Alireza Par

Mottaghizadeh

Vincent Bouchiat

Professeur Massimiliano Marangolo

Javier E Villegas

Hervé Aubin

Maître Alexandre Zimmers

Non-Conventional Insulators : Metal-Insulator Transition and Topological Protection

Keywords: vi

This manuscript presents an experimental study of unconventional insulating phases, which are the Anderson insulator, induced by disorder, the Mott insulator, induced by Coulomb interactions, and topological insulators.

In a first part of the manuscript, I will describe the development of a method to study the charge response of nanoparticles through Electrostatic Force Microscopy (EFM). This method has been applied to magnetite (Fe 3 O 4 ) nanoparticles, a material that presents a metal-insulator transition, i.e. the Verwey transition, upon cooling the system below a temperature T V ∼ 120 K.

Résumé

Ce manuscript présente une étude expérimentale de phase isolantes nonconventionnelles, l'isolant d'Anderson, induit par le désordre, l'isolant de Mott, induit par les interactions de Coulomb, et les isolants topologiques.

Dans une première partie du manuscript, je décrirais le développement d'une méthode pour étudier la réponse de charge de nanoparticules par Microscopie à Force Electrostatique (EFM). Cette méthode a été appliquée à des nanoparticules de magnétite (Fe 3 O 4 ), un matériau qui présente une transition métal-isolant, i.e. la transition de Verwey, lors de son refroidissement en dessous d'une température T V ∼ 120 K.

Dans une seconde partie, ce manuscript présente une étude détaillée de l'évolution de la densité d'états au travers de la transition métal-isolant entre un isolant de type Anderson-Mott et une phase métallique dans le matériau SrTiO 3 , et ceci, en fonction de la concentration de dopants, les lacunes d'oxygènes. Nous avons trouvé que dans un dispositif memoresistif de type Au-SrTiO 3 -Au, la concentration de dopants pouvait être ajustée par migration des lacunes d'oxygènes a l'aide d'un champ électrique. Dans cette jonction tunnel, l'évolution de la densités d'états au travers de la transition métal-isolant peut être étudiée de façon continue.

Finalement, dans une troisième partie, le manuscript présente le développement d'une méthode pour la microfabrication d'anneaux de Aharonov-Bohm avec l'isolant topologique, Bi 2 Se 3 , déposée par épitaxie à jet moléculaire. Des résultats préliminaires sur les propriétés de transport quantique de ces dispositifs seront présentés.

In a second part, this manuscript presents a detailed study of the evolution of the Density Of States (DOS) across the metal-insulator transition between an Anderson-Mott insulator and a metallic phase in the material SrTiO 3 , and this, as function of dopant concentration, i.e. oxygen vacancies. We found that in this memristive type device Au -SrTiO 3 -Au, the dopant concentration could be fine-tuned through electric-field migration of oxygen vacancies. In this tunnel junction device, the evolution of the DOS can be followed continuously across the metal-insulator transition.

Finally, in a third part, the manuscript presents the development of a method for the microfabrication of Aharonov-Bohm rings with the topological insulator material, Bi 2 Se 3 , grown by molecular beam epitaxy. Preliminary results on the quantum transport properties of these devices will be presented. 2.9 Top) Electrons in an insulator are bound in localized orbitals (left) and have an energy gap (right) separating the occupied valence band from the empty conduction band. bottom) A two-dimensional quantum Hall state in a strong magnetic field has a bulk energy gap like an insulator but permits electrical conduction along the sample boundary [START_REF] Charles | Physics. A new spin on the insulating state[END_REF]. . . . 17

N (ε) = N (0)[1 + | ε ∆ | 1/2 ],

Generalities

As described in any introductory book of solid state physics, the A.H. Wilson classification of solids teaches us that one can sort solids between metals and insulators according to their band filling, which provides an understanding of the metallic/insulating properties of many materials. Despite its success, a large number of materials do not fit into this classification scheme.

As we will see in this manuscript, disorder and interactions can drive a metallic system into insulators. A material driven into an insulator by disorder is called an Anderson insulator, a material driven into an insulator by Coulomb interactions is called a Mott insulator. While a good theoretical understanding exists for these unconventional insulators, the description of the phase transitions between the metal and these insulating phases remain highly debated theoretical problems. The theoretical description of these phase transitions remains among the most complex issues of the physics of strongly correlated electronic systems.

From an experimentalist perspective, one of the most important issue is to identify model systems which could be employed for the study of these metal-insulator transitions. In this thesis, we present two systems where a metal-insulator transition can be observed. For Fe 3 O 4 the metal-insulator transition is driven by the temperature. For the second one, SrTiO 3 , the metal-insulator transition is driven by the carrier concentration.

Introduction

If metallic systems are found to be insulators because of disorder and interactions, a new category of insulators have recently been predicted theoretically and intensively studied experimentally, i.e. the topological insulators, which should be insulators but actually becomes metallic because of the presence of topologically protected surface states. The identification and characterization of these surface states constitute one important goal of condensed matter physics.

Plan of manuscript

This manuscript is composed of seven chapters. The first and current chapter is an introduction and motivation for this study.

The second chapter described the theoretical concepts employed in this manuscript. In particular, I provide a description of the Anderson, Mott and topological insulators.

The third chapter describes the development of a method to study the charge response of magnetite Fe 3 O 4 nanoparticles using Electrostatic Force Microscopy (EFM).

The fourth chapter presents a study of metal-SrTiO 3 -metal memristors where we find that the doping concentration in SrTiO 3 can be fine-tuned through electric field migration of oxygen vacancies. In this tunnel junction device, the evolution of the Density Of States (DOS) can be studied continuously across the metal-insulator transition.

The fifth chapter presents my preliminary works on topological insulators. The goal of this work is to identify the topological surfaces through transport measurements, either through the observation of the Aharonov-Bohm effect or the Josephson effect.

The sixth chapter is a short summary of this manuscript in french.

The seventh chapter contains the conclusions and perspectives.

Chapter 2 Theory

This chapter is dedicated to the description of some basic concepts about unconventional insulators and the metal-insulator transition.

Metals and insulators

Most materials can be classified into two principal categories: metals and insulators according to their band filling, following a classification introduced by A.H. Wilson. In a periodic lattice, the electronic states are organized into energy bands separated by an energy gap which is called the band gap. A material composed only of empty or filled bands is an insulator while a material with partially filled band is metallic. Figure 2.1 shows the filling of the energy bands for typical metals and insulators. When the Fermi level lies in the band gap, the material is an insulator and when it lies within a band, the material is a metal. An insulator with a small band gap is also called a semiconductor.

Thus, simple electron counting should allow one to predict easily whether a material is an insulator or a metal. For example, magnesium oxide (MgO) is an insulator because it has only filled p bands but iron (Fe) is a metal because of its partially filled d bands. Beyond this classification, different mechanisms of driving a metallic state into an insulator have been identified. In metallic thin films, strong disorder leads to the localization of wave functions, the so-called Anderson localization. In doped semiconductors, metal-insulator transition have been observed as a function of dopant concentration [1]. Finally, even ordered crystalline materials can display an insulating phase while band theory predicts a metallic phase. For example, in 1937, De Boer and Verwey reported that [2] NiO is an insulator. NiO is made of Ni 2+ and O 2-ions. The electronic configuration of Ni is 3d 8 . Thus, according to the conventional band theory just described, NiO should be metallic. The reason of this disagreement is that the conventional band theory ignores the effect of electron-electron interaction on the materials bands structures. Thus, given a hypothetical material where it would be possible to tune the interactions between the electrons, one should observe a phase transition between a metallic phase, for weak Coulomb interactions, and a Mott insulator, for strong Coulomb interactions. Another way to drive a metal into an insulating phase is to increase disorder, this is the Anderson localization. We provide now a brief description of metal-insulator transition in different systems of the two types.

Theory

Anderson insulators

In the description of insulators provided in section 2.1, the system is assumed to be a perfect crystal. However, experimentally, crystals always have some disorder like impurities, dislocations, vacancies, etc., which disturb the lattice periodicity. If the amount of disorder exceeds a certain value, the system is no longer metallic. This is the Anderson localization, which is one of the most studied metal-insulator transitions. This metal-insulator transition has been mostly studied in amorphous metallic thin films and semiconductors.

Amorphous metallic thin films are usually obtained by thermal evaporation under vacuum. As the thickness of the film increases, a transition from an insulating to a metallic state is observed. In this kind of experiments, the thickness is the parameter controlling the amount of disorder in the films. This regime is reached at low temperature when the thickness of the film is smaller than the coherence length, L φ . This length characterizes the length on which the wave function remains phase coherent. When the film thickness is smaller than this coherence length, the diffusion coefficient D ∼ t 2 /τ , where τ is the elastic scattering time, for electron diffusion, depends on the thickness of the film. Consequently, the film thickness becomes a parameter that can be used to tune the metal-insulator transition at low temperature.

In semiconductors, which are essentially insulators with a small band gap, the transition to the metallic state is reached by doping the semiconductor with foreign doping atoms. The dopants can either add electrons to the material (n-type) or remove electrons (p-type). At low doping, the additional electronic states are localized on the doping sites. Upon increasing the density of doping atoms, the wave function of the doped electrons overlaps and a transition to a metallic state occurs. In three dimensions, the metal-insulator transition observed in phosphorous doped silicon [1].

Anderson transition to localization

In 1957, Anderson has shown [START_REF] Yu | Verwey transition in a single magnetite nanoparticle[END_REF] that the electron wave function can be localized by disorder, when the disorder, which creates a random potential for electrons is large enough. In this case an electron wave function localized at r 0 decays exponentially from this point as:

2. Theory | ψ(r) |∼ exp(-| r -r 0 | /ξ) (2.1)
where ξ is the localization length, as illustrated in figure 2.2. For weak disorder, where only weak potential fluctuations exist, nearby localized orbitals are in the same energy range and therefore will overlap with each other to give extended wave functions. However, for strong disorder, where large fluctuations in potential exist, the probability of overlap between two localized states become very weak. Indeed, the probability that nearby states are close in energy is weak, however, states with similar energies are likely very far from each other in space. In this situation, no overlap occurs and no extended state can be formed. In between the insulating and the metallic phase, a metal-insulator transition is expected. According to the scaling theory, section 2.2.3, the transition should be continuous with a diverging correlation length. This scaling theory describes the critical behavior of the metal-insulator transition. While the critical behavior is expected to be universal, the exact location of the critical point, i.e. the critical dopant density for doped semiconductors or the critical thickness for amorphous metallic films, depends on numerous microscopic details of the material. In amorphous metallic thin films, the critical thickness is reached when the film thickness is small enough so that the diffusion coefficient becomes much smaller than its value in the corresponding bulk material. As discussed above, this implies that the film thickness be smaller than the coherence length.

Theory

Mott criterion

In doped semiconductors, the value of the critical density for the metal-insulator transition is around the value set by the Mott criterion. This criterion is obtained by considering that the potential around a positive charge screened by electrons is given by:

V (r) = - e 2 r exp(-r/λ T F ) (2.2)
where λ T F is the Thomas-Fermi screening length and is given by:

λ T F = 2π E F /3e 2 n (2.3)
where E F is the Fermi energy, is the dielectric constant and n is the electron density. Every positive ion is characterized by an effective Bohr radius, a * H , which is the distance of the outermost bound electron experiencing the Coulomb potential screened by free carriers. If the Bohr radius is bigger than λ T F , the electric field will be screened by electrons and the farther electrons will not feel the Coulomb attractive potential and so the system becomes metallic. In contrast, if λ T F becomes larger than a * H , the free charges can not screen the attractive potential and therefore they are bound and the system becomes insulating. This leads to Mott's criterion for the value of the critical density. Thus, this critical point of the metal-insulator transition is reached when the Thomas-Fermi screening length is equal to the effective Bohr radius, λ T F ≈ a * H . Knowing the relation between the Thomas-Fermi screening length and the carrier density n:

λ 2 T F = a * H /4n -1/3 (2.4)
one obtains:

a * H ≈ 0.25n -1/3 (2.5)
Thus, the Mott criterion can be written as:

a * H n 1/3 c ≈ 0.25 (2.6)

Theory

In conclusion, if a * H n 1/3 c > 0.25 the system is metallic, if a * H n 1/3 c < 0.25 the system is insulating.

The critical transition point can also be discussed with respect to the position of the Fermi level in the band structure. Upon doping a semiconductor, the dopant states will form an impurity band at the top and bottom of the conduction band as depicted in figure 2.3. In the un-doped regime, the density of states curve is represented by the blue curve. Upon doping, new localized states appear at the top and bottom of the conduction band. The states far from the center of the conduction band are localized, while the states close to the center of the conduction band are delocalized. The energy scale that separates the localized states form the delocalized states is called the mobility edge, E m . While the Mott criterion provides the value of the critical density, the description of the critical behavior around this critical density is provided by the scaling theory of localization, which is described below.

DOS

E E E v c E c E c E m E c E m (a) (b)

Theory

Scaling theory of localization

In the vicinity of a metal-insulator transition, the scaling theory describes the evolution of conductance G as a function of system size L. It shows that the conductance is a measure of microscopic disorder. In the limit of weak scattering, the electron wave function is extended and the mean free path, l, is larger than the Fermi wavelength, k -1 F . In this regime, the conductivity is given by:

σ = ne 2 τ /m = ne 2 l/ k F (2.7)
where n is the electron density and τ is the relaxation time, τ = l/v. It is clear that the conductivity is independent of the system size L, provided the system is large enough, i.e. L >> l. In this Ohmic regime, the conductance G(L), of a diffusive d-dimensional system of size L is related to the conductivity through the equation:

G(L) = σL d-2
(2.8)

In highly disordered systems, i.e. the limit of strong scattering, the electron wave function is localized within the localization length ξ which is in general larger than the mean free path l. As mentioned in previous section, the localized states with energies close to each other are far in space, this implies that the hopping amplitude is exponentially small. In this regime L >> ξ the conductance is given by:

G(L) ∝ exp(-L/ξ) (2.9)
For a given disorder, as the size of the system L increases, the conductance changes and reaches one of the limiting cases given by equation 2.8 or equation 2.9. The limit reached depends on the amount of disorder. To describe the transition from one limit to the other, Abrahams, Anderson, Licciardello and Ramakrishnan [START_REF] Yu | Signature of molecular cleavage in field emission current[END_REF] introduced the scaling theory in 1979. A simple description of the theory follows. Considering the Thouless energy E T = D/L 2 , the diffusion coefficient D, the mean spacing level δ = 1/[n( F )L d ] where d is the dimension and L is the system size, the Einstein relation G(L) = 2e 2 n( F )D for conductance, we find that E T /δ = g(L)/2π where g(L) = G(L)/G 0 and G 0 = 2e 2 /h is the quantum of conductance. This formula shows that only states within the Thouless energy shell contribute to electronic transport. The 2. Theory hypothesis of the scaling theory is that the conductance g(L) is the only parameter that controls the evolution of quantum states of the system upon changing the size of the system. Mathematically, this implies that the conductance g(L ) of a system with size L where L = bL can be described as a function of g(L) and L/L , i.e. g(L ) = f (L /L, g(L)). This implies the relation ∂ ln g/∂ ln L = β(g) where β(g) is Gell-Mann and Low function. In three dimensions, β reaches unity at very large g and is negative at small g. It crosses zero at the point called g c which is a unstable fixed point. If g is larger than g c : upon increasing the length scale, L, g increases and β reaches the value β = 1. This value β = 1 implies that g ∝ L, which means that the system is described by the Ohm's law. If g is smaller than g c , β is negative and upon increasing the length scale L, β follows the form s ln g/g c which implies that g ∝ e -L/ξ and that the states are localized. To resume, a system with the g > g c is a metal and a system with g < g c is an insulator.

In two dimensions, β(g) is always negative. This implies that at a length scale large

Theory

enough, a two dimensional system is always localized. Thus, one believes that there is no metallic phase in two dimensions even for small microscopic disorder. However, in presence of spin-orbit coupling, the scaling function can be positive which leads to metallic behavior in some systems [START_REF] Hikami | Spin-Orbit Interaction and Magnetoresistance in the Two Dimensional Random System[END_REF].

In one dimension, β(g) < 0 and decreases at increasing length scale. Consequently all states are localized and the localization length is of the order of the back scattering mean free path.

Anderson-Mott insulators

The introduction of electron-electron interactions modifies the description of metalinsulator transitions [START_REF] Belitz | The Anderson-Mott transition[END_REF]. Two regimes can be considered. In the regime of weak Coulomb interactions, the standard theory of localization can be corrected to include the effects of electron-electron interactions [START_REF] Lee | Disordered electronic systems[END_REF]. In the regime of strong electron-electron interactions, a completely different theoretical description of the system is used, where the insulating phase is described as a Mott-Hubbard or charge transfer insulator.

