doped case. The dotted line corresponds to the ambient pressure situation.

Figure 5 . 1 :

 51 Figure 5.1: ThCr 2 Si 2 -type crystal structure of YbRh 2 Si 2 . The lattice parameters are a = 4.010 A and c = 9.841 A.

Figure 5 . 2 :

 52 Figure 5.2: Magnetic susceptibility for both crystallographic directions (a and c) in YbRh 2 Si 2 . The inset shows the sharp AF transition occurring at 70 mK [66].

Figure 5 . 3 :

 53 Figure 5.3: Electronic specific heat of YbRh 2 Si 2 divided by the temperature at zero and small fields below 0.06 T[67].

Figure 5 . 4 :

 54 Figure 5.4: Ambient pressure phase diagram of YbRh 2 Si 2 for a field in the (ab) plane. The quantum critical point (QCP) lies between the antiferromagnetic (AF) and Landau Fermi liquid (FL) regimes. Regions of small and larg Fermi surface (FS) are labelled, with dashed arrows indicating supposed continuous Fermi surface crossovers. T 0 is identified with the Kondo temperature, T K = 25 K, and H 0 = 10 T is the field at which Kondo fluctuations supposedly are suppressed. The T * line indicates a crossover in the Hall effect and other quantities [73].

Figure 5 . 5 :

 55 Figure 5.5: Electronic specific heat of YbRh 2 Si 2 divided by the temperature from 0.06 T and up to 1.5 T[67].

Figure 5 . 6 :

 56 Figure 5.6: Small and Large Fermi surfaces calculated with the Local density approach (LDA) + spin-orbit coupling method. The sheets are labelled donut D, jungle-gym J and pillbox P.From an electron point of view, the dark blue side of each sheet is the occupied side and the light yellow side is the unoccupied side. Therefore, D is a hole surface and P is an electron surface. Note that D is a torus in the small Fermi surface case only[72].

Figure 5 . 7 :

 57 Figure 5.7: Field dependence of coefficients A (blue triangles, left axis) and γ (red circles, right axis) from FL behaviour in the electrical resistivity and specific heat, respectively for a field in the easy (ab) plane. The inset shows ρ versus T for 8 T and 16 T. Solid lines represent the T 2 behaviour [65].

Figure 5 . 8 :

 58 Figure 5.8: Magnetization M of YbRh 2 Si 2 measured in pulsed fields at 1.6 K on oriented powder samples (B ⊥ c). [65].

Figure 5 . 9 :

 59 Figure 5.9: Thermoelectric power S and magnetoresistance ρ(H) of YbRh 2 Si 2 for H [110] and J [110] in the basal plane as a function of magnetic field measured at 120 mK and 30 mK respectively. All the different anomalies (extrema and inflection points) are indicated by double-vertical arrows underlying the excellent agreement between thermopower and resistivity measurements. The inset shows the ρ(H) at various temperatures in the field range around H 0 in an enlarged view. [42].

Figure 5 . 10 :

 510 Figure 5.10: (a) dHvA oscillations for a field applied in the [100] direction at 25 mK in YbRh 2 Si 2 and (b) the corresponding FFT spectrum in the field range from 9.5 T to 15 T. The inset of (a) shows a zoom of the region between 1/15 T and 1/13 T. The arrows denote the detected frequencies.

Figure 5 . 11 :Figure 5 . 12 :

 511512 Figure 5.11: Angular dependence from the a-axis [100] to the c-axis [001] in YbRh 2 Si 2 .

Table 5 . 1 :

 51 11.6m 0 m * f 5 = 10.5m 0 m * f 6 = 17.2m 0 non detected m * f 2 S = 7.0m 0 m * f 4 S = 8.9m 0 m * f 5 S = 10.5m 0 m * f 6 S = 13.2m 0 Effective masses of the frequencies detected when the field is along the [100] direction. The last line corresponds to value obtain by Suton et al. in their dHvA measurements [71]. in the figure 5.11 displaying the angular dependence from [100] to [001]. Two regions could be distinguished: • A low frequency region with three different signals. A signal with a frequency of 2945 T (f 1) that disappears after a rotation of 27 • from [100], another at 3200 T (f 2) that vanishes at 13.5 • and a last frequency that appears at 22.5 • with a value of 2945 T (f 3) to disappear at 45 • with a value of 4080 T.

Figure 5 . 13 :Table 5 . 2 :

 51352 Figure 5.13: Evolution of the dHvA signal in field at different angles starting from a magnetic field aligned to the a-axis and rotating toward the c-axis.

Figure 5 . 14 :Table 5 . 3 :Figure 5 . 15 :

 51453515 Figure 5.14: Angular dependence of the position of the transition occurring at H 0 . The black solid line is a guide to the eyes.

Figure 5 . 17 :Table 5 . 4 :

 51754 Figure 5.17: Angular dependence from the a-axis [110] to [001] in YbRh 2 Si 2 .

5. 2 . 6

 26 Conclusion: Field dependence of the frequencies and the effective masses through H 0 The figure 5.18 summarizes the angular dependence realized in this study. The figure 5.19 (a) displays the field dependence of the dHvA frequencies when the field is along the a-axis. Each one of these five frequencies have been normalized by the value of their respective frequency at 13.5 T and the results are shown in the figure 5.19 (b).

Figure 5 . 18 :Figure 5 .Figure 5 . 20 :

 5185520 Figure 5.18: Angular dependence of the frequencies in YbRh 2 Si 2 .

Figure 5 . 21 :

 521 Figure 5.21: Comparison of the pressure dependence of the characteristic spin fluctuation temperature T 0 (left axis) with that of the characteristic field B * (right axis) for YbRh 2 Si 2 in a semi-logarithmic representation [65].

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Field dependence of the specific heat in the form of C/T vs H at different temperatures in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 . H c (T ) and H m (T ) are indicated by small arrows [56]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Field dependence of the Sommerfeld coefficient γ in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 and CeRu 2 Si 2 normalized at H m and the value of γ at H m for the field and γ respectively.[56]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Magnetization curve extrapolated to 0 K obtained by the temperature dependence at different fields in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 . The inset shows the field derivative of the magnetization curve.[56]. . . . . . . . . . . . . . . . . . . . 4.8 (H, T ) phase diagram obtained by magnetostriction measurements in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 .[56]. 4.9 Field dependence of the A coefficient of resistivity at different pressures for H c-axis in Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 .[56]. . . . . . . . . . . . . . . . . . . . . . . 4.10 Schematic magnetic phase diagrams of (a) Ce 0 .8La 0 .2Ru 2 Si 2 and (b) Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 .[56].

  The figure 4.23 displays the (H, P ) phase diagram of Ce(Ru 0.92 Rh 0.08 ) 2 Si 2 representing the evolution of H c and H m with respect to the pressure. When no pressure is applied to the sample, a distinctive decoupling between the two critical fields is observed. The pressure effect on the compound acts principally on H c , and thus the AF state, and H m as it could be observed in the figure4.23. The AF state is continuously destroyed when the pressure is applied until it vanishes at P c = 1.9 kbar. Concerning the evolution in pressure of H m , it first increases rapidly al low pressures going from H m (0) = 5.8 T at ambient pressure to a value around 8 T at 1.6 kbar. At higher pressure H m seems to persist in this trend.

He allows to overcome the limitation imposed by refrigerators using only liquid[START_REF] Kittel | Introduction to solid state physics seventh[END_REF] He, that can only reach temperatures of about 1.3 K due to the low vapour pressure below this temperature. Three main stages can be identified: the mixing chamber, the still and the