Weak interactions 2. Theory

N (E) = N (0)(1 + (E/E T ) 1/2 ) (2.10)
where N(0) vanishes at the transition and E T is Thouless energy, E T = D/l 2 which depends on the diffusion constant, D and electron mean free path, l. The vanishing of the density of states was observed in different materials e.g. Au x Ge 1-x by McMillan and Mochel [START_REF] Mcmillan | Electron Tunneling Experiments on Amorphous Ge {1-x}Au {x}[END_REF] and granular aluminum by Dynes and Garno [START_REF] Dynes | Metal-Insulator Transition in Granular Aluminum[END_REF].

On the insulating side, Efros and Shklovskii [START_REF] Shklovskii | Coulomb gap and low temperature conductivity of disordered systems[END_REF] have shown in 1975 that the Coulomb interactions open a gap that goes as N ( ) ∼ 2 , in three dimensions, at the Fermi level, the so-called Coulomb gap or soft gap. Experimentally, the soft gap was observed by tunneling spectroscopy by different groups. Massey and Lee [START_REF] Massey | Electron Tunneling Study of Coulomb Correlations across the Metal-Isulator Transition in Si:B[END_REF][START_REF] Mark Lee | Coulomb gap in a doped semiconductor near the metal-insulator transition: Tunneling experiment and scaling ansatz[END_REF] observed a soft gap in Si:B and Bielejec et al [START_REF] Bielejec | Hard Correlation Gap Observed in Quench-Condensed Ultrathin Beryllium[END_REF] observed a soft gap in ultra thin beryllium near the metal-insulator transition.

Mott-Hubbard and charge transfer insulators

In 1937, Peierls noted that "It is quite possible that the electrostatic interaction between the electrons prevents them from moving at all. At low temperatures the majority of the electrons are in their proper places in the ions. The minority which have happened to cross the potential barrier find therefore all the other atoms occupied, and in order to get through the lattice have to spend a long time in ions already occupied by other electrons. This needs a considerable addition of energy and so is extremely improbable at low temperatures." [START_REF] Mott | Discussion of the paper by de Boer and Verwey[END_REF].

This was the first prediction about the role of electron-electron interactions in the electronic properties of materials. Another influential researcher was N. F. Mott. He was one of the first to investigate how the electron-electron interactions can change the electronic structure of materials. His model is based on a lattice with a single electron at each site. Ignoring electron-electron interactions, the overlap of atomic orbitals would lead to the formation of a single band half occupied which would make the system metallic. In the presence of electron-electron interactions, the Coulomb energy associated with double-occupancy of a single site leads to splitting of the band into a lower band of singly occupied site and upper band of doubly occupied site. For one electron per site, the lower band is full and the upper band is empty and the system becomes insulating, the so-called Mott insulator.

Theory

One of the theoretical approaches to understand Mott insulators is the Hubbard model which considers only electrons in a single band. The Hamiltonian is written as:

H H = -t <i,j>,σ (c † iσ c j,σ + c † jσ c i,σ ) + U n i=1 n i↑ n i↓ (2.11)
where c † i,σ (c i,σ ) is the creation (annihilation) operator of an electron with spin σ on site i, n i,σ = c † iσ c i,σ is the number operator, t is the hopping integral between neighboring sites given by equation 2.12 and U is the on-site Coulomb interaction given by equation 2.13.

t = drφ * i,σ (r)[- 2 2m ∇ 2 + V ]φ j,σ (r) (2.12) U = e 2 drdr φ * iσ (r)φ iσ (r) 1 | r -r | φ * i-σ (r )φ i-σ (r ) (2.13)
where φ i,σ is the wave function of an electron on site i with spin σ. Strongly correlated electron systems are mostly obtained from the transition metal elements with partially filled d-bands, rare earth elements with 4f orbitals and actinide elements with 5f orbitals. For transition metal elements, schematically illustrated in 2. Theory figure 2.5, in presence of Coulomb interactions, the d-band with the initial width W splits into two bands called the upper Hubbard band with the width W 1 and the lower Hubbard band with the width W 2 . The ratio U/W determines the electronic properties of the system. If U/W >> 1 the electron-electron interactions dominate which drives the system into the insulating regime where the energy gap is given by:

E g = U - 1 2 (W 1 + W 2 ) (2.14)
If the ratio U/W << 1, the second term in Eq. 2.11 is negligible which leads to the tight-binding Hamiltonian where the band is partially filled and the system becomes metallic. Figure 2.6 illustrates the band structure of a normal metal(a) and a Mott-Hubbard insulator(b). A detail description of the Mott insulators can be find in [START_REF] Imada | Metal-insulator transitions[END_REF]. A second category of strongly correlated electron system is the charge transfer insulators where the energy separation between d bands and p bands, ∆, is larger than Coulomb repulsion energy, U . Figure 2.7 illustrates the band structure of a charge transfer insulator.

-----------E DOS E F P band d band +U -----------E DOS E F U Mott-Hubbard Insulator Normal Metal (a) (b)
Here, the gap is between upper Hubbard band and filled p band. More details about charge transfer insulators could be found in the article by Zaanen, Sawatzky and Allen [START_REF] Zaanen | Band gaps and electronic structure of transition-metal compounds[END_REF]. ----------- To resume, we have seen that ordinary metals with partially filled band can become insulating either by increasing disorder, which leads to Anderson insulator, or by increasing Coulomb interactions, which leads to the Mott-Hubbard or charge transfer insulators.

Theory

In this manuscript, I present an experimental study of two different types of metalinsulator transitions. In the chapter 3, I will present a study of the Verwey transition in magnetite (Fe 3 O 4 ) where the transition is crossed by changing the temperature. In magnetite, electronic states are localized and the transition is believed to be driven by strong electron-electron interactions. In chapter 4, I will present a study of the metalinsulator transition in SrTiO 3 as a function of carrier concentration. In this study, the dopant concentration is changed by applying an electric field. we believe that this transition is of the Mott-Anderson type.

Topological insulators

In the past 10 years, a new field has emerged in condensed matter physics; the topologically protected insulating electronic phases. A topological insulator, like an ordinary insulator, has a bulk energy gap which separates the bottom of the conduction band from the top of the valence band. The surface of a topological insulator in 3D and the edges in 2D have gapless states which are protected topologically as shown theoretically by Kane and Mel [START_REF] Kane | Z {2} Topological Order and the Quantum Spin Hall Effect[END_REF]. A 2D topological insulator has some similarities to quantum Hall effect and can be understood in this framework.

Upon applying a large magnetic field perpendicular to a 2D electron gas, the electron are confined along closed circular trajectories, which leads to the opening of an energy gap, and ultimately, to the quantum Hall effect. At the edge of this 2D gas, the electrons have a different type of motion which leads to electronic states that conducts along the edge in only one direction without back scattering as shown in figure 2.9.

Similarly, topological insulators presents edge states, which do not need the presence of a magnetic field to form. Instead, the topology of the band structure guarantee that conducting electronic states should exist at the edge. To illustrate this, figure 2.10a shows the evolution of the band structure of a binary semiconductor MA, going from Figure 2.9: Top) Electrons in an insulator are bound in localized orbitals (left) and have an energy gap (right) separating the occupied valence band from the empty conduction band. bottom) A two-dimensional quantum Hall state in a strong magnetic field has a bulk energy gap like an insulator but permits electrical conduction along the sample boundary [START_REF] Charles | Physics. A new spin on the insulating state[END_REF].

the inside of the semiconductor to the outside across the edge. Here, M represents a metallic cation whose last filled orbital has s symmetry, A is the ligating anion whose first empty orbital has p symmetry. With weak spin-orbit coupling, the Fermi level is located between the filled cationic orbitals of s-symmetry and the empty anionic orbitals of p-symmetry. Upon going from inside to the outside, the band levels shift to their vacuum values without crossing the Fermi energy. With strong spin-orbit coupling, the cationic s-orbitals and the anionic p-orbitals reverse their role. The cationic s orbitals form the empty conduction band. The anionic p orbitals form the filled valence band. In this situation, the Fermi level is located between the filled anionic orbitals and the empty cationic orbitals. Upon going from inside to the outside, the band levels must cross to reach their values in the vacuum. At this crossing point, the band are not either completely filled or empty and so leads to the formation of a conducting state. Furthermore, because of strong spin-orbit coupling, the edge states are spin polarized, meaning that the electrons with spin up propagate in one direction and the electrons with spin down propagate in the opposite direction. These states are insensitive to disorder because there are no states available for back scattering. In 2006, Bernevig et al. [START_REF] Bernevig | Quantum spin Hall effect and topological phase transition in HgTe quantum wells[END_REF] predicted theoretically the existence of the quantum spin Hall effect, a property of 2D topological insulator, in HgTe/CdTe quantum well. Soon after, this effect was observed experimentally by Konig et al. [START_REF] König | Quantum spin hall insulator state in HgTe quantum wells[END_REF], in 2007.

Theory

The possibility of topological protection in 3D was also predicted by [START_REF] Liang Fu | Topological Insulators in Three Dimensions[END_REF][START_REF] Moore | Topological invariants of time-reversal-invariant band structures[END_REF][START_REF] Roy | Z2 classification of quantum spin Hall systems: An approach using time-reversal invariance[END_REF] in 2006. Fu et al. predicted this phase in some real materials such as Bi 1-x Sb x , HgTe, PbTe and SnTe [START_REF] Fu | Topological insulators with inversion symmetry[END_REF].

The first experimental observation of topologically protected states in 3D was reported by Hsieh et al in 2008 in Bi 1-x Sb x [START_REF] Hsieh | A topological Dirac insulator in a quantum spin Hall phase[END_REF]. Soon after, some groups reported the observation of surface states in Bi 2 Se 3 [START_REF] Xia | Observation of a large-gap topologicalinsulator class with a single Dirac cone on the surface[END_REF], Sb 2 Te 3 [START_REF] Hsieh | Observation of Time-Reversal-Protected Single-Dirac-Cone Topological-Insulator States in Bi2Te3 and Sb2Te3[END_REF] and Bi 2 Te 3 [START_REF] Y L Chen | Experimental realization of a three-dimensional topological insulator[END_REF].

Figure 2.10b shows the typical band diagram of 3D topological insulators displaying the bulk band gap, the conduction band and the valence band. Topological surface states appear in this band diagram as pairs of states that disperse linearly with energy. Indeed, the surface states remain degenerate even in the presence of strong spin-orbit coupling. This degeneracy is imposed by Kramers theorem. According to this theorem, time-reversal symmetry requires that all states come in degenerate pairs at k and -k.

Chapter 3 Metal-Insulator Transition in Magnetite

Magnetite Fe 3 O 4 , one form of iron oxide, is one of the strongly correlated electron systems which is well known since ancient times as lodestone. It was used to locate the North Pole by early navigators. Magnetite is ferrimagnetic at T < T c = 858 K where the spins of sub-lattices A and B are anti-parallel. Beside its magnetic behavior, it is known that bulk magnetite undergoes metal to insulator transition at T V = 120 K, the so-called Verwey transition.

In our group we have developed two methods to study nanoparticles, the fisrt one is based on the measure of electrostatic charge response with an EFM setup and the second one is based on tunneling spectroscopy where a nanoparticle is trapped in a nanogap.

The goal of this chapter is to describe the AFM/EFM experiment on magnetite nanoparticles. The second method is explained in more details in [START_REF] Yu | Transport electronique dans les nanoparticules: du reseau a la nanoparticule individuelle[END_REF]. At room temperature, magnetite is a bad metal. The resistivity of bulk magnetite increases by two orders of magnitude by decreasing the temperature below T V = 120 K where a first order phase transition occurs, figure 3.2. In 1939, E.J. Verwey suggested that, below T V = 120 K, charge ordering occurs and leads to the formation of alternated rows of Fe 3+ and Fe 2+ cations.

Magnetite-Fe

The Verwey transition is altered by size effects. For magnetite thin films, the transition is broad [35,[START_REF] Li | Transport and magnetic properties of epitaxial and polycrystalline magnetite thin films[END_REF] and only a crossover between a high temperature metallic phase and a low temperature insulating phase is observed. In nanoparticles, the size effects are expected to be more important and the Verwey transition becomes even broader. The Verwey transition is also altered by the oxidation state. The oxidation transforms the magnetite Fe 3 O 4 , to hematite Fe 2 O 3 , which is insulating with no metallic phase. Considering all these issues, the prediction of the nature of electronic state, metallic or insulating state, of magnetite nanoparticles is difficult.

There have been many theoretical and experimental studies to understand the nature and mechanism of the Verwey transition. Cullen and Callen [START_REF] Cullen | Collective Electron Theory of the Metal-Semiconductor Transition in Magnetite[END_REF] suggested that charges are transported by band-like delocalized states for T > T V and that a collective Coulomb gap opens for T < T V . In contrast, Mott [START_REF] Mott | Materials with mixed valency that show a verwey transition[END_REF][START_REF] Austin | Polarons in crystalline and non-crystalline materials[END_REF], Chakraverty [START_REF] Chakraverty | Verwey ordering on magnetite as a co-operative Jahn-Teller transition[END_REF] and Yamada [START_REF] Yamada | Molecular polarons and valence fluctuations in Fe 3 O 4[END_REF] suggested that polarons, formed because of strong electron-phonon interactions, are responsible for charge transport. In this polaronic model, N.F. Mott [START_REF] Mott | The metal-nonmetal transition[END_REF], suggested that the Verwey transition has much similarities with the Wigner cristallization [43]. At low temperature, Coulomb interactions localize the polarons, while at high temperature, the polarons crystal melts, allowing charge transport through hopping.

For T < T V , DC conductivity, infra-red optical conductivity [START_REF] Park | Charge-gap formation upon the Verwey transition in Fe3O4[END_REF][START_REF] Gasparov | Infrared and Raman studies of the Verwey transition in magnetite[END_REF] and terahertz conductivity [START_REF] Pimenov | Terahertz conductivity at the Verwey transition in magnetite[END_REF] decrease with temperature. DC transport follows an Arrhenius law with an activation energy about ∼ 120 meV [START_REF] Kuipers | Electrical transport in magnetite near the Verwey transition[END_REF]. In optical conductivity spectrum, a peak at mid-infrared, ε ∼ 0.5 eV , has been interpreted as the possible signature of polarons [START_REF] Austin | Polarons in crystalline and non-crystalline materials[END_REF][START_REF] Jozef | Fröhlich polaron and bipolaron: recent developments[END_REF][START_REF] Basov | Electrodynamics of correlated electron materials[END_REF]. This spectral feature is also visible in photoemission [START_REF] Park | Single-particle gap above the Verwey transition in Fe {3}O {4}[END_REF][START_REF] Schrupp | High-energy photoemission on Fe 3 O 4 : Small polaron physics and the Verwey transition[END_REF] and STM spectroscopy measurements [START_REF] Jordan | Scanning tunneling spectroscopy study of the electronic structure of Fe {3}O {4} surfaces[END_REF]. In this insulating phase, photoemission [START_REF] Park | Single-particle gap above the Verwey transition in Fe {3}O {4}[END_REF][START_REF] Schrupp | High-energy photoemission on Fe 3 O 4 : Small polaron physics and the Verwey transition[END_REF][START_REF] Chainani | Highresolution photoemission spectroscopy of the Verwey transition in Fe3O4[END_REF] and tunnel spectroscopy [START_REF] Jordan | Scanning tunneling spectroscopy study of the electronic structure of Fe {3}O {4} surfaces[END_REF] indicate the disappearance of the density of states within an energy range ε ∼ 100 -300 meV around the Fermi level.

For T > T V , the situation is more ambiguous. DC transport also follows an Arrhenius law with a smaller activation energy about 10 -100 meV [START_REF] Gasparov | Infrared and Raman studies of the Verwey transition in magnetite[END_REF][START_REF] Pimenov | Terahertz conductivity at the Verwey transition in magnetite[END_REF][START_REF] Kuipers | Electrical transport in magnetite near the Verwey transition[END_REF]. The activated temperature dependence of infra-red optical conductivity was described within a polaronic model [START_REF] Gasparov | Infrared and Raman studies of the Verwey transition in magnetite[END_REF], however, terahertz measurements by A. Pimenov et. al. [START_REF] Pimenov | Terahertz conductivity at the Verwey transition in magnetite[END_REF] suggest the presence of a Drude peak, implying band-like transport. As the conductivity can be generally described as σ = neµ, where n is the electron density and µ is the electronic mobility, the Arrhenius law could be either due to an activated carrier density, n = n 0 exp(-∆/k B T ), as expected for a semiconductor with a gap ∆, or an activated mobility, µ = µ 0 exp(W B /2k B T ), as expected for polarons of binding energy W B .

Charge response from Electrostatic Force Microscopy (EFM)

Electrostatic Force Microscopy (EFM) is a very sensitive probe of electric field which is well adapted to study the charge response of materials. This technique has been used for discriminating insulating from metallic nanotubes [54,[START_REF] Lu | A scanning probe microscopy based assay for single-walled carbon nanotube metallicity[END_REF], measuring their charging capacity [START_REF] Mélin | Electric force microscopy of individually charged nanoparticles on conductors: An analytical model for quantitative charge imaging[END_REF][START_REF] Mélin | Probing Nanoscale Dipole-Dipole Interactions by Electric Force Microscopy[END_REF][START_REF] Paillet | Electrostatics of Individual Single-Walled Carbon Nanotubes Investigated by Electrostatic Force Microscopy[END_REF] and measuring the dielectric coefficient of insulating nanoparticles and viruses [START_REF] Fumagalli | Label-free identification of single dielectric nanoparticles and viruses with ultraweak polarization forces[END_REF]. EFM is also of fundamental interest as it could be used to measure the charge compressibility κ = dn/dµ of conducting nanomaterials. The charge compressibility is a fundamental thermodynamic property of electronic systems [START_REF] Dultz | Thermodynamic Signature of a Two-Dimensional Metal-Insulator Transition[END_REF][START_REF] Li | Very large capacitance enhancement in a two-dimensional electron system[END_REF] whose determination requires measurements of the charge response on the scale of the screening length. For simple metals, the charge compressibility is large and is simply related to the density of states, n ∝ µ 3/2 ⇒ dn/dµ ∝ µ 1/2 ; in contrast, for band insulators, the charge compressibility is close to zero. In correlated electronic materials, the charge compressibility is expected to be altered by electron-electron interactions as illustrated by the transition from correlated insulators to metal [START_REF] Wu | Electric-fielddriven phase transition in vanadium dioxide[END_REF] where it has been predicted to diverge [START_REF] Watanabe | Thermodynamic Relations in Correlated Systems[END_REF].

One of the limitation of EFM is the capacitive coupling of tip to not only the nanoparticles but also the substrate. This coupling makes difficult the extraction of quantitative information from the EFM data.

Sample preparation

Undecanoic acid-capped magnetite nanoparticles of an average size of about 10 nm are obtained by the reaction of iron(III) acetylacetonate in benzyl alcohol at 175 • C followed by a ligand exchange process as described in [START_REF] Pinna | Magnetite Nanocrystals: Nonaqueous Synthesis, Characterization, and Solubility â[END_REF]. Figure 3.3 shows TEM images of magnetite nanoparticles. The substrate is a silicon where we evaporated 10 nm of Pd to make a clean and flat metallic substrate. To prepare the samples for EFM, we depose ∼ 100 µL of nanoparticles onto the substrate. In order to remove the organic ligands around the nanoparticles, we bake the sample at 300 • C in high vacuum (10 -6 mbar).

Principles of Electrostatic Force Microscopy

The electrostatic energy is given by:

U el = (1/2C)∆φ 2 (3.1)
In this equation, C is the total capacitance and ∆φ is the electrochemical potential difference between the tip and the sample added to the applied voltage V .

∆φ = φ tip -φ sample + V (3.2)
The electrostatic force exerted on the tip is spatial derivative of electrostatic energy:

F = - dU el dz = - 1 2 ∂C ∂z ∆φ 2 (3.3)
Atomic Force Microscopy (AFM) and EFM measurements are carried out with a home-built variable temperature (30 K -400 K) microscope installed in ultra high vacuum (10 -9 mbar), figure 3.4. The microscope is based on the so-called Pan design. Six piezoelectric shear motors control the coarse approach of the tip to the sample, figure 3.4a. The force detection is based on measurement of the frequency shift f of a quartz tuning fork which depends on the force F exerted on the tip through the relation f = -dF/dz. More details about tuning fork based microscopes can be found in [START_REF] Giessibl | Advances in atomic force microscopy[END_REF]. Typical value for the oscillation amplitude of the tuning fork is about 5 nm. One prong of tuning fork is glued to ceramic holder with stycast. The metallic Pt/Ir tip is prepared by electrochemical etching and glued to the other prong which is free to oscillate as showed in figure 3.4b. Figure 3.4c shows a SEM image of etched tip. The setpoint of the frequency shift f is set to -4 Hz. To perform EFM measurement, a voltage has to be applied on the tip. The applied voltage V has DC and AC parts. We set the DC voltage to V DC = 1 V while the AC part is V AC ∼ 500 mV , ω = 30 Hz which gives a total applied voltage:

V = V DC + V AC cos(ωt).
By substituting the applied voltage in Eq. 3.3 we obtain:

F = - 1 2 ∂C ∂z (V Dc + V AC cos(ωt)) 2 (3.4)
Using f = -dF/dz gives:

f (Hz) = f DC + f ω cos(ωt) + f 2ω cos(2ωt) f DC = 1 2 C (z)[V 2 DC + V 2 AC 2 ] (3.5) f ω = C (z)V DC V AC and f 2ω = C (z)V 2 AC Where C = ∂ 2 C ∂z 2 .
We used a a lock-in amplifier to measure the first f ω and the second f 2ω harmonics.

A schematic EFM measurement circuit is shown in figure 3.6. The measurements have been done in a constant amplitude mode. The electronic circuit is based on a Phase Locked Loop (PLL) from Nanosurf and a Proportional-Integral-Derivative (PID) controller from Nanonis. The working principle of the circuit is the following. The nanosurf PLL contains a voltage controlled oscillator (VCO) that is locked in frequency and phase with the tuning fork. The excitation voltage from the VCO is applied to the tuning fork, which generates a piezo-electric current. This current is converted to voltage thanks to a current to voltage amplifier from Femto, which has a gain G = 10 8 A/V . This voltage is feed into the PLL whose internal PID loop locks the VCO in frequency and phase to this input voltage. Thus, the VCO provides a measure of the frequency of the tuning fork. The frequency shift is the difference between the resonant frequency of the tuning fork far from the surface and close to the surface. This frequency shift is filtered by a low pass filter (30 Hz) to remove the frequency modes due to the EFM excitation oscillation, then, this signal is introduced in a Z PID loop which controls the tip-sample distance. In this PID loop, we set the frequency shift setpoint. The PID adjusts the tip-sample distance to maintain a constant frequency shift. As mentioned above, in order to perform an EFM measurement, a DC + AC voltage has to be applied. For that matter, a lock-in provides the AC bias voltage at 30 Hz while a constant voltage source is used to apply the DC bias voltage. These bias voltages are summed up in a summing amplifier and are applied to the sample. This EFM excitation modulates the frequency shift at ω = 30 Hz and 2ω = 60 Hz. Connecting the frequency shift output from the PLL to the lock-in allows to obtain the first harmonic f ω , and the second harmonic f 2ω . Equation 3.5 shows that the frequency shift depends on partial derivative of tipsample distance z. In the EFM measurement, the following multipass sequence is used as shown in figure 3.7. For each scan line, the height z 0 is recorded. Then, the tip is re-scanned at the distance z j+1 = z 0 + h j where h j is a constant offset, during which all electrostatic force components (f ω ,f 2ω ,f DC ) are acquired. This sequence is repeated for 9 different height offsets h j , where j = 1..9. This led to the formation of a three dimensional map of the electrostatic force, which takes about 10 hours to acquire.

Results

Now we show that by acquiring 3D maps of the EFM signal, it becomes possible to extract with the minimum of assumptions the capacitive contribution of the nanoparticles, making EFM measurements more quantitative. Figure 3.8 shows the 3D maps of single well-separated nanoparticles at three different temperatures T = 300 K, 160 K and 70 K. It shows only f ω because the signal to noise ratio for f 2ω is smaller than f ω in these measurements. Since the tip scans nanoparticles and its surroundings, we can extract f sub and f N P which are the frequency shifts measured far and above the nanoparticle respectively. Figure 3.9 shows f N P and f sub as a function of h j which are recorded by averaging the frequency shift values on the nanoparticle and out of the nanoparticle respectively. The frequency shifts, f N P and f sub are increasing as the tip-sample distance is decreased. This is due to the increasing capacitance between tip and surface. The slopes of these curves are different and are temperature dependent because of changes in the sensibility of tuning fork. However, these changes have no influence on the extraction of nanoparticle contribution on the EFM signal.

Figure 3.10 a-c shows the EFM profiles ∆f ω (X) = f ω -f sub as the nanoparticle is scanned for different heights and temperatures. In these profiles, |f ω | decreases as the tip scans the nanoparticle.

In next figure, 3.11, we present the different capacitive contributions in the EFM signal. C sub is the capacitance between the tip and the metallic substrate and C N P is the capacitance between the tip and the nanoparticle where it includes the quantum capacitance or the charge response of nanoparticles to applied electric field. C N P depends on electronic state of nanoparticles, for insulating nanoparticles, it depends on dielectric coefficient of the nanoparticles while for metallic nanoparticles, it depends on quantum capacitance, C Q of the nanoparticles, which is related to electronic com- As the tip scans the nanoparticle in the X direction with z = z 0 + h j , the relative change in the capacitance δC can be written as:

δC C sub = 1 C sub [ ∂C sub ∂z .D + C N P ] (3.6)
where D is the diameter of the nanoparticle. The first term in Eq. 3.6 represents the change in the tip-substrate capacitance as the tip rises above the nanoparticle position. Indeed, as the nanoparticle shows up as a bump of height D 10 nm in the topography scan, when the tip is re-scanned with a constant offset h j , it will also rise by the amount D 10 nm at the vertical of the nanoparticle position. Thus, it is important to notice that even a nanoparticle with no charge response, i.e. = 1 and κ = 0, will be visible in the EFM image because of this first term, which is only related to the z-dependence of C sub and does not include the charge response of the nanoparticle. Equation 3.5 shows that the frequency shift is proportional to the second derivative of capacitance, C . Considering that h(z) ∝ z α , we obtain:

dh /h = (α -2)/α(dh/h) (3.7)
and we will get: We find that this ratio goes to zero at large distance, as expected, because the nanoparticle contribution decreases as the height offset increases monotonically. This observation confirms the relevance of our analysis. As the height offset decreases, we find that the nanoparticle contribution to the total capacitance increases, reaching a maximum of 0.3%, for the two temperatures T = 70 K and T = 160 K (T = 300 K data could not be analyzed due to poor signal/noise ratio seen in the figure 3.8). Within the experimental uncertainty, this shows that the charge response of the nanoparticle does not change with temperature. If the nanoparticles were metallic, we would see a large change between the two temperatures as the Verwey temperature is between these two temperatures. This implies that the nanoparticle is insulating and that the main response of the nanoparticle to the electric field is through a dielectric response with > 1. This conclusion was corroborated by XPS spectra of the nanoparticles prepared under the same conditions have shown that the nanoparticle were probably of the Fe 2 O 3 stoichiometry shown in figure 3.14. The presence of the satellite peaks suggest that the nanoparticles are in an oxidation state close to the hematite, Fe 2 O 3 .

C N P C sub = α (α -2)f sub [ ∂f sub ∂z .D + ∆f N P ] (3.8)

S a t e l l i t e p e a k

Tunnel spectroscopy of single magnetite nanoparticles

A second approach in our group to study nanoparticles is to perform the tunnel spectroscopy of single magnetite nanoparticles trapped between nanometer-spaced electrodes, so-called nanogap. This experiment was a part of Qian Yu PhD thesis and is presented in more detail in [START_REF] Yu | Transport electronique dans les nanoparticules: du reseau a la nanoparticule individuelle[END_REF].

Fe 3 O 4 nanoparticles are obtained by the reaction of iron (III) acetylacetonate in benzyl alcohol at 175 • C followed by ligand exchange with citric acid. The citric acid allows the electrostatic stabilization of nanoparticles in water and contributes to the formation of the tunnel barrier. This synthesis provides populations of nanoparticles within the size range d = 6-12 nm, as shown by the TEM picture in figure 3.15a. While better nanoparticle size dispersions can be obtained with synthesis in organic solvents, as studied in EFM experiment presented in the previous section, they present the major inconvenient of employing long alkyl chains surfactants for colloidal stability, which produce thick insulating barrier on the nanoparticles and make them inappropriate for the fabrication of tunnel junctions. We trapped single magnetite nanoparticles between d 10 nm spaced gold electrodes deposited on a chip circuit. In comparison to scanning tunnel spectroscopy, such junctions are highly stable and can be measured as function of temperature. In comparison to planar junctions, where an inert tunnel barrier is often difficult to fabricate, the presence of small ligands at the surface of chemically synthesized nanoparticles allows for the formation of high impedance tunnel barriers, as required for tunnel spectroscopy. The detailed method of nanoparticle trapping in described in [START_REF] Yu | In-vacuum projection of nanoparticles for on-chip tunneling spectroscopy[END_REF].

By ebeam lithography, we fabricate chip circuits, each one contains 32 nanogaps, where the drain and source electrodes are separated by a distance of about 10 nm, figure 3.15b. On this chip, we trap a single nanoparticle within a nanogap following a method described previously [START_REF] Yu | In-vacuum projection of nanoparticles for on-chip tunneling spectroscopy[END_REF]. In this method, the chip is maintained in high vacuum, 10 -6 mbar, and the nanoparticles are projected through a fast pulsed valve. After each projection, where a small amount of nanoparticles is deposited, the tunnel current is measured to check for the presence of a nanoparticle. The projection is repeated hundred of times until a nanoparticle is detected. Representatives high resolution SEM images of nanoparticle devices are shown in figure 3.15c. Five chip circuits have been measured from T = 300 K to T = 4.2 K, which have similar IV curves.

The projection and measurement technique has significant advantages. First, because the sample is fabricated in high vacuum and the process is started with insulating junctions, i.e. unbridged nanogaps, there is no doubt that the conducting channels are formed with the magnetite nanoparticles. Second, the setup is implemented in a glove box under argon, so after fabrication, the sample can be kept free from oxidation indefinitely. Indeed, magnetite oxidize into the insulating hematite, Fe 2 O 3 , in presence of oxygen. We found that only magnetite nanoparticles can lead to the formation of conducting junctions. No conducting junction could be obtained with oxidized magnetite solutions. Furthermore, a conducting junction turns into an insulating junction when kept a few hours in oxygen gas.

Conclusion

We have shown that by acquiring 3D EFM maps of a single nanoparticle, it is possible to extract the tip-nanoparticle contribution to the capacitive EFM signal by subtracting the tip-substrate capacitive contribution, which is known from the z dependence of the EFM signal. Applying this technique to nominally Fe 3 O 4 nanoparticles, we could establish that the nanoparticles behave as insulating dielectric materials, actually indicating that the nanoparticles are oxidized, probably into the insulating Fe 2 O 3 . In a second part, we showed that, by preparing samples in an argon glovebox, conducting tunnel junctions could be fabricated. In order to study the magnetite nanoparticles by means of EFM, we need to prepare the samples in the glovebox and transfer it to EFM chamber under high vacuum to avoid the oxidation. This experiment should be repeated in these conditions to obtain the quantum capacitance of magnetite nanoparticles.

------------------------

Chapter 4

Metal-Insulator Transition in SrTiO 3

Introduction

This chapter presents a study of metal-SrTiO 3 -metal memristors where we find that the doping concentration in SrTiO 3 can be fine-tuned through electric field migration of oxygen vacancies. In this tunnel junction device, the evolution of the Density Of States (DOS) can be followed continuously across the metal-insulator transition. At very low dopant concentration, the junction displays characteristic signatures of discrete dopants levels. As the dopant concentration increases, the semiconductor band gap fills in but a soft Coulomb gap remains. At even higher doping, a transition to a metallic state occurs where the DOS at the Fermi level becomes finite and Altshuler-Aronov corrections to the DOS are observed. At the critical point of the metal-insulator transition, the DOS scales linearly with energy N (ε) ∼ ε, the possible signature of multifractality. In this chapter, we will first explain briefly the memristive devices and their operation mechanism. In the second part we show in detail the use of memristive devices to study the metal-insulator transition in SrTiO 3 .

Memristors

The three fundamental passive circuit elements are the resistor, the capacitor and the inductor. However, Leon Chua in 1971 [START_REF] Chua | Memristor-The missing circuit element[END_REF] demonstrated the existence of a fourth element, the memristor, as shown in figure 4.1a. The memristor is a short term for Memory-Resistor, where the M emristance, M , represents a functional relation between charge and flux [START_REF] Strukov | The missing memristor found[END_REF]. These materials attracted considerable attention due to their possible application as memory-elements [START_REF] Yang | Memristive devices for computing[END_REF]. The simplest configuration for these devices consists in a piece of insulator or semiconductor stacked between two metallic electrodes. These devices are characterized by the presence of a switching event in the current-voltage characteristics, figure 4.1b. This switching event leads to a transition from an insulating state to a conducting state. The memristive devices can be classified according to the switching mechanism. The two main categories of memristive devices are anion-based devices and cation-based devices. A brief explanation of these two categories follows.

Anion-based memristive devices

The anion based memristors can be made from a wide range of binary oxide insulators such as MgO, TiO x , ZrO x , HfO x , VO x and perovskites as SrTiO 3 , BiFeO 3 , etc. The most probable explanation for the switching is the creation and movement of oxygen vacancies upon applying a bias voltage between the two electrodes. Figure 4.2 shows a Cr-doped SrTiO 3 [START_REF] Janousch | Role of Oxygen Vacancies in Cr-Doped SrTiO3 for Resistance-Change Memory[END_REF] where an oxygen vacancy path is formed upon applying an electric field. This is an infrared thermal image of this memory cell device collected by applying 5 mA current and 30 V bias voltage. The false-color image reflects the temperature distribution between the electrodes while the current is flowing through the device. In the vicinity of the anode, temperature is highest indicating the highest resistive part of the device.

Cation-based memristive devices

In cation-based devices, the insulating material is an electrolyte such as sulfides, iodides, selenides based ionic crystals. This electrolyte is sandwiched between two elec-trodes. One electrode is made from an electrochemically active material such as Cu or Ag and the counter electrode in an electrochemically inert metallic material such as W, Pt, Au, Mo, etc. In these devices, the mobile ionic species is believed to be the metallic cation. Figure 4.3 shows SEM images of a cation-based memristive device [START_REF] Guo | Understanding the switching-off mechanism in Ag[sup +] migration based resistively switching model systems[END_REF] which is constituted of a Pt/H 2 O/Ag cell. These images show the electrochemical growth of metallic filament under the application of a voltage. By applying -1 V to the Pt electrode, the metallic Ag is oxidized to Ag + and migrates through H 2 O to the Pt electrode. At this electrode, Ag + is reduced back to metallic Ag. As a consequence, metallic Ag dendrites grow from the Pt electrode toward the Ag electrode. [START_REF] Farrel | X-Ray Diffractometry of Low-Temperature Phase Transformations in Strontium Titanate[END_REF]. One of the important characteristic of SrTiO 3 is its high dielectric constant, = 300, at room temperature and = 24000 at low temperature [START_REF] Müller | SrTiO {3}: An intrinsic quantum paraelectric below 4 K[END_REF].

SrTiO 3 is an insulator with the band gap E g = 3.2 eV where the conduction band is constituted by t 2g titanium orbitals (3d xy , 3d xz , 3d yz ) and the valence band is constituted by the oxygen 2p orbitals [START_REF] Mattheiss | Energy Bands for KNiF {3}, SrTiO {3}, KMoO {3}, and KTaO {3}[END_REF].

One of the remarkable property of SrTiO 3 is the appearance of superconductivity A first category of methods to change the carrier concentration in SrTiO 3 is to create oxygen vacancies, which are negatively charged centers and play the role of ntype dopants. A variety of methods can be employed to create oxygen vacancies in SrTiO 3 such as sample heating [START_REF] Lin | Fermi Surface of the Most Dilute Superconductor[END_REF], ion bombardment [START_REF] David | Highly conductive nanolayers on strontium titanate produced by preferential ion-beam etching[END_REF] and cleaving under UHV [START_REF] Santander-Syro | Twodimensional electron gas with universal subbands at the surface of SrTiO(3)[END_REF].

Another method to create a 2DEG at the surface of SrTiO 3 is to use a Field-Effect Transistor (FET) geometry. In standard FET geometry, the electric field is applied with a gate electrode separated from SrTiO 3 interface by a dielectric [START_REF] Biscaras | Two-Dimensional Superconducting Phase in LaTiO {3}/SrTiO {3} Heterostructures Induced by High-Mobility Carrier Doping[END_REF]. Recently, another technique employing an electrolyte has been used by [START_REF] Ueno | Electrostatic charge accumulation versus electrochemical doping in SrTiO[sub 3] electric double layer transistors[END_REF][START_REF] Ueno | Electric-field-induced superconductivity in an insulator[END_REF][START_REF] Lee | Phase Diagram of Electrostatically Doped SrTiO {3}[END_REF][START_REF] Lee | Electrolyte Gate-Controlled Kondo Effect in SrTiO {3}[END_REF] to create a large electric field through the formation of an electrostatic double layer. In this device, applying a gate voltage to the electrolyte leads to the migration of ions from the electrolyte toward the electrodes. Consequently, an electrostatic double layer forms and creates a large electric field. This electric field leads to the accumulation of electrons at the surface of the sample. Figure 4.6 shows the observation of superconductivity in an electrostatic double layer transistor fabricated with SrTiO 3 . Thus, we can notice that a memristive device is a mixed conductor, where ions and electrons are mobile. As SrTiO 3 is also an electrolyte for oxygen vacancies, applying a voltage on two closely spaced electrodes deposited on a SrTiO 3 substrate should lead to vacancies motion and so to a change in the electronic density of states. This is the principle of the experiment described in this chapter.

Sample preparation

The (110) SrTiO 3 substrates are purchased from Crystal GmbH. After a standard cleaning process (5 minutes in acetone and 5 minutes in isopropyl alcohol), a thin layer of PMMA, 150 nm thick, is spin coated and baked at 160 • C during 15 minutes. Then, a thin layer, 10 nm, of Al is evaporated on the surface of PMMA to avoid charging effects. This step is required as SrTiO 3 is insulating.

The patterning of the electrodes is done by E-beam lithography. Figure 4.7 shows the design of the electrodes. After the e-beam patterning, the samples are dipped in potassium hydroxide (KOH), to remove the Al layer. Then, the samples are developed in MIBK solution during 70 seconds and washed in isopropanol alcohol during 30 seconds. The next step is to evaporate the gold electrodes, 70 nm thick, which is followed by a lift-off process in acetone. The last step of sample preparation is the wire bonding. All steps of the sample fabrication were done in the clean room of ENS, which is part of the center of micro fabrication of Paris-centre.

Figure 4.8 shows SEM images of samples. All the measurements were performed in a 3 He fridge with the base temperature of T = 260 mK.

Results

Forming step

After preparation, the sample is placed in the 3 He fridge, which is pumped at room temperature down to low pressure, P = 10 -6 mbar. At room temperature, a conducting filament is formed, as illustrated in figure 4.9, by ramping up the voltage until a resistance switching is observed, figure 4.10. This is the so-called forming step where oxygen vacancies accumulate along some percolating path between the two electrodes. This leads to a junction of resistance of only a few hundred Ohms.

As soon as the switching occurs, the sample is cooled down to T = 260 mK to avoid thermal migration effects.

Superconductivity

The resistance of the formed junction is nearly temperature independent down to low temperature where it starts to decrease below T ∼ 340 mK indicating the apparition of superconductivity as shown in figure 4.11 for two different samples. Here the resistance does not decrease to zero because of two-wire measurement configuration.

Doped STO is known to become superconducting in this temperature range [START_REF] Koonce | Superconducting Transition Temperatures of Semiconducting SrTiO {3}[END_REF], with T c ∼ 350 mK for a carrier concentration of the order of 10 20 e cm -3 [START_REF] Ueno | Electric-field-induced superconductivity in an insulator[END_REF].

Electric field induced motion of oxygen vacancies

As illustrated in figure 4.12, the electrodes are connected by a conducting filament. By applying a high voltage bias between cathode and anode, several percolation paths made of oxygen vacancies are formed but as soon as one filament reaches the anode, As the junction is now insulating, applying negative voltage pulses moves the vacancies in the opposite direction and leads now to an increase of the conductance as showed in left panel of figure 4.13. Upon reaching a high differential conductance, G(0) ∼ 2 mS, a single positive voltage pulse (∼ 5 V ) is applied again, which switches the junction insulating once more. This protocol can be executed many times -up to 10 times for this junction. For each run, the junction is tuned from insulating to the conducting state by applying negative voltage pulses of small magnitude. Between each run, a positive pulse is applied to turn the junction insulating. These series of voltage pulses and differential conductance curves are showed in figure 4.14.

dI/dV(µS) After each run a positive pulse of V = 5 V is applied in order to return the system to the insulating state. The magnitude of applied pulses are shown in right panel of each set of curves.

temperature where thermal activated migration of ions is completely inhibited and vacancy motion is due to electric-field activated migration of ions. As electric-field induced migration of ions can only occur on a length of the order of the Debye screening length, and because the screening length is short in conducting filaments, ions migration only occurs on a short length scale. This leads to a device with characteristics depicted in figure 4.12, where a nanosized region of weakly doped SrTiO 3 is located between two conducting electrodes. This picture is consistent with previous explanations of memristive behaviors in titanium oxide based devices where the resistance switching is believed to occur in the immediate vicinity of the electrodes as illustrated in figure 4.15. The weakly doped SrTiO 3 nanosized region is connected on one side by an Ohmic contact to a highly doped-SrTiO 3 electrode, and, on the other side, is connected by a Schottky barrier to the gold electrode, where the Schottky barrier makes a tunnel barrier for electrons [START_REF] Wolf | Principles of Electron Tunneling SpectroscopySecond Edition[END_REF]. At low doping, the barrier height is given by the energy difference between the works functions of SrTiO 3 φ ST O = 4.2 eV [87] and gold φ gold = 5.2 eV [START_REF] Huber | THE EFFECT OF MERCURY CONTAMINATION ON THE WORK FUNCTION OF GOLD[END_REF], which gives φ ∼ 1 eV . As we will see below, features of interest in the DOS are located at low energy, ε < 0.1 eV , implying that the IV curves can be interpreted as DOS curves. As the doping concentration increases in this n-doped semiconductor, the barrier height increases, while its width decrease [START_REF] Sze | Physics of Semiconductor Devices[END_REF], and so the junction remains in the tunneling regime. In this geometry, the differential conductance curves shown in figure 4.14 represent, at low voltage, V < 100 meV , the DOS of the weakly doped region:

OFF ON

G(V ) G 0 = N (ε) N 0 ∂f (ε -eV, T ) ∂eV dε (4.1)
Thus, this method allows following precisely the evolution of the DOS in the nanosized region of SrTiO 3 located in the immediate vicinity of the gold electrode as the oxygen vacancies' concentration is tuned by applying voltage pulses.

When a voltage of positive polarity is applied, oxygen vacancies are repelled at the junction and the DOS decreases, for a voltage pulse of negative polarity, oxygen vacancies accumulate at the junction and the DOS increases. These jumps do not appear to be correlated with any specific values of the conductance. As the 6 different runs shown in figure 4.14 are plotted together, figure 4. [START_REF] Mott | Discussion of the paper by de Boer and Verwey[END_REF], we can see, first, that similar conductance curves can be reproduced even after a full cycle from insulating to conducting, which indicates that doping can be tuned reversibly in those junctions; second, that the junction conductance can be tuned continuously to any value as shown by the plot of conductance at zero bias for all measured conductance curves in figure 4.18.

Evolution of the density of states

A detailed look at the differential conductance curves taken at different conductance values allows to clearly characterize the evolution of the DOS in SrTiO 3 as a function of doping as shown in figure 4. [START_REF] Bocquet | Electronic structure of early 3d-transitionmetal oxides by analysis of the 2p core-level photoemission spectra[END_REF].

At the lowest doping, G(V ) shows a sharp gap and well resolved peaks at voltage values (-41 mV , 24 mV , 78 mV ). These peaks, which have been observed in a second -0.1 0.0 0.1 0 1 2 dI/dV (Arb. Units)

V Drain (V) doping level a b c d e 2 ( ) Ñ ε ε 1/2 ( ) (0) 1 N N ε ε   = +   ∆     ( ) Ñ ε ε
N (ε) = N (0)[1 + ε E T 1/2
], as shown by the dotted lines. E T ∼ 0.1 eV for curve e) E T ∼ 0.003 eV for curve d). Finally, just at the critical point of the Metal-Insulator Transition, the DOS has a linear dependence with energy, N (ε) ∼ ε , dotted line. Applying a magnetic field leads to a Zeeman splitting of the peaks, as shown in figure 4.21, which can be fitted approximately by the relation δε = m µ B B + 2m s B, using for the orbital quantum numbers m = -2, +1, -1 and for the spin quantum numbers m s = ±1/2. As the orbital number m = -2 has to be included to fit the Zeeman splitting, this indicates that this discrete doping level arises from doped electronic states into the t 2g conduction band composed of the 3d xy , 3d xz and 3d yz titanium orbitals [START_REF] Mattheiss | Energy Bands for KNiF {3}, SrTiO {3}, KMoO {3}, and KTaO {3}[END_REF]. This is further confirmed by the observation that this state is located just below the gap edge as expected for n-type dopants. Only samples annealed at high temperature [START_REF] Lee | Optical absorption and transport in semiconducting SrTiO {3}[END_REF][START_REF] Spinelli | Electronic transport in doped SrTiO {3}: Conduction mechanisms and potential applications[END_REF] have hydrogenic donor levels located very close to the conduction band. In non-annealed samples, as in our case, from an analysis of the temperature dependence of the Hall coefficient [START_REF] Lee | Optical absorption and transport in semiconducting SrTiO {3}[END_REF][START_REF] Tufte | Electron Mobility in Semiconducting Strontium Titanate[END_REF] and the observation of sharp peaks in optical absorption spectrum [START_REF] Lee | Optical absorption and transport in semiconducting SrTiO {3}[END_REF], it has been concluded that the oxygen vacancies levels may be located almost anywhere from nearly zero to about 0.1 eV below the conduction band, as observed in our tunneling spectrum.

While the observation of individual dopant states was attempted through the formation of nano-sized Schottky diodes [START_REF] Smit | Conductance distribution in nanometer-sized semiconductor devices due to dopant statistics[END_REF][START_REF] Calvet | Electron transport measurements of Schottky barrier inhomogeneities[END_REF] and nanosized field effect transistors [START_REF] Calvet | Fano resonance in electron transport through single dopant atoms[END_REF][START_REF] Calvet | Observation of the Linear Stark Effect in a Single Acceptor in Si[END_REF], we find that the use of oxide semiconductors, with its possibility of fine-tuning the dopant concentration in-situ at low temperature, offers a remarkable tool for the study of individual dopant states. This method allows studying the metal-insulator transition as a function of doping with unprecedented resolution. As the doping concentration increases, the gap fills in and sharp features disappear, however, a soft gap remains where the DOS vanishes at the Fermi level with the quadratic form N (ε) ∼ ε 2 in 3D, due to Coulomb interactions as shown by Efros and Shklovskii [START_REF] Shklovskii | Coulomb gap and low temperature conductivity of disordered systems[END_REF]. Such a soft gap was observed previously in doped semiconductors through tunneling spectroscopy [START_REF] Mcmillan | Electron Tunneling Experiments on Amorphous Ge {1-x}Au {x}[END_REF][START_REF] Massey | Electron Tunneling Study of Coulomb Correlations across the Metal-Isulator Transition in Si:B[END_REF][START_REF] Mark Lee | Coulomb gap in a doped semiconductor near the metal-insulator transition: Tunneling experiment and scaling ansatz[END_REF][START_REF] Bielejec | Hard Correlation Gap Observed in Quench-Condensed Ultrathin Beryllium[END_REF][START_REF] Lee | Phase Diagram of Coulomb Interactions across the Metal-Insulator Transition in Si:B[END_REF]. As the dopant concentration increases, the soft gap closes and a transition to a metallic regime occurs. In the metallic regime, the DOS is finite at the Fermi level and a cusp,

N (ε) = N (0)[1 + ε E T 1/2
] in 3D, is observed, where the correlation energy parameter E T = D/ 2 , i.e. the Thouless energy, depends on the diffusion constant D and the electron mean free path . This cusp is due to the exchange part of the Coulomb interactions as shown by McMillan [START_REF] Mcmillan | Scaling theory of the metal-insulator transition in amorphous materials[END_REF], Altshuler and Aronov [START_REF] Altshuler | Zero bias anomaly in tunnel resistance and electron-electron interaction[END_REF]. Fitting the data shows that E T decreases at the approach of the transition, as expected when the electronic states at the Fermi level gets more and more localized. The maximum value found for E T ∼ 0.1 eV can be explained assuming the reasonable values for D ∼ 2 cm 2 s -1 and ∼ 1 nm. Finally, just at the transition, the critical DOS follows a linear behaviour, N (ε) ∼ ε . The question of the energy dependence of the critical DOS remains one of the major theoretical question of the Mott-Anderson metal-insulator transition [START_REF] Belitz | The Anderson-Mott transition[END_REF]. A theoretical prediction suggests [START_REF] Mark Lee | Coulomb gap in a doped semiconductor near the metal-insulator transition: Tunneling experiment and scaling ansatz[END_REF] that the DOS should follow N c (ε) ∼ ε (3/η)-1 ; this implies that η ∼ 1.5, which is within the bounds 1< η <3 , as theoretically expected [START_REF] Mcmillan | Electron Tunneling Experiments on Amorphous Ge {1-x}Au {x}[END_REF]. This power law suppression of the DOS at criticality is a signature of multifractality, a remarkable property of electronic wavefunctions near the metal-insulator transition [100]. Recent theoretical works have found that multifractality persists even in presence of Coulomb interactions [START_REF] Burmistrov | Multifractality at Anderson Transitions with Coulomb Interaction[END_REF][START_REF] Amini | Multifractality and quantum-to-classical crossover in the Coulomb anomaly at the MottâAnderson metalâinsulator transition[END_REF] and multi-fractal characteristics have been found in STM spectroscopy measurements of Ga 1-x Mn x As [START_REF] Richardella | Visualizing critical correlations near the metal-insulator transition in Ga(1-x)Mn(x)As[END_REF]. Note that the DOS curves were compared with theoretical predictions assuming tridimensionality. Indeed, while previous works [START_REF] Yang | Memristive devices for computing[END_REF][START_REF] Waser | Nanoionics-based resistive switching memories[END_REF] have shown that the vacancies are distributed along filamentary paths, we don't expect that quantum confinement effects would alter the 3D band structure of SrTiO 3 along the filament. For this reason, we can assume that the effect of doping is to inject carriers into the 3D band structure of SrTiO 3 .

While we don't have a direct measure of the oxygen vacancy concentration, we can estimate that this concentration changes from nearly zero, where single impurity levels are observed, to 10 20 e cm -3 , where the superconducting transition temperature T c ∼ 350 mK is maximum [START_REF] Ueno | Discovery of superconductivity in KTaOâ by electrostatic carrier doping[END_REF]. The metal-insulator transition should be at concentrations below n ∼ 10 16 e cm -3 according to results from Spinelli et al. [START_REF] Spinelli | Electronic transport in doped SrTiO {3}: Conduction mechanisms and potential applications[END_REF].

This method should apply to a variety of other oxides, including materials presenting strong electron correlations such as the cuprates or the manganites where oxygen could be moved and leads to the transition from insulator to metal and from metal to insulator.

Conclusion

We have found that oxygen vacancies could be migrated by an electric field at very low temperature in nanosized metal-oxide-metal junctions. As the undoped region of the junction is connected by a Schottky tunnel barrier to the metal electrode, the current-voltage characteristics provides the DOS. Upon applying voltage pulses, we find possible to tune continuously the DOS across the metal-insulator transition where all the characteristics expected in the vicinity of the metal-insulator transition are found: the soft Coulomb gap on the insulating side and the Altshuler -Aronov corrections on the metallic side.

Chapter 5 Topological Insulators

Introduction

Topological insulators have attracted a lot of attention in the last 10 years. This interest is due to the prediction of topologically protected states at the edges (surfaces) of 2D (3D) materials. While signature of these topological protected states have been identified in the 2D topological insulator HgTe [START_REF] Roth | Nonlocal transport in the quantum spin Hall state[END_REF][START_REF] König | Quantum spin hall insulator state in HgTe quantum wells[END_REF], one of the main issue in the field is to identify signatures of the topological protection in transport measurements of 3D topological insulators such as Bi 2 Se 3 . In these materials, the topologically protected states have been observed and characterized by STM and photoemission measurements [START_REF] Hasan | Colloquium: Topological insulators[END_REF]. Given that the existence of these surface states has been demonstrated, if the conductance of the sample was determined by the propagation of these states alone the edges of the sample, we could expect that this conductance should be of the order of the quantum of conductance. As a matter of fact, the conductance measured in these 3D topological insulators is usually much smaller than the quantum of conductance. The reason most often invoked for this behavior is the presence of impurity states, such as Se vacancies, that dope the bulk of the semiconductor with electrons, which will dominate electronic transport even at low temperature, thus hiding the transport properties of the edge states.

While numerous works exist on trying to find more suitable materials, such as Bi 2 Se 3-x Te x [START_REF] Xiong | High-field Shubnikovâde Haas oscillations in the topological insulator Bi {2}Te {2}Se[END_REF] or SmB 6 [START_REF] Kim | Surface hall effect and nonlocal transport in SmBâ: evidence for surface conduction[END_REF][START_REF] Wolgast | Low-temperature surface conduction in the Kondo insulator SmB {6}[END_REF], with lower residual bulk conductivity, it is also important to develop methods to identify the surface states through transport mea-surements. The approach that we decided to employ is to identify typical quantum signatures of electronic transport and to study these quantum signatures as function of carrier density and film thickness.

The most obvious quantum signatures would be:

-The observation of an anomalous period for the Aharonov-Bohm effect [START_REF] Jens | Quantum interference and Aharonov-Bohm oscillations in topological insulators[END_REF] -The observation of anomalous Josephson currents and/or Andreev reflection in hybrid superconducting/topological insulator devices [START_REF] Orlyanchik | Signature of a topological phase transition in the Josephson supercurrent through a topological insulator[END_REF][START_REF] Kurter | Dynamical gate-tunable supercurrents in topological Josephson junctions[END_REF][START_REF] Kurter | Evidence for an anomalous current-phase relation in topological insulator Josephson junctions[END_REF][START_REF] Finck | Phase Coherence and Andreev Reflection in Topological Insulator Devices[END_REF] Recent works by the group of Van Harlingen suggest that it is indeed possible to identify signatures of the topological states in Josephson junctions fabricated with Bi 2 Se 3 films. Furthermore, Aharonov-Casher oscillations have also been observed in ring interferometers fabricated with Bi 2 Se 3 films [START_REF] Qu | Aharonov-Casher Effect in Bi {2}Se {3} Square-Ring Interferometers[END_REF].

Thank to a collaboration with the group of M. Marangolo (INSP), we have had access to Bi 2 Se 3 thin films grown by molecular beam epitaxy. The motivation of this work is to find experimental signatures of the surface states in transport measurements, through either detection of the Aharanov-Bohm effect or through measurements of the Josephson effect across the Bi 2 Se 3 thin films.

These measurements could be realized as function of a gate voltage and as function of the film thickness. This chapter describes the first steps in this direction. I developed the microfabrication method to fabricate the first devices, Hall bars and rings, from Bi 2 Se 3 thin films.

Bi 2 Se 3

The first experimental observation of the surface states in Bi 2 Se 3 has been done by Xia et al. using angle-resolved photoemission spectroscopy (ARPES) [START_REF] Xia | Observation of a large-gap topologicalinsulator class with a single Dirac cone on the surface[END_REF].

The surface states of Bi 2 Se 3 appear as a single Dirac cone with linear energy dispersion as shown in figure 5.1. While Bi 2 Se 3 has a large band gap of ∼ 0.3 eV , implying that high purity Bi 2 Se 3 could exhibit topological insulating behavior at room temperature [START_REF] Hsieh | A tunable topological insulator in the spin helical Dirac transport regime[END_REF], the samples usually grown have too many impurities that dope the semiconductor and prevent the observation of surface states in electrical transport experiments.

Aharanov-Bohm effect

In 1959, Aharonov and Bohm [START_REF] Aharonov | Significance of Electromagnetic Potentials in the Quantum Theory[END_REF] predicted that the wave function of a charged particle enclosing an electrostatic or magnetic potential experience a phase shift. If an electron passes through a ring as illustrated in figure 5.2, its wave function splits in two parts along each arms of the ring. The wave function on each arm of the ring experience a phase shift, which are of opposite signs. Consequently, a phase difference is established between the two arms of the rings.

The phase accumulated by an electron passing through one arm in presence of a magnetic field is given by:

ϕ = - e - → A .dx (5.1)
where e is the electron charge, is the Planck constant and -→ A is magnetic vector potential. The phase difference between the two arms is the difference between two integrals that can be expressed as a closed loop integral given by equation 5. where φ B is the magnetic flux enclosed by the two arms and is equal to φ B = B.S where S is the ring area. Thus, the phase difference can be tuned with the amplitude of the magnetic field. The conductance of the ring oscillates with the periodicity of the flux quantum φ 0 = h/e, as a consequence of the cycling pattern of constructive and destructive interferences.

In 1985, Webb et al. [START_REF] Webb | Observation of h/e Aharonov-Bohm Oscillations in Normal-Metal Rings[END_REF] reported the first experimental observation of oscillations in the magnetotresistance of a normal metal ring. Later in 1987, this experiment was reproduced in GaAs/AlGaAs hetero-structure [START_REF] Timp | Quantum transport in one-dimensional GaAs/AlGaAs microstructures[END_REF].

This interference effect is measurable if electrons keep their phase coherence while passing through the ring. The maximum distance on which an electron keeps its phase coherence is called the phase coherence length, l ϕ . This implies that, to observe the Aharanov-Bohm effect, the characteristic size of the sample has to be of the order of l ϕ . of PMMA on the surface of as-grown samples followed by baking at 160 • C during 15 minutes. The PMMA is patterned with an electron beam of 20 keV energy and a dose of 250 µC/cm 2 . After patterning, the sample is developed in MIBK solution during 70 sec and washed by isopropanol during 30 sec. After development, we evaporate the gold electrodes, 70 nm thick, followed by lift-off in acetone. Since the surface of Bi 2 Se 3 is highly conducting, an etching process is needed. To do so, a second e-beam lithography patterning is done by using an etching mask as illustrated in figure 5.4. We use Reactive Ion Etching (RIE) to etch the sample. The etching parameters are as follows: Argon flow=40 sccm, working pressure=50 mT, forward power=80 W and etching time=30 sec. After etching, the sample is wire bonded to a ceramic sample holder, which can be installed on the cryostat to be measured at low temperature. Figure 5.5 shows a SEM images of two Aharanov-Bohm rings. In a topological insulator, strong spin-orbit coupling prevents the usual weak localization phenomena; instead, weak antilocalization is observed. At low temperature T < 15 K a sharp cusp is observed which is consistent with the results of previous magnetotransport measurements of Bi 2 Se 3 thin films [START_REF] Matsuo | Weak antilocalization and conductance fluctuation in a submicrometer-sized wire of epitaxial Bi {2}Se {3}[END_REF][START_REF] He | Impurity Effect on Weak Antilocalization in the Topological Insulator Bi {2}Te {3}[END_REF]. The weak antilocalization can be explained by Hikami-Larkin-Nagaoka (HLN) model [START_REF] Hikami | Spin-Orbit Interaction and Magnetoresistance in the Two Dimensional Random System[END_REF]. In this model the conductance correction is given by:

Sample preparation

Results

δG = G(B) -G(0) = - αe 2 2π 2 ln( B φ B ) -ψ( 1 2 + B φ B ) (5.3)
where α is a coefficient and is equal to 1/2, B φ = /4el 2 φ is the magnetic field characteristic of the phase coherence length l φ and ψ is the digamma function. We used the Eq. 5.3 to fit the experimental data. Figure 5.8 shows the magnetoresistance data at 2 K, together with the calculated conductance correction for several values of the characteristic field B φ . One finds that the data are best described by the HLN equation with the characteristic field value B φ = 0.1 T . This gives for the coherence length, the value l φ = 40 nm.

Figure 5.9 shows the magnetoresistance data as a function of magnetic field for a 9 QL thick Bi 2 Se 3 Aharanov-Bohm ring at 2 K. The expected period of oscillation is given by ∆B = φ 0 /S where φ 0 = h/e is the flux quanta and S is the averaged area of the ring. This leads to ∆B = 0.6 mT for a ring of 2.7 µm size. No clear signatures of the Aharanov-Bohm oscillations could yet be observed. At this point, it is not yet surprising, given that the coherence length obtained from the HLN gives a value much smaller than the size of the ring we fabricated. Experiments at lower temperature are required to test the existence of these Aharanov-Bohm oscillations.

Conclusion

In conclusion, we have been able to fabricate Hall bar and Aharanov-Bohm rings with MBE grown Bi 2 Se 3 films. The microfabrication process is well understood and the parameters for each microfabrication step is optimized. We characterized the 9 QL thick sample at low temperature. As shown by recent results of the Van Harlingen group [START_REF] Orlyanchik | Signature of a topological phase transition in the Josephson supercurrent through a topological insulator[END_REF][START_REF] Kurter | Dynamical gate-tunable supercurrents in topological Josephson junctions[END_REF][START_REF] Kurter | Evidence for an anomalous current-phase relation in topological insulator Josephson junctions[END_REF][START_REF] Finck | Phase Coherence and Andreev Reflection in Topological Insulator Devices[END_REF] the question of the possible identification of topological surfaces states in Bi 2 Se 3 through measurements of quantum transport properties such as Aharonov-Bohm oscillations or Josephson current, is still an open subject. Much work remains to be done in this direction. In particular, the question of the evolution of Aharonov-Bohm oscillations or the Josephson current with the carrier density, which would be modified by applying a gate voltage, or the film thickness, remains to be explored.

Chapter 6

Summary in French

Transition métal-isolant

Les mécanismes à l'origine de la localisation des états électroniques impliquent des effets quantiques collectifs dont la transition métal-isolant constitue une manifestation macroscopique remarquable.

La transition d'un système métallique vers un état isolant va dépendre essentiellement de la dimensionnalité du système, du désordre et des interactions de Coulomb entre électrons. L'identification et l'étude de systèmes pouvant présenter une transition métal-isolant reste un enjeu majeur de la physique des électrons corrélés [START_REF] Belitz | The Anderson-Mott transition[END_REF][START_REF] Lee | Disordered electronic systems[END_REF][START_REF] Imada | Metal-insulator transitions[END_REF]100,[START_REF] Mott | The metal-nonmetal transition[END_REF][START_REF] Abrahams | Metallic behavior and related phenomena in two dimensions[END_REF].

Nous avons étudié la transition métal-isolant dans deux oxydes differents, la magnétite Fe 3 O 4 et le titanate de strontium SrTiO 3 .

Réponse de charges de nanoparticule de magnetite par EFM

En tant que sonde très sensible du champ électrique, la Microscopie à Force Electrostatique (EFM) est bien adaptée à l'étude de la réponse de charges de matériaux à l'échelle nanométrique. Cette technique a été utilisée pour discriminer des nanotubes isolants de nanotubes métalliques [54,[START_REF] Lu | A scanning probe microscopy based assay for single-walled carbon nanotube metallicity[END_REF], pour mesurer la capacité électrique de ces mêmes nanotubes [START_REF] Mélin | Electric force microscopy of individually charged nanoparticles on conductors: An analytical model for quantitative charge imaging[END_REF][START_REF] Mélin | Probing Nanoscale Dipole-Dipole Interactions by Electric Force Microscopy[END_REF][START_REF] Paillet | Electrostatics of Individual Single-Walled Carbon Nanotubes Investigated by Electrostatic Force Microscopy[END_REF] ainsi que pour mesurer le coefficient diélectrique de nanoparticules isolantes et de virus [START_REF] Fumagalli | Label-free identification of single dielectric nanoparticles and viruses with ultraweak polarization forces[END_REF].

La microscopie EFM est aussi d'un intérêt fondamental pour mesurer la compressibilité de charges κ = dn/dµ de matériaux conducteurs. La compressibilité de charge est une grandeur thermodynamique fondamentale des systèmes électroniques [START_REF] Dultz | Thermodynamic Signature of a Two-Dimensional Metal-Insulator Transition[END_REF][START_REF] Li | Very large capacitance enhancement in a two-dimensional electron system[END_REF] dont la détermination demande le mesure de la réponse de charge sur des longueurs de l'ordre de la longueur d'écrantage.

Pour les métaux simples, la compressibilité est large, elle est simplement reliée à la densité d'états, n ∝ µ 3/2 ⇒ dn/dµ ∝ µ 1/2 ; en contraste, dans les isolants de bande, la compressibilité de charge est proche de zero. Dans les matériaux corrélés, on attend à ce que la compressibilité de charge soit altérée par les interactions de Coulomb, comme cela est illustré par la prédiction d'une divergence de cette compressibilité à la transition métal-isolant [START_REF] Watanabe | Thermodynamic Relations in Correlated Systems[END_REF].

Les résultats de mesure de réponse de charges de nanoparticule de magnetite par EFM sont présenté au chapitre 3.

Transition métal-isolant dans SrTiO 3

La recherche de méthodes permettant le dopage des matériaux et leur contrôle au voisinage de la transition métal-isolant reste toujours un thème essentiel de la physique des électrons corrélés [START_REF] Ahn | Electrostatic modification of novel materials[END_REF]. Inspiré par des travaux récents par HPLabs [START_REF] Strukov | The missing memristor found[END_REF][START_REF] Yang | Memristive devices for computing[END_REF] sur la fabrication de memristors à partir d'oxyde de titane, TiO 2 , lesquels sont basés sur des nanofilaments conducteurs formés par la migration de lacunes d'oxygène sous champ électrique [START_REF] Dearnaley | Electrical phenomena in amorphous oxide films[END_REF], nous avons développé une nouvelle méthode pour étudier la formation de ces nanofilaments conducteurs, avec une résolution telle que même les effets de la diffusion d'un faible nombre de lacunes d'oxygène peuvent être caractérisés. Cette méthode permet d'étudier l'évolution de la densité d'états avec le taux de dopage avec une très grande résolution.

Évolution de la densité d'états au travers de la transition métal-isolant

Les composants memristifs ont attiré une attention considérable avec la découverte que les résistances électro-actives [START_REF] Strukov | The missing memristor found[END_REF] constituent une réalisation possible de memristors, dont l'existence avait été prédite par L. Chua en 1971 [START_REF] Chua | Memristor-The missing circuit element[END_REF]. Depuis la première revue sur les sauts électro-résistifs il y a environ une cinquantaine d'années [START_REF] Dearnaley | Electrical phenomena in amorphous oxide films[END_REF], de nombreux travaux sur les oxydes de métaux de transitions MO x et SrTiO 3 , voir revues [START_REF] Yang | Memristive devices for computing[END_REF][START_REF] Waser | Nanoionics-based resistive switching memories[END_REF], ont démontré que ces sauts électro-résistifs étaient dues à la migration des anions oxygènes ou, de manière équivalente, de lacunes d'oxygènes chargées positivement, activée par le champ électrique et/ou la température. Cette migration des anions conduit à un changement de valence des cations métalliques et, par conséquent, à un changement du taux d'occupation de la bande d [START_REF] Yang | Memristive switching mechanism for metal/oxide/metal nanodevices[END_REF]. Dans les oxydes à base de titane, tels que TiO 2 ou SrTiO 3 , le nombre d'oxydation du cation titane change de +IV à +III, ce qui correspond à un dopage de type n. Ces développements récents pourraient éventuellement mener à l'émergence d'une nouvelle branche de l'électronique où la manipulation contrôlée des ions dans les composants électroniques pourra être utilisée pour contrôler leur état, pour ouvrir ou fermer un canal conducteur, pour stocker un élément de mémoire, ou peut être, pour changer l'ordre électronique du composant d'un état isolant vers un état conducteur ou même un supraconducteur. Ces composants électro-ioniques sont également envisagés pour la réalisation de nouveaux types de composants comme des synapses artificielles [START_REF] Pickett | A scalable neuristor built with Mott memristors[END_REF], une étape nécessaire au développement d'architectures informatiques neuromorphiques.

Comme nous l'avons vu dans la partie théorie, l'étude des semiconducteurs en fonction du dopage est essentielle pour la compréhension du rôle du désordre et des interactions de Coulomb dans la transition métal-isolant [START_REF] Belitz | The Anderson-Mott transition[END_REF][START_REF] Lee | Disordered electronic systems[END_REF]. Dans ce domaine, la majorité des études sont réalisées avec des échantillons multiples dont le dopage est fixé lors de la fabrication de l'échantillon [START_REF] Mcmillan | Electron Tunneling Experiments on Amorphous Ge {1-x}Au {x}[END_REF][START_REF] Massey | Electron Tunneling Study of Coulomb Correlations across the Metal-Isulator Transition in Si:B[END_REF][START_REF] Mark Lee | Coulomb gap in a doped semiconductor near the metal-insulator transition: Tunneling experiment and scaling ansatz[END_REF][START_REF] Bielejec | Hard Correlation Gap Observed in Quench-Condensed Ultrathin Beryllium[END_REF][START_REF] Lee | Phase Diagram of Coulomb Interactions across the Metal-Insulator Transition in Si:B[END_REF]. Comme nous allons le voir, la migration des lacunes d'oxygène par le champ électrique permet d'ajuster avec une très grande précision la concentration des dopants du régime isolant au régime métallique, et cela, à basse température. Ceci transforme les résistances électro-actives en des échantillons remarquables pour l'étude de transitions métals-isolants.

Cette partie présente une étude de la transition isolant-métal dans SrTiO 3 via la migration des lacunes d'oxygène induite par le champ électrique à très basse température, T ∼ 260mK. En effet, SrTiO 3 est un matériau avec une mobilité en oxygène élevée dont les propriétés élecro-résistives ont été beaucoup étudiées [START_REF] Janousch | Role of Oxygen Vacancies in Cr-Doped SrTiO3 for Resistance-Change Memory[END_REF][START_REF] Waser | Redox-Based Resistive Switching Memories -Nanoionic Mechanisms, Prospects, and Challenges[END_REF][START_REF] Yang | Memristive switching mechanism for metal/oxide/metal nanodevices[END_REF][START_REF] Andreasson | Resistive switching in Cr-doped SrTiO3: An X-ray absorption spectroscopy study[END_REF][START_REF] Andreasson | Spatial distribution of oxygen vacancies in Cr-doped SrTiO[sub 3] during an electric-field-driven insulator-to-metal transition[END_REF][START_REF] Szot | Switching the electrical resistance of individual dislocations in single-crystalline SrTiO3[END_REF][START_REF] Meijer | Valence states of Cr and the insulator-to-metal transition in Cr-doped SrTiO3[END_REF], dans lequel il a été montré que des nanofilaments conducteurs pouvaient être obtenus [START_REF] Szot | Switching the electrical resistance of individual dislocations in single-crystalline SrTiO3[END_REF].

Sur un substrat de SrTiO 3 orienté (110), deux électrodes séparées par une distance de 200 nm sont déposés par les techniques standards de lithographie électronique et liftoff, comme montré figure 6.1. Dans cet échantillon, un filament conducteur est formé à la température ambiante en augmentant la tension jusqu'à ce qu'un saut électro-résistif ait lieu, i.e. le courant augmente abruptement ce qui conduit à une chute de la tension de polarisation en raison de la limite de courant imposé sur la source. Il s'agit de l'étape de formation du filament conducteur, où les lacunes d'oxygène s'accumulent le long d'un chemin de percolation entre les deux électrodes, comme cela est illustré figure 6.1a. Cela conduit à une résistance de seulement quelques centaines d'Ohms, laquelle est indépendante de la température jusqu'à très basse température, T ∼ 350 mK, en dessous de laquelle elle décroît, indiquant l'apparition de la supraconductivité. En effet, SrTiO 3 dopé est connu pour être supraconducteur dans cette gamme de température [START_REF] Koonce | Superconducting Transition Temperatures of Semiconducting SrTiO {3}[END_REF]. L'apparition de la supraconductivité indique que la densité de porteurs doit être de l'ordre de 10 20 e cm -3 .

À basse température, T ∼ 260 mK, la conductance de cette jonction décroît de 5 ordres de grandeur après avoir appliqué un pulse de tension unique, d'amplitude 5 V et de durée 1 ms, figure 6.2a. Dans cette figure, chaque courbe de conductance différentielle G(V ) = dI/dV (V ) montrée est mesurée juste après qu'un pulse de tension ait été appliqué. Jusqu'à 1600 impulsions, suivies de la mesure de conductance, ont été appliquées sur cette jonction. Comme la jonction est maintenant isolante, en appliquant maintenant des impulsions de tensions négatives, la conductance augmente, jusqu'à 5 ordres de grandeur, figure 6.2. Lorsque la conductance atteint une valeur élevée, G(0) ∼ 2 mS, une impulsion de tension positive est appliquée à nouveau, ce qui rend la jonction isolante à nouveau. Ce cycle peut être répété de nombreuses fois, nous l'avons répété 10 fois pour cette jonction. Pour chaque cycle, la jonction évolue progressivement de l'état isolant à l'état conducteur en appliquant des pulses de tensions négatives de faible amplitude. Entre chaque cycles, une impulsion de tension positive est appliquée pour rendre la jonction isolante à nouveau.

Comme discuté en détails dans la chapitre 4, les impulsions de tension font migrer les lacunes d'oxygènes dans une région étroite à proximité de l'une des électrodes. Le nanofilament conducteur dans SrTiO 3 est connecté d'un coté par un contact ohmique à une région de SrTiO 3 très dopé, et, de l'autre coté, il est connecté par une barrière Schottky à une électrode en or. Cette barrière Schottky produit une barrière tunnel Avant chaque mesure, une impulsion de tension d'une durée d'environ t = 1 ms est appliquée. Pour chaque courbe, l'amplitude de l'impulsion est indiquée sur le panneau de droite correspondant. a) La premièère courbe (i = 0) montre que la jonction est hautement conductrice. Une impulsion d'amplitude 5 V rend la jonction isolante, comme montré par la courbe (i = 1). Ensuite, des impulsions négatives, de -2 V à -3 V , rendent à nouveau la jonction conductrice, comme montré par l'augmentation de la conductance entre chaque courbe. Entre les panneaux a) et b), une impulsion positive a été appliquée à nouveau pour rendre la jonction isolante. À nouveau, des impulsions négatives conduisent à une augmentation de la conductance. Ce protocol a été répété six fois, panneaux a) à f). La transition de l'état isolant à l'état conducteur peut être répétée de façon reproductible, comme montré sur le panneau g), où toutes les courbes de conductance différentielle (i = 0..749) ont été tracées. Le panneau h) montre que la conductance à V Drain = 0 V augmente de 5 ordres de grandeur de l'état isolant à l'état conducteur. Les flèches, fig3abc, fig3d et fig3e, indiquent la conductance à V Drain = 0 V pour les jonctions dont la densité d'états est montrée.

[86] au travers de laquelle à lieu l'essentiel de la chute de potentiel.

Dans cette géométrie, la conductance différentielle montrée figure 6.2 représente, à basse tension, < 100 meV , la densité d'états de la région faiblement dopée.

G(V ) G 0 = N (ε) N 0 ∂f (ε -eV, T ) ∂eV dε (6.1)
Ainsi, cette méthode permet de suivre précisément l'évolution de la densité d'états dans la région du nanofilament localisée à proximité immédiate de l'électrode en or, pendant que sa concentration en oxygène est modifiée en appliquant des impulsions de tension.

La figure 6.3 montre une sélection de quelques courbes de conductance entre l'isolant et le métal, montrant l'évolution de la densité d'états dans le nanofilament en fonction du dopage. À faible dopage, G(V ) montre un gap abrupte et des pics bien résolus aux tensions (-41 mV , 24 mV , 78 mV ). Ces pics ont été retrouvés dans un second échantillon, ils correspondent à des états électroniques discrets de dopants. Au fur et à mesure que le dopage augmente, le gap se remplit et les caractéristiques s'adoucissent, i.e. les états de dopants disparaissent, toutefois un gap mou subsiste où la densité d'états diminue au niveau de Fermi avec la forme quadratique N ( ) ∼ 2 . Cette forme de la densité d'états a été prédite par Efros and Shklovskii [START_REF] Shklovskii | Coulomb gap and low temperature conductivity of disordered systems[END_REF] du coté isolant de la transition métal-isolant. Un tel gap avait déjà été observé dans les semiconducteurs dopés par spectroscopie tunnel [START_REF] Mcmillan | Electron Tunneling Experiments on Amorphous Ge {1-x}Au {x}[END_REF][START_REF] Massey | Electron Tunneling Study of Coulomb Correlations across the Metal-Isulator Transition in Si:B[END_REF][START_REF] Mark Lee | Coulomb gap in a doped semiconductor near the metal-insulator transition: Tunneling experiment and scaling ansatz[END_REF][START_REF] Bielejec | Hard Correlation Gap Observed in Quench-Condensed Ultrathin Beryllium[END_REF][START_REF] Lee | Phase Diagram of Coulomb Interactions across the Metal-Insulator Transition in Si:B[END_REF].

En augmentant encore le dopage, ce gap mou se ferme et une transition vers l'état métallique a lieu. Dans l'état métallique, la densité d'états est finie au niveau de Fermi mais une dépression subsiste, qui peut être décrit par la formule N (ε) =

N (0)[1 + ε E T 1/2
], valide à trois dimensions, et dépend de l'énergie de Thouless. Cette dépression est due aux termes d'échanges des interactions de Coulomb comme montré par McMillan [START_REF] Mcmillan | Scaling theory of the metal-insulator transition in amorphous materials[END_REF], Altshuler et Aronov [START_REF] Altshuler | Zero bias anomaly in tunnel resistance and electron-electron interaction[END_REF]. Cette énergie de Thouless est finie dans la phase métallique et décroît vers zéro à l'approche de la transition métal-isolant. Juste à la transition métal-isolant, la densité d'états dans ce régime critique suit une loi linéaire N ( ) ∼ . La question de la dépendance en énergie de la densité d'états au point critique est l'une des questions essentielles de la recherche sur les transitions de Mott-Anderson [START_REF] Belitz | The Anderson-Mott transition[END_REF]. Une prédiction théorique [START_REF] Mark Lee | Coulomb gap in a doped semiconductor near the metal-insulator transition: Tunneling experiment and scaling ansatz[END_REF] suggère que la densité d'états devrait suivre la relation N c ( ) ∼ (3/η)-1 ; ce qui impliquerait que η ∼ 1.5 dans notre système, ce qui est compatible avec les limites théoriques 1< η <3 attendues pour ce coefficient [START_REF] Mcmillan | Electron Tunneling Experiments on Amorphous Ge {1-x}Au {x}[END_REF].

Chapter 7

Conclusions and perspectives

The goal of this thesis was to study unconventional insulators and metal-insulator transitions. Metal-insulator transition constitute a complex quantum problem that remains the subject of much theoretical works. Metal-insulator transitions are not only of fundamental interest but also of practical interest. We could argue that the transition from the insulating to the conducting state of the channel in a field effect transistor constitutes a metal-insulator transition. In memristors, the transition from one state to another also constitute a metal-insulator transition, as shown by our experiment realized on SrTiO 3 .

The work on SrTiO 3 provided the evolution of the DOS as the doping concentration is tuned across the metal-insulator transition. The observation of the Efros-Shklovskii gap as well as the Altshuler-Ahoronov corrections indicated the presence of Coulomb interactions. Thus, even in a simple semiconductor such as SrTiO 3 , Coulomb interactions have measurable consequence in the vicinity of the metal insulator transition.

When Coulomb interactions become very large, a new type of insulators emerge, the so-called Mott insulators. In these insulators, the electron are localized on their sites and control the transport properties of the material.

Many of the binary transition metal oxides are Mott insulators where no electronic transport is possible. Among these, however, a few of them such as the magnetite Fe 3 O 4 , display a conducting phase. Consequently, this material allows to study electronic transport in a regime where strong Coulomb repulsion is present.

In this thesis, I focused on magnetite nanoparticles, where I have shown that EFM microscopy allows studying the charge response, i.e. the dielectric constant or the electronic compressibility, of single nanoparticles. In our work, we have found that the nanoparticles actually behaved as insulating dielectric materials, indicating their oxidation into the insulating phase Fe 2 O 3 . Future works in this direction is to repeat the experiment after carefully transferring the sample from the glovebox to the microscope, avoiding any contact of the sample with ambient air.

Finally, given the rising interest for topological insulators and their potential importance for applications, I developed the methods in the laboratory to fabricate devices out of MBE grown Bi 2 Se 3 thin films. In future works, these devices will be employed to study quantum transport signatures of topological surface states, as function of the film thickness and carrier concentration.

4. 1 64 5. 1

 1641 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.2 Memristors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.2.1 Anion-based memristive devices . . . . . . . . . . . . . . . . . . 42 4.2.2 Cation-based memristive devices . . . . . . . . . . . . . . . . . . 42 4.3 A system for the study of metal-insulator transition . . . . . . . . . . . 44 4.4 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.5.1 Forming step . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.5.2 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . .48 4.5.3 Electric field induced motion of oxygen vacancies . . . . . . . . 48 4.5.4 Evolution of the density of states . . . . . . . . . . . . . . . . . 57 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Topological Insulators Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.2 Bi 2 Se 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.3 Aharanov-Bohm effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.4 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2 . 31 vii 3 . 52 ix 4 . 58 x 4 . 1 / 2 ]

 231352458412 10 a) Evolution of the band structure of a binary semiconductor MA, going from he inside of the semiconductor to the outside across the edge. b) Electronic band structure of a 3D topological insulator. . . . . . . . . . 18 3.1 Inverse spinel structure of magnetite. . . . . . . . . . . . . . . . . . . . 20 3.2 Electrical conductivity vs. temperature in Magnetite. . . . . . . . . . . 21 3.3 TEM images of magnetite nanoparticles. Inset: power spectrum of the particle indicated by the arrow. . . . . . . . . . . . . . . . . . . . . . . 23 3.4 Home build AFM-EFM setup. a) Photo and schematic of microscope head showing the piezoelectric motor. b) Ceramic tuning fork holder and a tip glued on one prong. c) SEM image of electrochemically etched Pt/Ir tip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.5 AFM topography image of the sample. Inset: Zoom on the AFM topography image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.6 EFM measurement circuit. . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.7 Sketch of the multipass sequence used to acquire the 3D EFM map. . . 28 3.8 Panels (a): Topography images of magnetite nanoparticles taken at different temperatures. The full size width of the images are respectively W = 180 nm, 270 nm and 570 nm, from left to right. Panel (b): First harmonic of EFM signal, where each image column is acquired at T = 70 K, 160 K and 300 K, respectively. . . . . . . . . . . . . . . . 29 3.9 Frequency shift f sub , measured far from the nanoparticle, and f N P , measured at the vertical of the nanoparticle . . . . . . . . . . . . . . . . . . 30 3.10 Panels a,b,c) Frequency shift profile -∆f = -(f N P -f sub ) extracted from the images shown at figure 3.8, taken at different height offset h j and different temperatures. The minimum in the curves correspond to the location of the nanoparticle. . . . . . . . . . . . . . . . . . . . . . . 11 Main panel: Sketch of the capacitive contributions to the EFM signal. Inset: Schematic description of the evolution of the voltage from the tip to the substrate across the nanoparticle. . . . . . . . . . . . . . . . . . 32 3.12 Frequency shift difference ∆f N P , obtained from the profile shown figure 3.10, taken at different height offset h j and different temperatures. . . . 33 3.13 Contribution C N P /C sub of the nanoparticle to the capacitive signal at 160K and 70K, showing that the nanoparticle contributes to 0.3% of the full capacitive signal between the tip and the substrate. This nanoparticle contribution goes to zero at large tip-sample distance and does not change with temperature, as expected for an insulating dielectric nanoparticle. The metallic and insulating limits are shown as dotted lines at low tip/sample distance for comparison. . . . . . . . . . . . . . 34 3.14 XPS spectra of the Fe 3 O 4 nanoparticles. . . . . . . . . . . . . . . . . . 35 3.15 a) TEM image of Fe 3 O 4 nanoparticles. b) Optical microscopy images of circuits. c) High resolution SEM image, ×800000, of a nanoparticle trapped between two electrodes. . . . . . . . . . . . . . . . . . . . . . . 37 3.16 a) IV curves of a single nanoparticle junction. b) dV /dI as function of voltage. c) dV /dI as function of temperature at selected drain voltages. The dashed line shows the Verwey transition temperature T V ∼ 120 K. d) Color plot of dV /dI displaying the out-of-equilibrium phase diagram for the Verwey transition. . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.1 a) The four fundamental two-terminal circuit elements: resistor, capacitor, inductor and memristor [68]. b) A typical switching curve of a memristor. Starting from an insulating device, upon increasing the voltage, an abrupt jump in current indicates the switching from the insulating to the conducting state of the memristor. Upon decreasing the voltage to negative values, the current sharply drops to zero below some voltage threeshold, indicating the switching back from the conducting to the insulating state. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 Infrared thermal image of the memristive device with a current of +5 mA at an applied voltage of 30 V . In the color scale, blue and red represent room temperature and elevated temperature, respectively [70]. . . . . . 42 viii 4.3 a) Scanning electron microscopy (SEM) image showing the high-resistance state of the device with shorter and smaller Ag dendrites. b) SEM image of the same cell at low-resistance state with longer and larger Ag dendrites, obtained by applying a positive voltage on the Ag electrode side [71]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.4 The perovskite structure of SrTiO 3 . . . . . . . . . . . . . . . . . . . . . 44 4.5 Resistivity and Hall coeffcient as a function of 1/T in SrTiO 3 [75]. . . . 45 4.6 Temperature dependence of resistance of an electrostatic double layer transistor fabricated on SrTiO 3 . Inset: Schematic diagram of an electric double layer formed at an electrolyte-SrTiO 3 interface. Cations (K + ) in the electrolyte are electrostatically adsorbed on the SrTiO 3 surface on application of a positive gate bias to the Pt electrode. A negative image charge is induced in the SrTiO 3 surface layer, forming an electricdouble-layer capacitor (about 1 nm thick) as a dielectric layer [82]. . . 46 4.7 E-beam lithography patterning design on a 10 mm × 10 mm SrTiO 3 substrate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.8 SEM images of sample after gold evaporation. . . . . . . . . . . . . . . 49 4.9 False color SEM image of gold electrodes deposited on the SrTiO 3 substrate and the schematic filament creation between two electrodes which are separated by a distance of 200 nm. . . . . . . . . . . . . . . . . . . 50 4.10 IV characteristic switching curve during (blue curve) and after switching (orange curve). After switching, the resistance drops and the sample becomes metallic due to the formation of conducting filament showed schamatically at figure 4.9. . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.11 Two-wire resistance of two different conducting filaments measured at low temperature shows the changes of resistance in the temperature range between 300 mK and 400 mK. . . . . . . . . . . . . . . . . . . . 51 4.12 Top) Schematic illustration of formed filament between cathode and anode. Bottom) The schematic energy diagram of the formed device. A Schottky tunnel barrier is formed between the weakly doped SrTiO 3 region and the Au electrode. . . . . . . . . . . . . . . . . . . . . . . . . 13 First set of differential conductance curves. i = 0 curve is measured after cooling down the system to T = 260 mK. The i = 1 curve is the differential conductance curve after the first pulse of V = 5V has been applied. The process is repeated hundered of times, leading to the series of conductance curves labelled by i = 2...n. . . . . . . . . . . . . . . . . 53 4.14 a-f) Six different sets of differential conductance measurement curves. After each run a positive pulse of V = 5 V is applied in order to return the system to the insulating state. The magnitude of applied pulses are shown in right panel of each set of curves. . . . . . . . . . . . . . . . . 54 4.15 ON and OFF state of a SrTiO 3 based memristive device. The low oxygen vacancy concentration region is in the vicinity of the electrode [85]. . . 55 4.16 All the differential conductance curves, i = 0...n(n = 749), plotted separately in panels a) to f), in figure 4.14 can be superimposed on each other and the transition from the insulating state to the conducting state can be reproducibly repeated. . . . . . . . . . . . . . . . . . . . . . . . 56 4.17 Switching IV curves in TiO 2 shows resistance changes. Inset: log-scale IV switching curve [90]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.18 The zero bias conductance, measured at V Drain = 0 V , increases by 5 orders of magnitude from the insulating to the conducting sample. Arrows labelled, figure 4.19abc, figure 4.19.d and figure 4.19.e, indicate the zero bias conductance for the junctions whose DOS in shown in figure 4.19. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 DOS at low bias voltage (-0.1 V to 0.1 V ) at various steps of the transition from the insulator to the metal. The curves have been normalized at V Drain = 0.1 V and displaced for clarity. The horizontal dashed lines indicate the level of zero conductance. Curve labelled a) is measured at very low doping concentration; it shows a hard gap and discrete electronic levels. At high carrier concentration, curve b), the DOS has a quadratic energy dependence, N (ε) ∼ ε 2 , (Efros and Shklovskii) , as shown by the dotted line. At very high carrier concentration, curves d) and e), the DOS becomes finite at zero energy with a cusp that can be fitted by Altshuler and Aronov law N (ε) = N (0)[1 + ε E T , as shown by the dotted lines. E T ∼ 0.1 eV for curve e) E T ∼ 0.003 eV for curve d). Finally, just at the critical point of the Metal-Insulator Transition, the DOS has a linear dependence with energy, N (ε) ∼ ε , dotted line. . 59 4.20 Differential conductance curve for a second sample showing a discrete doping level located close to the gap edge. . . . . . . . . . . . . . . . . 60 4.21 a) DOS at low bias voltage (18 mV -100 mV ) in the low doping regime showing a sharp discrete doping level. b): Zoom (18 mV -28 mV ) on the discrete electronic level. The peak splits in two peaks due to Zeeman splitting. The magnitude of the Zeeman splitting suggests that these electronic levels arises from a d ( = 2) orbital level, shown panel c). 61

  comme montrée par la courbe en pointillée. Finalement, juste au point critique de la transition, la densité d'états a une dépendance linéaire avec l'énergie, N (ε) ∼ ε, ligne en pointillée. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2 . 1 :

 21 Figure 2.1: Band structure of a) insulator and b) metal, where Density Of States (DOS) is represented as function of energy. E F is the Fermi energy level. The insulators have either completely filled or completely empty band whereas the metals have at least one partially filled band.

Figure 2 . 2 :

 22 Figure 2.2: Typical wave function of a) an extended state with mean free path l and b) a localized state with localization length ξ.

Figure 2 .

 2 Figure 2.3: a) Evolution of the DOS of a semiconductor with doping. In the undoped semiconductors, the DOS goes to zero sharply at the band edge (blue curve). Upon doping, additional states appear at the top and bottom of the band, shown as the black curve on panel (a) and depicted on the band structure on panel (b). A mobility edge, E m separates the localized states, which are far from the center of the band, from the delocalized states, which are close to the center of the band.

Figure 2 .

 2 [START_REF] Yu | Signature of molecular cleavage in field emission current[END_REF] shows the behavior of scaling function, β as a function of conductance, g.

Figure 2 . 4 :

 24 Figure 2.4: Scaling function β vs. conductance g for different dimensions[START_REF] Yu | Signature of molecular cleavage in field emission current[END_REF]

Figure 2 . 5 :

 25 Figure 2.5: The Mott-Hubbard model in presence of Coulomb interaction. U represents the on-site Coulomb interaction. W 1 and W 2 represents the width of the bands due to the hopping integral term.

Figure 2 . 6 :

 26 Figure 2.6: Band structure of a) normal metal and b) Mott-Hubbard insulator.

Figure 2 . 7 :

 27 Figure 2.7: Band structure of a) normal metal and b) charge-transfer insulator.

Figure 2 .

 2 Figure 2.8 shows several strongly correlated electron materials on the (U,∆) phase diagram.

Figure 2 . 8 :

 28 Figure 2.8: Zaanen-Sawatzky-Allen phase diagram showing different compounds. T ef f denotes the effective hybridization strength [19].

Figure 2 .

 2 Figure 2.10: a) Evolution of the band structure of a binary semiconductor MA, going from he inside of the semiconductor to the outside across the edge. b) Electronic band structure of a 3D topological insulator.

3 O 4 MagnetiteFigure 3 . 1 :

 3431 Figure 3.1: Inverse spinel structure of magnetite.

Figure 3 . 2 :

 32 Figure 3.2: Electrical conductivity vs. temperature in Magnetite.

Figure 3 . 3 :

 33 Figure 3.3: TEM images of magnetite nanoparticles. Inset: power spectrum of the particle indicated by the arrow.

Figure 3 . 4 :

 34 Figure 3.4: Home build AFM-EFM setup. a) Photo and schematic of microscope head showing the piezoelectric motor. b) Ceramic tuning fork holder and a tip glued on one prong. c) SEM image of electrochemically etched Pt/Ir tip.

Figure 3 .

 3 [START_REF] Hikami | Spin-Orbit Interaction and Magnetoresistance in the Two Dimensional Random System[END_REF] shows a topographic image of nanoparticles disposed on the substrate. The inset is a zoom of few well-separated single nanoparticles.

Figure 3 . 5 :

 35 Figure 3.5: AFM topography image of the sample. Inset: Zoom on the AFM topography image.

Figure 3 . 6 :

 36 Figure 3.6: EFM measurement circuit.

Figure 3 . 7 :

 37 Figure 3.7: Sketch of the multipass sequence used to acquire the 3D EFM map.

Figure 3 . 8 :

 38 Figure 3.8: Panels (a): Topography images of magnetite nanoparticles taken at different temperatures. The full size width of the images are respectively W = 180 nm, 270 nm and 570 nm, from left to right. Panel (b): First harmonic of EFM signal, where each image column is acquired at T = 70 K, 160 K and 300 K, respectively.

Figure 3 . 9 :Figure 3 . 10 :Figure 3 . 11 :

 39310311 Figure 3.9: Frequency shift f sub , measured far from the nanoparticle, and f N P , measured at the vertical of the nanoparticle

Figure 3 . 12 :

 312 Figure 3.12: Frequency shift difference ∆f N P , obtained from the profile shown figure 3.10, taken at different height offset h j and different temperatures.

Figure 3 .

 3 Figure 3.13: Contribution C N P /C sub of the nanoparticle to the capacitive signal at 160K and 70K, showing that the nanoparticle contributes to 0.3% of the full capacitive signal between the tip and the substrate. This nanoparticle contribution goes to zero at large tip-sample distance and does not change with temperature, as expected for an insulating dielectric nanoparticle. The metallic and insulating limits are shown as dotted lines at low tip/sample distance for comparison.

Figure 3 .

 3 Figure 3.14: XPS spectra of the Fe 3 O 4 nanoparticles.

Figure 3 .

 3 Figure 3.15: a) TEM image of Fe 3 O 4 nanoparticles. b) Optical microscopy images of circuits. c) High resolution SEM image, ×800000, of a nanoparticle trapped between two electrodes.

Figure 3 .Figure 3 .

 33 Figure 3.16a shows the IV curves of a typical sample as a function of temperature. Taking the numerical derivative of the IV curves provides the differential resistance R = dV /dI as a function of voltage V Drain , at different temperatures between T ∼ 75 K and T ∼ 240 K, figure 3.16b. From these curves, one can extract dV /dI as function of temperature, at different voltage values between V Drain = 0 V and V Drain = 2.5 V , figure3.16c. For V Drain = 0 V , the resistance increases at the Verwey temperature, T V ∼ 120 K, which is identical to the transition temperature observed in bulk magnetite. Upon increasing V Drain , the transition temperature decreases. A color plot of dV /dI curves as function of temperature and voltage, figure3.16d, provides a phase diagram for this nanoparticle and demonstrate that the phenomena of charge localization at the Verwey transition also occurs in a single nanoparticle.

Figure 4 .

 4 Figure 4.1: a) The four fundamental two-terminal circuit elements: resistor, capacitor, inductor and memristor [68]. b) A typical switching curve of a memristor. Starting from an insulating device, upon increasing the voltage, an abrupt jump in current indicates the switching from the insulating to the conducting state of the memristor. Upon decreasing the voltage to negative values, the current sharply drops to zero below some voltage threeshold, indicating the switching back from the conducting to the insulating state.

Figure 4 . 2 :

 42 Figure 4.2: Infrared thermal image of the memristive device with a current of +5 mA at an applied voltage of 30 V . In the color scale, blue and red represent room temperature and elevated temperature, respectively [70].

Figure 4 .

 4 Figure 4.3: a) Scanning electron microscopy (SEM) image showing the high-resistance state of the device with shorter and smaller Ag dendrites. b) SEM image of the same cell at low-resistance state with longer and larger Ag dendrites, obtained by applying a positive voltage on the Ag electrode side [71].

4. 3 A

 3 system for the study of metal-insulator transition SrTiO 3 is one of the most studied oxides with the perovskite structure. The perovskites have ABX 3 form where A and B are cations which bonds to an anion (X).

TiFigure 4 . 4 :

 44 Figure 4.4: The perovskite structure of SrTiO 3 .

  at very low carrier density (10 19 cm -3 ) below T = 400 mK, figure 4.5.[START_REF] Koonce | Superconducting Transition Temperatures of Semiconducting SrTiO {3}[END_REF][START_REF] Lin | Fermi Surface of the Most Dilute Superconductor[END_REF]. Most other superconductors, conventional superconductors and high temperature cuprate superconductors, have much higher carrier concentration around 10 21 -10 23 cm -3 . In recent years, SrTiO 3 attracted considerable attention after the observation of twodimensional superconductivity at the interface LaTiO 3 /SrTiO 3[START_REF] Biscaras | Two-dimensional superconductivity at a Mott insulator/band insulator interface LaTiO3/SrTiO3[END_REF].

Figure 4 . 5 :

 45 Figure 4.5: Resistivity and Hall coeffcient as a function of 1/T in SrTiO 3 [75].

Figure 4 . 6 :

 46 Figure 4.6: Temperature dependence of resistance of an electrostatic double layer transistor fabricated on SrTiO 3 . Inset: Schematic diagram of an electric double layer formed at an electrolyte-SrTiO 3 interface. Cations (K + ) in the electrolyte are electrostatically adsorbed on the SrTiO 3 surface on application of a positive gate bias to the Pt electrode. A negative image charge is induced in the SrTiO 3 surface layer, forming an electric-double-layer capacitor (about 1 nm thick) as a dielectric layer [82].

Figure 4 . 7 :

 47 Figure 4.7: E-beam lithography patterning design on a 10 mm × 10 mm SrTiO 3 substrate.

Figure 4 . 10 :Figure 4 . 11 :

 410411 Figure 4.10: IV characteristic switching curve during (blue curve) and after switching (orange curve). After switching, the resistance drops and the sample becomes metallic due to the formation of conducting filament showed schamatically at figure 4.9.

51 4 .Figure 4 . 12 :

 4412 Figure 4.12: Top) Schematic illustration of formed filament between cathode and anode. Bottom) The schematic energy diagram of the formed device. A Schottky tunnel barrier is formed between the weakly doped SrTiO 3 region and the Au electrode.

Figure 4 . 13 :Figure 4 .

 4134 Figure 4.13: First set of differential conductance curves. i = 0 curve is measured after cooling down the system to T = 260 mK. The i = 1 curve is the differential conductance curve after the first pulse of V = 5V has been applied. The process is repeated hundered of times, leading to the series of conductance curves labelled by i = 2...n.

Figure 4 . 15 :

 415 Figure 4.15: ON and OFF state of a SrTiO 3 based memristive device. The low oxygen vacancy concentration region is in the vicinity of the electrode [85].

Figure 4 . 16 :

 416 Figure 4.16: All the differential conductance curves, i = 0...n(n = 749), plotted separately in panels a) to f), in figure 4.14 can be superimposed on each other and the transition from the insulating state to the conducting state can be reproducibly repeated.

Figure 4 . 17 :

 417 Figure 4.17: Switching IV curves in TiO 2 shows resistance changes. Inset: log-scale IV switching curve [90].

Figure 4 . 18 :

 418 Figure 4.18: The zero bias conductance, measured at V Drain = 0 V , increases by 5 orders of magnitude from the insulating to the conducting sample. Arrows labelled, figure 4.19abc, figure 4.19.d and figure 4.19.e, indicate the zero bias conductance for the junctions whose DOS in shown in figure 4.19.

Figure 4 . 19 :

 419 Figure 4.19: DOS at low bias voltage (-0.1 V to 0.1 V ) at various steps of the transition from the insulator to the metal. The curves have been normalized at V Drain = 0.1 V and displaced for clarity. The horizontal dashed lines indicate the level of zero conductance. Curve labelled a) is measured at very low doping concentration; it shows a hard gap and discrete electronic levels. At high carrier concentration, curve b), the DOS has a quadratic energy dependence, N (ε) ∼ ε 2 , (Efros and Shklovskii) , as shown by the dotted line. At very high carrier concentration, curves d) and e), the DOS becomes finite at zero energy with a cusp that can be fitted by Altshuler and Aronov law

  sample, figure4.20, correspond to discrete doping levels.

Figure 4 . 20 :

 420 Figure 4.20: Differential conductance curve for a second sample showing a discrete doping level located close to the gap edge.

Figure 4 .

 4 Figure 4.21: a) DOS at low bias voltage (18 mV -100 mV ) in the low doping regime showing a sharp discrete doping level. b): Zoom (18 mV -28 mV ) on the discrete electronic level. The peak splits in two peaks due to Zeeman splitting. The magnitude of the Zeeman splitting suggests that these electronic levels arises from a d ( = 2) orbital level, shown panel c).

Figure 5 . 1 :

 51 Figure 5.1: Angle-resolved photoemission spectroscopy of Bi 2 Se 3 crystal [29].

2 :BFigure 5 . 2 :

 252 Figure 5.2: Aharanov-Bohm ring.

Bi 2

 2 Se 3 samples are grown by Molecular Beam Epitaxy (MBE) with different thicknesses (10 -30 nm). Two types of samples have been fabricated, Hall bars, figure 5.3 and rings. Hall bars are used to characterize the mobility and carrier concentration of the films. The rings are used to detect the Aharanov-Bohm effect. All microfabrication processes are done by e-beam lithography. First, we spin coat a thin layer ∼ 170 nm

Figure 5 . 3 :

 53 Figure 5.3: electrodes design for first e-beam lithography.

Figure 5 .

 5 Figure 5.6 presents the resistance of a 9 QL Bi 2 Se 3 thin film as a function of temperature. The resistance has a linear behavior at high temperature and saturates to a

Figure 5 . 4 :

 54 Figure 5.4: Second design for etching the surface of Bi 2 Se 3 except the electrodes.

Figure 5 . 5 :

 55 Figure 5.5: SEM images of Aharanov-Bohm ring in two different geometries.

Figure 5 . 6 :

 56 Figure 5.6: Resistance vs. temperature curve for a 9 QL Bi 2 Se 3 sample. Inset: Hall resistance as a function of magnetic field at 2 K.

Figure 5 .

 5 Figure 5.7 shows the magnetoresistance measurement of a 9 QL Bi 2 Se 3 sample at different temperatures. In a topological insulator, strong spin-orbit coupling prevents the usual weak localization phenomena; instead, weak antilocalization is observed. At low temperature T < 15 K a sharp cusp is observed which is consistent with the results of previous magnetotransport measurements of Bi 2 Se 3 thin films[START_REF] Matsuo | Weak antilocalization and conductance fluctuation in a submicrometer-sized wire of epitaxial Bi {2}Se {3}[END_REF][START_REF] He | Impurity Effect on Weak Antilocalization in the Topological Insulator Bi {2}Te {3}[END_REF]. The weak antilocalization can be explained by Hikami-Larkin-Nagaoka (HLN) model[START_REF] Hikami | Spin-Orbit Interaction and Magnetoresistance in the Two Dimensional Random System[END_REF]. In this model the conductance correction is given by:

Figure 5 . 7 :

 57 Figure 5.7: The magnetoresistance of 9 QL Bi 2 Se 3 sample as a function of magnetic field at different temperatures.

Figure 5 . 8 :

 58 Figure 5.8: Magnetoconductance as a function of magnetic field for a 9 QL Bi 2 Se3 thin film at T = 2 K and the HLN fit of the magnetoresistance for different magnetic fields.

Figure 5 . 9 :

 59 Figure 5.9: The magnetoresistance of 9 QL thick Bi 2 Se 3 Aharanov-Bohm ring.

  Figure 6.1: a) Image MEB en fausses couleurs de l'échantillon de SrTiO 3 avec les électrodes en or. La distance entre les électrodes est d'environ 200 nm. L'application d'une tension entre les deux électrodes mène à la formation de filaments conducteurs en raison de l'accumulation de lacunes d'oxygènes. La région hautement dopée agit comme une cathode virtuelle qui est formée à la température ambiante lors de la procédure d'initialisation, qui est identifiée par le saut de courant dans la caractéristique couranttension montrée panneau c). b) Structure perovskite de SrTiO 3 . c) Caractéristique courant-tension montrant le saut de courant due à la formation du filament conducteur, G ∼ 5 ms. d) Résistance de deux filaments conducteurs mesurés à basse température. La résistance est indépendante de la température jusqu'aux plus basses températures, T ∼ 0.35 K, en dessous de laquelle la résistance commence à diminuer. Cela indique l'apparence de la supraconductivité tel que attendu pour SrTiO 3 dopé. La résistance ne décroît pas à zéro en raison des résistances de contact dans cette mesure à deux fils.

Figure 6 .

 6 Figure 6.2: a) à f) 750 courbes de conductance G = dI/dV , indexées par i = 0..n (n = 749), mesurées à basse température, T = 260 mK, sont montrées en fonction V Drain . Avant chaque mesure, une impulsion de tension d'une durée d'environ t = 1 ms est appliquée. Pour chaque courbe, l'amplitude de l'impulsion est indiquée sur le panneau de droite correspondant. a) La premièère courbe (i = 0) montre que la jonction est hautement conductrice. Une impulsion d'amplitude 5 V rend la jonction isolante, comme montré par la courbe (i = 1). Ensuite, des impulsions négatives, de -2 V à -3 V , rendent à nouveau la jonction conductrice, comme montré par l'augmentation de la conductance entre chaque courbe. Entre les panneaux a) et b), une impulsion positive a été appliquée à nouveau pour rendre la jonction isolante. À nouveau, des impulsions négatives conduisent à une augmentation de la conductance. Ce protocol a été répété six fois, panneaux a) à f). La transition de l'état isolant à l'état conducteur peut être répétée de façon reproductible, comme montré sur le panneau g), où toutes les courbes de conductance différentielle (i = 0..749) ont été tracées. Le panneau h) montre que la conductance à V Drain = 0 V augmente de 5 ordres de grandeur de l'état isolant à l'état conducteur. Les flèches, fig3abc, fig3d et fig3e, indiquent la conductance à V Drain = 0 V pour les jonctions dont la densité d'états est montrée.

Figure 6 . 3 :

 63 Figure 6.3: Densité d'états à faible tension de polarisation (-0.1 V à 0.1 V ) mesurée à différentes étapes de la transition entre l'isolant et le métal. Les courbes ont été normalisées à V Drain = 0.1 V et déplacées pour la clarté. Les lignes horizontales pointillées indiquent le niveau de la conductance à tension de polarisation nulle. La courbe a) est mesurée à très faible concentration de dopants; elle montre un gap et des états électroniques discrets. À haute densité de porteurs, courbe b), la densité d'états a une dépendance quadratique avec l'énergie, N (ε) ∼ ε 2 , (Efros et Shklovskii), comme montré par la ligne en pointillée. À très haute densité de porteurs, courbes d) et e), la densité d'états devient finie à énergie nulle et montre un minimum qui peut être ajustée par la loi de Altshuler et Aronov, N(ε) = N (0)[1 + | ε ∆ | 1/2], comme montrée par la courbe en pointillée. Finalement, juste au point critique de la transition, la densité d'états a une dépendance linéaire avec l'énergie, N (ε) ∼ ε, ligne en pointillée.

  Second design for etching the surface of Bi 2 Se 3 except the electrodes. . 69 5.5 SEM images of Aharanov-Bohm ring in two different geometries. . . . . 69 5.6 Resistance vs. temperature curve for a 9 QL Bi 2 Se 3 sample. Inset: Hall resistance as a function of magnetic field at 2 K. . . . . . . . . . . . . . 70 5.7 The magnetoresistance of 9 QL Bi 2 Se 3 sample as a function of magnetic field at different temperatures. . . . . . . . . . . . . . . . . . . . . . . . 71 5.8 Magnetoconductance as a function of magnetic field for a 9 QL Bi 2 Se3 thin film at T = 2 K and the HLN fit of the magnetoresistance for The magnetoresistance of 9 QL thick Bi 2 Se 3 Aharanov-Bohm ring. . . . 72 6.1 a) Image MEB en fausses couleurs de l'échantillon de SrTiO 3 avec les électrodes en or. La distance entre les électrodes est d'environ 200 nm. L'application d'une tension entre les deux électrodes mène à la formation de filaments conducteurs en raison de l'accumulation de lacunes d'oxygènes. La région hautement dopée agit comme une cathode virtuelle qui est formée à la température ambiante lors de la procédure Drain . Avant chaque mesure, une impulsion de tension d'une durée d'environ t = 1 ms est appliquée. Pour chaque courbe, l'amplitude de l'impulsion est indiquée sur le panneau de droite correspondant. a) La premièère courbe (i = 0) montre que la jonction est hautement conductrice. Une impulsion d'amplitude 5 V rend la jonction isolante, comme montré par la courbe (i = 1). Ensuite, des Drain = 0.1 V et déplacées pour la clarté. Les lignes horizontales pointillées indiquent le niveau de la conductance à tension de polarisation nulle. La courbe a) est mesurée à très faible concentration de dopants; elle montre un gap et des états électroniques discrets. À haute densité de porteurs, courbe b), la densité d'états a une dépendance quadratique avec l'énergie, N (ε) ∼ ε 2 , (Efros et Shklovskii),

5.1 Angle-resolved photoemission spectroscopy of Bi 2 Se 3 crystal [29]. . . . . 66 5.2 Aharanov-Bohm ring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.3 electrodes design for first e-beam lithography. . . . . . . . . . . . . . . 68 5.4 different magnetic fields. . . . . . . . . . . . . . . . . . . . . . . . . . . 72 xi LIST OF FIGURES 5.9 d'initialisation, qui est identifiée par le saut de courant dans la caractéristique courant-tension montrée panneau c). b) Structure perovskite de SrTiO 3 . c) Caractéristique courant-tension montrant le saut de courant due à la formation du filament conducteur, G ∼ 5 ms. d) Résistance de deux filaments conducteurs mesurés à basse température. La résistance est indépendante de la température jusqu'aux plus basses températures, T ∼ 0.35 K, en dessous de laquelle la résistance commence à diminuer. Cela indique l'apparence de la supraconductivité tel que attendu pour SrTiO 3 dopé. La résistance ne décroît pas à zéro en raison des résistances de contact dans cette mesure à deux fils. . . . . . 78 xii 6.2 a) à f) 750 courbes de conductance G = dI/dV , indexées par i = 0..n (n = 749), mesurées à basse température, T = 260 mK, sont montrées en fonction V impulsions négatives, de -2 V à -3 V , rendent à nouveau la jonction conductrice, comme montré par l'augmentation de la conductance entre chaque courbe. Entre les panneaux a) et b), une impulsion positive a été appliquée à nouveau pour rendre la jonction isolante. À nouveau, des impulsions négatives conduisent à une augmentation de la conductance. Ce protocol a été répété six fois, panneaux a) à f). La transition de l'état isolant à l'état conducteur peut être répétée de façon reproductible, comme montré sur le panneau g), où toutes les courbes de conductance différentielle (i = 0..749) ont été tracées. Le panneau h) montre que la conductance à V Drain = 0 V augmente de 5 ordres de grandeur de l'état isolant à l'état conducteur. Les flèches, fig3abc, fig3d et fig3e, indiquent la conductance à V Drain = 0 V pour les jonctions dont la densité d'états est montrée. . . . . . . . . . . . . . . . . . . . . . . . 79 xiii 6.3 Densité d'états à faible tension de polarisation (-0.1 V à 0.1 V ) mesurée à différentes étapes de la transition entre l'isolant et le métal. Les courbes ont été normalisées à V comme montré par la ligne en pointillée. À très haute densité de porteurs, courbes d) et e), la densité d'états devient finie à énergie nulle et montre un minimum qui peut être ajustée par la loi de Altshuler et Aronov,

The first works on the Anderson localization ignored the Coulomb interactions. Today, it is well established that Coulomb interactions must be taken into account to describe properly the metal-insulator transition. On the metallic side, there are two types of quantum corrections to the classical conductivity of weakly disordered metal at low temperature. The first is weak localization (WL) correction, a phase coherent contribution that originates from constructive interference between reversed electron trajectories which exists even in absence of Coulomb interactions between electrons. The second correction is the Altshuler-Aronov correction. In 1979, Altshuler and Aronov[START_REF] Altshuler | Zero bias anomaly in tunnel resistance and electron-electron interaction[END_REF] showed that in three dimensional weakly disordered systems electron-electron interactions lead to a square-root (E 1/2 ) dip in tunneling density of states. This arises from the fact that, in a metal, electrons are not only scattered by disordered potential but also by electrostatic potential created by electron-electron interactions. In 1981, McMillan[START_REF] Mcmillan | Scaling theory of the metal-insulator transition in amorphous materials[END_REF] proposed that this dip should extend to zero at the metal-insulator transition. He suggested the following functional form: